risk_fundation.rmd

---
title: "Risk Fundation Formula Sheet"
author: "Alex Dou"
date: "March 10, 2019"
output:
  pdf_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```


### Regression

$$ \beta_1 = \frac{Cov(x, y)}{\sigma_x^2}  $$


### Risk Utility curve

$$ U=E(r) - \frac{1}{2}A\sigma^2   $$


### Capital allocation line

$$ E(r_p) = r_f+[\frac{E(r_x)-r_f}{\sigma_x}]\sigma_p  $$

### Capital Market Line

$$ E(r_p) = r_f + [\frac{E(r_m)-r_f}{\sigma_m}]\sigma_p  $$

### CAPM

$$\beta_i = \frac{Conv(i, m)}{\sigma_m^2} = \rho_{i,m}\frac{\sigma_i}{\sigma_m}{}$$

$$ E(r_i) = r_f + \beta_i[E(r_m)-r_f]  $$

$$ \beta_p = \sum{w_i\beta_i}  $$

### Performance Measurement

#### Sharpe Ratio

$$ S_p=\frac{E(r_p)-r_f}{\sigma_p}  $$

#### Treynor Ratio

$$ T_p=\frac{E(r_p)-r_f}{\beta_p}  $$


#### Jensen's Alpha

$$ \alpha_p = E(r_p) - r_{capm}  $$

#### Sortino Ratio

$$ SOR=\frac{E(r_p)-MAR}{SSD}  $$


#### Tracking error

$$TE=\sigma{(r_p-r_b)}$$


#### Information Ratio

$$ IR=\frac{E(r_p)-E(r_b)}{\sigma_{(r_p-r_b)}}  $$


### Multi-factor Model

$$ r_i = E(r_i) + \sum_{j=1}^{n}{\beta_{i,j}F_j} + \epsilon_i $$


### APT

$$ E(r_p) = r_f + \sum{\beta_i\lambda_i}  $$


#### Fama-French model

$$ r  = r_f + \alpha + \beta_m{(r_m-r_f)} + \beta_{smb}{SMB} + \beta_{hml}{HML} + \epsilon  $$