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risk_models.rmd
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---
title: "Risk Models Formula Sheet"
author: "Alex Dou"
date: "March 10, 2019"
output:
pdf_document: default
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
### Coherent Risk Measures
#### Monotonicity
$$ A \ge B\Rightarrow \rho(A) \le \rho(B) $$
#### Subadditivity
$$ \rho(A+B) \le \rho(B) + \rho(B) $$
#### Positive Mogogeneity
$$ \rho(xA) = x\rho(A) $$
#### Translation Invariance
$$ \rho(A+cash) = \rho(A) - cash $$
### Value at Risk
$$ VaR_{derivatives} = VaR_{underling}\times \Delta{} $$
### Model-building VaR
$$ VaR=N(X)\sigma{}\sqrt{T}P $$
$$ \sigma_{xy}=\sqrt{\sigma_x^2 + \sigma_y^2 + 2\rho \sigma_x \sigma_y}$$
$$ \sigma_B=DB\sigma_y $$
$$ VaR_{xy} = \sigma_{xy}\sqrt{T}N^{-1}(1-\alpha)=\sqrt{\sigma_x^2 + \sigma_y^2 + 2\rho \sigma_x \sigma_y}\sqrt{T}N^{-1}(1-\alpha)$$
### Delta-normal approach
$$ VaR_{option} = |\Delta{}|VaR_{stock} $$
### Delta-gamma approach
$$ VaR_{bond} = |-DP|VaR_y-\frac{1}{2}C\times P\times{VaR_y}^2 $$
$$ VaR_{option} = |\Delta{}|VaR_{stock}-\frac{1}{2}\Gamma{}\times {VaR_{stock}}^2$$
### Credit Risk Components
$$ Rate_{loss} = 1- Rate_{recovery} $$
$$ LGD=1-RR $$
$$ E(loss) = EA *Rate_{default}\times Rate_{loss} $$
$$ EL=EAD*PD*LGC $$
### Adjusted Exposure
$$ Exposure^* = Outstandings + \alpha (Commitments - Outstandings) $$
$$ Exposure^* = Outstandings + UGD (Commitments - Outstandings) $$
### Unexpected Loss
$$ Loss_{unexpected} = EA\sqrt{P_{default}\sigma_{LR}^2+LR^2\sigma_{PD}^2} $$
$$ \sigma_{pd}^2 = P_d(1-P_d) $$
$$ \sigma_{EDF}^2 = EDF(1-EDF) $$
$$PD=EDF$$
### Expected loss of a portfolio
$$ EL_p = \sum_{i=0}^{n}{EL_i} = \sum_{i=0}^{n}{EA_i \times PD_i \times LR_i} $$
### Unexpected loss of a portfolio
$$ UL_p=\sqrt{ \sum_j{\sum_i\rho_{ij}UL_iUL_j} } $$
### Unexpected loss of two assets
$$ UL_{12} = \sqrt{UL_1^2 + UL_2^2 + 2\rho_{12}UL_1UL_2} $$
### Unexpected loss contribution
$$ UL_p = \sum_iUL_i $$
$$ ULC_i = \frac{UL_i\sum_jULj\rho_{ij}}{UL_p} $$
### ULC of two assets
$$ ULC1=\frac{UL_1^2+\rho_{12}UL_1UL_2}{UL_p} $$
$$ ULC2=\frac{UL_2^2+\rho_{12}UL_1UL_2}{UL_p} $$
### Economic capital for credit risk
$$ Capital_p = UL_p\times CM $$
### Basic indicator approach (BIA)
$$ K_{BIA}=(\frac{GI_1+GI_2+GI_3}{3})\alpha $$
$$ \alpha = 15\% $$
### Standardized approach (SA)
$$ K_{SA} = \frac{\sum_{t=1}^{3}{max(\sum_{i=1}^{8}{GI_i\beta_i}, 0)}}{3} $$
### Poisson Distribution
$$ P(k) = \frac{(\lambda{}T)^k}{k!}e^{-\lambda{}T} $$
### Scale adjustment
$$ Loss_a=Loss_b(\frac{Revenue_a}{Revenue_b})^{0.23} $$
### Power Law
$$ P(v>x)=Kx^{-\alpha} $$
### Prepayment
$$ Prepayment = Payment_{actual} - Payment_{scheduled} $$
$$ SMM = \frac{Prepayment}{Balance_0 - Payment_{scheduled}} $$
$$ (1-SMM)^{12} = 1-CPR $$