Understand Learn Apply the Formulas

Effective Annual Rate (EAR)

$$Effective\;annual\;rate\;=\;\left(1+\frac{stated\;annual\;rate}m\right)^m-1$$

Single Cash Flow (simplified formula)

$$FV_N=PV\;\times\;\left(1+r\right)^N$$

$$PV=\frac{FV_N}{\left(1+r\right)^N}$$

$$r=inretest\;rate\;per\;period\\PV=persent\;value\;of\;the\;investment$$

$$FV_N=\;future\;value\;of\;the\;investment\;N\;periods\;from\;today$$

Investments paying interest more than once a year

$$FV_N=\;PV\;\times\left(1+\frac{r_s}m\right)^{mN}\\PV=\frac{FV_N}{\left(1+{\displaystyle\frac{r_s}m}\right)^{mN}}\\$$

$$r_s=\;stated\;annual\;interest\;rate\\m=Number\;of\;compounding\;periods\;per\;year\\N=Number\;of\;year\\$$

Future Value (FV) of an Investment with Continuous Compounding

$$FV_N=PV\;e^r{}_sN\\$$

Ordinary Annuity

$$FV_N=A\times\left[\frac{\left(1+r\right)^N-1}r\right]\\PV=A\times\left[\frac{1+{\displaystyle\frac1{\left(1+r\right)^N}}}r\right]\\\;\\$$

$$A=\;Annuity\;amount\\\;\\$$

$$r=the\;interest\;rate\;per\;period\;corresponding\;to\;the\;frequency\;of\;anuity\;\\payments\;(for\;example,\;annual,\;quarterly,\;or\;monthly)\\\;\\$$

$$N=the\;number\;of\;annuity\;payments\\\;\\$$

Present Value (PV) of a Perpetuity

$$PV_{Perpetuity}=\frac Ar\\$$

$$A=Annuity\;amount\\$$

Future value (FV) of a series of unequal cash flows

$$FV_N=cash\;flow_1\;\left(1+r\right)^1+\;cash\;flow_2\;\left(1+r\right)^2\dots\;cash\;flow_N\;\left(1+r\right)^N\\$$

Net Present Value (NPV)

$$NPV=\sum_{t=0}^N\frac{CF_t}{\left(1+r\right)^t}$$

$$CF_t=Expected\;net\;cash\;flow\;at\;time\;t\\N=\;the\;investment's\;projected\;life\\r=\;The\;discount\;rate\;opportuntity\;\cos t\;of\;capital$$

Holding Period Return (HPR)

$$No\;cash\;flows$$

$$HPR=\frac{Ending\;valu-Beginning\;value}{Beginning\;value}$$

Holding Period Return (HPR)

$$Cash\;flows\;occur\;at\;the\;end\;of\;the\;period$$

$$HPR=\frac{Ending\;value-Beginning\;value+cash\;flows\;received}{Beginning\;value}$$

$$=\frac{P_1\;-P_0+D_1}{Beginning\;value}$$

$$P_1=\;Ending\;value\\P_0=Beginning\;value\\D=cash\;flow/divided\;received$$

Yield on a Bank Discount Basis (BDY)

$${}^rBD=\frac DF\times\frac{360}t$$

$${}^rBD=Annualized\;yield\;on\;a\;bank\;discount\;basis\\D=Dollar\;discount,\;which\;is\;equal\;to\;the\;defference\;bettwen\;the\;face\;value\;\\of\;the\;bill\;(F)and\;its\;purchase\;price\left(P_0\right)\\F=\;face4\;value\;of\;the\;T-bill\\t=\;Actual\;number\;of\;days\;remaining\;to\;maturity\\$$

Effective Annual Yield (EAY)

$$EAY=\left(1+HPR\right)^\frac{360}t-1\\$$

$$t=Time\;unit\;maturity\\HPR=Holding\;period\;retrun\\$$

Money Market Yield (CD Equivalent Yield)

$$Money\;market\;yield=HPR\times\left(\frac{360}t\right)=\frac{360\times r_{Bank\;Discount}}{360-\left(t\times r_{Bank\;Discount}\right)}\\$$

Interval Width

$$Interval\;width=\frac{Range}k\\$$

$$Range=Largest\;observation\;number-Smallest\;observation\;or\;number\\k=Number\;of\;desired\;intervals$$

Population Mean

$$\mu=\overset N{\underset{\frac{i=1\dots n}N=\frac{x_1+x_2+x_3+\dots+x_n}N}{\sum\chi i}}$$

$$N=Number\;of\;observation\;in\;the\;entire\;population\\x_i=the\;i^{th}\;observation$$

Sample Mean

$$\overline x=\overset n{\underset{\frac{i=1\dots n}n=\frac{x_1+x_2+x_3+\dots X_n}n}{\sum x^i}}$$

Geometric Mean

$$G=\sqrt[n]{x_1x_2x_{3\dots}x_n}$$

$$n=\;Number\;of\;observations$$

Harmonic Mean

$$\overline{x_n}=\frac n{{\displaystyle\sum_{i=1\dots n}^n}\left({\displaystyle\frac1{x_i}}\right)}$$

Median for odd numbers

$$Medium=\left\{\frac{\left(n+1\right)}2\right\}$$

Median for even numbers

$$Median=\left\{\frac{\left(n+2\right)}2\right\}\\Median=\frac n2$$

Weighted Mean

$$\overline{x_w}=\sum_{i=1\dots n}^nw_ix_i$$

$$w=Weights\\x=Observations\\Sum\;of\;all\;weights=1$$

Portfolio Rate of Return

$$r_p=w_a\;r_a\;+\;w_b\;r_b\;+\;w_c\;r_c\;+_\cdots\;+\;w_n\;r_n$$

$$w=\;Weights\\r=\;Retunrs$$

Position of the Observation at a Given Percentile 

y

 

$$L_y=\left\{\left(n+1\right)\frac y{100}\right\}$$

$$y=The\;percetage\;point\;at\;which\;we\;are\;diving\;the\;distribution\\L_y=The\;location\;\left(L\right)\;of\;the\;perecentile\;\left(p_y\right)\;in\;the\;array\;sorted\;in\;ascending\;order\;$$

Range

$$Range=Maximum\;value\;-\;Minimum\;value$$

Mean Absolute Deviation

$$MAD=\frac{{\displaystyle\sum_{i=1_\cdots n}^n}\vert X_i\;-\;\overline X\vert}n$$

$$x=\;The\;sample\;mean\\n=\;Number\;of\;observations\;in\;the\;sample$$

Population Variance

$$\sigma^2=\sum_\frac{i=1_\cdots n}n^n\left(x_{i-\mu}\right)^2$$

$$\mu=\;Population\;mean\\N=\;Size\;of\;the\;population$$

Population Standard Deviation

$$\sigma=\sqrt{\frac{{\displaystyle\sum_{i=1_\cdots n}^N}\left(x_i-\mu\right)^2}N}$$

$\mu=\;Popiulation\;mean\\N=\;Size\;of\;the\;population$

Sample Variance

$$s^2=\frac{{\displaystyle\sum_{i=1}^n}\left(X_i-\overline X\right)^2}{n-1}$$

$$x\;=\;Sample\;mean\\n\;=\;Number\;of\;observations\;in\;the\;sample$$

Sample Standard Deviation

$$s=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left(X_i-\overline X\right)^2}{n-1}}$$

$$x\;=\;Sample\;mean\\n=Number\;of\;observations\;in\;the\;sample$$

Semi-Variance

$$Semi-\;variance=\frac1n\;\sum_{r_t<Mean}^n\left(Mean-\;r_t\right)^2$$

$$n=Total\;number\;of\;observations\;below\;the\;mean\\r_t=\;Observed\;value$$

Chebyshev Inequality

$$Percentage\;of\;observations\;within\;k\;s\tan daed\;devitions\;of\;the\;arithmetic\;mean>1-\frac1{k^2}$$

$$k\;=\;Number\;of\;s\tan dard\;deviations\;from\;the\;mean$$

Coefficient of Variation

$$CV=\frac S{\overline X}$$

$$s=\;Sample\;s\tan dard\;deviation\\\overline X=Sample\;mean$$

Sharpe Ratio

$$Sharpe\;ratio=\frac{R_p-R_f}{\sigma_p}$$

$$R_p=\;Mean\;return\;to\;the\;protfolio\\R_f=\;Mean\;rtetun\;to\;a\;risk\;-\;free\;asset\\\sigma_p=\;Stan dard\;devaiation\;of\;return\;on\;the\;protfolio$$

Skewness

$$s_k=\left[\frac n{\left(n-1\right)\left(n-2\right)}\right]\times\frac{{\displaystyle\sum_{i=1_\cdots n}^n}\left(X_i-\overline X\right)^3}{s^3}$$

$$n=\;Number\;of\;observations\;in\;the\;sample\\s=\;Sample\;s\tan dard\;deviation$$

Kurtosis

$$K_E=\left[\frac{n\left(n+1\right)}{\left(n-1\right)\left(n-2\right)\left(n-3\right)}\times\frac{{\displaystyle\sum_{i=1_\cdots n}^n}\left(X_i-\overline X\right)^4}{s^4}\right]\times\frac{3\left(n-1\right)^2}{\left(n-2\right)\left(n-3\right)}$$

$$n=\;Sample\;size\\s=\;Sample\;standard\;deviaton$$

Odds FOR E

$$Odds\;FORE=\frac{P\left(E\right)}{1-P\left(E\right)}$$

$$E=\;Odds\;for\;event\\P\left(E\right)=Probabiloity\;of\;event$$

Conditional Probability

$$P\left(A\vert B\right)=\frac{P\left(A_{\;\cap}\;B\right)}{P\left(B\right)}$$

$$where\;P\left(B\right)\neq0$$

Additive Law (The Addition Rule)

$$P\left(A\;_\cup\;B\right)=P\left(A\right)+P\left(B\right)-P\left(A\;_\cap\;B\right)$$

The Multiplication Rule (Joint Probability)

$$P\left(A\;_\cap\;B\right)=P\left(A\vert B\right)\times P\left(B\right)$$

The Total Probability Rule

$$P\left(A\right)=P\left(A\vert S1\right)\times P\left(S_1\right)+P\left(A\vert S_2\right)\times P\left(S_2\right)+_\cdots+P\left(A\vert S_n\right)\times P\left(S_n\right)$$

$$S_{1,}\;S_{2,\cdots,}S_n\;are\;mutually\;exclusive\;and\;exhastive\;scenarios\;or\;events$$

Expected Value

$$E\left(X\right)=P\left(A\right)X_A+P\left(B\right)X_B+\;\cdots\;+P\left(n\right)X_n$$

$$P\left(n\right)=probability\;of\;an\;variable\\X_n=Value\;of\;the\;variable$$

Covariance

$$COV_{xy}=\;\frac{\left(x-\overline x\right)\left(y-\overline y\right)}{n-1}$$

$$x=Value\;of\;x\\\overline x=Mean\;of\;x\;values\\y=Value\;of\;y\\\overline y=Mean\;of\;y\\n=\;Total\;number\;of\;values$$

Correlation

$$\rho=\frac{cov_{xy}}{\sigma_x\sigma_y}$$

$$\sigma_x=S\tan dard\;Deviation\;of\;x\\\sigma_y=S\tan dard\;Deviation\;of\;y\\cov_{xy}=Covariance\;of\;x\;and\;y$$

Variance of a Random Variable

$$\sigma^2x=\sum_{i=1_\cdots n}^n\left(x-E\left(x\right)\right)^2\times P\left(x\right)$$

$$The\;sum\;is\;taken\;over\;all\;values\;of\;x\;for\;which\;p\left(x\right)>0$$

Portfolio Expected Return

$$E\left(R_P\right)=E\left(w_1r_1+w_2r_2+w_3r_3+_\cdots+w_nr_n\right)$$

$$w=Cons\tan t\\r=Random\;variable$$

Portfolio Variance

$$Var\left(R_P\right)=E\left[\left(R_p-E\left(R_p\right)\right)^2\right]=\\\left[w_1^2\sigma_1^2+w_2^2\sigma_2^2+w_3^2\sigma_3^2+2w_1w_2Cov\left(R_1R_2\right)+2w_2w_3Cov\left(R_2R_3\right)+2w_1w_3Cov\left(R_1R_3\right)\right]\\$$

$$R_p=\;Return\;on\;Portfolio\\$$

Bayes’ Formula

$$P\left(A\vert B\right)=\frac{P\left(B\vert A\right)\times P\left(A\right)}{P\left(B\right)}\\$$

The Combination Formula

$$nC_r=\begin{pmatrix}n\\c\end{pmatrix}=\frac{n!}{\left(n-r\right)!r!}\\n\;=\;Total\;objects\\r=Selected\;objects\\\\$$

The Permutation Formula

$$nP_r=\frac{n!}{\left(n-r\right)!}\\\\$$

The Binomial Probability Formula

$$P\left(x\right)=\frac{n!}{\left(n-x\right)!x!}p^x\times\left(1-p\right)^{n-x}\\\\$$

$$n\;=\;Number\;of\;trials\\x\;=\;Up\;moves\\p^x=Proability\;of\;up\;moves\\\left(1-p\right)^{n-x}=Probability\;of\;down\;moves\\\\$$

Binomial Random Variable

$$E\left(X\right)=np\\Variance=\;np\left(1-p\right)\\\\$$

$$n\;=\;Number\;of\;trials\\p\;=\;Probability\\\\$$

For a Random Normal Variable X

$$90\%\;confidance\;interval\;for\;X\;is\;\overline x-1.{65}_{s;}\;\overline x+1.{65}_s\\95\%\;confidance\;interval\;for\;X\;is\;\overline x-1.{96}_{s;}\;\overline x+1.{96}_s\\99\%\;confidance\;interval\;for\;X\;is\;\overline x-2.{58}_{s;}\;\overline x+2.{58}_s\\\\$$

$$s\;=\;S\tan dard\;error\\1.65=Reliability\;factor\\x\;=\;Point\;estimate\\\\$$

Safety-First Ratio

$$SF_{Ratio}=\left[\frac{E\left(R_p\right)-R_L}{\sigma_p}\right]\\$$

$$R_p=Portfolio\;Return\\R_L=Threshhold\;level\\\sigma_p=S\tan dard\;Deviation\\$$

Continuously Compounded Rate of Return

$$FV=PV\times e^{i\times t}\\$$

$$i\;=\;Interest\;rate\\t\;=\;Time\\l_{n\;e}\;=\;1\\e\;=\;the\;exponential\;function,\;equal\;to\;2.71828$$

Sampling Error of the Mean

$$Sample\;mean\;-\;Population\;mean$$

Standard Error of the Sample Mean (Known Population Variance)

$$SE=\frac\sigma{\sqrt n}$$

$$n\;=\;Number\;of\;samples\\\sigma\;=\;Standard\;deviation$$

Standard Error of the Sample Mean (Unknown Population Variance)

$$SE=\frac S{\sqrt n}$$

$$S\;=\;Standard\;deviation\;in\;unknown\;population’s\;sample$$

Z-score

$$Z=\frac{x-\mu}\sigma$$

$$x\;=\;Observed\;value\\\sigma\;=Standard\;deviation\\\mu\;=\;Population\;mean$$

Confidence Interval for Population Mean with z

$$\overline X-Z_\frac a2\times\frac\sigma{\sqrt n};\;\overline X+Z_\frac a2\times\frac\sigma{\sqrt n}$$

$$Z_\frac a2=Reliability\;factor\\X\;=\;Mean\;of\;sample\\\sigma\;=\;Standard\;deviation\\n\;=\;Number\;of\;trials/size\;of\;the\;sample\\$$

Confidence Interval for Population Mean with t

$$\overline X-t_\frac a2\times\frac S{\sqrt n};\;\overline X+t_\frac a2\times\frac S{\sqrt n}\\$$

$$t_\frac a2=\;Reliability\;factor\\n\;=\;Size\;of\;the\;sample\\S\;=\;Standard\;deviation\\$$

 or t-statistic?

$$Z\;\rightarrow\;known\;population,\;standard\;deviation\;\sigma\;,\;no\;matter\;the\;sample\;size\\t\;\rightarrow\;unknown\;population,\;standard\;deviation\;s,\;and\;sample\;size\;below\;30\\Z\;\rightarrow unknown\;population,\;stndard\;deviation\;s,\;and\;sample\;size\;above\;30\\$$

Test Statistics: Population Mean

$$z_a=\frac{\overline X-\mu}{\displaystyle\frac\sigma{\sqrt n}};\;t_{n-1,a}=\frac{\overline X-\mu}{\displaystyle\frac s{\sqrt n}}\\$$

$$t_{n-1}=\;t-\;statistic\;with\;n-1\;degrees\;of\;freedom\;(n\;is\;the\;sample\;size)\\\overline X=Sample\;mean\\\mu\;=The\;hypothesized\;value\;of\;the\;population\;mean\\s\;=\;Sample\;s\tan dard\;deviation\\\\$$

Test Statistics: Difference in Means – Sample Variances Assumed Equal (independent samples)

$$t-statistic=\frac{\left(\overline{X_1}-\overline{X_2}\right)-\left(\mu_1-\mu_2\right)}{\left({\displaystyle\frac{s_p^2}{n_1}}+{\displaystyle\frac{s_p^2}{n_2}}\right)^{\displaystyle\frac12}}\\\\s_p^2=\frac{\left(n_1-1\right)\;s_1^2\;+\left(n_2+1\right)\;s_2^2}{n_1+n_2-2}\\\\$$

$$Number\;of\;degrees\;of\;freedom\;=\;n_1+n_2-2\\$$

Test Statistics: Difference in Means – Sample Variances Assumed Unequal (independent samples)

$$t-statistic=\frac{\left({\overline x}_1-{\overline x}_2\right)-\left(\mu_1-\mu_2\right)}{\left({\displaystyle\frac{s_1^2}{n_2}}+{\displaystyle\frac{s_2^2}{n_2}}\right)^{\displaystyle\frac12}}\\$$

$$degrees\;of\;freedom=\frac{\left({\displaystyle\frac{s_1^2}{n_1}}+{\displaystyle\frac{s_2^2}{n_2}}\right)^2}{{\displaystyle\frac{\left({\displaystyle\frac{s_1^2}{n_1}}\right)^2}{n_1}}+{\displaystyle\frac{\left({\displaystyle\frac{s_2^2}{n_2}}\right)^2}{n_2}}}\\$$

$$s\;=\;Standard\;deviation\;of\;respective\;sample\\n\;=\;Total\;number\;of\;observations\;in\;the\;respective\;population$$

Test Statistics: Difference in Means – Paired Comparisons Test (dependent samples)

$$t=\frac{\overline d-{\mu_d}_῁}{S_d},\;where\;\overline{d\;}=\;\frac1n\;\sum_{i=1_\cdots n}^nd_i$$

$$degrees\;of\;freedom:\;n–1\\n\;=\;Number\;of\;paired\;observations\\d\;=\;Sample\;mean\;difference\\S_d=Standard\;error\;of\;dd$$

Test Statistics: Variance Chi-square Test

$$X_{n-1}^2=\frac{\left(n-1\right)^{s^2}}{\sigma_0^2}$$

$$degrees\;of\;freedom\;=n-1\\s^2=sample\;variance\\\sigma_0^2=hypothesized\;variance$$

Test Statistics: Variance F-Test

$$F\;=\;\frac{s_1^2}{s_2^2},\;where\;s_1^2>s_2^2$$

$$degrees\;of\;freedom\;=n_1-1\;and\;n_2-1\\s_1^2=\;larger\;sample\;variance\\s_2^2=smaller\;sample\;variance$$

Long risk-free asset (lending)

$$Long\;risk–free\;asset\;(lending)=Long\;asset+Short\;derivative$$

Long derivative

$$Long\;derivative=Long\;asset+Short\;risk–free\;asset\;(borrowing)$$

Short asset

$$Short\;asset=Short\;derivative+Short\;risk–free\;asset\;(borrowing)$$

 

FRA settlement to the long

$$FRA\;settlement\;to\;the\;long\;=\;\\\frac{Notional\;proncipal\;\times\;\left(Floating\;rate\;-\;Forward\;rate\right)\left({\displaystyle\frac{Days}{360}}\right)}{1+Floating\left({\displaystyle\frac{Days}{360}}\right)}$$

$$Days\;=\;Number\;of\;days\;in\;floating\;rate\;term\\Floating\;=\;Floating\;rate\\Forward\;=\;Forward\;rate$$

Forward rates

$$F_0\left(T\right)=S_0\left(1+r\right)^T$$

$$F_0\left(T\right)=Price\;of\;the\;Forward\\S_0=Spot\;price\;of\;the\;underlying\;asset\\r\;=\;Risk-free\;interest\;rate\\T=Time\;of\;the\;contract$$

Value

$$V_T\left(T\right)=S_T-F_0\left(T\right)$$

$$V_T\;\left(T\right)=Value\;of\;forward\;at\;time\;T\\S_T=Spot\;price\;of\;the\;underlying\;at\;time\;T\\F_0\left(T\right)=Price\;of\;the\;Forward\\$$

Net cost of Carry

$$Net\;\cos t\;of\;Carry\;=\gamma-\theta\\\gamma=Benefits\\\theta=Costs$$

Forward Price with Net cost of Carry

$$F_0\left(T\right)=\left(S_0-\gamma+\theta\right)\left(1+r\right)^T$$

$$F_0\left(T\right)=Price\;of\;the\;Forward\\S_0=Spot\;price\;of\;the\;underlying\;asset\\r=Risk-free\;interest\;rate\\T=Time\;of\;the\;contract\\\gamma=Benefits\\\theta=Costs$$

Value at any point during the contract (time t)

$$V_t\left(T\right)=S_t-F_0\left(T\right)\left(1+r\right)^{-\left(T-t\right)}$$

$$V_t\left(T\right)=Value\;at\;time\;t\\S_t=Spot\;price\;if\;the\;undrlying\;asset\;at\;time\;t$$

 

Call Options

$$In-the-money:\;S_T>X\\At-the-money:\;S_T=X\\Out-of-the-money:\;S_T<X\\$$

$$Call\;option\;buyer\\C_T=Max\left(0,S_T-X\right)\\\pi=C_T-P\\$$

$$Call\;option\;seller\\C_T=\;-Max\left(0,S_T-X\right)\\\mathrm\pi={\mathrm C}_{\mathrm T}+\mathrm P\\$$

$$C_T=Call\;option’s\;value\;at\;expiration\left(T\right)\\S_T=Stock\;price\;at\;expiration\;\left(T\right)\\X=Option’s\;exercise/strike\;price\\\mathrm\pi=\mathrm{Profit}\\\mathrm P=\mathrm{Option}’\mathrm s\;\mathrm{premium}\;\mathrm{paid}\\$$

Put Options

$$In-the-money:\;S_T<X\\At-the-money:\;S_T=X\\Out-of-the-money:\;S_T>X$$

$$Put\;option\;Buyer\\P_T=Max\left(0,X-S_T\right)\\\mathrm\pi={\mathrm P}_{\mathrm T}-\mathrm P\\\mathrm{Put}\;\mathrm{option}\;\mathrm{seller}\\{\mathrm P}_{\mathrm T}=-\mathrm{Max}\left(0,\mathrm X-{\mathrm S}_{\mathrm T}\right)\\\mathrm\pi=-{\mathrm P}_{\mathrm T}+\mathrm P$$

$$P_T=Put\;option’s\;value\;at\;expiration\left(T\right)\\S_T=Stock\;price\;at\;expiration\left(T\right)\\X=Option’s\;exercise/strike\;price\\\mathrm\pi=\mathrm{Profit}\\\mathrm P=\mathrm{Option}’\mathrm s\;\mathrm{premium}\;\mathrm{paid}$$

Put-Call Parity

$$S_0+p_0=c_0+\frac X{\left(1+r\right)^T}$$

$$S_0=Spot\;price\;of\;the\;underlying\;asset\;at\;time\;0\\p_0=Value\;of\;put\;option\;at\;time\;0\\c_0=Value\;of\;call\;option\;at\;time\;0\\T=Option’s\;duration\\X=Exercise\;price\;of\;the\;option\\r=Risk-free\;interest\;rate$$

Put-Call Forward Parity

$$\frac{F_0\left(T\right)}{\left(1+r\right)^T}+p_0=c_0+\frac X{\left(1+r\right)^T}$$

$$F_0\left(T\right)=Forward\;price\\p_0=Value\;of\;put\;option\;at\;time\;0\\c_0=Value\;of\;call\;option\;at\;time\;0\\T=Option’s\;duration\\X=Exercise\;price\;of\;the\;option\\r=Risk-free\;interest\;rate$$

Up-factor

$$u=\frac{S_1^+}{S_0}$$

$$u=Up-factor\\S_1^+=Upward\;value\;of\;the\;underlying\;asset\;after\;first\;period\\S_0=\;Value\;of\;underlying\;at\;time\;0$$

Down-factor

$$d=\frac{S_1^-}{S_0}$$

$$d=Down-factor\\S_1^-=Downward\;value\;of\;the\;underlying\;asset\;after\;first\;period\\S_0=Value\;of\;underlying\;at\;time\;0$$

Value of option on upward movement

$$c_1^+=Max\left(0,S_1^+-X\right)=S_1^+-X$$

$$X=Exercise\;price\;of\;the\;option\\c_1^+=Option’s\;value\;after\;upward\;movement$$

Value of option on downward movement

$$c_1^-=Max\left(0,S_1^--X\right)=0$$

$$X=Exercise\;price\;of\;the\;option\\c_1^-=Option’s\;value\;after\;downward\;movement$$

Synthetic probabilities

$$c_0=\frac{\mathrm{πc}_1^++\left(1-\mathrm\pi\right)\mathrm c_1^-}{1+r}$$

$$c_0=Value\;of\;call\;option\\1-\mathrm\pi=\mathrm{Synthetic}\;\mathrm{probability}\;\mathrm{of}\;\mathrm{downward}\;\mathrm{move}$$

Plain vanilla interest rate swap

$$Fixed-rate\;payment\left(t\right)=\\\left(Swap\;FR-LIBOR\right)\times\frac T{360}\times NP\;Plain\;vanilla\;interest\;rate\;swap$$

$$FR=Fixed\;rate\\T=Number\;of\;days\;in\;the\;settlement\;period\\NP=Notional\;principal$$

Conversion ratio

$$Conversion\;ratio=\frac{Par\;value}{Conversion\;price}$$

Conversion value

$$Conversion\;value=Share\;price\times Conversion\;ratio$$

Conversion premium/discount

$$Conversion\;premium/discount=Convertible\;bond\;price-Conversion\;value$$

Fixed-rate bonds

$$\\PV=\frac{PMT}{\left(1+r\right)^1}+\frac{PMT}{\left(1+r\right)^2}+...\frac{PMT+FV}{\left(1+r\right)^N}$$

$$PV=Present\;value(price)\\PMT=Coupon\;payment\;amount\;per\;period\;\\r=Discount\;rate\\N=Number\;of\;periods\;to\;maturity\\FV=Face\;value/par\;value/future\;value$$

$$\\PV=\frac{PMT}{\left(1+Z_1\right)^1}+\frac{PMT}{\left(1+Z_2\right)^2}+...\frac{PMT+FV}{\left(1+Z_N\right)^N}$$

$$PV=Present\;value(price)\\PMT=Coupon\;payment\;amount\;per\;period\;\\Z_n=Spot\;rate\;per\;period\\N=Number\;of\;periods\;to\;maturity\\FV=Face\;value/par\;value/future\;value$$

$$PV\;Flat=\;PV\;Full-AI\\\;PV\;Full=\left[\frac{PMT}{{(1+r)}^{1-{\displaystyle\frac tT}}}+\frac{PMT}{{(1+r)}^{2-{\displaystyle\frac tT}}}+...+\frac{PMT}{{(1+r)}^{N-{\displaystyle\frac tT}}}\right]$$

$$PV\;Full=PV\times\left(1+r\right)^\frac tT\\AI=\frac tT\times PMT$$

$$PV\;Full=Full\;price\;of\;a\;bond\\PV\;Flat=Flat\;price\;of\;a\;bond\\AI=Accured\;interest\\PMT=Coupon\;payment\;amount\;per\;period\\N=Number\;of\;periods\;to\;maturity\\T=Number\;of\;days\;within\;a\;coupon\;payment\;period\\t=Number\;of\;days\;from\;the\;last\;coupon\;payment\;to\;the\;settlement\;date$$

Fixed-rate bonds

$$\\\left(1+\frac{APR_m}m\right)^m\boldsymbol=\left(1+\frac{APR_n}n\right)^n$$

$$APR_m=Annual\;percentage\;rate\;for\;m\\m=Periodicity\;that\;you\;are\;converting\;from\\APR_n=Annual\;percentage\;rate\;for\;n\\n=Periodicity\;that\;you\;are\;converting\;to$$

Current yield

$$\\Current\;yield=\frac{Total\;PMT\;in\;a\;year}{Flat\;Price}$$

Floating Rate Notes (FRNs)

$$\\PV=\frac{\displaystyle\frac{\left(Index+QM\right)\times FV}m}{\left(1+\frac{Index+DM}m\right)^1}+\frac{\displaystyle\frac{\left(Index+QM\right)\times FV}m}{\left(1+\frac{Index+DM}m\right)^2}+...+\frac{\displaystyle\frac{\left(Index+QM\right)\times FV}m+FV}{\left(1+\frac{Index+DM}m\right)^N}$$

$$PV=Present\;value\;(price)\;of\;a\;floating-rate\;note\\Index=Reference\;rate\;(stated\;as\;an\;annual\;percentage\;rate)\\QM=\;Quoted\;margin\;(stated\;as\;an\;annual\;percentage\;rate)\\FV=Future\;value\;paid\;at\;maturity\;(par\;value)\\m=Periodicity\;of\;the\;floating-\;rate\;note,\;or\;the\;number\;of\;payment\;periods\;per\;year\\DM=\;Discount/required\;margin\;(stated\;as\;an\;annual\;percentage\;rate)\\N=Number\;of\;evenly\;spaced\;periods\;to\;maturity$$

Money market instruments

$$\\PV=FV\times\left(1-\frac{Days}{Year}\times DR\right)\\FV=PV+\left(PV\times\frac{180}{365}\times AOR\right)$$

$$PV=Present\;value\;(price)\;of\;the\;money\;market\;instrument\\FV=\;Future\;value\;(face/par\;value)\;of\;the\;money\;market\;instrument\\Days=Number\;of\;days\;between\;settlement\;and\;maturity\\Year=Number\;of\;days\;in\;the\;year\\DR=Discount\;rate\;(stated\;as\;an\;annual\;percentage\;rate)\\AOR=Add-on\;rate\;(stated\;as\;an\;annual\;percentage\;rate)\\\\$$

Forward rates

$$\\\left(1+Z_A\right)^A+\left(1+IFR_{A,B-A}\right)^{B-A}=\left(1+Z_B\right)^B\\Z_N=Spot\;rate\\IFR=Implied\;forward\;rate\\\\$$

Leverage ratio

$$\\Leverage\;ratio=\frac1{Initial\;Margin\;Requirement}\\1=100\%\\Initial\;margin\;requirement=x\%\\\\$$

Margin call price

$$\\Margin\;call\;price=P_0\left(\frac{1-Initial\;margin\;requirement}{1-Maintenance\;margin\;requirement}\right)\\P_0=Initial\;purchase\;price\\\\$$

Value of price return index

$$\\V_{PRI}=\frac{\sum_{i=1}^Nn_iP_i}D\\\\$$

$$V_{PRI}=Value\;of\;the\;price\;return\;index\\n_i=Number\;of\;units\;of\;constituent\;security\;ii\;held\;in\;the\;index\;portfolio\\N=Number\;of\;constituent\;securities\;in\;the\;index\\P_i=Unit\;price\;of\;constituent\;security\\D=Value\;of\;the\;divisor$$

Price return of the index portfolio

$$\\PR_I=\frac{V_{PRI1}-V_{PRI0}}{V_{PRI0}}=\sum_{i=1}^Nw_iPR_i=\sum_{i=1}^Nw_i\left(\frac{P_{i1}-P_{i0}}{P_{i0}}\right)\\\\\\$$

$$V_{PRI1}=\;Value\;of\;the\;price\;return\;index\;at\;the\;end\;of\;the\;period\\V_{PRI0}=Value\;of\;the\;price\;return\;index\;at\;the\;beginning\;of\;the\;period\\PR_i=Price\;return\;of\;constituent\;security\;i\\N=Number\;of\;individual\;securities\;in\;the\;index\\w_i=Weight\;of\;security\;ii\;(the\;fraction\;of\;the\;index\;portfolio\;allocated\;to\;security\;i\\P_{i1}=Price\;of\;constituent\;security\;i\;at\;the\;end\;of\;the\;period\\P_{i0}=Price\;of\;constituent\;security\;i\;at\;the\;beginning\;of\;the\;period$$

Total return of an index

$$\\TR_I=\frac{V_{PRI1}-V_{PRI0}+I}{V_{PRI0}}=\sum_{i=1}^Nw_iTR_i=\sum_{i=1}^Nw_i\left(\frac{P_{1i}-P_{0i}+Inc_i}{P_{0i}}\right)\\\\\\$$

$$TR_I=Total\;return\;of\;the\;index\;portfolio\\V_{PRI1}=Value\;of\;the\;price\;return\;index\;at\;the\;end\;of\;the\;period\\V_{PRI0}=Value\;of\;the\;price\;return\;index\;at\;the\;beginning\;of\;the\;period\\Inc_i=Total\;income\;(dividends\;and/or\;interest)\;from\;all\;securities\;in\;the\;index\;held\;over\;the\;period\\TR_i=Total\;return\;of\;constituent\;security\;i\\w_i=Weight\;of\;security\;i(the\;fraction\;of\;the\;index\;portfolio\;allocated\;to\;security\;i)\\N=Number\;of\;securities\;in\;the\;index$$

