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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

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

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