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$$E(R_P)=\Sigma W_iR_i=\;W_AE(R_A)+W_BE(R_B)\\\sigma_P^2=W_A^2\sigma_A^2+W_B^2\sigma_B^2+2W_AW_BCov(A,B)\\Where,\;Cov\;(A,\;B)\;=\;\sigma_A\sigma_{B\;}r_{(A,B)}\\\rho_{xy}=\frac{\;Cov_{A,B}}{\sigma_A\sigma_B}$$

$$When\;\rho=1,\;\sigma_P=W_A\sigma_A+W_B\sigma_B\\When\;\rho\;=\;0,\;\sigma_P^2=W_A^2\sigma_A^2+W_B^2\sigma_B^2$$

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:

$$RC_1=\frac{UL_1+(\;\rho_{1,2}\times\;UL_2)}{UL_p}\times UL_1$$