Value of price return index (Multiple periods)

$$V_{PRIT}=V_{PRI0}(1+PR_{I1})(1+PR_{I2})...(1+PR_{IT})$$

$$V_{PRI0}=Value\;of\;the\;price\;return\;index\;at\;inception\\V_{PRIT}=Value\;of\;the\;price\;index\;at\;time\;t\\PR_{IT}=Price\;return\;on\;the\;index\;over\;period\;t.t=1,2,...,T$$

Value of the total return index (Multiple periods)

$$V_{TRIT}=V_{TRI0}(1+TR_{I1})(1+TR_{I2})...(1+TR_{IT})$$

$$V_{TRI0}=Value\;of\;the\;index\;at\;inception\\V_{TRIT}=Value\;of\;the\;total\;return\;index\;at\;time\;t\\TR_{IT}=Total\;return\;on\;the\;index\;over\;period\;t.t=1,2,...,T$$

Price weighting

$$\\w_i^P={\textstyle\frac{P_i}{\sum_{i=1}^NP_i}}$$

$$w_i=\;Weight\;of\;security\;i\\P_i=Share\;price\;of\;security\;i\\N=Number\;of\;securities\;in\;the\;index$$

Equal weighting

$$\\w_i^E={\textstyle\frac1N}\\w_i=\;Weight\;of\;security\;i\\N=Number\;of\;securities\;in\;the\;index$$

Market-capitalization weighting

$$\\w_i^M={\textstyle\frac{Q_iP_i}{\sum_{j=1}^NQ_jP_j}}\\w_i=\;Weight\;of\;security\;i\\Q_i=\;Number\;of\;shares\;outs\tan ding\;of\;security\;i\\P_i=Share\;price\;of\;security\;i\\N=Number\;of\;securities\;in\;the\;index$$

Float-adjusted market-capitalization weighting

$$\\w_i^M={\textstyle\frac{f_iQ_iP_i}{\sum_{j=1}^Nf_iQ_jP_j}}\\f_i=Fraction\;of\;shares\;outs\tan ding\;in\;the\;market\;float\\w_i=\;Weight\;of\;security\;i\\Q_i=\;Number\;of\;shares\;outs\tan ding\;of\;security\;i\\P_i=Share\;price\;of\;security\;i\\N=Number\;of\;securities\;in\;the\;index$$

Fundamental weighting

$$\\w_i^F={\textstyle\frac{F_i}{\sum_{j=1}^NF_j}}\\w_i=\;Weight\;of\;security\;i\\F_i=Fundamental\;size\;measure\;of\;company\;i$$

Diversification ratio

$$\\Diversification\mathit\;ratio\mathit=\frac{\mathit\sigma\mathit\;\mathit o\mathit f\mathit\;\mathit e\mathit q\mathit u\mathit a\mathit l\mathit l\mathit y\mathit\;\mathit w\mathit e\mathit i\mathit g\mathit h\mathit t\mathit e\mathit d\mathit\;\mathit p\mathit o\mathit r\mathit t\mathit f\mathit o\mathit l\mathit i\mathit o\mathit\;\mathit o\mathit f\mathit\;\mathit n\mathit\;\mathit s\mathit e\mathit c\mathit u\mathit r\mathit i\mathit t\mathit i\mathit e\mathit s}{\mathit\sigma\mathit\;\mathit o\mathit f\mathit\;\mathit s\mathit i\mathit n\mathit g\mathit l\mathit e\mathit\;\mathit s\mathit e\mathit c\mathit u\mathit r\mathit i\mathit t\mathit y\mathit\;\mathit s\mathit e\mathit l\mathit e\mathit c\mathit t\mathit e\mathit d\mathit\;\mathit a\mathit t\mathit\;\mathit r\mathit a\mathit n\mathit d\mathit o\mathit m}\\\sigma\mathit=Volatility\mathit\;\mathit(Standard\mathit\;deviation\mathit)$$

Net asset value per share

$$\\Net\mathit\;asset\mathit\;value\mathit\;per\mathit\;share\mathit=\frac{\mathit F\mathit u\mathit n\mathit d\mathit\;\mathit A\mathit s\mathit s\mathit e\mathit t\mathit s\mathit-\mathit F\mathit u\mathit n\mathit d\mathit\;\mathit L\mathit i\mathit a\mathit b\mathit i\mathit l\mathit i\mathit t\mathit i\mathit e\mathit s}{\mathit N\mathit u\mathit m\mathit b\mathit e\mathit r\mathit\;\mathit o\mathit f\mathit\;\mathit S\mathit h\mathit a\mathit r\mathit e\mathit s\mathit\;\mathit O\mathit u\mathit t\mathit s\mathit{tan}\mathit d\mathit i\mathit n\mathit g\mathit\;}$$

Holding Period Return (HPR) – No cash flows

$$\\HPR\mathit=\frac{\mathit E\mathit n\mathit d\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e\mathit-\mathit B\mathit e\mathit g\mathit i\mathit n\mathit n\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e}{\mathit B\mathit e\mathit g\mathit i\mathit n\mathit n\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e}$$

Holding Period Return (HPR) – Cash flows occur at the end of the period

$$HPR\mathit=\frac{\mathit E\mathit n\mathit d\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e\mathit-\mathit B\mathit e\mathit g\mathit i\mathit n\mathit n\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e\mathit+\mathit C\mathit a\mathit s\mathit h\mathit\;\mathit f\mathit l\mathit o\mathit w\mathit s\mathit\;\mathit r\mathit e\mathit c\mathit e\mathit i\mathit v\mathit e\mathit d}{\mathit B\mathit e\mathit g\mathit i\mathit n\mathit n\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e}\mathit=\frac{{\mathit P}_{\mathit1}\mathit-{\mathit P}_{\mathit0}\mathit+{\mathit D}_{\mathit1}}{\mathit B\mathit e\mathit g\mathit i\mathit n\mathit n\mathit i\mathit n\mathit g\mathit\;\mathit v\mathit a\mathit l\mathit u\mathit e}$$

Holding Period Return (HPR) – Multiple years

$$HPR=\left[\left(1+R_1\right)\times\left(1+R_2\right)\right]-1\\R_1=Holding\;period\;return\;in\;year\;1\\R_2=Holding\;period\;return\;in\;year\;2$$

Arithmetic mean return

$$\\\overline{R_i}=\frac{{\mathrm R}_{\mathrm i1}+{\mathrm R}_{\mathrm i2}+...+{\mathrm R}_{\mathrm{iT}-1}+{\mathrm R}_{\mathrm{iT}}}{\mathrm t}=\frac1T\sum_{t=1}^T{\mathrm R}_{\mathrm{it}}\\\overline{{\mathrm R}_{\mathrm i}}=\mathrm{Arithmetic}\;\mathrm{mean}\;\mathrm{return}\\{\mathrm R}_{\mathrm{it}}=\mathrm{Return}\;\mathrm{in}\;\mathrm{period}\;\mathrm t\\\mathrm T=\mathrm{Total}\;\mathrm{number}\;\mathrm{of}\;\mathrm{periods}$$

Geometric mean return

$$\\\overline{R_{Gi}}=\sqrt{\left(1+R_{i1}\right)\times\left(1+R_{i2}\right)\times...\times\left(1+R_{i,T-1}\right)\times\left(1+R_{i,T}\right)-1}=\sqrt[T]{\prod_{t=1}^T\left(1+R_{iT}\right)-1}$$

Internal Rate of Return (IRR)

$$\\\sum_{t=0}^N\frac{CF_t}{\left(1+IRR\right)^t}=0\\t=Number\;of\;periods\\CF_t=Cash\;flow\;at\;time\;t$$

Time-weighted rate of return

$$\\r_{TW}=\left[\left(1+r_1\right)\times\left(1+r_2\right)\times...\times\left(1+r_N\right)\right]^\frac1N-1\\r_N=Holding\;period\;return\;in\;year\;n$$

Annualized return

$$\\r_{annual}=\left(1+r_{period}\right)^c-1\\r=Periodic\;return\\c=Number\;of\;periods\;in\;a\;year$$

Nominal rate of return

$$\mathit(\mathit1\mathit+r\mathit)\mathit=\mathit(\mathit1\mathit+r_{\mathit r\mathit F}\mathit)\mathit\times\mathit(\mathit1\mathit+\pi\mathit)\mathit\times\mathit(\mathit1\mathit+RP\mathit)\\r_{\mathit r\mathit F}\mathit=Real\mathit\;risk\mathit-free\mathit\;rate\mathit\;of\mathit\;return\\\pi\mathit=\mathit\;Inflation\\RP\mathit=Risk\mathit\;premium\\\boldsymbol \\\\$$

Real rate of return

$$\\\mathit(\mathit1\mathit+r_{\mathit r\mathit e\mathit a\mathit l}\mathit)\mathit=\mathit(\mathit1\mathit+r_{\mathit r\mathit F}\mathit)\mathit\times\mathit(\mathit1\mathit+RP\mathit)\mathit=\frac{\mathit(\mathit1\mathit+\mathit r\mathit)}{\mathit(\mathit1\mathit+\mathit\pi\mathit)}\\r_{\mathit r\mathit F}\mathit=Real\mathit\;risk\mathit-free\mathit\;rate\mathit\;of\mathit\;return\\\pi\mathit=\mathit\;Inflation\\RP\mathit=Risk\mathit\;premium\\\\\\$$

Population variance

$$\\\sigma^{\mathit2}\mathit=\frac{{\displaystyle\overset{\mathit N}{\underset{\mathit i\mathit=\mathit1\mathit.\mathit.\mathit.\mathit n}{\mathit\sum}}}{\mathit({\mathit x}_{\mathit i}\mathit-\mathit\mu\mathit)}^{\mathit2}}{\mathit N}\\x_{\mathit i}\mathit=Return\mathit\;for\mathit\;period\mathit\;i\\N\mathit=\mathit\;Total\mathit\;number\mathit\;of\mathit\;periods\\\mu\mathit=Mean\\\\\\$$

Population standard deviation

$$\\\sqrt{\mathit\sigma\mathit=\frac{\mathit{\displaystyle\sum_{i=1...n}^N{(x_i-\mu)}^2}}{\mathit N}}\\x_{\mathit i}\mathit=Return\mathit\;for\mathit\;period\mathit\;i\\N\mathit=\mathit\;Total\mathit\;number\mathit\;of\mathit\;periods\\\mu\mathit=Mean\\\\\\$$

Sample variance

$$\\S^{\mathit2}\mathit=\frac{\mathit{\displaystyle\sum_{i=1...n}^N{(x_i-\overline x)}^2}}{\mathit n\mathit-\mathit1}\\x_{\mathit i}\mathit=Return\mathit\;for\mathit\;period\mathit\;i\\N\mathit=\mathit\;Total\mathit\;number\mathit\;of\mathit\;periods\\\overset{\mathit¯}{\mathit x}\mathit=Mean\mathit\;of\mathit\;n\mathit\;returns\\\\\\$$

Sample standard deviation

$$\\s\mathit=\frac{{\displaystyle\overset{\mathit n}{\underset{\mathit i\mathit=\mathit1\mathit.\mathit.\mathit.\mathit n}{\mathit\sum}}}\mathit{\left({x_i-\overline x}\right)}^{\mathit2}}{\mathit n\mathit-\mathit1}\\x_{\mathit i}\mathit=Return\mathit\;for\mathit\;period\mathit\;i\\N\mathit=Total\mathit\;number\mathit\;of\mathit\;periods\\\overset{\mathit¯}{\mathit x}\mathit=\mathit\;Mean\mathit\;of\mathit\;n\mathit\;returns\\$$

Covariance

$$\\COV_{\mathit1\mathit,\mathit2}\mathit=\frac{\overset{\mathit n}{\underset{\mathit t\mathit=\mathit1}{\mathit\sum}}\mathit\{\mathit\lbrack{\mathit R}_{\mathit t\mathit,\mathit1}\mathit-\overline{{\mathit R}_{\mathit1}}\mathit\rbrack\mathit\lbrack{\mathit R}_{\mathit t\mathit,\mathit2}\mathit-\overline{{\mathit R}_{\mathit2}}\mathit\rbrack\mathit\}}{\mathit n\mathit-\mathit1}\mathit=\rho_{\mathit1\mathit,\mathit2}\sigma_{\mathit1}\sigma_{\mathit2}\\R_{\mathit t\mathit,\mathit1}\mathit=Return\mathit\;on\mathit\;Asset\mathit\;\mathit1\mathit\;in\mathit\;period\mathit\;t\\R_{\mathit t\mathit,\mathit2}\mathit=Return\mathit\;on\mathit\;Asset\mathit\;\mathit2\mathit\;in\mathit\;period\mathit\;t\\\rho\mathit=Correlation\\\overline{\mathit R}\mathit=Mean\mathit\;of\mathit\;respective\mathit\;assets\\$$

Correlation

$$\\\rho_{\mathit1\mathit,\mathit2}\mathit=\frac{\mathit C\mathit O{\mathit V}_{\mathit1\mathit,\mathit2}}{{\mathit\sigma}_{\mathit1}{\mathit\sigma}_{\mathit2}}\\\sigma\mathit=S\mathit{tan}dard\mathit\;deviation\\$$

Utility function

$$\\U\mathit=E\mathit(r\mathit)\mathit-\frac{\mathit1}{\mathit2}A\sigma^{\mathit2}\\U=Utility\;of\;an\;investment\\E\mathit(r\mathit)\mathit=\mathit\;Expected\mathit\;return\\\sigma^{\mathit2}\mathit=Variance\mathit\;of\mathit\;the\mathit\;investment\\A\mathit=Risk\mathit\;aversion\mathit\;level$$

Portfolio return (Many risky assets)

$$\\R_{\mathit P}\mathit=\overset{\mathit N}{\underset{\mathit i\mathit=\mathit1}{\mathit\sum}}w_{\mathit i}R_{\mathit i}\mathit,\overset{\mathit N}{\underset{\mathit i\mathit=\mathit1}{\mathit\sum}}w_{\mathit i}\mathit=\mathit1\\R_{\mathit i}\mathit=Return\mathit\;of\mathit\;asset\mathit\;i\\w_{\mathit i}\mathit=Weight\mathit\;within\mathit\;the\mathit\;portfolio$$

Portfolio variance

$$\\\sigma_{\mathit P}^{\mathit2}\mathit=\overset{\mathit N}{\underset{\mathit i\mathit,\mathit j\mathit=\mathit1}{\mathit\sum}}w_{\mathit i}w_{\mathit j}COV\mathit(R_{\mathit i}\mathit,R_{\mathit j}\mathit)\\w\mathit=Weights\\R\mathit=Returns$$

Portfolio variance (Two-asset portfolio)

$$\\\sigma_{\mathit P}^{\mathit2}\mathit=w_{\mathit1}^{\mathit2}\sigma_{\mathit1}^{\mathit2}\mathit+w_{\mathit2}^{\mathit2}\sigma_{\mathit2}^{\mathit2}\mathit+\mathit2w_{\mathit1}w_{\mathit2}COV\mathit(R_{\mathit1}\mathit,R_{\mathit2}\mathit)$$

Portfolio standard deviation (Two-asset portfolio)

$$\\\sigma_{\mathit P}\mathit=\sqrt{\mathit w_{\mathit1}^{\mathit2}\mathit\sigma_{\mathit1}^{\mathit2}\mathit+\mathit w_{\mathit2}^{\mathit2}\mathit\sigma_{\mathit2}^{\mathit2}\mathit+\mathit2{\mathit w}_{\mathit1}{\mathit w}_{\mathit2}\mathit C\mathit O\mathit V\mathit({\mathit R}_{\mathit1}\mathit,{\mathit R}_{\mathit2}\mathit)}$$

Portfolio return of two assets (when one asset is the risk-free asset)

$$\\E\mathit(R_{\mathit P}\mathit)\mathit=w_{\mathit1}R_{\mathit f}\mathit+\mathit(\mathit1\mathit-w_{\mathit1}\mathit)E\mathit(R_{\mathit i}\mathit)\\R_{\mathit f}\mathit=Returns\mathit\;of\mathit\;respective\mathit\;asset\\R_{\mathit i}\mathit=Returns\mathit\;of\mathit\;respective\mathit\;asset\\w_{\mathit1}\mathit=Weight\mathit\;in\mathit\;asset\mathit\;\mathit1\\\mathit1\mathit-w_{\mathit1}\mathit=w_{\mathit2}$$

Portfolio standard deviation of two assets (when one asset is the risk-free asset)

$$\\\sigma_P=\sqrt{w_1^2\sigma_f^2+{(1-w_1)}^2\sigma_i^2+2w_1(1-w_1)\rho_{1,2}\sigma_f\sigma_i}=(1=w_1)\sigma_i\\f=Risk-free\;asset\\i=Asset\\\sigma=S\tan dard\;deviation\\w=Weight$$

Capital Asset Pricing Model (CAPM)

$$E\mathit(R_{\mathit i}\mathit)\mathit=R_{\mathit F}\mathit+\beta_{\mathit i}\mathit{\left[{E(R_M)-R_F}\right]}\\\beta_{\mathit i}\mathit=Return\mathit\;sensitivity\mathit\;of\mathit\;stock\mathit\;i\mathit\;to\mathit\;changes\mathit\;in\mathit\;the\mathit\;market\mathit\;return\\E\mathit(R_{\mathit M}\mathit)\mathit=Expected\mathit\;return\mathit\;on\mathit\;the\mathit\;market\\E\mathit(R_{\mathit M}\mathit)\mathit-R_{\mathit F}\mathit=Expected\mathit\;market\mathit\;risk\mathit\;premium\\R_{\mathit F}\mathit=Risk\mathit-free\mathit\;rate\mathit\;of\mathit\;interest$$

Capital allocation line

$$E\mathit(R_{\mathit p}\mathit)\mathit=R_{\mathit f}\mathit+\mathit{\left(\frac{E(R_M)-R_f}{\sigma_m}\right)}\mathit\times\sigma_{\mathit p}\\E\mathit(R_{\mathit M}\mathit)\mathit=Expected\mathit\;return\mathit\;of\mathit\;the\mathit\;market\mathit\;portfolio\\R_{\mathit f}\mathit=Risk\mathit-free\mathit\;rate\mathit\;of\mathit\;return\\\sigma_{\mathit m}\mathit=S\mathit{tan}dard\mathit\;deviation\mathit\;of\mathit\;the\mathit\;market\mathit\;portfolio\\\sigma_{\mathit p}\mathit=S\mathit{tan}dard\mathit\;deviation\mathit\;of\mathit\;the\mathit\;portfolio\mathit\;P$$

Expected return (multifactor model)

$$E\mathit(R_{\mathit i}\mathit)\mathit-R_{\mathit f}\mathit=\beta_{\mathit i\mathit1}\mathit\times E\mathit(Factor\mathit\;\mathit1\mathit)\mathit+\beta_{\mathit i\mathit2}\mathit\times E\mathit(Factor\mathit\;\mathit2\mathit)\mathit+\mathit.\mathit.\mathit.\mathit+\beta_{\mathit i\mathit k}\mathit\times E\mathit(Factor\mathit\;k\mathit)\\\beta_{\mathit i\mathit k}\mathit=Stock\mathit\;i\mathit’s\mathit\;sensitivity\mathit\;to\mathit\;changes\mathit\;in\mathit\;the\mathit\;k^{\mathit t\mathit h}\mathit\;factor\\\mathit(Factor\mathit\;k\mathit)\mathit=Expected\mathit\;risk\mathit\;premium\mathit\;for\mathit\;the\mathit\;k^{\mathit t\mathit h}\mathit\;factor$$

Beta of an asset

$$\beta_{\mathit i}\mathit=\frac{\mathit C\mathit o\mathit v\mathit({\mathit R}_{\mathit i}\mathit,{\mathit R}_{\mathit m}\mathit)}{\mathit\sigma_{\mathit m}^{\mathit2}}\mathit=\frac{{\mathit\rho}_{\mathit i\mathit,\mathit m}{\mathit\sigma}_{\mathit i}{\mathit\sigma}_{\mathit m}}{\mathit\sigma_{\mathit m}^{\mathit2}}\mathit=\frac{{\mathit\rho}_{\mathit i\mathit,\mathit m}{\mathit\sigma}_{\mathit i}}{{\mathit\sigma}_{\mathit m}}\\\sigma\mathit=S\mathit{tan}dard\mathit\;deviation\\m\mathit=Market\mathit\;portfolio\\i\mathit=Asset\mathit\;portfolio$$

Portfolio beta

$$\beta_{\mathit p}\mathit=\overset{\mathit n}{\underset{\mathit i\mathit=\mathit1}{\mathit\sum}}w_{\mathit i}\beta_{\mathit i}\mathit\;\mathit\;\mathit\;\mathit\;\mathit\;\mathit\;\mathit\;\mathit\;\mathit\;\overset{\mathit n}{\underset{\mathit i\mathit=\mathit1}{\mathit\sum}}w_{\mathit i}\mathit=\mathit1\\$$

Sharpe ratio

$$Sharpe\mathit\;ratio\mathit=\frac{{\mathit R}_{\mathit p}\mathit-{\mathit R}_{\mathit f}}{{\mathit\sigma}_{\mathit p}}\\R_{\mathit p}\mathit=Portfolio\mathit\;return\\R_{\mathit f}\mathit=Risk\mathit-free\mathit\;rate\mathit\;of\mathit\;return\\\sigma_{\mathit p}\mathit=S\mathit{tan}dard\mathit\;deviation\mathit\;\mathit(volatility\mathit)\mathit\;of\mathit\;portfolio\mathit\;return\\$$

M2 ratio

$$M^{\mathit2}\mathit\;ratio\mathit=\mathit(R_{\mathit p}\mathit-R_{\mathit f}\mathit)\frac{{\mathit\sigma}_{\mathit m}}{{\mathit\sigma}_{\mathit p}}\mathit-\mathit(R_{\mathit m}\mathit-R_{\mathit f}\mathit)\\m\mathit=Market\mathit\;portfolio\\$$

Treynor ratio

$$Treynor\mathit\;ratio\mathit=\frac{\mathit E\mathit({\mathit R}_{\mathit p}\mathit)\mathit-{\mathit R}_{\mathit f}}{{\mathit\beta}_{\mathit p}}\\\beta_{\mathit p}\mathit=Portfolio\mathit\;beta\\R_{\mathit p}\mathit=\mathit\;Portfolio\mathit\;return\\R_{\mathit f}\mathit=\mathit\;Risk\mathit-free\mathit\;rate\mathit\;of\mathit\;return\\$$

Jensen’s alpha

$$\alpha_{\mathit p}=R_p-\left[R_f+\beta_p\left(R_m-R_f\right)\right]\\$$

 

 

Leverage Ratio

$$Leverage=\frac{Total\;debt}{Total\;equity}$$

$$This\;is\;one\;of\;several\;definitions\;and\;formulas\;for\;leverage,\\also\;known\;as\;Debt-to-Equity\;ratio$$

Volatility (standard deviation of returns) – population

$$\sigma=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left(R_i-Ravg\right)}n}$$

$$R_i=Individual\;returns\;data\;points\\R_{avg}=Average\;of\;all\;return\;data\;points\;in\;the\;set\\n=Number\;of\;data\;points$$

Volatility (standard deviation of returns) – sample

$$\sigma=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left(R_i-R_{avg}\right)^2}{n-1}}$$

$$R_i=Individual\;returns\;data\;points\\R_{avg}=Average\;of\;all\;return\;data\;points\;in\;the\;set\\n=Number\;of\;data\;points$$

Sharpe Ratio

$$Sharpe\;Ratio=\frac{R_p-R_f}{\sigma_p}$$

$$R_p=Portfolio\;return\\R_f=Risk-free\;rate\;of\;return\\\sigma_p=S\tan dard\;deviation\;(volatility)\;of\;portfolio\;return$$

Sortino Ratio

$$Sortino\;Ratio=\frac{R_p-R_f}{\sigma_d}$$

$$R_p=Portfolio\;return\\R_f=Risk-free\;rate\;of\;return\\\sigma_d=S\tan dard\;deviation\;(volatility)\;of\;the\;downside\;(“downside\;risk”)$$

Downside Risk (semi-deviation) – population

$$\sigma_d=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left(R_i-R_{treshold}\right)^2}n}$$

$$R_i=Individual\;returns\;data\;points\\R_{treshold}=Return\;threshold\;(determined\;by\;the\;user,\\for\;example\;the\;risk-free\;rate,\;hard\;target\;return\;or\;0\%\;can\;be\;used)\\n\;=\;Number\;of\;data\;points$$

Downside Risk (semi-deviation) – sample

$$\sigma_d=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left(R_i-R_{treshold}\right)^2}{n-1}}$$

$$R_i=Individual\;returns\;data\;points\\R_{treshold}=Return\;threshold\;(determined\;by\;the\;user,\;for\;example\;the\\risk-free\;rate,\;hard\;target\;return\;or\;0\%\;can\;be\;used)\\n\;=\;Number\;of\;data\;points$$

Discounted Cash Flow (DCF) = Net Present Value (NPV) of an investment

$$DCF=NPV=\sum_{t=0}^n\frac{CF_t}{\left(1+r\right)^t}\\CF_t=Cash\;flow\;in\;time\;t\\r\;=\;Discount\;rate$$

Capitalization Rate (Cap Rate)

$$Cap\;rate=\frac{Net\;Operating\;Income\;(NOI)}{Market\;value\left(or\;purchase\;price\;of\;property\right)}$$

Funds From Operations (FFO)

$$FFO=Net\;Income+Depreciation\left(and\;other\;non-cash\;items\right)-\\Gains/Losses\;from\;property\;sales\;\left(and\;other\;non-recurring\;items\right)$$

Adjusted Funds From Operations (AFFO)

$$AFFO=FFO-Recurring\;Capital\;Expenditures\;\left(CAPEX\right)\;$$

Net Asset Value per share (NAV per share)

$$NAV\;per\;share=\frac{NAV}{Total\;number\;of\;share\;outstanding}$$

Price Elasticity

$$Price\mathit\;Elasticity\mathit=\frac{\mathit\%\mathit\triangle\mathit Q\mathit u\mathit a\mathit n\mathit t\mathit i\mathit t\mathit y\mathit\;\mathit d\mathit e\mathit m\mathit a\mathit n\mathit d\mathit e\mathit d\mathit({\mathit Q}_{\mathit x}\mathit)}{\mathit\%\mathit\triangle\mathit P\mathit r\mathit i\mathit c\mathit e\mathit({\mathit P}_{\mathit x}\mathit)}\\\mathit0\mathit>e\mathit>\mathit-\mathit1\mathit\rightarrow\mathit\;inelastic\mathit\;demand\\\mathit-\mathit1\mathit>e\mathit>\mathit-\mathit\infty\mathit\rightarrow\mathit\;elastic\mathit\;demand\\e\mathit=\mathit-\mathit1\mathit\rightarrow\mathit\;unit\mathit\;elastic\mathit\;demand\\e\mathit=\mathit0\mathit\rightarrow perfectly\mathit\;inelastic\mathit\;demand\\e\mathit=\mathit-\mathit\infty\mathit\rightarrow\mathit\;perfectly\mathit\;elastic\mathit\;demand\\$$

Income Elasticity

$$Income\mathit\;Elasticity\mathit=\frac{\mathit\%\mathit\triangle\mathit Q\mathit u\mathit a\mathit n\mathit t\mathit i\mathit t\mathit y\mathit\;\mathit d\mathit e\mathit m\mathit a\mathit n\mathit d\mathit e\mathit d\mathit({\mathit Q}_{\mathit x}\mathit)}{\mathit\%\mathit\triangle\mathit I\mathit n\mathit c\mathit o\mathit m\mathit e\mathit({\mathit I}_{\mathit x}\mathit)}\\e\mathit>\mathit0\mathit\rightarrow\mathit\;normal\mathit\;goods\\e\mathit<\mathit0\mathit\rightarrow inferior\mathit\;goods\\\varepsilon_{\mathit Y}\mathit=Income\mathit\;elasticity\\$$

Cross-price Elasticity

$$Cross\mathit-price\mathit\;Elasticity\mathit=\frac{\mathit\%\mathit\triangle\mathit Q\mathit u\mathit a\mathit n\mathit t\mathit i\mathit t\mathit y\mathit\;\mathit d\mathit e\mathit m\mathit a\mathit n\mathit d\mathit e\mathit d\mathit\;\mathit({\mathit Q}_{\mathit x}\mathit)}{\mathit\%\mathit\triangle\mathit P\mathit r\mathit i\mathit c\mathit e\mathit\;\mathit o\mathit f\mathit\;\mathit a\mathit\;\mathit r\mathit e\mathit l\mathit a\mathit t\mathit e\mathit d\mathit\;\mathit g\mathit o\mathit o\mathit d\mathit\;\mathit({\mathit P}_{\mathit y}\mathit)}\\e\mathit>\mathit0\mathit\rightarrow the\mathit\;related\mathit\;product\mathit\;is\mathit\;a\mathit\;substitute\\e\mathit<\mathit0\mathit\rightarrow the\mathit\;related\mathit\;product\mathit\;is\mathit\;a\mathit\;complement\\y\mathit=Related\mathit\;product\\\varepsilon_{\mathit p\mathit y}\mathit=Cross\mathit-price\mathit\;elasticity\\$$

For all market structures, $$Max\mathit\;Profit\mathit\longrightarrow when\mathit\;MC\mathit=MR\\$$

$$MC\mathit=Marginal\mathit\;\mathit{cos}t\\MR\mathit=Marginal\mathit\;revenue\\$$

Breakeven points:

$$AR\mathit=ATC\mathit\;\mathit(perfect\mathit\;competition\mathit)\\TR\mathit=TC\mathit\;\mathit(imperfect\mathit\;competition\mathit)\\ATC\mathit=\mathit\;Average\mathit\;Total\mathit\;Cost\\AR\mathit=\mathit\;Average\mathit\;Revenue\\TR\mathit=\mathit\;Total\mathit\;Revenue\\TC\mathit=\mathit\;Total\mathit\;Cost\\AR\mathit=ATC\mathit\;holds\mathit\;true\mathit\;in\mathit\;imperfect\mathit\;competition$$

Short-run shutdown points:

$$AR\mathit<AVC\mathit\;\mathit(perfect\mathit\;competition\mathit)\\TR\mathit<TVC\mathit\;\mathit(imperfect\mathit\;competition$$

Market structures:

$$Perfect\mathit\;Competition\\Monopolistic\mathit\;Competition\\Oligopoly\\Monopoly$$

Total GDP = final value of goods and services produced (market value)+ government services (at cost) + rental value of owner-occupied housing (an estimate)

$$GDP\mathit\;Deflator\mathit=\mathit\;\frac{\mathit N\mathit o\mathit m\mathit i\mathit n\mathit a\mathit l\mathit\;\mathit G\mathit D\mathit P}{\mathit R\mathit e\mathit a\mathit l\mathit\;\mathit G\mathit D\mathit P}\mathit\;\mathit​\mathit\;\mathit\;\mathit\times\mathit{100}$$

$$Nominal\mathit\;GDP_t=P_t\times Q_t\\Real\mathit\;GDP_t=P_b\times Q_t\\t=Current\;year\\b=Base\;year\\P_t=Prices\;in\;year\;\{\_t\}\\P_b=Prices\;in\;base\;year\\Q_t=Quantity\;produced\;in\;year\;\{\_t\}$$

Expenditure Approach

$$Real\;GDP=Consumption\;spending\;(C)+Investment\;(I)+Government\;spending\;(G)+Net\;exports\;(X-M)$$

$$X=Exports\\M=Imports$$

Income Approach

$$Real\;GDP=National\;income+Capital\;consumption\;allowance+Statistical\;discrepancy\\Real\;GDP=Consumption\;spending\;(C)+Savings\;(S)+Taxes\;(T)\\Savings\;(S)=Investments\;(I)+Fiscal\;Balance\;(G-T)+Trade\;Balance\;(X-M)\\S–I=Fiscal\;Balance\;(G-T)+Trade\;Balance\;(X-M)$$

National Income =Employees’ compensation

+ Corporate and government profits before taxes

+Interest income

+Unincorporated business net income (business owners’ incomes)

+ Rent

+ Indirect business taxes

− Subsidies

Personal Income = National income

+ Transfer payments (social insurance, unemployment or disability payments)

− Indirect business taxes
− Corporate income taxes
− Undistributed corporate profits

Personal Disposable Income = Personal income – Personal taxes

Potential GDP = Aggregate hours worked × Labor productivity

⟶Aggregate hours worked = Labor force × Average hours worked per week

 Growth in Potential GDP = Growth in labor force + Growth in labor productivity

The Production Function

$$Y=A\times f(K,L)\\Y=Aggregate\;output\\A=Total\;Factor\;Productivity\;(TFP)\\K=Capital\\L=\;Labor$$

