Got any Questions Query Inquisition

$$Profit\;or\;loss\;data=P_n+CF-P_0\\Arithmetic\;Return\;=\frac{P_n-P_0+CF}{P_0}\\Geometric\;Return=\ln\left(\frac{P_n+CF}{P_0}\right)\\VaR:\left[\left(\sigma\times n\right)+1\right]\;th\;observation$$

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:

$$\left[b_1R_{B1}+b_2R_{B2}+\cdots+b_nR_{Bn}\right]-R_B\\R_P-\left[b_1R_{B1}+b_2R_{B2}+\cdots+b_nR_{Bn}\right]\\$$