Growth in Potential GDP = Growth in technology + WL × (growth in labor) + WC × (growth in capital)
WL = Labor’s percentage share of national income
WC = Capital’s percentage share of national income

$$Unemployment\;Rate=\frac{Number\;of\;unemployed\;people\;​\;\;}{Total\;labor\;force\;}\\Participation\;Rate\;(Activity\;Ratio)=\frac{Total\;labor\;force\;​}{\;Total\;working–age\;population\;}\\Labor\;Force=Unemployed\;people+Employed\;people\\Unemployed\;=\;Looking\;for\;job$$

$$Consumer\;Price\;Index=\frac{Cost\;of\;basket\;at\;current–year\;prices}{Cost\;of\;basket\;at\;base–year\;prices}\times100\\Laspeyres’Index=\frac{\Sigma\;(Current–year\;price\times Base–year\;quantity)\;​}{\Sigma\;(Base–year\;price\times Base–year\;quantity)\;​}\\Fisher’s\;Index=\sqrt{(Laspeyres’\;Index)\times(Paashe\;Price\;Index)}\\Paashe\;Price\;Index=\frac{\Sigma\;(Current–year\;price\times Current–year\;quantity)\;​}{\Sigma\;(Base–year\;price\times Base–year\;quantity)\;​}$$

$$Money\;Multiplier=\frac1{\;Reserve\;requirement}\\Fiscal\;Multiplier=\frac1{1-MPC\times(1-t)}\\MPC=Marginal\;propensity\;to\;consume\\t=Tax\;rate$$

Equation of Exchange

$$MV=PY\;(Money\;supply\times Velocity=Price\times Real\;output)$$

Fisher Effect

$$Nominal\;Interest\;Rate=Real\;interest\;rate+Expected\;inflation\;rate$$

Neutral Interest Rate

$$Neutral\;interest\;rate=Real\;trend\;rate\;of\;economic\;growth+inflation\;target$$

GDP

$$GDP=C+I+G+X-M\\C=Consumption\\I=Investments\\G=Government\;Spending\\X=Export\\M=\;Import$$

Balance of Payments

$$Current\;Account+Capital\;Account+Financial\;Account=0$$

Trade Balance

$$X-M=Private\;Savings+Government\;Savings-Investments\;in\;domestic\;capital$$

$$Real\;Exchange\;Rate=Nominal\;exchange\;rate\times\frac{CPI\;base\;currency}{CPI\;price\;currency}$$

Net present value (NPV)

$$NPV=\sum_{t=0}^n\frac{CF_t}{\left(1+r\right)^t}$$

$$CF_t=After-tax\;cash\;flow\;at\;time\;t\\r\;=\;Required\;rate\;of\;return\;for\;the\;investment$$

Internal Rate of Return (IRR)

$$\sum_{t=0}^N\frac{CF_t}{\left(1+IRR\right)^t}=0$$

Average Accounting Rate of Return (AAR)

$$AAR=\frac{Average\;net\;income}{Average\;book\;value}$$

Profitability Index (PI)

$$PI=\;\frac{\;Initial\;Investment\;PV\;of\;future\;cash\;flows}{Initial\;investment}=1+\frac{NPV}{Initial\;investment}\;​\;\;$$

Weighted Average Cost of Capital (WACC)

$$WACC=w_dr_d\left(1-t\right)+w_pr_p+w_er_e\\w_d=Proportion\;of\;debt\;that\;the\;company\;uses\;when\;\\it\;raises\;new\;funds\\r_d=Before-tax\;marginal\;\cos t\;of\;debt\\t\;=\;Company’s\;marginal\;tax\;rate\\r_p=The\;marginal\;\cos t\;of\;preferred\;stock\\w_e=Proportion\;of\;equity\;that\;the\;company\;uses\;when\\it\;raises\;new\;funds\\r_e=Marginal\;\cos t\;of\;equity$$

Tax Shield

$$Tax\;shield=Deducation\times Tax\;rate\\$$

Cost of Preferred Stock

$$r_p=\frac{D_p}{P_p}\\$$

$$P_p=Current\;preferred\;stock\;price\;per\;share\\D_p=Preferred\;stock\;dividend\;per\;share\\r_p=Cost\;of\;preferred\;stock\\$$

Cost of Equity (Dividend discount model approach)

$$r_e=\frac{D_1}{P_0}+g$$

$$P_0=Current\;market\;value\;of\;the\;equity\;market\;index\\D_1=Dividends\;expected\;next\;period\;on\;the\;index\\r_e=\;Required\;rate\;of\;return\;on\;the\;market\\g=Expected\;growth\;rate\;of\;dividends$$

Growth Rate

$$g=\left(1-\frac D{EPS}\right)\times ROE\\\frac D{EPs}=Assumed\;stable\;dividend\;payout\;ratio\\ROE=Historical\;return\;on\;equity$$

Cost of Equity (Bond yield plus risk premium)

$$r_e=r_d+Risk\;premium\\Risk\;premium\;=\;the\;additional\;yield\;on\;a\;company’s\;stock\;relative\;\\to\;its\;bonds$$

Capital Asset Pricing Model (CAPM)

$$E\left(R_i\right)=R_F+\beta_i\left[E\left(R_M\right)-R_F\right]\\\beta_i=The\;return\;sensitivity\;of\;stock\;i\;to\;changes\;in\;the\;market\;return\\E\left(R_M\right)=The\;expected\;return\;on\;the\;market\\E\left(R_M\right)-R_F=\;The\;expected\;market\;risk\;premium\\R_F=\;Risk-free\;rate\;of\;interest$$

Beta of a Stock

$$\beta_i=\frac{Cov\left(R_{i,}R_M\right)}{Var\left(R_M\right)}$$

$$R_M=Average\;expected\;rate\;of\;return\;on\;the\;market\\R_i=Expected\;return\;on\;an\;asset\;i\\Cov=Covariance\\Var=Variance$$

Pure-play Method Project Beta (De-lever)

$$\beta_{Unlevered\left(Comparable\right)}=\frac{\beta_{Levered,Comparable}}{\left[1+\left(1-t_{Comparable}\right){\displaystyle\frac{D_{Comparable}}{E_{Comparable}}}\right]}$$

$$t\;=\;Tax\;rate\\D\;=\;Debt\\E\;=\;Equity$$

Pure-play Method for Subject Firm (Re-lever)

$$\beta_{Levered,Project}=\beta_{Unlevered,Comparable}\left[1+\left(\left(1-t_{project}\right)\frac{D_{Project}}{E_{project}}\right)\right]$$

Adjusted CAPM (for country risk premium)

$$E\left(R_i\right)=R_F+\beta_i\left[E\left(R_M\right)-R_F+Country\;risk\;premium\right]$$

Country Risk Premium

$$CRP=Sovereign\;yield\;spread\times\\\left(\frac{\sigma\;of\;equity\;index\;of\;the\;developing\;country}{\sigma\;of\;sovereign\;bond\;market\;in\;terms\;of\;the\;developed\;market\;currency}\right)$$

$$\sigma\;=\;S\tan dard\;deviation$$

Break Point

$$Break\;point=\frac{Amount\;of\;capital\;at\;which\;the\;source’s\;\cos t\;of\;capital\;changes}{Proportion\;of\;new\;capital\;raised\;from\;the\;source}$$

Degree of Operating Leverage

$$Degree\;of\;Operating\;Leverage=\frac{Percentage\;change\;in\;operating\;income\;​}{Percentage\;change\;in\;unit\;sold}$$

Degree of Financial Leverage

$$Degree\;of\;Financial\;Leverage=\frac{Percentage\;change\;in\;Net\;Income}{Percentage\;change\;in\;EBIT}$$

Return on Equity (ROE)

$$Return\;on\;Equity=\frac{Net\;income}{Shareholders'\;Equity}$$

The Breakeven Quantity of Sales

$$Q_{Breakeven}=\frac{F+C}{P-V}\\P=Price\;per\;unit\\V=Variable\;\cos t\;per\;unit\\F=Fixed\;operating\;\cos ts\\C=Fixed\;financial\;\cos t\\Q=Quantity\;of\;units\;produced\;and\;sold$$

Operating Breakeven Quantity of Sales

$$Q_{Operating\;Breakeven}=\frac F{P-V}$$

$$P\;=\;Price\;per\;unit\\V\;=\;Variable\;\cos t\;per\;unit\\F\;=\;Fixed\;operating\;costs$$

Current Ratio

$$Current\;Ratio=\frac{Current\;assets}{current\;liabilities}$$

Quick Ratio

$$Quick\;Ratio=\frac{Cash+Receivables+Short-trem\;marketable\;investment}{Current\;liabilities}$$

Accounts Receivable Turnover

$$Accounts\;Receivable\;Turnover=\frac{Credit\;sales}{Average\;receivables\;}$$

Number of Days of Receivables

$$Number\;of\;days\;of\;recebvables=\frac{365}{Accounts\;receivable\;turnover}$$

Inventory Turnover

$$Inventory\;Turnover=\frac{Cost\;of\;goods\;sold}{Average\;inventory}$$

Number of Days of Inventory

$$Number\;of\;days\;of\;Inventory=\frac{365}{Inventory\;turnover}$$

Payables Turnover

$$Payables\;Turnover\;Ratio=\frac{Purchases}{Average\;accounts\;payables}$$

Number of Days of Payables

$$Number\;of\;days\;of\;Payables=\frac{365}{Payables\;turnover\;ratio}$$

Net Operating Cycle

$$Net\;operating\;cycle=\\Number\;of\;days\;of\;inventory+Number\;of\;days\;of\;receivables-\\Number\;of\;days\;of\;payables$$

Yield on a Bank Discount Basis (BDY)

$$r_{BD}=\frac DF\times\frac{360}t$$

 $$D\;=\;Dollar\;discount,\;which\;is\;equal\;to\;the\;difference\;between\;the\\face\;value\;of\;the\;bill\;(F)\;and\;its\;purchase\;price\;(P_0)\\F\;=\;Face\;value\;of\;the\;T-bill\\t\;=\;Actual\;number\;of\;days\;remaining\;to\;maturity\\r_{BD}=\;Annualized\;yield\;on\;a\;bank\;discount\;basis\\$$

Effective Annual Yield (EAY)

$$EAY=\left(1+HPR\right)^\frac{360}t-1$$

Holding Period Return

$$Cost\;oftrade\;credit=\left(1+\frac{\%Discount}{1-\%Discount}\right)^\frac{360}{Number\;of\;days\;past\;discount}-1\;\;$$

Cost of Borrowing

$$Cost\;of\;borrowing=\frac{Interest+Dealer's\;commission+other\;costs}{Loan\;amount-Interest}$$

Basic Accounting Equation

$$Assets=Liabilities+Equity$$

Net income

$$Net\;Income=Revenue-Expenses$$

Gross profit (income)

$$Gross\;profit\;(income)=Revenue-Cost\;of\;goods\;sold$$

Operating profit (income)

$$Operating\;profit\;(income)=Profit\;before\;interest\;and\;tax\;(PBIT)=Gross\;profit-Operating\;expenses$$

Profit Before Tax (PBT)

$$Profit\;before\;tax\;(PBT)=PBIT-Interest\;expense$$

Net profit (income)

$$Net\;profit\;(income)=PBT-Tax\;expense=Operating\;profit-Interest\;Expense-Tax\;expense$$

Basic Earnings per Share (EPS)

$$Basic\;EPS=\frac{Net\;Income-Preferred\;Dividends}{Weighted\;average\;number\;of\;common\;shares\;outs\tan ding}$$

Diluted Earnings per Share (DEPS)

$$Diluted\;EPS=\frac{Adjusted\;income\;available\;for\;common\;shares}{Weighted\;average\;common\;and\;potential\;common\;shares\;outs\tan ding}$$

$$\\Diluted\;EPS=$$

$$\frac{\;\lbrack Net\;Income-Preferred\;dividends\rbrack+\lbrack Convertible\;preferred\;dividends\rbrack+\lbrack Convertible\;debt\;interest\rbrack(1-t)}{(Weighted\;average\;shares)+(Shares\;from\;conv.\;pfd.\;shares)+(Shares\;from\;conversion\;of.\;conv.\;debt)+(Shares\;issuable\;from\;stock\;options)}$$

Gross profit margin

$$Gross\;profit\;margin=\frac{Gross\;profit\;​}{\;\;Revenue\;}$$

Net profit margin

$$Net\;profit\;margin=\frac{Net\;profit\;​}{\;\;Revenue\;}$$

Liquidity Ratios

Current ratio

$$Current\;ratio=\frac{Current\;assets\;​\;​}{Current\;liabilities\;}$$

Quick ratio

$$Quick\;ratio=\frac{Cash+Short–term\;marketable\;securities+Receivables\;​​\;​}{Current\;liabilities\;}$$

Cash ratio

$$Cash\;ratio=\frac{Cash+Short–term\;marketable\;securities}{Current\;liabilities\;}$$

 

Solvency Ratios

Long-term debt-to-equity

$$Long–term\;debt–to–equity=\frac{Long–term\;debt\;}{Total\;equity\;​}\;$$

Total debt-to-equity

$$Total\;debt-to-equity=\frac{Total\;debt\;}{Total\;equity\;​}\;$$

Debt ratio

$$Debt\;ratio=\frac{Total\;debt\;}{Total\;assets}\;$$

Financial leverage

$$Financial\;leverage=\frac{Total\;\;assets}{Total\;equity}\;$$

 

Free Cash Flow Measures

$$FCFF=CFO+\lbrack Int\times(1-tax\;rate)\rbrack-FCInv\\CFO=Cash\;flow\;from\;operations\\Int=Cash\;interest\;paid\\FCInv=Fixed\;capital\;investment\;(net\;capital\;expenditures)\\FCFF=NI+NCC+\lbrack Int\times(1–tax\;rate)\rbrack-FCInv-WCInv\\NI=Net\;income\\NCC=Non-cash\;charges\;(depreciation\;and\;amortization)\\Int=Cash\;interest\;paid\\FCInv=Fixed\;capital\;investment\;(net\;capital\;expenditures)\\WCInv=Working\;capital\;investment\\FCFE=CFO-FCInv+Net\;borrowing\\CFO=\;Cash\;flow\;from\;operations\\FCInv=Fixed\;capital\;investment\;(net\;capital\;expenditures)\\Net\;borrowing=\;Debt\;issued\;–\;debt\;repaid$$

 

Cash Flow Ratios

Performance Ratios

Cash flow-to-revenue

$$Cash\;flow–to–revenue=\frac{Cash\;flow\;from\;operations}{\;\;Revenue\;​}$$

Cash-to-income

$$Cash-to–income=\frac{Cash\;flow\;from\;operations}{Operating\;income}$$

Cash return-on-assets

$$Cash\;return-on-assets=\frac{Cash\;flow\;from\;operations}{Average\;total\;assets}$$

Cash return-on-equity

$$Cash\;return-on-equity=\frac{Cash\;flow\;from\;operations}{Average\;total\;equity}$$

Cash flow per share

$$Cash\;flow\;per\;share=\frac{CFO-Preferred\;dividends}{Weighted\;average\;number\;of\;common\;shares}\\\;\;\;$$

Cash Flow Ratios

Coverage Ratios

Debt Coverage

$$Debt\;coverage=\frac{Cash\;flow\;from\;operations}{Total\;debt\;\;​}\\\;\;\;$$

Interest Coverage

$$Interest\;Coverage=\frac{CFO+Interest\;paid+Taxes\;paid\;​}{Interest\;paid}\\\;\;\;$$

Reinvestment ratio

$$Reinvestment\;ratio=\frac{Cash\;flow\;from\;operations\;​\;​}{Cash\;paid\;to\;acquire\;long–term\;assets\;}\\\;\\\;\;\;$$

Debt payment

$$Debt\;payment=\frac{Cash\;flow\;from\;operations\;​\;​}{Cash\;paid\;to\;repay\;long–term\;debt\;}\\\;\\\;\;\;$$

Dividend payment

$$Dividend\;payment=\frac{Cash\;flow\;from\;operations\;​\;​}{Dividends\;paid\;}\\\;\\\;\;\;$$

Investing and financing ratio

$$Investing\;and\;financing\;ratio=\frac{Cash\;flow\;from\;operations\;​\;​}{Cash\;outflows\;from\;investing\;and\;financing\;activities\;\;}\\\\\;\\\;\;\;$$

Activity Ratios

$$Receivables\;turnover=\frac{Annual\;sales\;​}{\;Average\;receivables\;}$$

Meaning: The efficiency of a company in collecting its trade receivables

$$Days\;of\;sales\;outs\tan ding=\frac{365\;​}{Receivables\;turnover\;\;}$$

Meaning: The average number of days a company takes to collect its receivables from clients

$$Inventory\;turnover=\frac{Cost\;of\;goods\;sold}{\;\;Average\;inventory}$$

Meaning: The efficiency of a company in terms of inventory management

$$Days\;of\;inventory\;on\;hand=\frac{365}{Inventory\;turnover}$$

Meaning: The average inventory processing period

$$Payables\;turnover=\frac{Purchases}{\;\;Average\;trade\;payables}$$

Meaning: The efficiency of a company in allowing trade credit to suppliers

$$Number\;of\;days\;of\;payables=\frac{365}{\;\;Payables\;turnover\;ratio}$$

Meaning: The average number of days a company takes to pay its suppliers

$$Fixed\;assets\;turnover=\frac{\;Revenue}{\;\;Average\;net\;fixed\;assets}$$

Meaning: The efficiency of a firm in utilizing its fixed assets

$$Working\;capital\;turnover=\frac{\;Revenue}{\;\;Average\;working\;capital}$$

Meaning: The efficiency of a firm in managing its working capital (current assets – current liabilities)

$$Total\;assets\;turnover=\frac{\;Revenue}{\;\;\;Average\;total\;assets}$$

Meaning: The efficiency of a firm in using its total assets to create revenue

$$Cash\;conversion\;cycle=Days\;of\;sales\;outs\tan ding+\\Days\;of\;inventory\;on\;hand-Number\;of\;days\;of\;payables$$

Meaning: The number of days a company takes to convert its investments in inventory and other resources into cash flows from sales

$$Equity\;turnover=\frac{\;Revenue}{\;\;\;\;Average\;total\;equity}​$$

Meaning: The efficiency of a firm in utilizing equity to create revenue

 

Liquidity Ratios

$$Current\;ratio=\frac{Current\;assets\;​}{\;Current\;liabilities}\\\;$$

Meaning: Ability to meet current liabilities (with total current assets)

$$Quick\;ratio=\frac{Cash+Marketable\;securities+Receivables}{Current\;liabilities}\\\;$$

Meaning: Ability to meet current liabilities (with total current assets, excluding inventory)

$$Cash\;ratio=\frac{Cash+Marketable\;securities}{Current\;liabilities}\\\;$$

Meaning: Ability to meet current liabilities (with cash and marketable securities only)

$$Defensive\;interval=\frac{Cash+Marketable\;securities+Receivables}{Average\;daily\;expenditure}\\\;$$

Meaning: The number of days a company can cover its average daily expenses with the use of current liquid assets only

 

Solvency Ratios

$$Debt–to–equity=\frac{Total\;debt\;​}{\;Total\;shareholder’s\;equity\;}\;\\\;$$

Meaning: Debt as a percentage of total equity

$$Debt–to–capital=\frac{Total\;debt\;​}{\;Total\;debt+Total\;shareholder’s\;equity\;}\;\\\;$$

Meaning: Debt as a percentage of total capital

$$Debt–to–assets=\frac{Total\;debt\;​}{\;Total\;assets\;}\\\;$$

Meaning: Debt as a percentage of total assets

$$Financial\;leverage=\frac{Average\;total\;assets\;​}{Average\;total\;equity\;\;}$$

Meaning: An indicator of a company’s debt financing usage

$$Interest\;coverage=\frac{Earnings\;before\;interest\;and\;taxes\;\;​}{Interest\;payments\;​}$$

Meaning: The ability to cover interest expenses

$$Fixed\;charge\;coverage=\frac{Earnings\;before\;interest\;and\;taxes+Lease\;payments​}{Interest\;payments+Lease\;payments}$$

Meaning: The ability to cover interest and lease expenses

 

Profitability Ratios

$$Gross\;profit\;margin=\frac{Gross\;profit}{Revenue}$$

Meaning: Gross profitability as a percentage of total revenue

$$Operating\;profit\;margin=\frac{Operating\;income\;(EBIT)}{Revenue}$$

Meaning: Operating profitability (before interest and tax) as a percentage of total revenue

$$Pre–tax\;margin=\frac{EBT}{Revenue}$$

Meaning: Operating profitability (before tax) as a percentage of total revenue

$$Net\;profit\;margin=\frac{Net\;income}{Revenue}$$

Meaning: Net profitability as a percentage of total revenue

$$Return\;on\;assets\;(ROA)=\frac{Net\;income}{Average\;total\;assets}$$

Meaning: Net profitability (excluding interest and taxes) as a percentage of total invested funds

$$Operating\;return\;on\;assets\;(ROA)=\frac{Operating\;profit\;(EBIT)}{Average\;total\;assets}$$

Meaning: Net profitability (including interest and taxes) as a percentage of total invested funds

$$Return\;on\;total\;capital=\frac{Operating\;profit\;(EBIT)}{Average\;total\;capital}$$

Meaning: Operating profitability as a percentage of total capital

$$Return\;on\;equity\;(RoE)=\frac{Net\;income}{Average\;equity}$$

Meaning: Net profitability as a percentage of total equity

 

Valuation Ratios

$$Earnings\;per\;Share\;(EPS)=\frac{Net\;Income-Preferred\;dividends}{\;Outs\tan ding\;number\;of\;common\;shares}$$

Meaning: Income earned per 1 common share outstanding

$$Earnings\;per\;Share\;(EPS)=\frac{Net\;Income-Preferred\;dividends}{Earnings\;per\;share\;(EPS)}$$

Meaning: The price that investors are willing to pay per $1 of earnings

$$P/E\;ratio\;(company\;wide)=\frac{Market\;capitalization}{\;Net\;income}$$

Meaning: Total price that investors are willing to pay for a company’s Net income

$$Dividend\;yield=\frac{Dividend\;per\;share}{Current\;share\;price\;}$$

Meaning: The “portion “of a share price that is distributed as dividends

$$Retention\;rate\;(RR)=\frac{Net\;income-Dividends\;declared}{Net\;income}$$

Meaning: The “portion” of Net income that is reinvested in the company

$$Dividend\;payout=\frac{Dividends\;declared}{Net\;income}$$

Meaning: The “portion” of Net income that is distributed as dividends

$$Sustainable\;growth\;rate\;(g)=RR\times ROE$$

Meaning: Equity growth rate

 

DuPont Analysis

$$Return\;on\;Equity(ROE)=Net\;profit\;margin\times Asset\;turnover\times\\Financial\;leverage\;ratio\\\\Net\;profit\;margin=Tax\;burden\times Interest\;burden\times Operating\;profit\;margin\\\times Asset\;turnover\times Financial\;leverage\;ratio$$

 

Inventories

Where:
FIFO = First-in, First-out method
LIFO = Last-in, First-out method

$$Ending\;inventory=Beginning\;inventory+Purchases-Cost\;of\;goods\;sold\;(COGS)$$

$$Cost\;of\;goods\;sold\;(COGS)=Beginning\;inventory+Purchases-Ending\;inventory$$

$$FIFO\;inventory=LIFO\;inventory+LIFO\;reserve$$

$$Delta\;Cash=LIFO\;reserve\times Tax\;rate$$

$$Delta\;Cash\;=\;Excess\;cash\;saved\;on\;the\;valuation\;method$$

$$FIFO\;retained\;earnings=LIFO\;retained\;earnings+LIFO\;Reserve\times(1-Tax\;rate)$$

$$FIFO\;COGS=LIFO\;COGS-(Ending\;LIFO\;reserve-Beginning\;LIFO\;reserve)$$

 

Long-lived Assets

$$Straight–line\;depreciation\;expense=\frac{Cost-Salvage\;(residual)\;value}{\;\;Useful\;life}$$

$$Double–declining\;balance(DDB)\;depreciation\;expense=\frac{2\times(Cost-Accumulated\;depreciation)}{\;\;Useful\;life}$$

Double-declining balance (DDB) depreciation expense = Double the straight-line depreciation rate

$$Units\;of\;production\;depreciation\;expense=\frac{Cost-Salvage\;value\;​}{\;Life\;in\;output\;units}\times Output\;units\;in\;the\;period$$

$$Ending\;PPE\;net\;book\;value=(Original)\;Cost-Accumulated\;depreciation$$

$$Average\;age=\frac{Accumulated\;depreciation\;​}{Annual\;depreciation\;expense}$$

$$Total\;useful\;life=\frac{Historical\;\cos t}{Annual\;depreciation\;expense}$$

$$Remaining\;useful\;life=\frac{\;Ending\;PPE\;net\;book\;value\;​}{Annual\;depreciation\;expense}​$$

 

Income Taxes

$$Income\;tax\;expense=Taxes\;Payable+\\\Delta\;Deferred\;Tax\;Liabilities\;(DTL)-\\\Delta\;Deferred\;Tax\;Assets\;(DTA)\\Effective\;tax\;rate=\frac{Income\;tax\;expense\;​}{\;\;Pre–tax\;income\;}$$

 

Non-Current Liabilities

$$Interest\;expense=Market\;rate\;at\;Issuance\times\\Balance\;sheet\;value\;of\;the\;liability\;at\;the\;beginning\;of\;the\;period\\$$

$$Coupon\;interest\;payment=Coupon\;rate\;(as\;per\;contract)\times Par\;Value$$

 

 

 

Sample Covariance

$$Cov_{XY}=\frac{\sum_{i=1}^n(X_i-\overline X)\left(Y-\overline Y\right)}{n-1}$$

Sample Correlation Coefficient

$$r_{x,y}=\frac{Cov\left(x,y\right)}{\sigma_x,\sigma_y}$$

t-test statistics with

$$d_f=n-2=\sqrt{\frac{n-2}{1-r^2}}\\H_0:\rho=0\\H_0:\rho\neq0\\t_{stat}=\sqrt{\frac{n-2}{1-r^2}}\;Accept\;H_0\;if\;-t_{\;critical}<\;t\;_{stat}<\;+t_{critical}\\$$

$$Y_i\;=b_o+b_1X_{1\;}+\varepsilon_i\\\overset\frown Y=b_o+b_1X_\;\\$$

Slope Coefficient

$$\frac{Cov(x,y)}{\sigma x^2}=\frac{r(x,y)\sigma y}{\sigma x}\\$$

Intercept Term

$$\widehat{b_0}=\overline Y-\widehat{b_1}\overline X\\$$

SSE = Squared vertical distance between estimated Y value and actual Y value. (Regression line minimizes the SSE)

$$Slope\;Term(b_1)=\frac{r\times\sigma y}{\sigma x}=\frac{Cov(x_1,y)}{\sigma^2x}$$

$$\widehat{b_0}=\overline y-\widehat{b_1}\overline x\\$$

SEE (Standard error of Estimates)

$$Degree\;of\;variability\;of\;actual{\;y}_i\;value\;relative\;to\;estimated\widehat{\;y}\;value.\\SEE=\sqrt{MSE}=\sqrt{\frac{SSE}{n-k-1}}\\\lbrack SEE\downarrow\;relationship\;strong\rbrack\\\lbrack SEE\uparrow\;relationship\;weak\rbrack\\$$

$$R^2=\frac{RSS}{TSS}=\frac{ESS}{TSS}=\frac{TSS-SSE}{TSS}\\Confidence\;Interval\;for\;b_1={\widehat b}_1\pm(t_c\times s_{{\widehat b}_1})\\Confidence\;Interval\;for\;Y=\widehat Y\pm(t_c\times s_f)\\s_f=Standard\mathit\;error\mathit\;of\mathit\;forecast\\$$

Testing of Hypothesis

$$t_{b_1}=\frac{\widehat{b_1}-b_1}{S_\widehat{b_1}}\\S_f^2={SEE}^2\left[1+\frac1n+\frac{\left(x-\overline x\right)^2}{(n-1)s_x^2}\right]\\$$

The General Multiple Linear Regression Model

$$Y_i=b_0+b_1X_{1i}+b_2X_{2i}+...+b_kX_{ki}+\varepsilon_i$$

Hypothesis Testing of Regression Coefficient

$$t\;=\frac{b_j-b_i}{S_{{\widehat b}_j}}\;\;(n\;–\;k\;–\;1)\;degrees\;of\;freedom\\$$

$$P\;value: Smallest\;level\;of\;significance\;for\;which\;the\;null\;hypothesis\;can\;be\;rejected$$

$$P\;value\;>\;a;\;H_0\;accept\;(a\;=\;Significance\;level)$$

$$P\;Value\;<\;a;\;H_0\;rejected$$

Confidence Interval

$$SSE=\sum{(y_i\;\;-\widehat y)}^2=\sum i^2\\\sigma\varepsilon^I\;=\;\sigma(y_i\;\;-\widehat y)\\SEE/SER=\sqrt{\frac{\sum{(x\;-\overline x)}^2}{n-2}}=\sqrt{\frac{\sum{(\varepsilon_i\;-\overline{\varepsilon_l})}^2}{n-2}}$$

$$If\;\overline{\varepsilon_l}=0;\;then:\sqrt{\frac{\sum\overline{\varepsilon_i}^2}{n-2}}=\sqrt{\frac{SSE}{n-2}}=\sqrt{MSE}$$

Confidence Interval for A Regression Coefficient

$${\widehat b}_j\pm(t_c\times s_{{\widehat b}_j})\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;s\tan dard\;error\;of\;b_i\;\\{\widehat b}_j\rightarrow use\;(n-k-1)\;d_f\\where\;K\;=\;no\;of\;independent\;variables\\S_{bI\;}=f_n\;\lbrack SEE\rbrack\\$$

Confidence interval for forecasted variables

$$Y=\widehat Y\pm(t_c\times s_f)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;S.E\;of\;\widehat Y\\TSS=SSE+ESS\\SSR\;\;\;\;\;\;\;RSS\\\sum{(y_i-\overline y)}^2=\sum{(y_i-\widehat y)}^2+\sum{(\widehat y-\overline y)}^2$$

$$ESS\;/\;RSS\;=\;the\;difference\;that\;is\;explained\;by\;independent\;variable\\R^2=\frac{ESS}{TSS}=\frac{TSS-SSR}{TSS}=1-\frac{SSE/SSR}{TSS}\;\\R^2\;is\;Always\;+ve$$

Predicting the dependent variable

$${\widehat Y}_i={\widehat b}_0+{\widehat b}_1{\widehat X}_{1i}+{\widehat b}_2{\widehat X}_{2i}+...+{\widehat b}_k{\widehat X}_{ki}$$

The F – Statistics

$$F-stat=\frac{MSR\;}{MSE}(one\;–\;tailed\;test)\\(RSS/ESS)\\Where\;MSR=mean\;regression\;sum\;of\;squares\\MSE=mean\;squared\;error\\(SSR/SSE)$$

Coefficient of Determination

$$R^2=\frac{RSS}{TSS}=\frac{ESS}{TSS}=\frac{TSS-SSE}{TSS}$$

Adjusted R2

$$R_a^2=1-\left[\left(\frac{n-1}{n-k-1}\right)\times\left(1-R^2\right)\right]$$

Anova Table

$$R^2=\frac{RSS}{TSS}\\F=\frac{MSR}{MSE}with\;k\;and\;n-k-1\;degrees\;of\;freedom\\MSR=\frac{RSS}K=\frac{RSS}1;\;K=1\;for\;simple\;linear\;regression\\MSE=\frac{SSE}{n-K-1}=\frac{SSE}{n-2}\\If\;K\;\geq\;1;\;R^2>{R_a\;}^2\;(No\;comparison\;of\;R^2\;\&\;{R_a\;}^2\;)\\{R_a\;}^2=1-\left[\left(\frac{n-1}{n-k-1}\right)\left(1-R^2\right)\right]$$

 

Linear Trend Model

$$Y_t\;=\;b_0+b_1(t)+\varepsilon_t$$

Ordinary Least Squares (OLS) regression

$${\widehat Y}_i=\;{\widehat b}_0+{\widehat b}_1(t)$$

Log - Linear Trend Model

$$Y_t\;=\;e^{b_0+b_1(t)}$$

Autoregressive Model (AR)

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon_t$$

AR (p) Model

$$X_t\;=\;b_0+b_1X_{t-1}+b_2X_{t-2}+b_pX_{t-p}...+\varepsilon_t$$

Autocorrelation & Model Fit

$$t=\frac{\rho_{(e_{t,}e_{t-k})}}{\displaystyle\frac1{\sqrt T}}$$

Random Walk with A Drift

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon_t$$

Covariance Stationarity

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon_t$$

Unit Root Testing for Non-Stationary

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon\\X_t-X_{t-1}\;=\;b_0+b_1X_{t-1}-X_{t-1}+\varepsilon$$

First Differencing

$$Y_t=X_t-X_{t-1}\Rightarrow Y_t=\varepsilon_t\\Y_t=b_0+b_1Y_{t-1}+\varepsilon_t$$

ARCH (1) Regression Model

$${\widehat\varepsilon}_t^2=a_0+a_1{\widehat\varepsilon}_{t-1}^2+\mu_t$$

Predicting the Variance of a Time Series

$${\widehat\sigma^2}_{t+1}={\widehat a}_0+{\widehat a}_1{\widehat\varepsilon}_t^2\\$$

$$Mark\;To\;Market\;Value\;=\frac{\left(Forward\;Price\;New-F.P\;locked\;oldሻ\;ሺሺContract\;Size\right)}{\left[1+r\;\left({\displaystyle\frac n{360}}\right)\right]}\\\\$$

Covered Interest Rate Parity

$$F_{A/B}=S_{A/B}\left(\frac{1+iA}{1+iB}\right)\\F\left(disc/premium\right)=S_A+\left(\frac{\left(iA-iB\right)\left({\displaystyle\frac n{360}}\right)\rightarrow if\;Libor\;rates}{\left(1+iB\right)n/360)}\right)$$

Uncovered Interest Rate Parity

$$E{(\%\Delta S)}_{\;(A/B)}=R_A-R_B\\$$

International Fisher Relation

$$R_{nominal\;A}-R_{nominal\;A}=\;+\;E\;(Inflation_A)\;-\;E\;(Inflation_B)\\$$

Purchasing Power Parity

$$F=S\left(\frac{1+Inflation_A}{1+Inflation_B}\right)\\$$

Absolute PPP

$$S(A/B)=\frac{CPI(A)}{CPI(B)}\\$$

Relative PPP

$$\%\triangle S(A/B)\;=\;Inf_A\;-\;Inf_B\\$$

Real Exchange Rate

$$S_t\times\left(\frac{1+r_B}{1+r_A}\right)^T\\$$

$$BOP\Rightarrow Current\;A/c+Capital\;A/c+Official\;Reserve\;A/c=0\\$$

$$Real\;exchange\;rate\;A/B=(Equilibrium\;Real\;Exchange\;Rate\;A/B)\\(B\;affricates)\;\uparrow+(\uparrow Real\;Int.\;rate_B\;–\;Real\;Int.\;rate_A)\\-(Risk\;Premium_B\;-\;\uparrow Risk\;Premium_A)\\$$

Taylor’s Rule

$$R=r_n+\pi+\alpha(\pi-\pi\ast)+\beta(y-y\ast)\\$$

Real Interest Rate

$$Real\;Interest\;Rate=r_n+\pi+\alpha(\pi\;–\;\pi\ast)+\beta(\gamma-\gamma\ast)\\$$

 

$$\triangle P=\triangle GDP+\triangle(E/GDP)+\triangle(P/E)\\$$

Cobb – Douglas Function

$$Y=TK^aL^{(1-a)}\\$$

$$Output\;Per\;Worker=Y/L=T(K/L)\;a\\$$

$$Marginal\;Product\;of\;Capital=\frac{\triangle Y}{\triangle K}=\frac{\alpha Y}K\\(Cons\tan t)\\Marginal\;Productivity=\frac{\triangle Y/L}{\triangle K/L},K\uparrow L=Cons\tan t\\(Diminishing)\\$$

$$MP\;K=r(Marginal\;\cos t\;of\;K)\rightarrow rental\;price\;of\;capital\\$$

$$\downarrow\\\\$$

Growth Accounting Relation

$$\triangle Y/Y=\triangle A/A+\alpha(\triangle K/K)+(I-\alpha)(\triangle L/L)\\\\$$

$$Growth\;in\;Potential\;GDP=i)\;Long\;Term\;Growth\;of\;Techno\log y\\+\alpha(Long\;Term\;Growth\;of\;K)+(I-\alpha)\;(Long-Term\;Growth\;of\;L)\\ii)\;Long\;Term\;Growth\;of\;Labour\;Force+Long\;Term\;Growth\;in\;Labour\;Productivity\\(Output\;Per\;Worker)\;Both\;Capitals\;Depending+Techno\log y\;Process.\\\\$$

$$Labour\;Force\;Participation=\frac{Labour\;Force}{\;Working\;Age\;Population}\\Where\;Labour\;Force=Employed+Unemployed\;Available\;to\;Work\\\\$$

$$G\ast\;(Growth\;of\;Output\;Per\;Capita)=\frac\theta{\;I-\alpha}\\G\ast\;(Growth\;of\;Output)\;=\frac\theta{\;I-\alpha}+\triangle L\\\\$$

 

$$Full\;Goodwill=(Fair\;Value\;of\;Equity\;of\;Whole\;Subsidiary)\\-(Fair\;Value\;of\;Net\;Identifiable\;Assets\;of\;The\;Subsidiary)\\-Allowed\;under\;both\;IFRS\;\&\;USGAAP$$

$$Partial\;goodwill=Purchase\;Price–(\%\;owned\;\times\;\;FV\;of\;Net\;Identifiable\;Assets)\\-Allowed\;under\;only\;IFRS\\\\$$

Goodwill Impairment

US GAAP
1) CA > FV of reporting unit
2) CA of g/w implied FV of g/w

FV of unit
net identifiable asset

IFRS

1) CA> RA => loss in P/L

$$Years\;to\;Repay\;Debt\;from\;CFO=\frac{Total\;Debt}{\;Operating\;CF-Reinvestment}$$

Pension

Plan Asset  Plan Assets
FV at beginning of year PBO at beginning of year
+ Contributions
+ Actual Return
− Benefits paid

= FV at end of year

PBO at beginning of year
(+) Current service cost
(+) Interest cost
(+) PSC
(+) Actual Loss/ (-) gain
(-)Benefits

= PBO at end of year

Plan Asset > PBO Overfunded Plan
Plan Asset < PBO Underfunded Plan

$$Funded\;Status=Fair\;Value\;of\;Plan\;Asset-PBO\\TPPC=Employer\;contributions-\lbrack Ending\;Funded\;status–Beginning\;Funded\;status\rbrack\\=Employer\;contribution–\lbrack(End\;Plan\;Asset–End\;PBO)–(Beginning\;plan\;Asset–Beginning\;PBO)\rbrack\\=Employer\;contribution–\lbrack(Ending\;Plan\;Asset–Beginning\;Plan\;Asset)–(Ending\;PBO-Beginning\;PBO)\rbrack\\=Actual\;Return-\;(Current\;SC\;+\;\ln t\;\cos t\;+\;PSC\;\pm\;Actual\;gain/\;loss)\\TPPC=Current\;SC+Int\;\cos t+Past\;SC\pm Actuarial\;gain/loss–Actual\;Return$$

IFRS
i. Current Service cost in P /L
ii. Net Interest income in P/L
-income if overfunded (A > L)
-expense if underfunded (A < L)
iii. PSC in P/L
(recognized immediately)
iv. Actuarial g/l in OCI
(expected return actual return)

Not amortized over

US. GAAP
v. Current Service cos t in P/L
vi. Interest cost in P/L
Expected Return in P/L
vii. PSC in OCI
(amortized over remaining life)
viii. Actuarial g/l in OCI
(expected return-actual return)
Corridor approach
Amortize amount that is above 10% x max (A, L) over the remaining life

$$Expected\;Return=Expected\;rate\times Beginning\;plan\;Asset\\Net\;Int\;\cos t/\;income=Disc.\;Rate\times Beginning\;funded\;status\;(A-L)\\For\;IFRS,\;disc\;rate\;\&\;expected\;return\;is\;same\\Periodic\;pension\;\cos t\;in\;OCI=TPPC–periodic\;pension\;\cos t\;in\;P/L\\Or\;Actuarial\;Gains/\;losses\;+(Actual\;–\;expected)\;return$$

To reclassify

  • Op. income + Full pension exp current SC
  • Add int cost to int exp.
  • Add Actual return to other (non operating) income.

Cont^n>> TPPC -> reduction in PBO

Cont^n< TPPC => source of borrowing

From
CFO -> (+)
CFF -> (-)

The Beneish Model (M-score)

$$DSRI:\frac{Days\;Rec_{t\;}}{Days\;Rec_{t-1\;}}\uparrow X\\GMI:\frac{Gross\;Margin_{t-1\;}}{Gross\;Margin_{t\;}}\uparrow X\\AQI:\frac{Noncurrent\;asset\;except\;PP\;\&ET_t/Total\;Assets_t}{NCA\;except\;PP\;\&ET_{t-1}/Total\;Assets_t}\uparrow X\\SGI:\frac{Sales_{t\;}}{Sales_{t-1\;}}\uparrow X\\DEP:\frac{Depriciation_{t-1\;}}{Depriciation_{t\;}}\uparrow X\\SGAI:\frac{\%SGA/Sales_{t\;}}{\%SGA/Sales_{t-1\;}}\uparrow X\\Accruals:\frac{Income\;Before\;EOI-CFO}{Assets}\uparrow X\\Leverage\;Index:\frac{D/A_t}{D/A_{t-1}}(\ast higher\;the\;better\downarrow)\;X$$

Gauging Earning Persistence

$$Earnings_{(t+1)}\;=\;\alpha+\beta_1earnings_{\;t}+\varepsilon\\Earnings_{(t+1)}\;=\;\alpha+\beta_1cash\;flow_{\;t}+\beta_2accruals_{\;t}+\varepsilon$$

Sources of Earnings and ROE

$$DoPont\;Decomposition-\\ROE=NI/EBT\times EBT/EBIT\times EBIT/Revenue\times Revenue/Averagge\;Asset\times Averagge\;Asset/Averagge\;Equity$$

$$AccrualsBS=NOA\:End-NOA\:Bgn$$

$$Accruals\;Ratio\;BS=\frac{Accruals^{BS}}{\left(NOA_{End}+NOA_{Beg}\right)/2}$$

$$Accruals\;CF=NI-CFO-CFI$$

$$Accruals\;Ratio\;CF=\frac{Accruals^{CF}}{\left(NOA_{End}+NOA_{Beg}\right)/2}$$

$$CGO=EBIT+non\;cash\;changes-increase\;in\;WC$$

                         IFRS     USGAAP
Int paid           CFO/CFF   CFO
Div. Paid          CFO/ CFF   CFF
Int/Div Recd.  CFO/CFI    CFO

Market Value decomposition

$$Implied\;value=Parents\;Pro-rate\;share\;in\;associate's\;MV\\\\$$

$$Initial\;Investment\;Outlay=FC\;Investment+NWC\;Investment\;Proceeds\;from\;Sale\;of\;Earlier\;Asset\;sold\;\lbrack After\;Tax\rbrack$$

$$Include\;in\;Terminal\;Value=Initial\;NWC\;Investment$$

$$After-Tax\;operating\;CF=(EBITDA\;Dep)\;(I-Tax)+Dep^n\\Or,\;(EBIT)\;(I-tax)+Dep^n=(EBITDA)\;(I-tax)\;+\;(Dep^n\times\;tax)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Dep^n\;tax\;savings$$

Terminal Value

$$Sal_T\;+\;NWC_{INV}\;-Tax\;Marginal\;(Sal_T-B_T)$$

For Replacement Project

$$Investment\;Outlay=FC\;Investment+NWC\;Investment-Sal_0\;+T(Sal_0-B_0)\\Operating=\Delta CF=(\;\Delta S-\Delta VC)\;(1-T)+\Delta D(1-T)\\Terminal\;value=\left[Sal_{T\;new}-Sal_{T\;old}+NWC\;Investment\right]-T(Sal_{T\;new}-B_{T\;new})\\Proj.\;NPV\;without\;option\;\&\;add\;value\;later=Proj.\;NPV\;(DCF)-option\;\cos t+option\;value\\PI=I\;+\frac{NPV}{outflow}=\frac{PV\;inflow}{\;PV\;outflow}\\$$

$$Economic\;Income:\;Cash\;Flow+(Ending\;Market\;Value-Beginning\;Market\;Value)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Economic\;Dep^n\\(Firm)\;@KC\\Economic\;\mathrm\pi=\;\;\;\;NOPAT\;-\;\;\;\;\;\;\$\;WACC\;\\\;\;\;\;\;\;\;\;\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\sum PV=NPV\;\;\;\;EBIT(I-t)\;\;\;\;\;\;\;\;WACC\;\times\;capital\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\Co.\;value=NPV+initial\;investment\;(capital)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\Residual\;Income=net\;income-equity\;charge\\\;\;\;\;\;\;\;\;\downarrow\\(Equity)\;@\;Ke\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Ke\times Bg^{\;n}\;BV_{eq}\\Co.\;value\;=NPV+BV\;of\;debt+BV\;of\;equity\\Claims\;valuation\;Approach=PV\;of\;CF\;to\;debt\;holders+PV\;of\;CF\;to\;equity\;holders\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Principal\;+\;Int)\;\;\;\;\;\;\;\;\;\;\;(FCFE)\;(DIV+share\;repurchase)$$

Merger Acquisition

$$FCFF=NI+NCC+INT\;(I-t)-CAPEX-\Delta\;WC\;I_{NV}+\Delta\;DTL-\Delta\;DTA\\CFO=NI+NCC–WC_{INV}\\FCFE=CFO–FC\;I_{NV}+net\;borrowing\\Takeover\;Premium=\frac{Deal\;Price/Share–target\;co.Stock\;price}{Target\;co.Stock\;price}$$

Cash VS Stock Payment

Cash

$$Value\;of\;Firm=Value_{TARGET}+V_{ACQUIRER}+Synergy–Cash\;Paid\\Gain\;to\;target\_\\Price\;Paid–Price\;Merger\;Value\;of\;Target\\Gain\;to\;acquirer\\Synergy\;-\;Gain\;to\;Target$$

Stock

$$Value\;of\;Firm=V_{T\;}+V_A+synergy\\calculate\;the\;new\;price\\=\frac{V\;firm}{O/S\;shares\;of\;acquirer}\\Then\;calculate\;gain$$

Mergers & Acquisitions

Steps: -
1) DCF Method
• After calculating FCFF
• Discount FCF -> calculate PV

$$V_{firm}=\frac{FCFF_1}{\left(I+Kc\right)}+\frac{FCFF_2}{\left(I+Kc_2\right)^2}+...+\frac{FCFF_5+V_5}{\left(I+Kc\right)^5}\\Where\;V_5=\frac{FCFF_6}{Kc-g}\\Target\;V_{firm}\neq No.\;of\;shares$$

2) Comparable Company
• First calculate P/E, P/B, P/S ratios
• Take mean of all ratios
• Then multiply the mean P/E, P/B, with E, BV of the target company
• Get the mean of that = Stock price / value

Calculate Takeover Premium
Takeover Price = Stock Value X Takeover Premium
3) Comparable transaction (not needed to calculate takeover premium separately)
Take the deal price & continue to (2) process

$$Terminal\;Vfirm_5=\frac{FCF_T(1+g)}{WACC_{adj}-g}$$

MM Proposition I

$$No\;Taxes\Rightarrow V_L=V_U\;;\;EBIT/WACC\\With\;Taxes\Rightarrow V_L=V_U(t\times d)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;tax\;shield\\\boldsymbol S\boldsymbol t\boldsymbol a\boldsymbol t\boldsymbol i\boldsymbol c\boldsymbol\;\boldsymbol t\boldsymbol r\boldsymbol a\boldsymbol d\boldsymbol e\boldsymbol\;\boldsymbol o\boldsymbol f\boldsymbol f\boldsymbol\;\boldsymbol t\boldsymbol h\boldsymbol e\boldsymbol o\boldsymbol r\boldsymbol y:\;(benefit\;of\;debt)\\V_L=V_U+(t\times d)-PV(\cos ts\;of\;financial\;distress)$$

MM Proposition II

$$No\;Taxes\Rightarrow R_e=R_0+D/E(R_0-R_D)\\With\;Taxes\Rightarrow R_e=R_0+D/E(R_0-R_D)(I-T_C)$$

Dividend Share Repurchases

$$\Delta P=\frac{D(I-T_D)}{(1-T_{CG})}\\-\Delta P=when\;stock\;goes\;from\;dividend\;to\;ex-dividend\\Effective\;Tax\;Rate=Corporate\;Tax\;Rate+(I-Corporate\;Tax\;Rate)\;(IndividuaL\;Tax\;Rate)\\$$

For double taxation system we use above

For split tax
Taxes for retained ↑ as retained > distributed
Effective calculated same as above

Under Imputation Tax System
Effective Tax Rate = Shareholder’s Marginal Tax Rate
Shareholder tax bracket < company rate
Shareholder receive tax credit & vice versa.

Target Pay Out Ratio Adjustment Model

$$Expected\;Dividend=Previous\;Dividend+\left[(Expected\;Increase\;in\;EPS)\times(target\;pay\;out\;ratio)\times Adjustment\;Factor\right]\\Where\;Adjustment\;factor=\frac1{Number\;of\;Years\;Over\;Which\;the\;Adjustments\;in\;Dividends\;will\;take\;place\;}\\Earnings\;yield>K_d(I-t)\Rightarrow EPS\;\uparrow\\(EPS/P_o)\\Earnings\;yield<K_d(I-t)\Rightarrow EPS\;\downarrow\\BVPS\downarrow\Rightarrow\;if\;repurchase\;price>old\;BVPS\\Repurchase\;price<old\;BVPS\Rightarrow BVPS\;\uparrow\\EPS\;After\;Buyback=\frac{Total\;Earnings-After\;Tax\;Cost\;of\;Funds\;}{No.\;of\;Shares\;O/S}$$

$$Interest\;Coverage\;ratio=EBIT/INT\\Fixed\;Interest\;Coverage\;Ratio=\frac{EBIT+lease}{Interest+lease}\\Div.\;Coverage=NI/Div\\FCFE\;Coverage=FCFE/(Div+Share\;Repurchase)\\Accounting\;income=(Operating\;Income\;Before-Interest)\;(I-t)$$

 

Equity Valuation: Application & Processes

$$IV_{analyst}-price=(IV_{actual}\;-price)+(IV_{analyst}-IV_{actual})\\Conglomerate\;Discount=Sum\;of\;individu\;al\;parts-Sum\;as\;a\;whole$$

$$Holding\;Period\;Return=\frac{P_1-P_0+CF_1}{P_0}\\Target\;Price=Price(R_e)\;D_i$$

Price Convergence

$$Expected\;return=Required\;Return+\left[\frac{V_0-P_0}{P_0}\right]\\Equity\;Risk\;Premium=Required\;Return\;on\;Equity\;Index-Risk\;Free\;Rate\\Required\;Return\;for\;Stock\;j=Risk\;free\;Return+\beta_j\times(Equity\;Risk\;Premium)+other\;risk\;premium$$

Forward Looking Estimates

Gordon Growth Model:

$$(Re-Rf)=ERP=\frac{D_I}{P_0}+g-RF\\Where\;RF=long\;term\;bond\;yield\\Equity\;Index\;Price=PV_{rapid}(r)+PV_{transition}(r)+PV_{mature}(r)$$

Supply Side Estimates

$$ERP=\left[\left(\;I+\widehat i\;\right)\times(I+r\widehat Eg)\times\left(I+P\widehat Eg\right)-1+\widehat Y\right]-\widehat{RF}$$

Capital Asset Pricing Model (CAPM)

$$R_e=R_F+(R_{equity\;index}-R_F\;)\times\beta$$

Multifactor Models

$$R_e=R_f+\beta_1F_1+\beta_2F_2+...\\Where,\;F_1=(R_{F_1}-R_F)$$

Fama French Model

$$Required\;return\;for\;stock\;j=RF+\beta_{mkt,j}\times(R_{mkt}-RF)+\beta_{SMB,j}\times(R_{small}-R_{big})\\+\beta_{HML,j}\times(R_{HBM}-R_{LBM})$$

Pastor Stambaugh Model

$$Liquidity\;+\;Fama\;French\;Model\\\downarrow\\(Liquidity\;premium)\;x\;\beta\\+ve\rightarrow\;less\;liquid\\-\;ve\rightarrow more\;liquid$$

Build Up Method

$$Required\;return=RF+equity\;risk\;premium+size\;premium+specific\;company\;premium$$

Bond yield + Risk Premium Method

$$R_e=YTM+risk\;premium\;for\;holding\;equity\\Adjusted\;\beta=(2/3)\;\times\;regression\;\beta)\;+\;(1/3\;\times\;1.0)$$

Country Spread Model

$$Premium=YTM_{Bond\;Emerging\;Market}-YTM_{Bond\;Developed\;Market}$$

Country risk rating model

$$Developed\;market\;returns\;(ERP)=a+b_1\;\times\;(ratings)\\WACC=\frac{Market\;value\;of\;debt}{Market\;value\;of\;debt\;and\;equity}\times r_d\times\left(1-Tax\;rate\right)+\\\frac{Market\;value\;of\;debt}{Market\;value\;of\;debt\;and\;equity}\times r_e$$

 

Cost of Goods Sold (COGS)

$$Forecast\;COGS=(Historical\;COGS/revenue)\;\times\;(Estimate\;of\;Future\;Revenue)$$

Financing Cost

$$Net\;debt=Gross\;debt-Cash,\;cash\;equivalents\;\&\;short\;term\;investment\\Net\;interest\;expense=Gross\;Interest\;Expense-Interest\;Income\;(on\;cash\;\&\;short\;term\;debt\;securities)$$

$$Gross(net)\;interest\;expense\;rate=gross(net)\;expense/\;gross(net)\;debt\\Yield\;on\;average\;cash=interest\;income/cash+ST\;securities\\Effective\;tax\;rate=\frac{Income\;tax\;expense}{PBT}\\Cash\;tax\;rate=\frac{Cash\;taxes\;paid}{PBT}\\Projected\;Accounts\;Receivables=Days\;Sales\;Outs\tan ding\times\left(\frac{Forecasted\;Sales}{365}\right)\\ROC=\frac{Net\;Operating\;Profit}{D+E}\;\rightarrow\;Not\;adjusted\;for\;taxes\\ROIC=\frac{NOPLAT}{D+E}\;\rightarrow\;Net\;operating\;profit\;taxes\\ROE=\frac{NI}E\;\rightarrow\;Not\;suitable\;for\;comparing\;companies\;with\;different\;capital\;structures\\Cannibalization\;rate=\frac{new\;product\;sales\;that\;replace\;existing\;product\;sales}{total\;new\;product\;sales}$$

DDM Model

One Period- $$P_0=\frac{D_1+P_1}{I+R_e}$$

Two period-

$$P_0=\frac{D_1}{I+R_e}+\frac{D_2+P_2}{\left(I+R_e\right)^2}$$

Multi period-

$$P_0=\frac{D_1}{I+R_e}+\frac{D_2}{\left(I+R_e\right)^2}+...+\frac{D_n+P_n}{\left(I+R_e\right)^n}$$

Gordon Growth Model

$$P_0=\frac{D_1}{R_e-g}$$

Present value of growth opportunities (PVGO)

$$P_0=\frac{E_1}{R_e}+PVGO$$

Justified trailing P/E

$$\frac{P_0}{E_0}=\frac{\left(I+g\right)\left(I-b\right)}{R_e-g}$$

Justified leading P/E

$$\frac{P_0}{E_1}=\frac{\left(I-b\right)}{R_e-g}$$

$$Value\;of\;perpetual\;preferred\;shares=\frac{D_P}{r_P}$$

Valuation using H Model

$$V_0=\frac{D_0(1+g_L)}{R_e-g_L}+\frac{D_0\times t/2\times(g_S-\;g_L)}{R_e-g_L}$$

Sustainable growth rate

$$SGR(g)=b\times ROE\\Where,\;ROE=\frac{NI}{Stockholders'\;Equity}=\frac{NP(NI)}{Sales}\times\frac{Sales}{Total\;Assets}\times\frac{Total\;Assets}{Stockholders'\;Equity}\\So,\;g=\left(\frac{Net\;Income-Dividends}{Net\;Income}\right)\times\frac{Net\;Income}{Sales}\times\frac{Sales}{Total\;Assets}\times\frac{Total\;Assets}{Stockholders'\;Equity}$$

$$Intrinsic\;value>market\;value\rightarrow undervalued\\Intrinsic\;value=market\;value\rightarrow fairly\;valued\\Intrinsic\;value<market\;value\rightarrow overvalued$$

FCFF 4 Approaches

$$FCFF=NI+NCC+Interest(1-TAX)-fCInvestment-WCInvestment\\FCFF=NI+NCC-WC\;Investment+Interest(1-TAX)-fC\;investment=CFO+Interest(1-TAX)-FCInvestment\\FCFF=EBIT(1-TAX)+Depreciation-fC\;investment-WC\;investment\\FCFF=EBITDA(1-TAX)+(Depreciation\times TAX)-fC\;investment-WC\;investment$$

FCFE 4 Approaches

$$FCFF=FCFF-Interest(1-TAX)+NET\;Borrowings\\FCFF=NI+NCC-WC\;Investment-FC\;investment+NET\;Borrowings\\FCFF=CFO-FC\;Investment+NET\;Borrowings\\FCFF=NI-(1-DR)\left[(FC\;Investment-Depreciation\;)+WC\;Investment\right]\\therefore,1-DR=1-\frac DA=\frac{A-D}A=\frac EA$$

Single Stage FCFF / FCFE Model

$$FCFF:\;Value\;of\;Firm=\frac{FCFF_1}{WACC-g}=\frac{FCFF_0\times(1+g)}{WACC-g}\\FCFF:\;Value\;of\;Equity=\frac{FCFF_1}{r-g}=\frac{FCFE_0\times(1+g)}{r-g}\\Terminal\;Value\;in\;year\;n=(trailing\;P/E)\times(earnings\;i\;n\;years\;n)\\Terminal\;Value\;in\;year\;n=(leading\;P/E)\times(forecasted\;earnings\;in\;year\;n+1)=\frac PE\times E_0(1+g)$$

$$Trailing\;P/E=\frac{Market\;price\;per\;share}{EPS\;over\;previous\;12\;months}\\Leading\;P/E=\frac{Market\;price\;per\;share}{Forecasted\;EPS\;over\;next\;12\;months}\\P/B\;Ratio=\frac{Market\;price\;of\;Equity}{Book\;value\;of\;Equity}\\P/S\;Ratio=\frac{Market\;price\;of\;Equity}{Total\;Sales}\\Dividend\;Yield:\;D/P\\Trailing\;D/P=D_o/P_o\\Leading\;D/P=D_I/P_o$$

Justified P/E Multiple

$$P_o=\frac{D_1}{R_e-g}\\Justified\;trailing\;P/E=\frac{P_0}{E_0}=\frac{\left(1-b\right)\left(1+g\right)}{r-g}\\Leading\;P/E=\frac{P_0}{E_0}=\frac{1-b}{r-g}$$

Justified P/B Multiple

$$Justified\;P/B\;Ratio=\frac{ROE-g}{r-g}$$

Justified P/S Multiple

$$Justified\;\frac{P_0}{S_0}=\frac{\left(\frac{E_0}{S_0}\right)\times\left(1-b\right)\left(1+g\right)}{r-g}\\$$

Justified P/CF Multiple

$$V_0=\frac{FCFE_0(1+g)}{r-g}\\$$

Justified Dividend Yield

$$\frac{D_0}{P_0}=\frac{r-g}{1+g}\\$$

Fed & Yardeni Model

Fed model:

$${(E/P)}_{S\&P}>{(E/P)}_{10\;yr\;T-Bond}\;\Rightarrow undervalued\\{(E/P)}_{S\&P}<{(E/P)}_{10\;yr\;T-Bond}\;\Rightarrow overvalued\\$$

Yardeni model:

$$Earnings\;yield\;of\;market\;(E/P)=yield\;on\;‘A’\;rated\;bond-k\times(Long\;term\;growth\;rate)\\$$

Peg Ratio

$$Peg\;ratio=\frac{P/E\;ratio}g\\CF=Net\;Income+depreciation+amortization\\FCFE=CFO-FC\;Inv+Net\;borrowing\\P/CF=\frac{MV\:of\;equity}{CF}=\frac{Market\;price\;per\;share}{CF\:per\;share}\\EV/EBITDA_{ratio}=\frac{enterprise\;value}{EBITDA}\\\\$$

Momentum indicator

$$Earnings\;Surprise=Reported\;EPS-Expected\;EPS\\S\tan dardized\;Unexpected\;Earnings\;(SUE)=\frac{earnings\;surprise}{SD\;of\;earnings\;surprise}\\$$

$$RI=Net\;Income-Cost\;of\;Equity\times Equity\;Capital(equity\;charge)\\=\;(ROE-r)\;BV_{equity\;(t-I)}\;where\;BV=beginning\;BV\\EVA=NOPAT-(WACC\times TOTAL\;CAPITAL)\rightarrow Beginning\;invested\;capital\;(D\;+\;E)\\=\left[EBIT-(1-tax)\right]-WACC\\MVA=Market\;Value-Total\;Capital\\RI_t\;=\;E_t-(r\;\times B_{t-1})\;=(ROE-r)\times B_{t-1}\\$$

Intrinsic Value

$$P_0=B_0+\left\{\frac{RI_1}{\left(1+r\right)^1}+\frac{RI_2}{\left(1+r\right)^2}+\frac{RI_3}{\left(1+r\right)^3}+...\right\}\\$$

Single stage Residual Model

$$P_0=B_0+\left[\frac{(ROE-r)\times\;B_0}{r-g}\right]\\$$

The growth rate implied by the market price in a single- stage residual income

$$g=r-\left[\frac{\;B_0\times(ROE-r)}{P_0-B_0}\right]\\$$

$$Tobin’s\;Q=\frac{market\;value\;of\;debt+market\;value\;of\;equity}{replacement\;\cos t\;of\;total\;asset}\\P_0=B_0+(PV\;of\;interim\;high\;growth\;RI)+(PV\;of\;continuing\;residual\;income)\\PV\;of\;Continuing\;Residual\;Income\;in\;year\;T-1=\frac{RI_T}{1+r-\omega}\\$$

 

$$V_F=\frac{FCFF_1}{WACC-g}\\V_E=\frac{FCFE_1}{K_e-g}\\Control\;premium=pro\;rata\;value\;of\;controlling\;interest-pro\;rata\;value\;of\;non\;controlling\;interest.\\Adjusted\;control\;premium\;(applicable\;for\;MVIC\;multiple)=(control\;premium\;on\;equity)\times(1-DR)\\\left[DR\;=\;Debt\;to\;asset\;ratio\right]\\DLOC=1-\left[1/(1+control\;premium)\right]\\Total\;discount\;for\;lack\;of\;market\;ability=1-\left[(1-DLOC)\;(1-DLOM)\right]\\$$

The Relationship Between the Discount Factor and the Spot Rate

$$P_T=\frac1{{(1+S_T)}^T}$$

Forward Rates

$$F_{(j,k)}=\frac1{\left[1+f(j,k)\right]^k}$$

The Forward Pricing Model

$$F_{(j,k)}=\frac{P_{(j,k)}}{P_j}$$

The Forward Rate Model

$$\left[1+S_{(j,k)}\right]^{(j+k)}=\left(1+S_j\right)^j\left[1+f(j,k)\right]^k$$

Swap Rate

$$Swap\;Rate=\frac{1-d_L}{\sum d_L}\\Swap\;Spread_t\;:\;=\;Swap\;Rates_t-Treasury\;yield_{t\;}(Same\;Maturity)\\I\;spread=Riskiness\;of\;Corporate\;Bond\;Over\;Banks\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\comp^n\;credit\;liquidity\;risk=corp\;Bond-swap\;rate\\TED\;Spread=(3months\;LIBOR\;Rate)-(3months\;T\;bill\;Rate)\\LIBOR-\;O/S\;Spread\;=\uparrow=banks\;unwilling\;to\;lend;\;\downarrow=liquidity\\LIBOR\rightarrow Includes\;credit\;risk\\O/S\;Spread\rightarrow minimal\;credit\;risk\\O/S\;Spread\rightarrow measure\;of\;money\;market\;securities\;risk\\\\$$
Cox Ingersoll Ross Model (CIR)

$$dr=\left[a(b-r)dt+\left(\sigma\sqrt r.dz\right)\right]$$

Vasicek Model

$$dr=\left[a(b-r)dt+\left(\sigma dz\right)\right]$$

Ho Lee Model

$$dr_t=\theta_t\;dt+\sigma dz_t$$

Sensitivity to Parallel, Steepness, and Curvature Movements

$$\frac{\Delta P}P=D_L\Delta X_L-D_S\Delta X_S-D_C\Delta X_C$$

Value of option embedded in a bond

$$V_{Call}\;=\;V_{straight}\;\;bond-V_{callable}\;bond\\V_{Put}\;=\;V_{Putable}\;\;bond-V_{Straight}\;bond\\OAS=Z-Call\;Risk\\OAS=Z+Put\;Risk\\Effective\;Duration=\frac{P_2-P_1}{2P_0\triangle y}\\Effective\;Convexity=\frac{P_2-P_1-2P_0}{P_0\left(\triangle y\right)^2}\\Market\;Conversion\;Premium\;Ratio=\frac{Conversion\;Premium\times Market\;Per\;Share}{Market\;Price\;of\;Convertible\;Stock}\\Conversion\;Value=Market\;Price\;of\;Stock\times Conversion\;Ratio\\Market\;Conversion\;Price=\frac{Market\;Price\;of\;Convertible\;bond}{Convertion\;Ratio}\\Market\;Conversion\;Premium\;Per\;Share=Market\;Conversion\;Price-Stock’s\;Market\;Price\\Premium\;Over\;Straight\;Value=\frac{Market\;Price\;of\;Convertible\;bond}{Straight\;Value}-1$$

Put Call Parity

$$C-P=PV\;(Forward\;price\;of\;the\;bond\;on\;exercise\;date)-PV(Exercise\;price)$$

$$Present\;Value\;of\;Expected\;Losses=Expected\;Loss+Risk\;Premium-Time\;Value\;Discount\\Value\;of\;Stock_{T\;}=\;Max\;(0,\;A_T-K)\\Value\;of\;Debt_{T\;}=\;Min\;(A_T,K)\\Probability\;of\;Default=I-N\;(e_2)\\e_1=\frac{\ln\left({\displaystyle\frac{A_t}K}\right)+\mu(T-t)+{\displaystyle\frac12}\sigma^2(T-t)}{\sigma\sqrt{T-t}}$$

Key rate duration total duration -> same effect if parallel shift

$$Present\;Value\;of\;Expected\;Losses=Expected\;Loss+Risk\;Premium-Time\;Value\;Discount\\Value\;of\;Stock_{T\;}=\;Max\;(0,\;A_T-K)\\Value\;of\;Debt_{T\;}=\;Min\;(A_T,K)\\Probability\;of\;Default=I-N\;(e_2)\\e_1=\frac{\ln\left({\displaystyle\frac{A_t}K}\right)+\mu(T-t)+{\displaystyle\frac12}\sigma^2(T-t)}{\sigma\sqrt{T-t}}\\Key\;rate\;duration\;total\;duration\;\rightarrow same\;effect\;if\;parallel\;shift\\Duration\;exposure=Add\;the\;duration\\Effective\;Duration=\frac{P_2-P_1}{2P_0\triangle y}\\Effective\;Convexity=\frac{P_2-P_1}{P_0\left(\triangle y\right)^2}\\\%\;\Delta\;Bond\;Price=-\Delta y\times ED=\frac12\times EC\times\left(\triangle y\right)^2\\VCB=VNCB-Call\;Price\\VPB=VN_{PB}\;+Put\;Price$$

$$Pay\;out\;Amount=pay\;out\;Ratio\times NP\\Pay\;out\;Ratio=I-(Recovery\;Rate)\;\%\\Hazard\;rate/conditional\;Prob.\;Of\;default=Prob.\;(PD/Default\;has\;not\;occurred)\\Expected\;Loss=Hazard\;Rate\times LGD\;(\%\;terms)\\Upfront\;payment=PV\;(protection\;leg)-PV\;(premium\;leg)\\\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\Based\;on\;CDS\;spread\;\;Based\;on\;coupon\;rate\\Upfront\;Premium=(CDS\;spread-CDS\;coupon)\times duration\;of\;spread\\Price\;of\;CDS=100-Upfront\;Premium\;(\%)$$

Valuation After Inception of CDS

$$Profit\;for\;protection\;buyer\approx(\triangle spread\times duration)\times Notional\;Principal\;Or,\\Profit\;for\;protection\;buyer\;(\%)\approx change\;in\;spread\;(\%)\times duration$$

$$Forward\;Price=Price\;That\;Prevents\;Profitable\;Riskless\;Arbitrage\;in\;Frictionless\;Markets$$

The Forward Contract Price

$$FP=S_0\times{(I+R_f)}^t$$

Forward Contracts with Discrete Dividends

$$FP(on\;An\;Equity\;Security)=(S0-PVD)\times(1+Rf)T\\FP(on\;an\;Equity\;Security)=\left[S_0\times{(1+R_f)}^T\right]-FVD\\$$

Value of the Long Position in A Forward Contract on A Dividend Paying Stock

$$FP(on\;An\;Equity\;Security)=(S0-PVD)\times(1+Rf)T\\FP(on\;an\;Equity\;Security)=\left[S_0\times{(1+R_f)}^T\right]-FVD\\V_t(long\;position)=(S_t-PVD_t)-\left[\frac{FP}{\left(I+R_f\right)^{T-t}}\right]\\$$

Equi ty Forward Contracts with A Continuous Dividends

$$FP(on\;An\;Equity\;Index)=S_0\times e^{(R_f^c-\delta^c)\times T}=(S_0\times e^{\delta^c\times T})\times e^{R_f^c\times T}\\F=\frac{Spot\times e^{interest\times t}}{e^{Dividends\times t}}\\$$

Forward Price on A Coupon Paying Bond

$$FP=\left[\left(Full\;price\right)\left(1+R_f\right)^T-AI_T-FVC\right]\\QFP=FP/CF=\left[\left(Full\;price\right)\left(1+R_f\right)^T-AI_T-FVC\right]\left(\frac1{CF}\right)\\$$

Covered Interest Rate Parity

$$FT(Currency\;Forward\;Contract)=S0\times\frac{\left(1+R_{PC}\right)^T}{\left(1+R_{BC}\right)^T}\\$$

Valuing Currency Forward Contracts After Initiation

$$FT(Currency\;Forward\;Contract)=S0\times\frac{\left(1+R_{PC}\right)^T}{\left(1+R_{BC}\right)^T}\\V_{t\;}=\frac{\left[FP_t-FP\right]\times\left(contract\;size\right)}{\left(1+r_{PC}\right)^{(T-t)}}=\left[\frac{S_t}{\left(1+R_{BC}\right)^{(T-t)}}\right]-\left[\frac{F_T}{\left(1+R_{PC}\right)^{(T-t)}}\right]\\Value\;of\;Futures\;Contract=Current\;Futures\;Price-Previous\;Mark\;to\;Market\;Price\\$$

Discount factors (zs)

$$Z=\frac1{\left[1+\left(LIBOR\times\frac{days}{360}\right)\right]}\\SFR\;(Periodic)=\frac{1-last\;discount\;factor}{sum\;of\;discount\;factors}$$

Market Value of An Interest Rate Swap

$$Value\;to\;The\;Payer=\sum Z\;\times\;(SFR_{New}-SFR_{Old}\;)\times\frac{days}{360}\times Notional\;Principal$$

Equity Swaps

$$SFR(Periodic)=\frac{1-Last\;Discount\;Factor}{Sum\;of\;Discount\;Factors}$$

Put Call Parity

$$C_0+\frac X{{(1+\;R_F)}^T}=P_0+S_D\\H=\frac{C^+-C^-}{S^+-S^-}$$

Black Scholes Model

$$C_0=\frac X{{(1+\;R_F)}^T}=P_0+S_D\\H=\frac{C^+-C^-}{S^+-S^-}\\C_o=S_0\;N(d_1)-e^{-rt}XN(d_2)\\P_o=e^{rt}XN(-d_2)-S_0\;N(-d_1)\\Where\\d_1=\frac{ln\lbrack{\displaystyle\frac SX}\rbrack+(r+{\displaystyle\frac{\sigma^2}2})T}{\sigma\sqrt T}\\d_2=d_1-\sigma\sqrt T$$

Options on Dividend Paying Stocks

$$C_o=S_0\;e^{-\delta t}N(d_1)-e^{-rt}N(d_2)\\P_o=e^{-rt}N(-d_2)-S_0\;e^{-\delta t}N(-d_1)\\Where\;\delta=\;Continuously\;Compounded\;Dividend\;Yield\\d_1=\frac{ln({\displaystyle\frac SX})+(r-\delta+{\displaystyle\frac{\sigma^2}2})T}{\sigma\sqrt T}\\d_2=d_1-\sigma\sqrt T$$

Options on Currencies

$$C_o=S_0\;e^{-rBT}N(d_1)-e^{-rPT}N(d_2)\\P_o=e^{-rPT}N(-d_2)-S_0\;e^{-rBT}N(-d_1)$$

The Black Model

$$C_o=\frac{F_T}{e^{rt}}N(d_1)-\frac X{e^{rt}}N(d_2)\\Where,\;d_1=\frac{\ln\left({\displaystyle\frac{F_T}X}\right)+\left({\displaystyle\frac{\sigma^2}2}\right)t}{\sigma\sqrt t}\\\;d_2=d_1-\sigma\sqrt t$$

Interest Rate Options

$$C_o=\frac{AP}{e^{r(N\times{\displaystyle\frac{30}{360}})}}\lbrack FRA(M\ast N)N(d_1)-XN(d_2)\rbrack\times NP\\Where,\;AP=Accrual\;Period=\frac{Actual}{365}=\left[\frac{\left(N-M\right)\times30}{360}\right]\\NP=Notional\;Principal\;on\;the\;FRA$$

Swaptions

$$V_{payer\;swaption}=PVA\;\lbrack SFR.\;N(d_1)-XN(d_2)\rbrack\times NP\times AP\\\Delta C=Call\;Delta\times\Delta s+\frac12Gamma\times\Delta s^2\\\Delta P=Put\;Delta\times\Delta s+\frac12Gamma\times\Delta s^2\\$$

$$V_{payer\;swap}=V_{floating\;rate\;note}-V_{fixed\;rate\;bond}\\Duration\;of\;Payer\;Swap=Duration\;of\;Floating\;Rate\;Bond-Duration\;of\;Fixed\;Rate\;Bond\\Long\;Futures\;+Risk\;Free\;Asset=Long\;Stock\\$$

Covered Call

$$Initial\;Investment=S_0-C_0\\Max\;Profit=C_0-(S_0-X)\\Max\;loss=Initial\;Inv\\Breakeven\;price=Initial\;Inv\\$$

Bull Call

$$Initial\;Investment=C_L-C_H\\Max\;loss=Initial\;Inv\\Breakeven=X_L+(C_L-C_H)\\Max\;\mathrm\pi=(X_H\;–\;X_L)\;–\;(C_L\;–\;C_H)\\$$

Bear Call

$$Initial\;Inflow=C_L-C_H\\Max\;\mathrm\pi=Initial\;Inflow\\Breakeven=X_L+(C_L-C_H)\\Max\;loss=(X_H\;–\;X_L)\;–\;(C_L\;–\;C_H)\\$$

Collar

$$Breakeven=S_0+P_0+C_0\\Max\;\mathrm\pi=(X_H-S_0)+C_0-P_0\\Max\;loss=(S_0\;–\;X_L)\;+(P_0-C_0)\\$$

Long Straddle

$$Max\;Profit=S_T\;–\;X-(C_0+P_0)\\Max\;Loss=C_0+P_0\\Breakeven=X-(C_0\;+\;P_0)\;and\;\;X+(C_0+P_0)\\$$

Breakeven Price Analytics

$$\sigma_{annual}=\%\triangle PX\sqrt{\frac{252}{trading\;days\;until\;maturity}}\\Where\;\%\;\triangle P\;=\frac{\vert breakeven\;price-current\;price\vert}{current\;price}\\$$

 

$$Capitalization\;Rate=\frac{Net\;Operating\;Income}{Property\;Value}\\Property\;Value=\frac{Net\;Operating\;Income}{Capitalization\;Rate}\\NAVPS\;(u\sin g\;current\;market\;values)=\frac{A-L}{Number\;of\;Shares}(here\;property\;value\;is\;added)\\$$

FFO and AFFO

Funds from Operations:

$$Accounting\;net\;earnings+Depreciation\;expense+Deferred\;tax\;expenses\;(i.e.\;deferred\;tax\;liabilities)\\-Gains\;from\;sale\;of\;property\;and\;debt\;restructuring+Losses\;from\;sale\;of\;property\;and\;debt\;restructure\\=Funds\;from\;Operations$$

Adjusted Funds from Operations:

$$Fund \;from \;Operations (FFO)- Non \;cash \;rent\; adjustment- Recurring\; maintenance \;type \;capital \;expenditure \;and \;leasing \;commissions\\
=Adjusted \;funds \;from \;operations$$

Income Approach

Net Operating Income:

$$Rental \;income \;(if \;fully \;occupied)+ other \;income\\= Potential \;gross \;income\;- vacancy \;and \;collection \;loss\\
= Effective \;gross \;income \;- Operating expense\;
= Net operating income$$

The Capitalization Rate:

$$Cap\;Rate=\frac{NOI_I}{Value}\\or,Value=V_0=\frac{NOI}{Cap.Rate}\\or,\;Cap.Rate=\frac{NOI}{Comparable\;Sales\;Price}\\P_0=\frac{D_1}{R_e-g}\\or,\;P_0=\frac{NOI}{Cap.Rate}\\or,\;Cap.Rate=\frac{NOI}{P_0}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\Value\;of\;property\;today$$

$$Value=\frac{Rent}{ARY}\\or,\;ARY=\frac{Rent}{Comparable\;Sale\;P_x}\\Gross\;Income\;Multiplier=\frac{Sales\;Price}{Gross\;Income}\\Value=Gross\;Income\times Gross\;Income\;Multiplier$$

Discounted Cash Flow Method:

$$Cap\;Rate=Discounted\;Rate-Growth\;Rate=R_e-g\\Discounted\;Rate=Cap\;Rate+Growth\;Rate\\Value=V_0=\frac{NOI}{(r-g)}=\frac{NOI}{Cap\;Rate}$$

Cost Approach

$$V_{property}=(Replacement\;\cos t-Cost\;of\;fixing\;curable\;items)-Depreciation-Incurable\;expense\;capitalised+Market\;value\;of\;land\\Where\;Depreciation=\frac{Effective\;Age}{Economic\;Life}\times\left[Replacement\;Cost-Curable\;Items\right]$$

Appraisal Based indices

$$Return=\frac{NOI–Capital\;Expenditure+End\;Market\;Value–Beginning\;Market\;Value}{Beginning\;Market\;Value}\;$$

Ratios to consider for evaluation

$$Debt\;Service\;Coverage\;Ratio.=\frac{First\;year\;NOI}{Debt\;Service}\left[higher\;the\;better\right]\\Loan\;to\;Value\;Ratio=\frac{Loan\;Amt}{\;Appraisal\;Value}\left[lower\;the\;better\right]\\Equity\;Dividend\;Rate=\frac{First\;year\;CF}{Equity}$$

$$Valuation\;After\;Renovation=\frac{Stabilised\;NOI}{Cap.Rate}\\Valuation\;After\;Renovation-PV\;of\;loss=Total\;Value$$

Mostly talking about LBO

$$\frac DE\rightarrow Public\\\frac D{EBITDA}\rightarrow P/E\;firm$$

LBO’s Exit Value

$$Investment\;Cost+Earning\;Growth+Increase\;in\;Price\;Multiple+Reduction\;in\;Debt=Exit\;Value$$

Valuation Issues in Venture Capital Investments

$$The\;Post\;Money\;Valuation\;of\;The\;Investment\;Company\;is:\\PRE+INV=POST\\The\;Ownership\;Proportion\;of\;The\;Venture(VC)Investor\;is:\\=INV/POST$$

Paid in capital (PIC) $$Paid\;in\;Capital(PIC)=\frac{Paid\;in\;Capital\;(Cumulative)}{Committed\;Capital}\\Distributed\;to\;paid\;in\;capital(DPI)=\frac{Total\;Distribution\;(Cumulative)}{Total\;Invested\;Capital}\\Residual\;Value\;to\;paid\;in\;capital(RVPI)=\frac{Value\;of\;fund}{Total\;Invested\;capital}\\Total\;Value\;to\;paid\;in\;capital(TVPI)=DPI+RVPI$$

NAV Before Distribution

$$NAV\;Before\;Distribution=NAV\;After\;Distributions\;in\;Prior\;Year+Capital\;Called\;Down-Management\;Fees+Operating\;Results$$

NAV After Distribution

$$NAV\;After\;Distribution=NAV\;Before\;Distributions-Carried\;Interest-Distributions$$

The Fraction of VC Ownership ( f ) for the VC Investment can be computed as

$$NPV\;Method:\;f=\frac{INV}{POST}\\IRR\;Method:\;f=\frac{FV(INV)}{Exit\;Value}$$

 

 

$$Total\;Return=Collateral\;Return(HPY\;on\;T\;bill)+Price\;Return\left(\frac{P_I-P_O}{P_O}\right)+Roll\;Return\\Price\;Return=\frac{Current\;Price-Previous\;Price}{Previous\;Price}\\Roll\;Return=\frac{Price\;of\;Expiring\;Futures\;Contract\;-\;Price\;of\;New\;Futures\;Contract}{Price\;of\;Expiring\;Futures\;Contract}$$

APT

$$E(R_A)=R_f+B_{A,1}\;\lambda_1+B_{A,2}\;\lambda_2+...$$

Multifactor

$$Macro-economic:\;R_i=E\left(R_i\right)+b_1F_1+b_2F_2+...+b_nF_n+e_t\\Fundamental\;Factor:\;R_i=a_i+b_{i,1}F_{P/E}+b_{j,2}F_{Size}+e_t$$

Standardized P/E Sensitivity

$$b_{i1}=\frac{\left({\displaystyle\frac PE}\right)i-\overline{P/E}}{\sigma P/E}$$

Active Return

$$Active\;Return=(R_P)-(R_B)\\Active\;Risk=\sigma_{(R_P-R_B)}\\IR=\frac{{\overline R}_P-{\overline R}_B}{\sigma_{(R_P-R_B)}}$$

Return Attribution

$$Active\;Return=Factor\;Return+Security\;Selection\;Return$$

Factor Return

$$Factor\;Return=\sum_{i=1}^k\left(\beta_{Pi}-\beta_{bi}\right)\times\lambda_i$$

Active Specific Risk

$$Active\;Specific\;Risk=\sum_{i=1}^n\left(W_{Pi}-W_{bi}\right)^2\times\sigma_{\varepsilon i^2}$$

Carhart Model

$$E(R)=R_F+\beta_1RMRF+\beta_2SMB+\beta_3HML+\beta_4WML$$

$$\sigma_{Portfolio}^2=W_A^2\sigma_A^2+W_B^2\sigma_B^2+2W_AW_BCov_{AB}\\E(R_i)=R_f+Beta_i\;\left[E(R_{MKT})-R_f\right]\\Change\;in\;Price=-Duration\;(\triangle Y)+\frac12Convexity\;{(\triangle Y)}^2\;Change\;in\;Call\;Price=delta\;(\triangle S)+\frac12gamma\;{(\triangle S)}^2+vega(\triangle V)\\Discount\;Rate=R+\pi+\theta+\gamma+K+\phi\\Inter\;temporal\;Rate\;of\;Substitution=\frac{U_t}{U_o}=\frac{Future}{Current}\\P_O=E(m_t)$$

Real Risk free rate of Return

$$R=\frac{1-\;P_0}{P_0}=\left(\frac I{E(m_t)}\right)-1$$

$$R=\frac{1-\;P_0}{P_0}=\left(\frac I{E(m_t)}\right)-1\\P_0=\frac{E\left(P_1\right)}{I+R}+cov(P_1,M_1)\\P_0\;is\;lower;\;Return\;\uparrow\;(Since\;Risk\;Taken)\\\frac{E\left(P_1\right)}{I+R}\;;When\;no\;risk=P_0\\Disc.\;Rate=R+\pi\;(short\;term)\\R+\pi+\theta\;(long\;term)$$

Taylor Rule

$$r=R_n+\pi\;+0.5(\pi-\pi\ast)+0.5(y-y\ast)$$

$$BEI=Yield\;on\;Non\;Inflation\;Indexed\;Bond-Yield\;on\;Non\;Inflation\;Indexed\;Bond\\BEI=\pi+\theta$$

$$Active\;Return\;E(R_A)=E(R_P)-E(R_B)$$

For an Active Portfolio of N Securities

$$E(R_A)==\sum\triangle w_iE(R_i)$$

Weighted Average of Securities Returns

$$E(R_P)=\sum w_{P,i}E(R_{P,j})\;and\;E(R_P)=\sum w_{B,i}E(R_{B,j})\;$$

Ex ante Active Return

$$E(R_A)=\sum\triangle w_{P,i}E(R_{P,j})-\sum w_{B,i}E(R_{B,j})\;$$

Security Selection Return

$$E(R_A)=\sum\triangle w_iE(R_{B,j})+\sum w_{P,i}E(R_{A,j})\;$$

Sharpe Ratio

$$Sharpe\;Ratio=\frac{R_P-R_F}{\sigma_P}$$

IR

$$IR=\frac{R_P-R_B}{\sigma_{(R_P-R_B)}}$$

With Optimal Level of Active Risk

$$SR_P=\sqrt{SR_B^2+IR^2}\\Total\;Risk\;of\;The\;Portfolio:\;\sigma_P^2\;=\sigma_B^2+\sigma_A^2$$

Unconstrained

$$IR\ast=IC\times\sqrt{BR}\\E(R_A)\ast=IC\sqrt{BR\sigma_A}$$

Constrained

$$IR=IC\times\sqrt{BR}\times TC\\E(R_A)=IC\times\sqrt{BR}\times TC\times\sigma_A\\SR_{pc}=\sqrt{SR_\beta^2+(IR^2\times TC)}\\\sigma_{CA}=\frac{TC.IR\ast}{SR_B}\times\sigma_B$$

Ex post Performance Measurement

$$E(R_A\vert IC_R)=TC\times IC_R\sqrt{BR}\sigma_A\\R_A=E(R_A\vert IC_R)+noise$$

The Expected Active Return for A Given Target Level of Active Risk

$$E(R_A)=IR\times\sigma_A\\IC=2(\%\;correct)-1\\\sigma_c=\left[\sigma_x^2+\sigma_y^2-2\sigma_x\sigma_yr_{x,y}\right]^{1/2}$$

Annualized Active Risk

$$\sigma_A=\sigma_c\times\sqrt{BR}$$

Annualized Active Return

$$E\left(R_A\right)=IC\sqrt{BR}\times\sigma_A\\BR=\frac N{1+(N-I)r}$$

Effective Spread

$$For\;Buy\;Order=2\times(Execution\;Price–Mid\;Quote)\\For\;Sell\;Order=2\times(Mid\;Quote\;–\;Execution\;Price)$$

Total Implementation Shortfall

$$Total\;Implementation\;Shortfall\;(\%)=\frac{Paper\;Gain-Real\;Gain}{Paper\;Investment}$$

Missed Trade

$$\;Missed\;Trade=\vert CP-DP\vert\times\;\#\;of\;shares\;canceled$$

Explicit Cost

$$\;Explicit\;Cost=\cos t\;per\;share\times\;\#\;of\;shares\;executed$$

Delay

$$Delay=\vert BP^\ast-DP\vert\times\#\;of\;shares\;later\;executed$$

Market Impact

$$\;Market\;Impact=\vert EP\;–\;DP\vert or\;BP^\ast\;\times\;\#\;of\;shares\;executed$$

Time – Weighted Rate of Return (TWRR)

$$R_P=(1+R_{S1})\;(1+R_{S2})\;(1+R_{S3})\;(1+R_{S4})\dots.\;(1+R_{Sk})–1$$

Money – Weighted Rate of Return (MWRR)

$$MV_1=MV_0{(1+R)}^m+\overset n{\underset{i=1}\sum}CF_i{(1+R)}^{L(i)}$$

$$P=M+S+A\\Where,\;P=portfolio\;return\\M=market\;index\;return\\S=return\;to\;style\\A=return\;due\;to\;active\;management$$

Asset Category Contribution

$$R_{AC}=\sum_{i=1}^Aw_i(R_i-R_F)$$

$$R_{IM}=\sum_{i=1}^A\;\sum_{j=1}^Mw_i\times w_{i,j}\times(R_{A,i,j}-R_{B,i,j})$$

Micro Performance Attribution

$$R_V=\overset S{\underset{j=1}{\sum(}}W_{P,j}-W_{B,j})\times(r_{B,j}-r_B)+\sum_{j=1}^SW_{B,j}\times(r_{P,j}-r_{B,j})+\overset S{\underset{j=1}{\sum(}}W_{P,j}-W_{B,j})\times(r_{P,j}-r_{B,j})$$

Ex-post Alpha

$$\alpha_A=R_{At}-{\widehat R}_A$$

Treynor’s Measure

$$T_A=\frac{{\overline R}_A-{\overline r}_f}{\beta_A}$$

Sharpe Ratio

$$Sharpe\;Ratio=\frac{{\overline R}_A-{\overline r}_f}{\sigma_A}$$

$$M^2={\overline r}_F+\left[\frac{{\overline R}_A-{\overline R}_F}{\sigma_A}\right]\sigma_M$$

Information Ratio

$$IR_A=\frac{{\overline R}_A-{\overline R}_B}{\sigma_{A-B}}$$

The Original Dietz method

$$R_{Dietz}=\frac{EMV-BMV-CF}{BMV+0.5CF}$$

Modified Dietz method

$$\frac{EMV-BMV-CF}{BMV+{\displaystyle\sum_{i=1}^n}W_i\times CF_i}$$

$$Downside\;Deviation=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left[Min\;(Return_t-threshold,0)\right]^2}{n-1}}\\Sharpe_{Hf}=\frac{Annualized\;return-Annual\;RiskFreeRate}{Annualized\;\sigma}$$

$$Analytical\;Method\;for\;Estimating\;VaR=\left[{\widehat R}_P-(z)(\sigma)\right]V_P$$

Measuring Risk Adjusted Performance

$$Sharpe\;Ratio=\frac{{\overline R}_P-{\overline R}_F}{\sigma_P}$$

$$Return\;over\;Maximum\;Drawdown(RoMAD)=\frac{R_P}{MaximumDrawdown}\\Sortino=\frac{{\overline R}_P-MAR}{DownsideDeviation}$$

$$\beta_i=\frac{Cov(i,m)}{\sigma^2m}\\Number\;of\;Contracts=\left(\frac{\beta_T-\beta_P}{\beta_f}\right)\left(\frac{V_P}{P_f(multiplier)}\right)\\Underlying\;Portfolio+Derivative=Target\;Portfolio\\V_p\times\;\beta_p+(n\times m\times F_p)\times\beta_f=V_p\times\beta_T\\n=V_p(\beta_T–\beta_P)/m\times F_p\times\beta_F\\Effective\;Beta=\frac{Percentage\;Change\;In\;Value\;of\;Portfolio}{Percentage\;Change\;in\;Index}\\Synthetic\;Equity=Cash+Futures\\Long\;Stock+Short\;Future=Long\;risk\;free\;Bond$$

Target Duration

$$Number\;of\;Contracts=(yield\;beta)\left(\frac{MD_T-MD_P}{MD_F}\right)\left(\frac{V_P}{P_f(multiplier)}\right)\\Number\;of\;Contracts=\left(\frac{D_T-D_P}{D_{CTD}P_{CTD}}\right)\;P_p$$

Covered Call

$$Covered\;Call=Long\;stock\;position+Short\;call\;position\\Value\;at\;expiration=V_T=S_T–max\;(0,\;S_T\;–\;X)\\Profit=V_T\;–S_0+c_0\\Maximum\;Profit=X\;–S_0+c_0\\Max\;loss\;(when\;S_T=0)=S_0–c_0\\Breakeven=S_T^\ast=S_0–c_0$$

Protective Put

$$Protective\;Put=Long\;stock\;position+Long\;Put\;position\\Value\;at\;expiration:V_T=S_T+max\;(0,X-S_T\;)\\Profit=V_T\;–S_0-p_0\\Maximum\;Profit=\infty\\Maximum\;loss=S_0+p_0-X\\Breakeven=S_T^\ast=S_0+p_0$$

Bull Call Spread

$$Bull\;Call\;Spread=Long\;Call\;(lower\;exercise\;price)+Short\;Call\;(higher\;exercise\;price)\\Initial\;value=V_0=c_1–c_2\\Value\;at\;expiration:VT=value\;of\;long\;call–Value\;of\;short\;call=max\;(0,\;S_T\;–\;X_1)-max\;(0,\;S_T\;–\;X_2)\\Profit=V_T-c_1+c_2\\Maximum\;Profit=X_2–X_1-c_1+c_2\\Maximum\;Loss=c_1-c_2\\Breakeven=S_T^\ast=X_1+c_1–c_2$$

Bull Put spread

$$Bull\;Put\;spread=Long\;Put\;(lower\;XP)+Short\;Put\;(higher\;XP)\\Identical\;to\;the\;sale\;of\;Bear\;Put\;Spread\;XP=exercise\;price$$

Bear Put Spread

$$Bear\;Put\;spread=Long\;Put\;(higher\;XP)+Short\;Put\;(lower\;XP)\\Initial\;value=V_0=p_2–p_1\\Value\;at\;expiration:\;V_T=value\;of\;long\;put–value\;of\;short\;put=max\;(0,\;X_2-S_T)-max\;(0,\;X_1-S_T)\\Profit=V_T–p_2+p_1\\Max\;Profit=X_2–X_1–p_2+p_1\\Max\;Loss=p_2-p_1\\Breakeven\;=S_T^\ast=X_2–p_2+p_1\\$$

Bear Call Spread

$$Bear\;Call\;Spread=Short\;Call\;(lower\;XP)+Long\;Call\;(higher\;XP)\\Identical\;to\;the\;sale\;of\;Bull\;Call\;Spread$$

Long Butterfly Spread

$$Long\;Butterfly\;Spread\;(Using\;Call)=Long\;Butterfly\;Spread\\=Long\;Bull\;call\;spread+Short\;Bull\;call\;spread\;(or\;Long\;Bear\;call\;spread)\\Long\;Butterfly\;Spread=(Buy\;the\;call\;with\;XP\;of\;X_1\;and\;sell\;the\;call\;with\;XP\;of\;X_2)\\+(Buy\;the\;call\;with\;XP\;of\;X_3\;and\;sell\;the\;call\;with\;XP\;of\;X_2)\\where,\;X_1<X_2<X_3\;and\;Cost\;of\;X_1(c_1)>Cost\;of\;X_2\;(c_2)>Cost\;of\;X_3\;(c_3)\\Value\;at\;expiration:\;V_T=max\;(0,S_T–X_1)–2\;max\;(0,\;S_T–X_2)+max\;(0,\;S_T–X_3)\\Profit=V_T–c_1+2c_2-c_3\\Max\;Profit=X_2–X_1–c_1+2c_2-c_3\\Maximum\;Loss=c_1-2c_2+c_3$$

Two breakeven points

$$Breakeven=ST\ast=X_1+net\;premium=X_1+c_1–2c_2+c_3\\Breakeven=ST\ast=2X_2–X_1–Net\;premium=2X_2–X_1–(c_1–2c_2+c_3)=2X_2–X_1–c_1+2c_2-c_3$$

Short Butterfly Spread

$$Short\;Butterfly\;Spread\;(U\sin g\;Call)=Selling\;calls\;with\;XP\;of\;X_1\;and\;X_3\;and\;buying\;two\;calls\;with\;XP\;of\;X_2\\Max\;Profit=c_1+c_3–2c_2$$

Long Butterfly Spread

$$Long\;Butterfly\;Spread\;(U\sin g\;Puts)=(Buy\;put\;with\;XP\;of\;X_3\;and\;sell\;put\;with\;XP\;of\;X_2)\\+(Buy\;the\;put\;with\;XP\;of\;X_{1\;}and\;sell\;the\;put\;with\;XP\;of\;X_2)\\where,\;X_1<X_2<X_3\;and\;Cost\;of\;X_1\;(p_1)<Cost\;of\;X_2\;(p_2)<Cost\;of\;X_3\;(p_3)$$

Short Butterfly Spread

$$Short\;Butterfly\;Spread\;(U\sin g\;Puts)=Short\;butterfly\;spread\\=Selling\;puts\;with\;XPs\;of\;X_1\;and\;X_3\;and\;buying\;two\;puts\;with\;XP\;of\;X_2\\Max\;Profit=p_3\;+\;p_1\;–\;2p_2$$

For zero-cost collar

$$Initial\;value\;of\;position=V_0=S_0\\Value\;at\;expiration:\;V_T=S_T+max\;(0,\;X_1\;-\;S_T)\;–\;max\;(0,\;S_T\;–\;X_2)\\Profit=V_T–V_0=V_T–S_0\\Max\;Profit=X_2–S_0\\Max\;Loss=S_0–X_1\\Breakeven=S_T\ast=S_0$$

Straddle

$$Straddle=Buying\;a\;put\;and\;a\;call\;with\;same\;strike\;price\;on\;the\;same\;underlying\;with\;the\;same\;expiration;\\;both\;options\;are\;at\;the\;money\\Value\;at\;expiration:\;V_T=max\;(0,\;S_T-X)+max\;(0,\;X–S{{}_T)}\\Profit=V_T-p_0-c_0\\Max\;Profit=\infty\\Max\;Loss=p_0+c_0\\Breakeven=ST\ast=X\pm(p_0+c_0)\\$$

Short Straddle: Selling a put and a call with same strike price on the same underlying with the same expiration; both options are at-the-money.

  • Adding call option to a straddle “Strap”.
  • Adding put option to a straddle “Strip”.

Long Strangle = buying the put and call on the same underlying with the same expiration but with different exercise prices.

Short Strangle = selling the put and call on the same underlying with the same expiration but with different exercise prices.

Box-spread = Bull spread + Bear spread

Long Box-spread

$$Long\;Box\;spread=(buy\;call\;with\;XP\;of\;X_1\;and\;sell\;call\;with\;XP\;of\;X_2)\\+(buy\;put\;with\;XP\;of\;X_2\;and\;sell\;put\;with\;XP\;of\;X_1)\\Initial\;value\;of\;the\;box\;spread=Net\;premium=c_1–c_2+p_2–p_1\\Value\;at\;expiration:\;V_T=X_2-X_1\\Profit=X_2-X_1-\left(c_1–c_2+p_2–p_1\right)\\Max\;Profit=same\;as\;profit\\Max\;Loss=no\;loss\;is\;possible\;given\;fair\;option\;prices\\Breakeven=ST\ast=no\;break\;even;\;the\;transaction\;always\;earns\;R_f\;rate,\;given\;fair\;option\;prices\\$$

Pay-off of an interest rate Call Option

$$Pay\;off\;of\;an\;interest\;rate\;Call\;Option=(NP)\times max(0,Underlying\;rate\;at\;expiration–X\;rate)\times\frac{Days\;in\;Underlying\;Rate}{360}$$

Pay-off of an interest rate Put Option

$$Pay\;off\;of\;an\;interest\;rate\;Put\;Option=(NP)\times max(0,X\;rate-Underlying\;rate\;at\;expiration)\times\frac{Days\;in\;Underlying\;Rate}{360}$$

Loan Interest payment

$$Loan\;Interest\;payment=NP\times(LIBOR\;on\;previous\;reset\;date+Spread)\times\frac{Days\;in\;Settlement\;Period}{360}$$

Cap Pay-Off

$$Cap\;Pay\;Off=NP\times(0,\;LIBOR\;on\;previous\;reset\;date–X\;rate)\times\frac{Days\;in\;Settlement\;Period}{360}$$

Floorlet Pay-Off

$$Floorlet\;Pay\;Off=NP\times(0,\;X\;rate-LIBOR\;on\;previous\;reset\;date)\times\frac{Days\;in\;Settlement\;Period}{360}$$

Effective Interest

$$Effective\;Interest=Interest\;received\;on\;the\;loan+Floorlet\;pay\;off$$

Delta

$$Delta=\frac{Days\;in\;Settlement\;Period}{360}=\frac{\triangle C}{\triangle S}$$

Size of the Long position

$$Size\;of\;the\;Long\;position=N_c/N_s=-1/(\triangle C/\triangle S)=-1/Delta\\where,\;N_c=No\;of\;call\;options\\N_s=No\;of\;stocks$$

Hedging using non-identical option

  • One option has a delta of Δ1.
  • Other option has a delta of Δ2.
  • Value of the position$$V=N_1c_1+N_2c_2\\where,\;N=option\;quantity\;\&\;c=option\;price$$
  • To delta hedge:$$Desired\;Quantity\;of\;option\;1\;relative\;to\;option\;2=\frac{Delta\;of\;option\;2}{Delta\;of\;option\;1}$$

$$\;Gamma=\frac{Change\;in\;Delta}{Change\;in\;Underlying\;Price}\\Gamma\;hedge=Position\;in\;underlying+Positions\;in\;two\;options\\Vega=\frac{Change\;in\;Option\;Price}{Change\;in\;Volatility\;of\;the\;Underlying}$$

For a pay floating counterparty in a swap, the duration can be expressed as:

$$D_{pay\;floating}=D_{fixed}–D_{floating}>0$$

The duration of portfolio plus a swap position:

$$V_P(MD_T)=V_P(MD_p)+N_P(MD_{swap})$$

NP of a swap:

$$NP=V_P\left(\frac{MD_T-MD_P}{MD_{Swap}}\right)$$

Callable Bond

$$Straight\;Bond\;Buy+Receiver\;Swaption\;Sell=Callable\;Bond$$

Putable bond

$$Straight \;Bond \;Buy + Payer \;Swaption \;Buy = Putable \;bond$$

 

Risk Objectives

$$UP={\widehat R}_P-0.005(A)(\sigma_P^2)$$

Roy’s Safety-First Measure (RSF)

$$RSF=\frac{{\widehat R}_P-R_{MAR}}{\sigma_P}$$

The Investor’s Utility from Investing A Portfolio

$$U=E(R)–0.5\times\lambda\times\sigma^2$$

$$MCTR_i=\beta_{i,p}\sigma_P$$

$$ACTR_i=Wi\times MCTR_i\\\%\;ACTR\;to\;Total\;Risk=\frac{ACTR_i}{\sigma_P}$$

An Investor’s Return in Domestic Currency

$$R_{DC}=(1+R_{FC})(1+R_{FX})–1\\RDC\;=\sum_{i=1}^nw_i\left(R_{DC,i}\right)$$

Risk Variance of a Two Asset of a Portfolio

$$\sigma^2(R_{DC})\approx w^2(R_{FC})\sigma^2(R_{FC})+w^2(R_{FX})\sigma^2(R_{FX})+2w(R_{FC})\;w(R_{FX})\;\sigma(R_{FC})\;\sigma(R_{FX})\;\rho(R_{FC},R_{FX})$$

Single FC denominated Asset

$$\sigma^2(R_{DC})\approx\sigma^2(R_{FC})+\sigma^2(R_{FX})+2\sigma(R_{FC})\;\sigma(R_{FX})\;\rho(R_{FC},R_{FX})\\\sigma^2(R_{DC})\approx\sigma^2(R_{FX})\;(1+R_{FC})$$

 

$$Utility\;(U)=w(p_1)\;v(X_1)+w(p_2)\;v(X_2)\\Subjective\;expected\;U\;of\;an\;individual=\Sigma\left[u(x_i)\times Prob(x_i)\right]\;\\Bayes’\;Formula=P(A\vert B)=\left[P(B\vert A)/P(B)\right]\times P(A)\\Risk\;Premium=Certainty\;Equivalent–Expected\;Value\\Perceived\;Value\;of\;Each\;Outcome=U=w(p_1)\;v(x_1)+w(p_2)\;v(x_2)+\dots+w(p_n)\;v(x_n)\\Abnormal\;Return\;(R)=Actual\;R–Expected\;R\\$$

$$After\;tax\;(AT)Real\;Required\;return\;(RR)\;\%=\\\frac{Client's\;Required\;Expenditure\;in\;Year\;n}{Net\;Investable\;Assets}=\frac{Projected\;Needs\;in\;Year\;n}{Net\;Investable\;Assets}\\AT\;Nominal\;RR\;\%=\frac{Projected\;Needs\;in\;Year\;n}{Net\;Investable\;Assets}+Current\;Annual(Ann)\;Inflation\;(Inf)\;\%\\=AT\;real\;RR\%+Current\;Ann\;Inf\%\;\;\;\\Or,\;\;AT\;Nominal\;RR\%=(1+AT\;Real\;RR\%)\times(1+Current\;Ann\;Inf\%)–1\\Total\;Investable\;Assets=Current\;Portfolio-Current\;Year\;Cash\;Outflows+Current\;Year\;Cash\;Inflows\\Pre-tax\;income\;needed=AT\;income\;needed/(1-tax\;rate)\\Pre-tax\;Nominal\;RR=(Pre-tax\;income\;needed/Total\;investable\;assets)+Inf\%$$

If Portfolio returns are tax-deferred:

$$Pre-tax\;projected\;expenditure\;\$=AT\;projected\;expenditure\;\$/(1–tax\;rate)\\Pre-tax\;real\;RR\;\%=Pre-tax\;projected\;expenditures\;\$/Total\;investable\;assets\\Pre-tax\;nominal\;RR=(1+Pre-tax\;real\;RR\%)\times(1+Inflation\;rate\%)–1$$

If Portfolio returns are NOT tax-deferred:

$$AT\;real\;RR\%=AT\;projected\;expenditures\;\$/Total\;Investable\;assets\\AT\;nominal\;RR\%=(1+AT\;real\;RR\%)\times(1+Inf\%)–1$$

Procedure of converting nominal, pre-tax figures into real, after-tax return:

$$Real\;AT\;R=\left[Expected\;total\;R–(Expected\;total\;R\;of\;Tax-exempt\;Invs\times\;wt\;of\;Tax-exempt\;Invst)\;\right]\\\times(1–tax\;rate)+(Expected\;total\;R\;of\;Tax-exempt\;Invst\times wt\;of\;Tax-exempt\;Invst)–Inf\;rate\\$$

Or

$$Real\;AT\;R=\left[(Taxable\;R\;of\;asset\;class\;1\times wt\;of\;asset\;class\;1)+(Taxable\;R\;of\;asset\;class\;2\times wt\;of\;asset\;class\;2)\\+\dots+(Taxable\;return\;of\;asset\;class\;n\times wt\;of\;asset\;class\;n)\right]\\\times(1–tax\;rate)+(Expected\;total\;R\;of\;Tax-exempt\;Invst\times wt\;of\;Tax-exempt\;Invst)–Inf\;rate$$

 

Accrual Taxes

$$FVIF_{AT}=\left[1+r\;(1-t)\right]^n$$

Deferred Capital Gain Taxation

$$FVIF_{AT}={(1\;+\;r)}^n\;(1-t_{cg})+t_{cg}\\FVIF_{AT}={(1\;+\;r)}^n\;(1-t_{cg})+t_{cg}\left(B\right)$$

Annual Wealth Basis Taxation

$$FVIF_{AT}=\left[(1+r)\;(1-t_w)\right]^n$$

Effective Annual After-tax Return of a Blended Tax Regime

$$r^\ast=r_T\;(1-p_it_i–p_dt_d+p_{cg}t_{cg})$$

Effective Capital Gains Tax Rate

$$r^\ast=r_T\;(1-p_it_i–p_dt_d+p_{cg}t_{cg})\\T^\ast=\left[\frac{1-(p_i+p_d+p_{cg})}{1-(p_it_i+p_dt_d+p_{cg}t_{cg})}\right]$$

Modified Version of The Standard Deferred Capital Gains FV Formula

Standard Formula:$$FVIF_{AT}={(1\;+\;r)}^n\;(1-t_{cg})+t_{cg}(B)\;$$

Modified Formula:$$Modified\;FVIF_{AT}={(1+r^\ast)}^n\times(1-T^\ast)+T^\ast-(1-B)t_{cg}$$

Tax – Deferred Accounts (TDA)

$$FVIF_{AT}={(1+r)}^n(1-t_n)$$

Tax – Exempt Account (TEA):

$$FVIF_{AT}={(1+r)}^n$$

After – Tax Return and Risk are calculated as:

$$r_{AT}=r(1-t)\\\sigma_{AT}=\sigma(1-t)\\$$

 

$$Core\;Capital\;(CC)\;Spending\;Needs=\sum_{j=1}^N\frac{p(Survival_j)\times Spending_j}{(1+r)^j}\\Expected\;Real\;Spending=Real\;Annual\;Spending\times Combined\;Probability\\CC\;needed\;to\;maintain\;given\;spending\;pattern=Annual\;Spending\;Needs\;/\;Sustainable\;Spending\;Rate$$

Tax-Free Gifts

$$\;RV_{Tax-free\;gift\;}=\frac{\left[1+r_g(1-t_{ig})\right]^n}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}$$

Taxable Gifts

$$\;RV_{Taxable\;gift\;}=\frac{\left[1+T_g\right]\left[(1-r_g)(1-t_{ig})\right]^n}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}$$

The relative after-tax value of the when the donor pays gift tax and when the recipient’s estate will not be taxable (assuming rg = re and tig = tie)

$$RV_{Taxable\;gift\;}=\frac{FV_{Gift}}{FV_{Bequest}}\;\\\frac{\left[1+r_g\left(1-t_{ig}\right)^n\right](1-T_g+T_gT_e)}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}\\FV_{no\;skipping}=PV\left[{(1+r)}^{n1}\right]\left[(1-t)\right]\left[{(1+r)}^{n2}(1-t)\right]\\FV_{skipping}=PV\left[{(1+r)}^N(1-T_e)\right]$$

Charitable Gratuitous Transfers

$$RV_{Charitable\;Donation\;}=\frac{FV_{Charitable\;Gift}}{FV_{Bequest}}\;\\\frac{\left(1+r_g\right)^n+T_a\left[1+r_e(1-t_e)\right]^n(1-T_e)}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}$$

 

Human Capital

$$Human\;Capital\;HC_O=\sum_{t=1}^N\frac{W_t}{\left(1+r\right)^t}$$

extended model

$$Extended\;Model\;HC_O\;=\sum_{t=1}^N\frac{p\left(S_t\right)W_{t-1}\left(1+g_t\right)}{\left(1+r_f+y\right)^t}$$

Income Yield

$$Income\;Yield\;(payout)=\frac{Total\;Ongoing\;Annual\;Income}{Initial\;Purchase\;Price}\\$$

Mortality Weighted Net Present Value of the Pension

$$mNPV_0=\sum_{t=1}^N\frac{p\left(S_t\right)b_t}{\left(1+r\right)^t}\\\\$$

 

Simple Spending Rule

$$\;Spending_t=S_{market\;value\;asset\;(t-1)}$$

Rolling 3 years Average Spending Rule

$$\;Spending_t=SpendingRate\left(\frac{MV_{t-1}+MV_{t-2}+MV_{t-3}}3\right)$$

Geometric Spending Rate

$$\;Spending_t=R(Spending_{t-1})\;(1+inflation_{t-1})+(1-\;R)\;S\;(MV_{t-1})\;\;\;\;\;\;\;$$

Leverage – Adjusted Duration Gap

$$\;LADG=D_{asset}–(L/A)\;D_{liabilities\;}\;$$

Market Volatility

$$\sigma_t^2=\beta\sigma_{t-1}^2+(1-\beta)\sigma_t^2$$

Multifactor Models

$$R_i=\alpha_i+\beta_{i,1}\;F_1+\beta_{i,2\;}F_2+\varepsilon_i$$

Gordon Growth Model

$$P_0=\frac{Div_1}{{\widehat R}_i-g}\Rightarrow{\widehat R}_i=\frac{Div_1}{P_0}+g$$

Grinold – Kroner Model

$${\widehat R}_i=\frac{D_1}{P_0}+i+g-\triangle S\triangle\left(\frac PE\right)$$

Expected Income Return

$$Expected\;Income\;Return=\left(\frac{D_1}{P_0}-\triangle S\right)$$

Expected Nominal Earnings Growth 

$$Expected\;Nominal\;Earnings\;Growth=i+g$$

Expected Repricing Return

$$\;Expected\;Repricing\;Return=\triangle\left(\frac PE\right)$$

Estimating Fixed Income Returns: Risk Premium Approach

$$R_B=Real\;Risk-Free\;Rate+Inflation\;Risk\;Premium+Default\;Risk\;Premium\\+Illiquidity\;Risk\;Premium+Maturity\;Risk\;Premium+Tax\;Premium$$

The equation for the CAPM

$${\widehat R}_i=R_F+\beta_i({\widehat R}_M-R_F)$$

The Covariance Between Two Markets Given Two Factors

$$The\;covariance\;between\;two\;markets\;given\;two\;factors\;is:\\Cov(i,j)=\beta_{i,1}\;\beta_{j,1}\;\sigma_{F_1}^2+\beta_{i,2}\;\beta_{j,2}\;\sigma_{F_2}^2+\left(\beta_{i,1}\;\beta_{j,2}+\beta_{i,2}\;\beta_{j,1}\right)Cov(F_1,F_2)\\If\;there\;is\;only\;one\;factor\;driving\;returns\\Cov(i,j)=\beta_i\;\beta_j\;\sigma_M^2\\$$

Target Interest Rate

$$R_{target}=r_{neutral}+\left[0.5\;(GDP_{expected}\;–\;GDP_{trend})+0.5\;(i_{expected}\;–\;i_{target})\right]\\$$

 

Cobb-Douglas Production Function

$$Y=A\times K^\alpha\times L^\beta\\\frac{\triangle Y}Y\approx\frac{\triangle A}A+\alpha\frac{\triangle K}K+(1-\alpha)\frac{\triangle L}L$$

Solow Residual 

$$Solow\;Residual=\%\triangle TEP=\;\%\triangle Y–\alpha\;(\%\triangle K)–(1–\alpha)\%\triangle L$$

Gordon Growth Model for Mature Economies

$$V_0=\frac{D_1}{r-\overline g}=\frac{D_0\left(1+\overline g\right)}{r-\overline g}$$

H – Model for Emerging Economies

$$V_0=\frac{D_0}{r-g_L}\left[(1+g_L)\frac N2(g_s-g_L)\right]\\$$

Fed Model Ratio

$$Fed\;Model\;Ratio=\frac{S\;\&\;P\;Earnings\;Yield}{10\;year\;Treasury\;Yield}\\$$

The Yardeni Model

$$P_0=\frac{E_1}{r-g}\\\frac{E_1}{r-g}=Y_B-d\;\left(LTEG\right)$$

P/10-Year MA

$$\;P/10-Year\;MA\;(E)=\frac{Market\;Price\;of\;S\&P\;500}{10\;year\;Average\;Real\;Reported\;Earnings}$$

Tobin’s q

$$Tobin’s\;q=\frac{Market\;Value\;of\;Debt+Equity}{Asset\;Replacement\;Cost}$$

Equity q

$$Equity\;q=\frac{Market\;Value\;of\;Equity}{Replacement\;Value\;of\;Assets-Liabilities}$$

Modelling Expected Return

$$Income\;Yield=\frac{Annual\;Coupon\;Payment}{Current\;Bond\;Portfolio\;Price}$$

Roll Down Return

$$Roll\;Down\;Return=\frac{\left(Bond\;Price_{End}-Bond\;Price_{Beginning}\right)}{Bond\;Price_{Beginning}}$$

Leverage Portfolio’s Return

$$r_p=\frac{Portfolio\;Return}{Portfolio\;Equity}$$

Leveraged Portfolio’s Return

$$\;(RoE)=R_A+\frac DE\left[R_A-K_d\right]$$

Leverage

$$\;Leverage=\frac{Notional\;Value\;of\;Contract-Margin\;Amount}{Margin\;Amount}$$

Rebate Rate

$$Rebate\;Rate=Collateral\;Earnings\;Rate–Security\;Lending\;Rate$$

Convexity

$$Convexity=\frac{Mac.Duration^2+Mac.Duration+Dispersion}{(1+periodic\;IRR)^2}$$

Basis Point Value

$$Basis\;Point\;Value\;(BPV)=MD\times V\times0.0001$$

Future BPV

$$FutureBPV\approx\frac{BPV_{CTD}}{CF_{CTD}}$$

Future Contracts

$$N_f=\frac{Liability\;Portfolio\;BPV-Asset\;Portfolio\;BPV}{Futures\;BPV}$$

Effective Duration

$$Effective Duration=\frac{(PV_+)-(PV_-)}{2\times\triangle Curve\times(PV_0)}$$

Projected Holding Period Return for each bond

$$Projected\;Holding\;Period\;Return\;for\;each\;bond\;can\;be\;approximated\;as:\\\left[ending\;MD\times(Manager’s\;Forecasted\;Yield–Implied\;Forward\;Yield)\right]+Coupon\;Yield$$

Portfolio PVBP

$$Portfolio\;PVBP=0.0001\times modified\;duration\times portfolio\;value$$

Excess Return

$$XR=(s\times t)–\triangle s\times SD$$

Expected XR

$$EXR=(s\times t)–\left(\triangle s\times SD\right)-\left(t\times p\times L\right)$$

Spread risk as the spread bond will underperform in relative price change when spread widens

$$\;\%\triangle\;Value=-MD\triangle y\\\;\%\triangle\;Relative\;Value=-D_S\triangle y\\Spread=y_{\;higher\;yield}–y_{\;government}$$

Information Ratio

$$\;Information\;Ratio\approx Information\;Coefficient\times\sqrt{Investor's\;Breadth}$$

Maximize Utility Through Manager Selection

$$U_A=r_A-\lambda_A\times\sigma_A^2$$

Portfolio Active Risk

$$Portfolio\;Active\;Risk=\sqrt{\sum_{i=1}^n{w^2}_{a,i}\;{\sigma^2}_{a,i}}$$

Total Active Risk

$$Total\;Active\;Risk=\sqrt{\left(True\;Active\;Risk^2\right)+\left(Misfit\;Active\;Risk^2\right)}$$

True Information Ratio

$$\;True\;Information\;Ratio=\frac{True\;Active\;Return}{True\;Active\;Risk}$$

$$Effective\;No.\;of\;Stocks=\frac1{HHI}\\HHI=\sum_{i=1}^nW_i^2$$

Return Based Style Analysis

$$R_P=a+b_1SCG+b_2\;LCG+b_3\;SCV+b_4\;LCV+\varepsilon$$

Ex-Post RA

$$Ex-Post\;R_A\;=\sum_{i=1}^n\triangle W_iR_i$$

Decomposition of Ex Post Active Return

$$R_A\;=\sum\left(\beta_{Pk}-\beta_{bk}\right)\times F_k+(\alpha+\varepsilon)\\E(R_A)=IC\sqrt{BR}\sigma_{R_A}TC$$

Active Share

$$Active\;Share=\frac12\sum_{i=1}^n\left|W_{pi}W_{bi}\right|$$

Active Risk

$$Active\;Risk=\sqrt{\frac{\sum_{t=1}^T\left(R_P-R_B\right)^2}{T-1}}$$

Decompostion of Active Risk

$$\sigma_{R_A}=\sqrt{\sigma^2((\beta_{Pk}-\beta_{bk})\times F_k)+\sigma^2e}$$

CVi

$$CV_i=\beta_i\sum_{j=1}^n\beta_jCV_{ij}=\beta_jCV_{ij}$$

CAVi

$$CAV_i=\left(W_{pi}-W_{bi}\right)\sum_{j=1}^n\left(W_{pj}-W_{bj}\right)RC_{ij}=\left(W_{pi}-W_{bi}\right)\times RC_{iP}$$

$$E(R_P)=\Sigma W_iR_i=\;W_AE(R_A)+W_BE(R_B)\\\sigma_P^2=W_A^2\sigma_A^2+W_B^2\sigma_B^2+2W_AW_BCov(A,B)\\Where,\;Cov\;(A,\;B)\;=\;\sigma_A\sigma_{B\;}r_{(A,B)}\\\rho_{xy}=\frac{\;Cov_{A,B}}{\sigma_A\sigma_B}$$

$$When\;\rho=1,\;\sigma_P=W_A\sigma_A+W_B\sigma_B\\When\;\rho\;=\;0,\;\sigma_P^2=W_A^2\sigma_A^2+W_B^2\sigma_B^2$$

weights of the minimum variance portfolio:

$$When\;\rho\;=\;0,\;w_A=\frac{\sigma_B^2}{\sigma_A^2+\sigma_B^2}\;\\w_B\;=\;1-w_A$$

weights of a ‘zero’ variance portfolio:

$$When\;\rho\;=-1,\;w_A=\frac{\sigma_B}{\sigma_A+\sigma_B}\;\\w_B\;=\;1-w_A$$

Capital Allocation Line:

$$E(R_P)\;=W_{RF}R_F+W_AE(R_A)\\\sigma_P=W_A\sigma_A\;\\E(R_P)=R_F+\frac{R_A-R_F}{\sigma_A}\times\sigma_P$$

Capital Market Line:

$$E(R_P)\;=W_{RF}R_F+W_ME(R_M)\\\sigma_P=W_M\sigma_M\;\\E(R_P)=R_F+\frac{R_M-R_F}{\sigma_M}\times\sigma_P$$

$$E(R_P)\;=R_F+\left[\frac{E(R_M)-R_F}{\sigma_M}\right]\times\sigma_P,\;\;as\;per\;CAL\\\beta_i=\frac{Cov_{(i,m)}}{\sigma_m^2}=\rho_{i.m}\times\frac{\sigma_i}{\sigma_m}\\\rho_{i.m}=\frac{Cov_{(i,m)}}{\sigma_i\sigma_m}$$

CAPM:

$$R_e=R_F+(R_M-\;R_F)\;\beta\;,\;as\;per\;CAPM.$$

For ‘n’ equally weighted asset:

$$\sigma_P^2=\frac{\overline{\;\sigma^2}}n+\frac{\;(n-1)}n\overline{cov}\;=\frac{var-cov}n+\;Cov\;,\;for\;unequally\;weighted\;assets.$$

Treynor Ratio:

$$Treynor\;Ratio=\left[\frac{E(R_P)-\;R_F}{\beta_P}\right]$$

Sharpe Measure:

$$Sharpe\;Measure=\left[\frac{E(R_P)-\;R_F}{\sigma_P}\right]$$

Jensen′s α:

$$Jensen′s\;\alpha=E(R_P)-\left[R_F+\left[E(R_M)-R_F\right]\;\beta_P\right]\\Extension\;to\;Jensen′\;s\;\alpha:\\E(R)=R_F+\left[E(R_M)-R_F\right]\left[\frac{\sigma_P}{\sigma_M}\right]\\The\;alpha\;in\;the\;case\;would\;be\;the\;portfolio’s\;return\;minus\;the\;reference\;return:\\\alpha=E(R_P)-\;E(R_{reference})$$

Information Ratio:

$$Information\;Ratio=\left[\frac{E(R_P-R_B)}{tracking\;error}\right]=\frac{active\;return}{active\;risk},\;Tracking\;error\;=\frac{\sqrt{\Sigma{(R_P-R_B)}^2}}{n-1}$$

Sortino Ratio:

$$Sortino\;Ratio=\frac{R_P-R_{min}}{downside\;deviation}=\frac{active\;return}{active\;risk}\\Where,\;MSD_{min}=\frac{\Sigma{(R_{Pt}-R_{min})}^2}N$$

$$Sharpe\;\approx\left[\frac{Treynor\;measure}{\sigma_M}\right]\;for\;well\;diversified\;portfolio.$$

Multifactor Model:

$$R_i=E(R_i)+\beta_{i1}F_1+\beta_{i2}F_2+...+\beta_{ik}F_k+e_i$$

Single Factor Security Market Line:

$$E\left(R_P\right)=\left[R_F+\beta_P\left[E\left(R_M\right)-R_F\right]\right]$$

As per the Arbitrage Pricing Theory:

$$E\left(R_i\right)=R_F+\beta_{i1}RP_1+\beta_{i2}RP_2+...+\beta_{ik}RP_k$$

Fama-French Three-Factor Model:

$$R_i-R_F=\alpha_i+\beta_{i,M}(R_M-\;R_F)+\beta_{i,SMB}\times SMB+\beta_{i,HML}\times HML+e_i$$

Single Cash Flow:

$$FV=PV\left(1+\frac rm\right)^{n\times m}\\PV=\frac{FV}{\left(1+\frac rm\right)^{n\times m}}$$

Perpetuity:

$$PV=\frac{PMT}{\displaystyle\frac IY}$$

Uneven Cash Flow:

$$PV=\frac{CF_1}{\left(1+r\right)}+\frac{CF_2}{\left(1+r\right)^2}+...+\frac{CF_n}{\left(1+r\right)^n}\\Outs\tan ding\;Loan\;at\;any\;point\;of\;time\;=\;PV\;of\;remaining\;PMTs$$

$$Probability=\frac{\;No.\;of\;favourable\;outcome}{Total\;no.\;of\;possible\;outcome}\\\Sigma P\;=1\;(For\;all\;exhaustive\;events)\\0\leq P\leq1\\P(A\cap B)=P(A\vert B)\times P(B)\\P(A\vert B)\;=\frac{P(AB)}{\;P(B)}$$

Addition Rule:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Multiplication Rule:

$$P(A\cap B)=P(A)\times P(B)$$

For mutually exclusive events:

$$P(A\cup B)=P(A)+P(B)$$

$$∵P(A\cap B)=0$$

Population:

$$\mu=\frac{\sum_{i=1}^N\;X_i}N$$

Sample:

$$\overline x=\frac{\sum_{i=1}^n\;x_i}n$$

Sum of mean deviation:

$${\textstyle\sum_{i=1}^n}\left(X_i-\overline X\right)=0$$

$$GM=\left[(1+r_1)(1+r_2)\dots\dots..(1+r_n)\right]^\frac1n-1\\AM\geq GM\geq HM\\E(X)=\Sigma\;P(X_1)X_1+P(X_2)X_2+...+P(X_n)X_n=\Sigma P(X).X\\\rho_{A,B}=\frac{cov(A,B)}{\sigma_A\sigma_B}$$

If c is any constant, then:$$E(cX)\;=\;cE(X)$$

If X and Y are any random variables, then:$$E(X+Y)=E(X)+E(Y)$$

If c and a are constant then:$$E\;(cX+a)=cE(X)+a$$

If X and Y are independent random variables, then:$$E(XY)=E(X)\times E(Y)$$

If X and Y are not independent, then:$$E(XY)\neq E(X)\times E(Y)$$

If X is a random variable, then:$$E(X^2)\neq\left[E(X)\right]^2$$

$$\sigma^2=E\left[\left(R-\mu\right)^2\right]$$

Properties of variance include:

$$Var(X)=E\left[\left(X-\mu\right)^2\right]=E\left(X^2\right)-\left[E{(X)}^2\right]$$

If c is any constant, then:$$Var(c)=0$$

If c is any constant, then:$$Var\;(cX)\;=\;c^2\times Var(X)$$

If c is any constant, then:$$Var\;(X\;+\;c)=var(X)$$

If a and c are any constant, then:$$Var\;(aX\;+\;c)=a^2\times Var(X)$$

If X and Y are independent random variables, then:$$Var\;(X\;+Y)=Var(X)+Var(Y)\\Var\;(X\;-\;Y)=Var(X)+Var(Y)$$

If X and Y are independent and a and c are constant, then:$$Var\;(Ax+cY)=a^2\times Var(X)+c^2\times Var(Y)$$

$$Cov(R_i,\;R_j)=E\left\{\left[R_i–E(R_i)\right]-\left[R_j–E(R_j)\right]\right\}\\Cov(R_i,\;R_j)=E\;(R_i,R_j)-E(R_i)\times E(R_j)$$

If X and Y are independent random variables, then:$$Cov\;(X,Y)=0$$

The covariance of random variable X with itself is the variance of X.$$Cov\;(X,X)=\sigma_x^2$$

If a, b, c, d are constant, then:$$Cov(a+bX,\;c+dY)=b\times d\times Cov(X,\;Y)$$

If X and Y are not independent, then:$$Var\;(X\;+Y)=Var(X)+Var(Y)+2Cov\;(X,\;Y)\\Var\;(X\;-Y)=Var(X)+Var(Y)-2Cov\;(X,\;Y)$$

$$E(R)=\mu\;\Sigma\;PX$$

$$Skewness=\frac{E\left[{(R-\mu)}^3\right]}{\sigma^3}\\Kurtosis=\frac{E\left[{(R-\mu)}^4\right]}{\sigma^4}\\Excess\;Kurtosis=Kurtosis–3$$

The Binomial Distribution:

$$n_{C_X}p^x{(1–p)}^{n-x}\;Where,\;n_{C_X}=\frac{\;n!}{(n-x)!x!}$$

$$E(X)=np\\\sigma_x^2=npq=np(1-p)$$

Poisson Distribution:

$$P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}$$

Normal Distribution:

$$Z=\frac{observation-population\;mean}{s\tan dard\;deviation}=\frac{x-\mu}\sigma$$

Chi - squared Distribution:

$$X^2=\frac{\left(n-1\right)S_x^2}{\sigma^2}$$

F – Distribution:

$$F=\frac{S_1^2}{S_2^2},\;S_1>S_2\;always$$

Central limit Theorem:

$$If\;n\;\geq30,\;E\;(\overline X)=\;\mu\\For\;S\tan dard\;Error:\;\sigma_\overline x=\frac{\;\sigma_\overline x}{\sqrt n}or\;\frac{\;S_x}{\sqrt n}\;if\;‘\sigma’\;not\;known.$$
Uniform distribution range:

$$P\;(x_1\leq X\leq x_2)=\frac{x_2-x_1}{b-a}$$

PDF of continuous uniform distribution:

$$f(x)\;=\frac1{b-a}\;for\;a\leq x\;\leq b,\;else\;f(x)=0$$

Mean of uniform distribution:

$$E(x)\;=\frac{a+b}2$$

Variance of uniform distribution:

$$Var(x)\;=\frac{\left(b-a\right)^2}{12}$$
Binomial probability function:

$$p(x)\;=\frac{n!}{\left(n-x\right)!\;x!}\;p^x$$

Expected value of binomial random variable:

$$expected\;value\;of\;X=E(X)=np$$

Variance of binomial random variable:

$$variance\;of\;X=np(1\;-\;p)$$

$$P\;(A\vert B)=\frac{P(B\vert A)\times P(A)}{\;P(B)}$$

$$Ly_t=y_{t-1}\\\triangle y_t=(1-L)y_t=y_t-y_{t-1}\\\overline y=\frac1T\overset T{\underset{t=1}{\;\sum}}y_t\\\widehat\rho\left(T\right)=\frac{\sum_{t=\zeta+1}^T\left[\left(y_t-\overline y\right)\left(t_{t-\zeta}-\overline y\right)\right]}{\sum_{t=1}^T\left(y_t-\overline y\right)^2}$$

EWMA:

$$\sigma_n^2=\lambda\sigma_{n-1}^2+(1-\lambda)\mu_{n-1}^2$$

GARCH (1,1):

$$\sigma_n^2=\omega+\alpha\mu_{n-1}^2+\beta\sigma_{n-1}^2\;;\;\omega=\gamma V_L\\Where,\;VL=long-run\;Variance=\frac\omega{1-\alpha-\beta}\\\alpha+\beta+\gamma=1$$

$$r_{x_1y}=\frac{Cov(x_1y)}{\sigma_x\sigma_y}$$

EWMA :

$$Cov_n=\lambda Cov_{n-1}+(1-\lambda)X_{n-1}y_{n-1}$$

GARCH (1,1):

$$Cov_n=\omega+\alpha X_{n-1}y_{n-1}+\beta Cov_{n-1}\\Where,\;\omega=\gamma\timesLong\;term\;covariance.$$

$$Z_{\alpha/2}=1.65\;(90\%\;Confidence\;Interval)\\Z_{\alpha/2}=1.96\;(95\%\;Confidence\;Interval)\\Z_{\alpha/2}=2.58\;(99\%\;Confidence\;Interval)\\Confidence\;Interval=Point\;Estimate\pm(Reliability\;Factor\times S\tan dard\;Error)\\Or,\;Confidence\;Interval=\overline X\pm Z_{\alpha/2}\frac\sigma{\sqrt n}\\H_a:\;Alt.\;Hypothesis;\\H_0:\;Null\;Hypothesis\\Test\;Statistic=\frac{Sample\;statistic-hypothesized\;value}{S\tan dard\;error\;of\;the\;sample\;statistics}$$

t-Test:

$$t_{n-1}=\frac{\overline x-\mu_0}{\displaystyle\frac s{\sqrt n}}$$

z-Test:

$$z_{test}=\frac{\overline x-\mu_0}{\displaystyle\frac\sigma{\sqrt n}}$$

Chi-square test:

$$H_0:\sigma^2=\sigma_0^2\;\;vs\;H_a:\sigma_1^2\neq\sigma_2^2$$

F- test:

$$H_0:\sigma_1^2=\sigma_2^2\;Vs\;H_a:\;\sigma_1^2\neq\sigma_2^2\;\\F_{Stat}=\frac{s_1^2}{s_2^2}$$

Type I error: The rejection of the null hypothesis when it is actually true. [HORN]

Type II error: The failure to reject the null hypothesis when it is actually false.

$$P(Type\;I\;error)=\alpha\\P(Type\;II\;error)\;=\;1\;-\;Power\;of\;the\;Test$$

$$Y=b_0+b_1X;\;where\;y\;is\;dependent\;variable.\;X\;is\;independent\;variable\;and\;b_0\;and\;b_1\;is\;regression\;coefficient\\E(Y_i\vert X_i)=B_0+B_1\times X_i\\\varepsilon_i=Y_i-E(Y_i\vert X_i)\;\;Or,\;\varepsilon_i=\left(Y_i-\widehat Y\right)\\Y_i=B_0+B_1\times X_i+\;\varepsilon_i$$

OLS: Minimize $$\Sigma e_i^2=\Sigma\left[Y_i-\left(b_0+b_1X_i\right)\right]^2\\or,\;\Sigma\left(Y_i-\widehat Y\right)^2=\Sigma\left[Y_i-\left(b_0+b_1X_i\right)\right]^2$$
$$b_1=\frac{Cov(x,y)}{Var(x)}\;and\;b_o=\overline Y-b_1\overline X$$

Total sum of squares:

$$Total\;sum\;of\;squares\;\lbrack TSS\rbrack=Explained\;sum\;of\;squares\;\lbrack ESS\rbrack+Sum\;of\;squared\;Residuals\;\lbrack SSE\rbrack\\Or,\;TSS=RSS+SSR\\Or,\;\Sigma\left(Y_i-\overline Y\right)^2=\;\Sigma\left(\widehat Y-\overline Y\right)^2+\Sigma\left(Y_i-\widehat Y\right)^2\\R^2=\frac{ESS}{TSS}=\frac{\;\Sigma\left(\widehat{Y_1}-\overline Y\right)^2}{\;\Sigma\left(Y_i-\overline Y\right)^2}\\Or,\frac{TSS-SSR}{\;TSS\;}\;or\;\;1-\frac{\;SSR}{TSS}$$

Confidence interval for the regression coefficient:

$$B_{1:}\;\;b_1\pm(t_c\times\sigma_{b_i})$$

Test statistics with n-2 degrees of freedom:

$$t=\frac{b_1-B_1}{s_{b_i}}$$

$$H_0∶B_1=0\;versus\;H_A∶B_1\neq0$$

The predicted value of Y:

$$Y=b_0+b_1X$$

Confidence interval for a predicted value of Y:

$$\widehat Y-(t_c\times\sigma_f)<Y<\widehat Y+(t_c\times\sigma_f)$$

$$t\;statistic=\frac{Estimated\;regression-hypothesized\;value}{coefficient\;st.\;error}\\The\;statistic\;has\;n-k-1\;degrees\;of\;freedom$$

Testing Statistical Significance:

$$H_0:\;b_j=0\;versus\;H_A:\;b_j\neq0$$

Confidence interval for the regression coefficient: $$b_j\pm\;(t_c\times s_{b_j})$$

Predicting the dependent variable:

$${\widehat Y}_i=b_0+b_1{\widehat{\;X}}_{1i}+b_2\;{\widehat X}_{2i}+\cdots+b_k\;{\widehat X}_{ki}$$

For Joint Hypothesis:

$$using\;F_{stat}\;:\;H_0∶B_1=B_2=B_3=B_4=0\;vs\;H_A∶at\;least\;one\;b_j\neq0$$

$$Homoskedasticity\;only\;f_{stat}=\frac{\displaystyle\frac{ESS}{df_n}}{\displaystyle\frac{SSR}{df_d}}\;\\df_{numerator}=k\\df_{denominator}=n-k-1$$

Coefficient of Determination:

$$R^2=\frac{ESS}{TSS}=\frac{\Sigma\left({\widehat Y}_1-\overline Y\right)^2}{\Sigma\left(Y_i-\overline Y\right)^2}\\Adjusted\;R^2=1-\left[\left(\frac{n-1}{n-k-1}\right)\times\left(1-R^2\right)\right]$$

 

A pure seasonal dummy model:

$$y_t=\sum_{i=1}^s\gamma_i\left(D_{i,t}\right)+\varepsilon_t$$

Adding a trend:

$$y_t=\beta_1\left(t\right)+\sum_{i=1}^s\gamma_i\left(D_{i,t}\right)+\varepsilon_t$$

Allowing for holiday variations (HDV) and trading day variations (TDV):

$$y_t=\beta_1\left(t\right)+\sum_{i=1}^s\gamma_i\left(D_{i,t}\right)+\sum_{i=1}^{v_1}\delta_i^{HDB}\left(HDB_{i,t}\right)+\sum_{i=1}^{v_2}\delta_i^{TDV}\left(TDV_{i,t}\right)+\varepsilon_t\\y_{T-h}=\beta_1\left(T+h\right)+{\textstyle\sum_{i=1}^s}\gamma_i\left(D_{i,T+h}\right)+{\textstyle\sum_{i=1}^{v_1}}\delta_i^{HDB}\left(HDB_{i,T+h}\right)+{\textstyle\sum_{i=1}^{v_2}}\delta_i^{TDV}\left(TDV_{i,T+h}\right)+\varepsilon_t$$

$$Combined\;Ratio=Loss\;ratio+Expense\;Ratio\\C.R\;after\;dividends=Combined\;Ratio+Dividends\\Operating\;Ratio=Combined\;Ratio\;after\;dividends–Investment\;Income$$

$$NAV=\frac{fund\;assets-fund\;liabilities}{total\;share\;outs\tan ding}$$

Call option payoff:

$$C_T=Max(0,S_T-x)\\Profit\;to\;option\;buyer\;=C_T-C_0\\Profit\;to\;option\;seller\;=C_0-C_T$$

Put option payoff:

$$P_T=Max(0,\;X-S_T)\\Profit\;to\;option\;buyer\;=P_T-P_0\\Profit\;to\;option\;seller\;=P_0-P_T$$

Forward contract payoff:

$$Payoff\;to\;a\;long\;position=S_T-k\\Pay\;off\;to\;a\;short\;position=k\;-S_T$$

$$Basis=Spot\;Price-Future\;price=S_T-F_0\\H.R=\rho_{s,f}\times\frac{\sigma_s}{\sigma_f}$$

Effectiveness of Hedge:

$$R^2=\rho^2$$

Correlation:

$$\rho=\frac{cov_{s.f}}{\sigma_s\sigma_f}\\and\;\frac{cov_{s.f}}{\sigma_s\sigma_f}\times\frac{\sigma_s}{\sigma_f}=\frac{cov_{s.f}}{\sigma_{f^2}}=\beta_{S,F}$$

Hedging with stock index futures:

$$N=\beta_P\times\left(\frac{Portfolio\;Value}{Value\;of\;the\;future\;contract}\right)\\=\frac{V_P(\beta_T-\beta_P)}{m\times F_P\times\beta_f}$$

Adjusting portfolio beta:

$$number\;of\;contracts=(\beta\ast-\beta)\frac PA$$

For hedging the tail:

$$\Rightarrow n\times\frac{daily\;spot}{daily\;futures}$$

Discrete:

$$FV=A\left(1+\frac Rm\right)^{m\times n}$$

Continuous:

$$FV=Ae^{R\times n}$$

Bond pricing:

$$B=\left(\frac c2\times{\textstyle\sum_{j=1}^N}e^{-\frac{z_j}2\times j}\right)\\where:\;c=the\;annual\;coupon\\N=the\;number\;of\;semiannual\;payment\;periods\\z_j=the\;bond\;equivalent\;spot\;rate\;that\;corresponds\;to\;j\;periods\;(j/2\;years)\;on\;a\;continuously\;compounded\;basis\\FV=the\;face\;value\;of\;the\;bond$$

Using above two equations:

$$A\left(1+\frac Rm\right)^{m\times n}=Ae^Rc^n\\ R_C=m\times l_n\left(1+\frac Rm\right)$$

Forward Rate Agreements:

$$Cash\;flow\;(if\;receiving\;R_K)=L\times(R_K-R)\times(T_2\;-\;T_1)\\Cash\;flow\;(if\;paying\;R_K)=L\times(R-R_K)\times(T_2\;-\;T_1)\\where:\;\\L=principal\\R_K=annualized\;rate\;on\;L,\;expressed\;with\;compounding\;period\;T_2\;-\;T_1\\R=annualized\;actual\;rate,\;expressed\;with\;compounding\;period\;T_2\;-\;T_1\\T_i\;=\;time\;i,\;expressed\;in\;years\\Payoff=\frac{(Mkt.rate-contract\;rate)\times\frac n{12}\times NP}{1+(Mkt.rate\;\times\frac n{12})}\\Percentage\;bond\;price\;change\approx duration\;effect+convexity\;effect$$

$$Forward\;Price=S\times{(1+\;r)}^t\\Forward\;Price:\;F=S_0e^{rt}\\With\;benefits:\;F\;=(S_0-I)e^{rt}\\With\;dividend:\;F\;=\;S\times\left[(1\;+\;r)/(1\;+\;q)\right]^T\\Currency\;Futures:\;F_0=S_0e^{(r_{DC}-r_{FC})T}\\With\;income:\;F=(S\;-\;I)\times{(1\;+\;r)}^T$$

$$A.I=coupon\times\frac{\#\;of\;days\;from\;last\;coupon\;to\;the\;settlement\;date}{\#\;of\;days\;in\;coupon\;period}\\Cash\;Price=Quoted\;Price+Accrued\;Interest\\Quoted\;Price=Cash\;Price-Accrued\;Interest\\Clean\;Price=Dirty\;Price-Accrued\;Interest\\Annual\;rate\;on\;a\;T-bill:\;T-bill\;discount\;rate=\frac{360}n(100-Y)\\BDY=\frac{FV-Cash\;Price}{FV}\times\frac{360}n\\C.T.D=QBP–(QFP\times CF)\\Conversion\;Factor=\frac{Bond\;Price-Accrued\;Interest}{Face\;Value}\\Cash\;received\;by\;the\;short=(QFP\times CF)+AI\\Euro\;dollar\;future\;prices=\$10,000\left[100-(0.25)(100-Z)\right]\\Duration-based\;H.R\\N=\frac{V_P(D_T-D_P)}{F\times D_F}$$

$$R_{forward}=R_2+(R_2-R_1)\frac{T_1}{T_2-T_1}$$

Interest rate swap value:

$$V_{swap}\;=\;Bond_{fixed}\;–\;Bond_{floating}$$

Currency swap value:

$$V_{swap}(USD)=B_{USD}\;–(S_0\times B_{GBP})$$

Put – Call Parity:

$$c+Xe^{-rt}=S+p\\S=c-p+Xe^{-rt}\\P=c-S+Xe^{-rt}\\C=S+p+Xe^{-rt}\\Xe^{-rt}=S+p-c$$

Relationship between American Call and Put Options:

$$S_0–X\leq C-P\leq S_0-Xe^{-rt}$$

Widest possible range after considering dividend:

$$S_0–X-D<C-P<S_0-Xe^{-rt}$$

Lower and Upper Bounds for Options:

$$Minimum\;Value\;of\;European\;call=c\geq max\left(0,\;S_{0\;}-Xe^{-rT}\right)\\Maximum\;Value\;of\;European\;call=S_{0\;}\\Minimum\;Value\;of\;American\;call=C\geq\;max\left(0,\;S_{0\;}-Xe^{-rT}\right)\\Maximum\;Value\;of\;American\;call=S_{0\;}\\Minimum\;Value\;of\;European\;put=p\geq\;max\left(0,\;Xe^{-rT}-S_{0\;}\right)\\Maximum\;Value\;of\;European\;put=Xe^{-rT}\\Minimum\;Value\;of\;American\;put=P\geq\;max\left(0,\;X-S_{0\;}\right)\\Maximum\;Value\;of\;American\;put=X$$

$$Cash-or-nothing=\frac{QN(d_2)}{e^{rt}}\\Asset-or-nothing\;=\frac{S_0N(d_2)}{e^{qt}}$$

Commodity Forward Price:

$$F_{O,T}=E(S_T)e^{(r-\alpha)T}$$

$$NPV=E(S_T)e^{-\alpha T}-S_0$$

Commodity Forward Price with active lease market:

$$F_{O,T}=S_0e^{(r-\delta)T}\;\;or\;S_0\times\left[\left(1+r\right)/\left(1+\delta\right)\right]^T$$

With Storage Cost:

$$F_{O,T}=S_0e^{(r+\lambda)T}\;\;or\;\left(S_0+U\right)\times\left(1+r\right)^T$$

With Convenience Yield:

$$F_{O,T}=S_0e^{(r-c)T}\;\;or\;F_{O,T}\geq\left(S_0+U\right)\times\left[\left(1+r\right)/\left(1+y\right)\right]^T$$

Combination of cost & benefits:

$$F_{O,T}=S_0e^{(r+\lambda-c)T}$$

Arbitrage free range of the forward price:

$$S_0e^{(r+\lambda-c)T}\leq F_{O,T}\leq S_0e^{(r+\lambda)T}$$

$$Net\;EUR\;exposure=(EUR\;assets–EUR\;liabilities)+(EUR\;bought–EUR\;sold)\\Net\;EUR\;exposure=Net\;EUR\;assets+net\;EUR\;bought\\Dollar\;gain/loss\;in\;EUR=Net\;EUR\;exposure\;(measured\;in\;\$)\times\;\%\;change\;in\;\$/FC\;rate$$

Purchasing power parity:

$$\%\triangle S=inflation(foreign)-inflation(domestic)\\where:\;\%\triangle S=change\;in\;the\;domestic\;spot\;rate$$

IRP:

$$Forward=Spot\left[\frac{\left(1+r_{YYY}\right)}{\left(1+r_{XXX}\right)}\right]^T\\Or,\;\\Forward=Spot\times e^{\left(r_{YYY}-r_{XXX}\right)T}\\where\;r_{YYY}=quote\;currency\\rate\;r_{XXX}=base\;currency\;rate$$

The nominal interest rate:

$$Exact\;methodo\log y:\;(1+Nominal\;Rate)=(1+Real\;Rate)\;(1+Inflation\;Rate)\\Linear\;approximation:\;Nominal\;Rate\approx Real\;Rate+Inflation\;Rate$$

$$Original-issue\;discount\;(OID)=face\;value–offering\;price\\Issue\;default\;rate=\frac{No.of\;issuers\;defaulted}{Total\;no.of\;issuers\;at\;the\;beginning\;of\;the\;year}\\Dollar\;default\;rate=\frac{Cumulative\;dollar\;value\;of\;all\;defaulted\;bonds}{(cumulative\;dollar\;value\;of\;all\;issuance)\times(weighted\;average\;\#\;of\;years\;outs\tan ding)}\\Or,\;\frac{Cumulative\;dollar\;value\;of\;all\;defaulted\;bonds}{cumulative\;dollar\;value\;of\;all\;issuance}\\Expected\;loss\;rate=probability\;of\;default\;\times(1-expected\;recovery\;rate)\\$$

$$SMM=1–{(1\;-\;CPR)}^\frac1{12}\\CPR=1–{(1\;-SMM\;)}^{12}\\Option\;\cos t=Zero\;volatility\;spread–\;OAS\\Value\;of\;a\;dollar\;roll=A-B+C-D\\A=Price\;at\;which\;pool\;is\;sold\;in\;month\;1,\;with\;accrued\;interest\\B=Price\;at\;which\;pool\;is\;bought\;in\;month\;2,\;with\;accrued\;interest\\C=Interest\;earned\;on\;funds\;from\;the\;sale\;for\;one\;month\\D=Coupon\;and\;principal\;payment\;that\;was\;foregone\;on\;the\;pool\;sold\;in\;month\;1\\$$

Bull call spread:

$$profit=max(0,\;S_T-X_L)-max(0,\;S_T-X_H)-C_{LO}+C_{HO}$$

Bear put spread:

$$profit=max(0,\;X_H-S_T)-max(0,\;X_L-S_T)-P_{HO}+P_{LO}$$

Butterfly spread with calls:

$$profit=max(0,\;S_T-X_L)-2max(0,\;S_T-X_M)+max(0,\;S_T-X_H)-C_{LO}+2C_{MO}-C_{HO}$$

Straddle:

$$profit=max(0,\;S_T-X)+max(0,\;X-S_T)-C_O-P_0$$

Strangle:

$$profit=max(0,\;S_T-X_H)+max(0,\;X_L-S_T)-C_O-P_0$$

$$VaR\;(\%)=\overline X-(Z_{stat}\times\sigma)$$

Mean: $$\mu_P=w_1\;\mu_1+w_2\;\mu_2$$

Standard deviation: $$\sigma P=\sqrt{w_1^2\;\sigma_1^2+w_2^2\;\sigma_2^2+\;2\;w_1\;w_2\;\sigma_1\;\sigma_2\;\rho}$$

$$VaR\;(\$)=VaR\;(\%)\times V_P$$

Delta-normal VAR:

$$VaR=\left[\mu-Z_{stat}.\sigma\right]\times portfolio\;value$$

Expected shortfall:

$$ES=\left(\mu+\sigma\frac{e^{-\left({\displaystyle\frac{z^2}2}\right)}}{\left(1-x\right)\sqrt{2\pi}}\right)$$

$$VaR=modified\;duration\times Z\times annualized\;yield\;volatility\times portfolio\;value$$

Delta: $$\delta=\frac{\triangle P}{\triangle S}$$

$$VaR(T,X)=VaR(1,X)\times\sqrt T$$

$$ES(T,X)=ES(1,X)\times\sqrt{\_T}$$

$$\sigma_{daily}\cong\frac{\sigma_{annual}}{\sqrt{250}}\;or,\\\sigma_{monthly}\cong\frac{\sigma_{annual}}{\sqrt{12}}$$

Parametric Approach:

$$r_{t-k,t-k-1}^2=\left(r_{t-3,t-2}^2+...+r_{t-2,t-1}^2+r_{t-1,t}^2\right)$$

Exponentially weighted moving average (EWMA) model:

$$\sigma_n^2=\lambda\sigma_{n-1}^2+\left(1-\lambda\right)r_{n-1}^{\;2}$$

GARCH (1, 1):

$$\sigma_t^2=\omega+\alpha r\;_{t-1,t}^2+\gamma\sigma_{t-1}^{\;2}$$

Risk Metrics Approach:

$$\sigma_t^2=\left(1-\lambda\right)\left(\lambda^0r\;_{t-1,t}^2+\lambda^1r\;_{t-2,t-1}^2+\lambda^2r\;_{t-3,t-2}^2+...+\lambda^Nr\;_{t-N-1,t-N}^2\right)$$

Hybrid Approach:

$$Weightage\;to\;‘t’\;return:\;r_t=\left[\frac{1-\lambda}{1-\lambda^k}\right]\lambda^{t-1}$$

MDE:

$$\sigma_t^2={\textstyle\sum_{i=1}^K}\overline\omega\left(x_{t-1}-i\right)r\;_{t-i}^2$$

 

Taylor Series approximation:

$$f(x)\approx f(x_0)+f^1(x-x_0)+\frac12\;f"(x_0)\;{(x-\;x_0)}^2$$

$$HR=\frac{C_U-\;C_D}{S_U-\;S_D}\\Call\;price=hedge\;ratio\times\left[stock\;price\;–\;PV\;(borrowing)\right]\\U=size\;of\;the\;up-move\;factor=e^{\sigma\sqrt t}\\D=size\;of\;the\;down-move\;factor=e^{-\sigma\sqrt t}=\frac1{e^{\sigma\sqrt t}}=\frac1U\\\pi_U=\frac{e^{rt}-D}{U-D}\;\\\pi_d=1-\pi_U\\\pi_U=\frac{e^{(r-q)t}-D}{U-D}\;,In\;case\;a\;stock\;pays\;dividend\\\pi_d=1-\pi_U\\\pi_U=\frac{e^{(r_{DC}-r_{FC})t}-D}{U-D},\;In\;case\;of\;currencies\\\pi_U=\frac{1-D}{U-D},\;In\;case\;of\;options\;on\;futures$$

$$A.I=Coupon\;payment\times\left(\frac{No.of\;days\;from\;last\;coupon\;settlement}{No.of\;days\;in\;coupon\;period}\right)\\P=\frac C{\left(1+Y\right)^W}+\frac C{\left(1+Y\right)^{1+W}}+\frac C{\left(1+Y\right)^{2+W}}+\frac C{\left(1+Y\right)^{n-1+W}}+\frac M{\left(1+Y\right)^{n-1+W}}\\Flat\;Price=Full\;Price–A.I\\Clean\;price=dirty\;price–accrued\;interest$$

$$E(S_T)=S_0e^{\mu T}$$

Valuation of warrants:

$$\frac N{N+M}\times value\;of\;regular\;call\;option$$

Continuously compounded returns:

$$u_i=\ln\left(\frac{S_i}{S_{i-1}}\right)$$

BSM Option Pricing Model:

$$C_0=\left[S\times N(d_1)\right]-\left[X\times e^{-R_f^c\times T}\times N(d_2)\right]\\P_0=\left\{X\times e^{-R_f^c\times T}\times\left[1-N(d_2)\right]\right\}-\left\{S_0\times\left[1-N(d_1)\right]\right\}\\Where,\\d_1=\frac{\left[\ln\left({\displaystyle\frac{S_0}X}\right)+\left\{R_f^c+\left(0.5\times\sigma^2\right)\right\}\right]T}{\sigma\sqrt T}\\d_1=d_1\;–(\;\sigma\times\sqrt T)$$

Delta:

$$\triangle=\frac{\delta_c}{\delta_s}$$

$$No.\;of\;options\;needed\;to\;delta\;hedge=\frac{No.of\;shares\;hedged}{delta\;of\;call\;option}\\\triangle Value\;of\;puts=\triangle Value\;of\;long\;stock\;option$$

Portfolio delta:

$$\triangle_p=\overset n{\underset{i=1}{\sum w_i}}\triangle_i$$

$$gamma=\frac{\partial_{C^2}}{\partial_{S^2}}\\vega=\frac{\partial_C}{\partial_\sigma}=S0N'\;(d1)\\rho=\frac{\partial_C}{\partial_r}\\theta:\;\theta=\frac{\partial_C}{\partial_t}$$

Relationship among delta, theta, and gamma:

$$\sqcap r=\theta+rS\triangle+0.5\;\sigma^2S^2\gamma$$

HPR:

$$R_{t-1,t}=\frac{BV_t+C_t-BV_{t-1}}{BV_{t-1}}\\PV\;of\;Perpetuity=\frac Cy$$

Bond Price:

$$P=\frac{c_1}{{(1+y)}^1}+\frac{c_2}{{(1+y)}^2}+\frac{c_3}{{(1+y)}^3}+...+\frac{c_n}{{(1+y)}^n}$$

$$FV_n=PV_0\times\left[1+\frac rm\right]^{m\times n}$$

Compounding frequencies:

$$R_2=\left[\left(1+\frac{R_1}{m_1}\right)^\frac{m_1}{m_2}-1\right]m_2$$

$$Holding\;Period\;Return=\frac{P_n-P_0+C.F}{P_0}$$

Forward rate:

$$F=\frac{R_2T_2-R_1T_1}{T_2-T_1}$$

$$EAY=m\left[\left(1+HPY\right)^{m\times n}-1\right]$$

Spot Rate:

$$Z_{\left(t\right)}=2\left[\left(\frac1{d\left(t\right)}\right)^\frac1{2t}-1\right]$$

Discount factor:

$$d\left(t\right)=\left(1+\frac{r\left(t\right)}2\right)^{-2t}$$

Discount Rate:

$$d(n)=\frac1{{(1+S_n)}^n}$$

Par Rate:

$$x=\frac{1-d_2}{\Sigma d}$$

$$DV01=\frac{\triangle P}{\triangle y}\\HR=\frac{\mathit D{\mathit V}_{\mathit{01}\mathit(\mathit\;\mathit p\mathit e\mathit r\mathit\;\mathit\$\mathit{100}\mathit\;\mathit o\mathit f\mathit\;\mathit i\mathit n\mathit i\mathit t\mathit i\mathit a\mathit l\mathit\;\mathit p\mathit o\mathit s\mathit i\mathit t\mathit i\mathit o\mathit n\mathit)}}{\mathit D{\mathit V}_{\mathit{01}\mathit(\mathit\;\mathit p\mathit e\mathit r\mathit\;\mathit\$\mathit{100}\mathit\;\mathit o\mathit f\mathit\;\mathit h\mathit e\mathit d\mathit g\mathit i\mathit n\mathit g\mathit\;\mathit i\mathit n\mathit s\mathit t\mathit r\mathit u\mathit m\mathit e\mathit n\mathit t\mathit)}}\\Macaulay’s\;Duration=\frac{\Sigma wx}{\Sigma x}\\Duration\;Gap=Macaulay’s\;Duration–Investment\;Horizon\\Effective\;Duration=\frac{\triangle P}{P\triangle y}\\Effective\;Convexity=\frac{\left(P_2+P_1-2P_0\right)}{P_0\left(\triangle y\right)^2}\\Effect\;of\;EC=\frac12\times EC\times\left(\triangle y\right)^2\\Modified\;Duration=\frac{Macaulay’s\;Duration}{1+periodic\;market\;yield}\\MD\simeq ED\left[for\;all\;option\;free\;bonds\right]\\DV01=Duration\times0.0001\times Bond\;Value\\Percentage\;Price\;Change\approx Duration\;effect+Convexity\;Effect\\=\left[-duration\times\triangle y\times100\right]+\left[\left(\frac12\right)\times convexity\times{(\triangle y)}^2\times100\right]\\D_{portfolio}={\textstyle\sum_{i=1}^k}W_iD_i\\\triangle P=-D\times P\times\triangle y+\frac12\times C\times P\times\triangle y^2$$

$$Key\;Rate\;01=-\frac1{10,000}\frac{\triangle BV}{\triangle y^{\;k}}\\Key\;Rate\;duration=-\frac1{BV}\frac{\triangle BV}{\triangle y^{\;k}}$$

$$E.L=PD\times LGD\times EAD$$

Standard deviation of credit loss:

$$\sigma=\sqrt{PD-PD^2}\times\left[L\left(1-RR\right)\right]$$

Standard deviation of credit loss as percentage of size:

$$\alpha=\frac{\sigma_P}{nL}=\frac{\sigma\sqrt{1+\left(n+1\right)\rho}}{\sqrt n\times L}$$

Unexpected loss:

$$UL=(WCDR–PD)\times LGD\times EAD$$

$$UL=EAD\times\sqrt{\left(PD\times\sigma_{LGD}^2\right)+\left(LGD^2\times\sigma_{PD}^2\right)}\\\sigma_{PD}^2=PD\times(1-PD)\\ELP=\Sigma\left(EAD_i\times PD_i\times LGD_i\right)\\ULP=\sqrt{UL_1^2+UL_2^2+2\rho_{1,2}.UL_1.UL_2}$$

For two assets portfolio:

$$RC_1=\frac{UL_1+(\;\rho_{1,2}\times\;UL_2)}{UL_p}\times UL_1$$

$$Profit\;or\;loss\;data=P_n+CF-P_0\\Arithmetic\;Return\;=\frac{P_n-P_0+CF}{P_0}\\Geometric\;Return=\ln\left(\frac{P_n+CF}{P_0}\right)\\VaR:\left[\left(\sigma\times n\right)+1\right]\;th\;observation$$

Delta normal VaR:

$$VaR\left(\alpha\%\right)=\left(\mu_r-\sigma_r\times z_\alpha\right)\;\;(In\;\%\;terms)\\VaR\left(\alpha\%\right)=P_0\times\left(\mu_r-\sigma_r\times z_\alpha\right)\;\;(In\;\$\;terms)\\Weights\;in\;Expected\;Shortfall=\left(\frac1{1-confidence\;level}\right)$$

Lognormal VaR:

$$VaR\left(\alpha\%\right)=P_0\times\left(1-e^{\mu_r-\sigma_r\times z_\alpha}\right)$$

$$Se\left(q\right)=\frac{\sqrt{p\left(1-p\right)/n}}{f\left(q\right)}\\SE_{Quantile}=\sqrt{Variance_{Quantile}}$$

Confidence Interval for VaR:

$$q+se\left(q\right)\times z_\alpha>VaR>q-se\left(q\right)\times z_\alpha$$

Age weighted Historical Simulation:

$$W(i)=\frac{\lambda^{i-1}\left(1-\lambda\right)}{1-\lambda^n}$$

Volatility weighted Historical Simulation:

$$r\ast=\left(\frac{\sigma_{T,i}}{\sigma_{t,i}}\right)r_{t,i}$$

$$R_i=\alpha_i+\beta_iR_M+\varepsilon_i\\R_P={\textstyle\sum_{i-1}^N}w_iR_i={\textstyle\sum_{i-1}^N}{\textstyle w_i}{\textstyle\beta_i}{\textstyle R_M}{\textstyle+}{\textstyle\sum_{i-1}^N}{\textstyle w_i}{\textstyle\varepsilon_i}{\textstyle\;}$$

$${\textstyle\beta_P}{\textstyle=}\sum_{i-1}^N{\textstyle{\scriptstyle w}_i}{\textstyle{\scriptstyle\beta}_i}{\textstyle\;}{\textstyle V}{\textstyle\left(R_P\right)}{\textstyle=}{\textstyle{\scriptstyle\beta}_P^2}{\textstyle\times}{\textstyle{\scriptstyle\sigma}_M^2}{\textstyle+}{\textstyle\sum_{i-1}^N}{\textstyle{\scriptstyle w}_i^2}{\textstyle\times}{\textstyle{\scriptstyle\sigma}_{\varepsilon,i}^2}{\textstyle\;}$$

$${\textstyle U}{\textstyle n}{\textstyle d}{\textstyle i}{\textstyle v}{\textstyle e}{\textstyle r}{\textstyle s}{\textstyle i}{\textstyle f}{\textstyle i}{\textstyle e}{\textstyle d}{\textstyle\;}{\textstyle V}{\textstyle a}{\textstyle R}{\textstyle=}{\textstyle\sum_{i=1}^N}{\textstyle\left|x_i\right|}{\textstyle\times}{\textstyle{\scriptstyle V}_i}{\textstyle\;}{\textstyle\;}$$

$${\textstyle D}{\textstyle i}{\textstyle v}{\textstyle e}{\textstyle r}{\textstyle s}{\textstyle i}{\textstyle f}{\textstyle i}{\textstyle e}{\textstyle d}{\textstyle\;}{\textstyle V}{\textstyle a}{\textstyle R}{\textstyle=}{\textstyle\alpha}{\textstyle\sqrt{x'\Sigma x}}{\textstyle=}{\textstyle\sqrt{\left(x\times V\right)'R\left(x\times V\right)}}{\textstyle\;}{\textstyle\;}$$

$${\textstyle T}{\textstyle r}{\textstyle a}{\textstyle c}{\textstyle k}{\textstyle i}{\textstyle n}{\textstyle g}{\textstyle\;}{\textstyle E}{\textstyle r}{\textstyle r}{\textstyle o}{\textstyle r}{\textstyle\;}{\textstyle V}{\textstyle a}{\textstyle R}{\textstyle=}{\textstyle\alpha}{\textstyle\sqrt{\left(x-x_0\right)'\Sigma\left(x-x_0\right)}}{\textstyle\;}{\textstyle\;}$$

$${\textstyle V}{\textstyle a}{\textstyle r}{\textstyle i}{\textstyle a}{\textstyle n}{\textstyle c}{\textstyle e}{\textstyle\;}{\textstyle i}{\textstyle m}{\textstyle p}{\textstyle r}{\textstyle o}{\textstyle v}{\textstyle e}{\textstyle m}{\textstyle e}{\textstyle n}{\textstyle t}{\textstyle=}{\textstyle1}{\textstyle-}{\textstyle{\scriptstyle\left(tracking\;error/benchmark\;VaR\right)}^2}{\textstyle\;}{\textstyle F}{\textstyle o}{\textstyle r}{\textstyle w}{\textstyle a}{\textstyle r}{\textstyle{\scriptstyle d}_t}{\textstyle=}{\textstyle\left(F_t-K\right)}{\textstyle{\scriptstyle e}^{-rt}}{\textstyle\;}{\textstyle O}{\textstyle n}{\textstyle e}{\textstyle\;}{\textstyle d}{\textstyle a}{\textstyle y}{\textstyle\;}{\textstyle r}{\textstyle i}{\textstyle s}{\textstyle k}{\textstyle\;}{\textstyle h}{\textstyle o}{\textstyle r}{\textstyle i}{\textstyle z}{\textstyle o}{\textstyle n}{\textstyle:}{\textstyle\;}{\textstyle-}{\textstyle{\scriptstyle z}_{\displaystyle\frac\alpha2}}{\textstyle\sigma}{\textstyle\sqrt T}{\textstyle\;}{\textstyle\;}$$

Model Accuracy Test:

$${\textstyle z}{\textstyle=}\frac{x-pT}{\sqrt{p\left(1-p\right)\times T}}{\textstyle\;}\\$$

$$\textstyle H_0\;:\;Model\;=\;Unbiased\\H_a\;:\;Model\;\neq\;Unbiased\\L.R_{CC}=L.R_{UC}+L.R_{ind}$$

Log likelihood ratio:

$$\textstyle L.R_{UC}=-2\ln\;\left[\left(1-P\right)^{T-N}P^N\right]+2\ln\left\{\left[\left(1-N/T\right)\right]^{T-N}P^N\right\}\;$$

Probability of exception:

$$p=1-c$$

$$Failure\;rate=\frac NT\\No.\;of\;exception=(1-p)\times T$$

$$\mu_P=w_x\mu_x+w_y\mu_y\;\\\sigma_P=\sqrt{w_x^2\sigma_x^2+w_x^2\sigma_x^2+2w_x\sigma_x.cov_{x,y}}\\cov_{x,y}=\frac{\sum_{t-1}^n\left(X_t-\mu_x\right)\left(Y_t-\mu_y\right)}{n-1}\\\rho_{\left(x,y\right)}=\frac{cov_{xy}}{\sigma_x\sigma_y}\\\sigma_E=\sqrt{\sigma_X^2+\sigma_Y^2-2cov_{XY}}\\\rho_{\;realized}=\frac2{n^2-n}{\textstyle\sum_{i>j}}\rho_{i,j}\\$$

Correlation Swap pay off:

$$Notional\;amount\times\left(\rho_{\;realized}-\rho_{\;fixed}\right)$$

VaR by variance covariance method :

$$VaR_p=\sigma_p\times\sqrt x$$

$$\sigma_p=\sqrt{B_h\times C\times B_v}$$

Joint probability of default:

$$P\left(AB\right)=P_{AB}\sqrt{PD_{A\left(1-PD_A\right)\times PD_B\left(1-PD_B\right)}}+PD_A\times PD_B$$

Mean Reversion Rate:

$$S_t-S_{t-1}=a\left(\;\mu-S_{t-1}\right)\triangle t+\sigma_S\varepsilon\sqrt{\triangle t}$$

Auto Correlation:

$$AC_{\left(\rho_t,\rho_{t-1}\right)}=\frac{Cov_{\left(\rho_t,\rho_{t-1}\right)}}{\sigma\left(\rho_t\right)\times\sigma\left(\rho_{t-1}\right)}$$

Pearson Correlation:

$$\rho_{\left(x,y\right)}=\frac{cov_{xy}}{\sigma_x\sigma_y}\\cov_{xy}=\frac{\sum_{t-1}^n\left(X_t-\mu_X\right)\left(Y_t-\mu_Y\right)}{n-1}\\E\left\{\left[X-E\left(X\right)\right]\left[Y-E\left(Y\right)\right]\right\}\\Or,\\E\left(XY\right)-E\left(X\right)E\left(Y\right)$$

Correlation with expected value:

$$\rho_{\left(x,y\right)}=\frac{E\left(xy\right)-E\left(x\right)E\left(y\right)}{\sqrt{E\left(x^2\right)-\left(E\left(x\right)\right)^2}\times\sqrt{E\left(y^2\right)-\left(E\left(y\right)\right)^2}}$$

Spearman’s Rank Correlation:

$$\rho_S=1-\frac{6\times\sum_{i=1}^nd_i^2}{n\left(n^2-1\right)}$$

Kendall’s T:

$$T=\frac{n_c-n_d}{n\left(n-1\right)/2}$$

Correlation copula:

$$C\left[G_1\left(u_1\right),...,G_n\left(u_n\right)\right]=F_n\left[F_1^{-1}\left(G_1\left(u_1\right)\right),...,F_n^{-1}\left(G_n\left(u_n\right)\right);\rho_F\right]$$

The Gaussian default time copula:

$$C_{GD}\left[Q_i\left(t\right),...,Q_n\left(t\right)\right]=M_n\left[N_1^{-1}\left(Q_1\left(t\right)\right),...,N_n^{-1}\left(Q_n\left(t\right)\right);\rho_M\right]$$

The Gaussian copula for the bivariate standard normal distribution:

$$C_{GD}\left[Q_B\left(t\right),...,Q_C\left(t\right)\right]=M_2\left[N_1^{-1}\left(Q_B\left(t\right)\right),...,N_n^{-1}\left(Q_C\left(t\right)\right);\rho\right]$$

Calculate the regression hedge adjustment factor, beta:

$$\triangle y_t^{\;N}=\alpha+\beta\triangle y_t^R+\varepsilon_t$$

$$F^R=F^N\times\left(\frac{DV01^N}{DV01^R}\right)\times\beta$$

Two variables regression Hedge: (For a combination of 10-and 30-year swap)

$$\triangle y_t^{\;20}=\alpha+\beta^{10}\triangle y_t^{10}+\beta^{30}\triangle y_t^{30}+\varepsilon_t$$

Regressing change on change:

$$\triangle y_t=\alpha+\beta\triangle x_t+\triangle\varepsilon_t$$

Nominal on real (not change):

$$y_t=\alpha+\beta\;x_t+\varepsilon_t$$

$$\varepsilon_t=\rho\;\varepsilon_{t-1}+v_t$$

 

2 year Short Rate:

$$\widehat{r\;}\left(2\right)=\sqrt[2]{\left(1+r_1\right)\left(1+r_2\right)}-1$$

3 year Short Rate:

$$\widehat{r\;}\left(3\right)=\sqrt[3]{\left(1+r_1\right)\left(1+r_2\right)\left(1+r_3\right)}-1$$

Jensen’s inequality:

$$E\left[\frac1{1+r}\right]>\frac1{E\left(1+r\right)}$$

Model 1:

$$dr=\sigma dw$$

Model 2:

$$dr=\lambda dt\pm\sigma dw$$

Ho-LEE Model:

$$dr=\lambda_1dt\pm\sigma dw\\Recombined\;middle\;node\;at\;t=2\\=r_0+\left(\lambda_1+\lambda_2\right)dt$$

VASICEK Model:

$$dr=k\left(\theta-r\right)dt\pm\sigma dw$$

$$\theta\approx r_L+\frac\lambda k$$

$$r_0e^{-kT}+\theta\left(1-e^{-kT}\right)\\\tau=\ln\frac2k$$

Model 3:

$$dr=\lambda(t)\;dt+\sigma(t)\;dw$$

Cox Ingersoll Ross (CIR) Model:

$$dr=k\left(\theta-r\right)\;dt+\sigma\sqrt r\;dw$$

Model 4:

$$dr=ar\;dt+\sigma_r\;dw$$

Lognormal Model with Deterministic Drift:

$$d\left[\ln\left(r\right)\right]=a\left(t\right)\;dt+\sigma\;dw$$

Lognormal Model with Mean Reversion:

$$d\left[\ln\left(r\right)\right]=k\left(t\right)\left[\ln\theta\left(t\right)-\ln\left(r\right)\right]\;dt+\sigma\left(t\right)\;dw$$

Interest Rate Tree with Lognormal Model (mean revision):

$$k\left(2\right)=\frac{\sigma\left(1\right)-\sigma\left(2\right)}{\sigma\left(1\right)\;dt}\\k\left(2\right)=\frac1{dt_2}\left[1-\frac{\sigma\left(2\right)\sqrt{dt_2}}{\sigma\left(1\right)\sqrt{dt_1}}\right]$$

Put Call Parity:

$$P_{mk}\;-P_{BSM}=C_{mkt}-C_{BSM}\\c-p=S=PV\left(x\right)\\Or,\\c-p=S-X.e^{-rt}\;\;or\;\;PV(X)=X.e^{-rt}\;$$

$$Expected\;Loss=EAD\times PD\times LGD\;$$

$$EAD=drawn\;amount\;+(limit-drawn\;amount)\times Loan\;Equivalency\;Factor\\ULC_i=\frac{\partial UL_{portfolio}}{\partial w_i}\times w_i\\ULC_i=\rho_{i,portfolio}\times w_i\times UL_{portfolio}$$

Marginal Risk Contribution:

$$\beta_i=\frac{\displaystyle\frac{ULC_i}{w_i}}{UL_{portfolio}}$$

$$EVA=\left(RARORAC-K_e\right)\times economic\;capital\\RARORAC=\frac{spread+fees–EL–\cos t\;of\;capital–\cos t\;of\;operation}{economic\;capital}$$

$$PD_k=\frac{defaulted\;_t^{t+k}}{names_t}$$

$$PD_k^{cumulative}=\frac{\sum_{i=t}^{i=t+k}defaulted_i}{names_t}\\PD_k^{Marginal}=PD_{t+k}^{cumulative}-PD_t^{cumulative}$$

Discrete:

$$ADR_t=1-\sqrt[t]{\left(1-PD_t^{cumulative}\right)}$$

Continuous:

$$ADR_t=-\frac{\ln\;\left(1-PD_t^{cumulative}\right)}t\\PD=N\left[\frac{\ln\left(F\right)-\ln\left(V_A\right)-\mu T+{\displaystyle\frac12}\sigma_A^2T}{\sigma_A\sqrt T}\right]\\DtD=\frac{\ln\left(V_A\right)-\ln\left(F\right)+\left(\mu_{risky}-{\displaystyle\frac{\sigma_A^2}2}\right)-"other\;payouts"}{\sigma_A}\cong\frac{\ln\left(V\right)-\ln\left(F\right)}{\sigma_A}\\\ln\left(\frac{q_{solv}}{q_{insolv}}\right)\\LOGIT\left({\mathrm\pi}_{\mathrm i}\right)=\log\frac{{\mathrm\pi}_{\mathrm i}}{1-{\mathrm\pi}_{\mathrm i}}\\\frac{{\mathrm\pi}_{\mathrm i}}{1-{\mathrm\pi}_{\mathrm i}}=e^{\left(\beta_0+\beta_1x_{i1}\right)}\\{\mathrm\pi}_S=\frac{\mathrm{Scaled}\;\mathrm{Odds}}{1+\mathrm{Scaled}\;\mathrm{Odds}}$$

Value of Equity at T:

$$S_T=Max(V_T-F,0)$$

Value of debt:

$$D_T=F-Max(F-V_T,0)$$

$$V_t=D_t+S_t\\\;S_{t\;}=V\times N(d)–Fe^{-r(T-t)}\;\times N(d-\sigma\sqrt{T-t})\\d=\frac{\left({\displaystyle\frac V{Fe^{-r(T-t)}}}\right)}{\sigma\sqrt{T-t}}+\frac12\sigma\sqrt{T-t}\\Credit\;Spread=-\left(\frac1{(T-t)}\right)\times ln\left(\frac DF\right)-R_f$$

Vasicek Model:

$$\triangle r_t=k(\theta-r_t)\triangle t+\sigma_r\varepsilon_t$$

$$LGD=F\times(PD)-V_e^{\mu(T-t)}\times N\left(\frac{ln(F)-ln(V)-\mu(T-t)-0.5\sigma^2(T-t)}{\sigma\sqrt{T-t}}\right)$$

Expected bond value: 

$$E(BV_p={\textstyle\sum_{i=1}^N}pi\;BV_i)$$

$$Vu\ln erable\;Option=\left[(1-PD)\times c\right]+(PD\times RR\times c)$$

 

 

Probability density:

$$f\left(x\right)=\frac1\beta\times e^\frac{-x}\beta\;,x\geq0$$

Poisson:

$$P\left(X=x\right)=\frac{\lambda^xe^{-\lambda}}{x!}$$

Cumulative PD:

$$P\left(t^\ast<t\right)=F\left(t\right)=1-e^{-\lambda t}$$

Marginal Default Probability:

$$\lambda e^{-\lambda t}$$

Hazard rate:

$${\lambda^\ast}_T\approx\frac{z_T}{1-RR}$$

Default Probability: (when hazard rate varies)

$$\pi_t=1-e^{-\int_0^t\lambda\left(s\right)ds}$$

$$\rho_{1,2}=\frac{\pi_{12}-\pi_1\pi_2}{\sqrt{\pi_1\left(1-\pi_1\right)}\;\sqrt{\pi_2\left(1-\pi_2\right)}}$$

Conditional Cumulative Default Probability:

$$P\left(m\right)=\phi\left(\frac{k_i-\beta_i\overline m}{\sqrt{1-\beta_i^2}\;}\right)$$

Single Factor Model:

$$a_i-\beta_i\overline m=\sqrt{1-\beta_i^2\varepsilon_i}$$

$$\rho=\frac{\phi\begin{pmatrix}k\\k\end{pmatrix}-\mathrm\pi^2}{\pi\left(1-\pi\right)}\\\rho_{1,2}=\frac{\phi\begin{pmatrix}k_i\\k_j\end{pmatrix}-{\mathrm\pi}_1{\mathrm\pi}_2}{\sqrt{\pi_i\left(1-\pi_i\right)}\sqrt{\pi_j\left(1-\pi_j\right)}}\\X\left(m\right)=P\left(m\right)=\phi\left(\frac{k-\beta\overline m}{\sqrt{1-\beta^2}}\right)\\\phi^{-1}\left(\overline x\right)=\left(\frac{k-\beta\overline m}{\sqrt{1-\beta^2}}\right)$$

Loan interest:

$$\left(N-\sum_{t=1}^Td_t\right)\times\left(LIBOR+spread\right)\times par$$

Proceeds (par) from redemption of surviving loans:

$$\left(N-\sum_{t=1}^Td_t\right)\times par$$

Recovery in final year:

$$R_t=0.4d_T\times\;loan\;amount$$

Residual in trust account:

$$\sum_{\tau=1}^T\left(1+r\right)^{t-\tau}OC_t$$

$$Netting\;factor=\frac{\sqrt{n+\left(n-1\right)n\overline p}}n\\E.E=\frac1{\sqrt{2\pi}}\times\sigma_E\times\sqrt{T_m}\approx0.4\times\sigma_E\times\sqrt{T_m}\\PEE=k\times\sigma_E\times\sqrt{T_m}$$

When there is no correlation between the volatility of the underlying exposure:

$$\sqrt{variance\;of\;non\;cash\;collateral+variance\;of\;underlying\;exposure}$$

Overall risk of the position as a function of correlation:

$$K\times effective\;volatility\times\sqrt{T_M}$$

$$Risk\;neutral\;default\;Probability=Liquidity\;Premium+default\;risk\;premium+real\;world\;default\;probability$$

$$F(u)=1-exp\left(-h\times u\right)\\F(u)=1-exp\left[\frac{-spread}{1-recovery}\times u\right]\\Marginal\;PD=q\left(t_{i-1},\;t_i\right)\approx exp\left[\frac{-spread_{t_{i-1}}}{1-recovery}\times t_{i-1}\right]-exp\left[\frac{-spread_{t_i}}{1-recovery}\times t_i\right]\\No.\;of\;defaults=n\left[\frac{x\%}{1-recovery}\right]$$

$$C.V.A=LGD\times{\textstyle\sum_{i=1}^m}d\left(t_i\right)\times EE\left(t_i\right)\times PD\left(t_{i-1,t_i}\right)$$

CVA as a spread:

$$\frac{CVA\left(t,T\right)}{CDS_{premium}\left(t,T\right)}=X^{CDS}\times EPE\\V(i)=\triangle CVA_{NS,i}=CVA\left(NS,i\right)-CVA\left(NS\right)\\BCVA=CVA+DVA\\CVA=+LGD_c\times{\textstyle\sum_{i=1}^m}EE\left(t_i\right)\times PD_C\left(t_{i-1,t_i}\right)\\DVA=-LGD_1\times{\textstyle\sum_{i=1}^mN}EE\left(t_i\right)\times PD_1\left(t_{i-1,t_i}\right)$$

BCVA Spread:

$$\frac{BCVA\left(t,T\right)}{CDS_{premium}\left(t,T\right)}=X_C^{{}^{CDS}}\times EPE-X_1^{{}^{CDS}}\times ENE$$

Loan Portfolios:

$$EL=\sum_{i-1}^NPD_i\times EAD_i\times LGD_i\\EL_S=\sum_{i-1}^NPD_i^S\times EAD_i\times LGD_i$$

Derivatives Portfolios:

$$EL=\sum_{i-1}^NPD_i\times\left[EPE_i\times\alpha\right]\times LGD_i\\EL_S=\sum_{i-1}^NPD_i^S\times\left[EPE_i^s\times\alpha\right]EAD_i\times LGD_i$$

$$CVA_n=LGD_n\times{\textstyle\sum_{i=1}^T}EE_n\left(t_j\right)\times PD^\ast\left(t_{j-1},t_j\right)\\CVA_S={\textstyle\sum_{n=1}^n}LGD_n^\ast\times{\textstyle\sum_{j=1}^T}EE_n^s\left(t_j\right)\times PD_n^s\left(t_{j-1},t_j\right)\\BCVA={\textstyle+\sum_{n=1}^n}LGD_n^\ast\times{\textstyle\sum_{j=1}^T}EE_n^\ast\left(t_j\right)\times PD^\ast\left(t_{j-1},t_j\right)\times S_I^\ast\left(t_{j-1}\right)\\-\sum_{n=1}^nLGD_I^\ast\times\sum_{j=1}^TNEE_n^\ast\left(t_j\right)\times PD^\ast\left(t_{j-1},t_j\right)\times S_n^\ast\left(t_{j-1}\right)$$

$$WAL=\Sigma\left(\frac a{365}\right)\times PF\left(t\right)\\CPR=1-{(1-SMM)}^{12}\\SMM=1-{(1-CPR)}^\frac1{12}$$

Basic Indicator approach:

$$K_{BIA}=\frac{\sum_{i=1}^nGI_i\times\alpha}n$$

The standard approach:

$$K_{TSA}=\frac{\left\{\sum_{3\;years}max\left[\Sigma\left(GI_{1-8}\times\beta_{1-8}\right),0\right]\right\}}3$$

Alternative Standardized Approach:

$$K_{RB}=\beta_{RB}\times LA_{RB}\times m$$

$$BI=ILDC_{avg}+SC_{avg}+FC_{avg}\\Internal\;Loss\;Multiplier=In\left(e^1-1+\frac{loss\;component}{BI\;component}\right)\;or\;\left(\frac{m\times LC+\left(m-1\right)\times BIC}{LC+\left(2m-2\right)\times BIC}\right)\\Where\;ILDC=Min\left[abs\left(II_{avg}-IE_{avg}\right),0.035^\ast IEA_{avg}\right]+abs\left(LI_{avg}-LE_{av}\right)+DI_{avg}\\SC=Max\left(OOI_{avg},OOE_{avg}\right)+max\left\{abs\left(FI_{avg}-FE_{avg}\right),min\left[max\left(FI_{avg},FE_{avg}\right),\\0.5^\ast uBI+0.1^\ast\left(max\left(FI_{avg},FE_{avg}\right)-0.5^\ast uBI\right)\right]\right\}\\FC=abs\left(net\;P\&\;LTB_{avg}\right)+abs\left(net\;P\&\;LBB_{avg}\right)$$

GEV distribution:

$$F\left(X\vert\xi,\;\mu,\;\sigma\right)=exp\left[-\left(1+\xi\times\frac{x-\mu}\sigma\right)^{-\frac1\xi}\right]\;if\;\xi\neq0\\F\left(X\vert\xi,\;\mu,\;\sigma\right)=exp\left[-exp\;\left(\frac{x-\mu}\sigma\right)\right]\;if\;\xi=0$$

Generalized Pareto Distribution:

$$1-\left[1+\frac{\xi x}\beta\right]^{-\frac1\xi}\;if\;\xi\neq0\\1-exp\left[-\frac x\beta\right]\;if\;\xi=0$$

POT:

$$F_u\left(x\right)=P\left\{X-u\leq x\vert X>u\right\}=\frac{F\left(x+u\right)-F\left(u\right)}{1-F\left(u\right)}$$

VaR using POT:

$$u+\frac\beta\xi\left[\left[\frac n{N_u}\left(1-confidence\;level\right)\right]^{-\xi}-1\right]$$

ES:

$$\frac{Var}{1-\xi}+\frac{\beta-\xi u}{1-\xi}$$

$$Economic\;capital=risk\;capital+strategic\;risk\;capital\\RAROC=\frac{after\;tax\;expected\;risk-adjusted\;net\;income}{economic\;capital};\\\frac{expected\;revenues-\cos ts-expected\;losses-taxes+return\;on\;economic\;capital\pm transfer}{economic\;capital}$$

Hurdle Rate :

$$h_{AT}=\frac{\left[\left(CE\times R_{CE}\right)+\left(PE\times R_{PE}\right)\right]}{\left(CE+PE\right)}\\Adjusted\;RAROC=RAROC-\beta_E\left(R_M-R_F\right)\\R_{CE}=R_f+\beta_{CE}\left(R_M-R_F\right)$$

$$LC=0.5\times\;V\times spread\\Spread=\frac{ask\;price-bid\;price}{\left(ask\;price+bid\;price\right)/2}$$

Liquidity adjusted VaR(constant spread):

$$LVaR=(V\times Z_\alpha\times\sigma)+(0.5\times V\times spread)\\LVaR=VaR+LC$$

Lognormal VaR:

$$VaR=\left[1-exp\left(\mu-\sigma\times z_\alpha\right)\right]\times v\\LVaR=VaR+LC=\left[1-exp\left(\mu-\sigma\times z_\alpha\right)+0.5\times spread\right]\times v$$

Ratio of LVaR to VaR assuming 𝜇=0:

$$\frac{LVaR}  {VaR}=1+\frac{spread}{2\times\left[1-exp\left(-\sigma\times z_\alpha\right)\right]}\\LVaR=VaR\times\left(1-\frac{\triangle P}P\right)=VaR\times\left(1-E\times\frac{\triangle N}N\right)\\{\left(\frac{LVaR}{VaR}\right)}_{combined}={\left(\frac{LVaR}{VaR}\right)}_{exogenous}\times{\left(\frac{LVaR}{VaR}\right)}_{endogenous}$$

Exogenous price approach:

$$E=\frac{\triangle P/P}{\triangle N/N}\\LVaR=\;VaR+0.5\left[\left(M_S+Z_\alpha\times\sigma_S\right)\right]\times V$$

Profit or loss based on data gamma approximation:

$$II=s\triangle s+\frac\gamma2\left(\triangle s\right)^2$$

Leverage ratio:

$$L=\frac AE=\frac{\left(E+D\right)}E=1+\frac DE$$

Leverage effect:

$$r_E=Lr_A-\left(L-1\right)r_D\\ROE=(leverage\;ratio\times ROA)-\left[(leverage\;ratio-1)\times\cos t\;of\;debt\right]$$

Effect of increasing leverage:

$$\frac{\delta r_E}{\delta L}=r_A-r_D$$

Transaction cost:

$$Transaction\;\cos t\;(99\%confidence\;interval):+/-P\times\frac1{2\;}(s+2.33\sigma_s)$$

Spread risk factor:

$$Spread\;risk\;factor:\frac12\left(s+2.33\sigma_s\right)$$

1 day position VaR:

$$1\;day\;position\;VaR:\;VaR_t\times\sqrt T$$

VaR when position can be liquidated for a period of days:

$$VaR_t\times\sqrt{\frac{\left(1+T\right)\left(1+2T\right)}{6T}}$$

Credit Equivalent Amount:

$$Credit\;Equivalent\;Amount:\;Max\;(V,0)+(a\times L)$$

Total RWA:

$$Total\;RWA:{\textstyle\sum_{i=1}^N}w_iL_i+{\textstyle\sum_{j=1}^M}{\textstyle w_j}{\textstyle C_j}$$

Market Risk Capital Requirement:

$${\textstyle Market\;Risk\;Capital\;Requirement:\;Max\;(VaR_{t-1},m_c\times VaR_{avg})+SRC}\\{\textstyle Total\;Capital=0.08\times(credit\;risk\;RWA\;+\;market\;risk\;RWA)}\\Market\;RWA=12.5\times\left(max\left(VaR_t,m_c\times VaR_{avg}\right)SRC\right)\\Credit\;RWA=\Sigma\left(RWA\:on\;balance\;sheet\right)+\Sigma\left(RWA\:off\;balance\;sheet\right)\\VaR_{99.9\%,1-year}\approx{\textstyle\sum_i}EAD_i\times LGD_i\times WCDR_i$$

Expected loss:

$$Expected\;loss:\;EL={\textstyle\sum_i}EAD_i\times LGD_i\times PD_i\\Required\;Capital=EAD_i\times LGD_i\times\left(WCDR_i-PD_i\right)\\\rho=0.12\times\left(1+e^{-50\times PD}\right)$$

From a counterparty′s perspective,the capital required for the counterparty incorporates a maturity adjustment as follows:

$$Required\;Capital=EAD\times LGD\times(\;WCDR-PD)\times MA\\Where\;MA=Maturity\;Adjustment\;;\\MA=\frac{\left(1+\left(M-2.5\right)\times b\right)}{\left(1-1.5\times b\right)};\;M\;=\;Maturity\;of\;the\;exposure\\RWA=12.5\times\left[EAD\times LGD\times\left(WCDR\right)\times MA\right]$$

Minimum Capital Requirements:

$$Total\;capital=0.08\times(credit\;risk\;RWA+market\;risk\;RWA+operational\;risk\;RWA)$$

$$Stressed\;VaR:\;max(VaR_{t-1},\;m_c\times VaR_{avg})+max(SVaR_{t-1},m_s\times SVaR_{avg})\\Liquidity\;Coverage\;Ratio=\frac{high\;quality\;liquid\;asset}{net\;cash\;flows\;in\;a\;30\;days\;period}\geq100\%\\Net\;stable\;funding\;ratio=NSFR=\frac{amount\;of\;available\;stable\;funding}{amount\;of\;required\;stable\;funding}\geq100\%$$

Investor Risk Premium:

$$E(R_M)-R_F=\overline\gamma\times\sigma_m^2$$

Security Market line:

$$E(R_i)-R_F=\frac{cov\left(R_I,R_M\right)}{VaR\left(R_M\right)}\times\left[E\left(R_M\right)-R_F\right]=B_i\times\left[E\left(R_M\right)-R_F\right]$$

Multifactor Model:

$$M=a+b_1f_1+b_2f_2+\cdots+b_kf_k$$

SDF model:

$$P_i=E\left[m\times payoff_i\right]$$

$$E(R_i)-R_F=\frac{cov\left(R_{I,M}\right)}{var\left(m\right)}\times\left(\frac{-var\left(m\right)}{E\left(m\right)}\right)=\beta_{i,m}\times\lambda_m\\E(R_i)=R_F+\beta_{i,1}\times E(f_1)+\beta_{i,2}\times E(f_2)+...+\beta_{i,k}\times E(f_k)$$

$$E(R_M)-R_F=\overline\gamma\times\sigma_m^2$$

Fama French three factor Model:

$$E(R_i)=R_F+\beta_{i,Mkt}\times E(R_m-R_f)+\beta_{i,SMB}\times E(SMB)+\beta_{i,HML}\times E(HML)$$

Fama French Model with Momentum Effect:

$$E(R_i)=R_F+\beta_{i,Mkt}\times E(R_m-R_f)+\beta_{i,SMB}\times E(SMB)+\beta_{i,HML}\times E(HML)+\beta_{i,WML}\times E(WML)$$

$$R_t^{ex}=R_t-R_t^B\\\alpha=\frac1T{\textstyle\sum_{t=1}^T}R_t^{ex}$$

Fundamental Law of Active Management:

$$IR=\frac\alpha\sigma\\\alpha=R_t-R_F\\Sharpe\;ratio=\frac{\overline{\overline{R_t}}-\overline{R_F}}\sigma\\IR\approx IC\times\sqrt{BR}\\E\left(R_i\right)=R_F+B\left[E\left(R_M\right)-R_f\right]\\Fama\;and\;French\;three\;factor\;model=R_i-R_F\\=\alpha+\beta_{i,Mkt}\times\left(R_M-R_F\right)+\beta_{i,SMB}\times\left(SMB\right)+\beta_{i,HML}\times\left(HML\right)$$

$$Alpha=volatility\times(information\;coefficient)\times(score)\\Risk\;Aversion=\frac{information\;ratio}{2\times active\;risk}\\Average\;alpha\;for\;the\;stocks\;with\;forecasts=\frac{\left(weighted\;average\;of\;the\;alphas\;with\;forecast\right)}{\left(value-weighted\;fraction\;of\;stocks\;with\;forecast\right)}$$

$$Marginal\;contribution\;to\;value\;added=(alpha\;of\;asset)-\left[2\times\left(risk\;aversion\right)\times(active\;risk)\\\times(marginal\;contribution\;to\;active\;risk\;of\;asset)\right]$$

Cost:

$$Cost:\;–(\cos t\;of\;selling)<(\;marginal\;contribution\;to\;value\;added\;)<(\cos t\;of\;purchase)$$

Range:

$$Range:\;\left[2\times(risk\;aversion)\times(active\;risk)\times(marginal\;contribution\;to\;active\;risk)\right]-(\cos t\;of\;selling)\\<alpha\;of\;asset<\left[2\times(risk\;aversion)\times(active\;risk)\times(marginal\;contribution\;to\;active\;risk)\right]+(\cos t\;of\;purchase)$$

Portfolio construction technique:

$$Portfolio\;construction\;technique:\;(Portfolio\;alpha)-(risk\;aversion)\times{(active\;risk)}^2–(transaction\;\cos t)$$

Diversified VaR: 

$$VaR_P=Z_C\times\sigma_P\times P$$

Individual VaR: 

$$VaR_P=Z_C\times\sigma_i\times\left|P\right|=Z_C\times\sigma_i\times\left|w_i\right|\times P$$

Standard deviation of a two-asset portfolio:

$$\sigma_P=\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{1,2}\sigma_1\sigma_2}$$

VaR of a two-asset portfolio:

$$VaR_P=Z_cP\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{1,2}\sigma_1\sigma_2}$$

Undiversified VaR: 

$$VaR_P=\sqrt{VaR_1^2+VaR_2^2+2VaR_1VaR_2}=VaR_1+VaR_2$$
Marginal VaR: 

$$MVaR_i=\frac{VaR}{portfolio\;value}\times\beta_i\;or\;Z_c\frac{cov(R_i,R_p)}{\sigma_p}$$

Component VaR:

$$CVaR_i=(MVaR_i)\times(w_i\times P)=VaR\times\beta_i\times w_i$$

$$\sigma_P\;of\;2\;asset\;portfolio\;with\;equal\;weights\;and\;same\;s.d\;and\;same\;\\correlations\;between\;each\;pair:\\\sigma_P=\sqrt[\sigma\times]{\frac1N+\left(1-\frac1N\right)\rho}$$

 

 

$$Surplus=Assets–Liabilities\\\triangle Surplus\;=\triangle Assets–\triangle Liabilities$$

Return on the surplus:

$$R_{surplus}=\frac{\triangle Surplus}{Assets}=\frac{\triangle Assets}{Assets}-\left(\frac{\triangle Liabilities}{Liabilities}\right)\left(\frac{Liabilities}{Assets}\right)=R_{asset}-R_{liabilities}\left(\frac{Liabilities}{Assets}\right)\\Weight\;of\;portfolio\;managed\;by\;manager\;i=\frac{IR_i\times(portfolio's\;tracking\;error)}{IR_P\times(manager's\;tracking\;error)}$$

 

Liquidity Duration:

$$LD=\frac Q{(0.10\times V)}$$

 

Sharpe Ratio: 

$$SA=\left[\frac{{\overline R}_A-{\overline R}_F}{\sigma_A}\right]$$

Treynor Measure:

$$\\T_A=\left[\frac{{\overline R}_A-{\overline R}_F}{\beta_A}\right]$$

 Jensen's Alpha:

$$\alpha_A=R_A-E(R_A)$$

Information Ratio:

$$IR_A=\left[\frac{{\overline R}_A-{\overline R}_B}{\sigma_{A-B}}\right]$$

$$M^2=R_P-R_M$$

$$Null(H_0)\;:\;True\;alpha\;is\;zero\\Alternate(H_A)\;:\;True\;alpha\;is\;not\;zero$$

Statistical significance of alpha returns: 

$$t=\frac{\alpha-0}{\sigma/\sqrt N}$$

Measuring Market Timing with Regression:

$$R_P-R_F=\alpha+\beta_P(R_M-R_F)+M_P(R_M-R_F)D+\varepsilon_P$$

Measuring Market Timing with a Call Option Model:

$$100\%\;invested\;in\;the\;S\;\&\;P\;500\;if\;E(R_M)>R_F\\100\%\;invested\;in\;Treasury\;bills\;if\;E(R_M)<R_F$$

Asset Allocation Attribution:

$$\left[b_1R_{B1}+b_2R_{B2}+\cdots+b_nR_{Bn}\right]-R_B\\R_P-\left[b_1R_{B1}+b_2R_{B2}+\cdots+b_nR_{Bn}\right]\\$$

 

 

 

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