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investment_management.rmd
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---
title: "FRM Investment Management"
author: "Alex Dou"
date: "November 3, 2019"
output: html_document
---
### Factor
#### CAPM
$$ E(r_i) = r_f + \beta_i (r_m - r_f) $$
#### Multifactor models
$$ E(r_i) = r_f + \sum{\beta_i (r_m - r_f)} $$
### Alpha
#### Active Management
* Excess return
$$ r^{ex}_t = r_t- r^{bmk}_t $$
* Grinold fundamental low
$$ IR \approx IC \sqrt{BR} $$
### Portofolio Construction
#### Refining alphas
$$ \alpha = Volatility \times IC \times Score $$
#### Active Risk aversion
$$ \lambda_A = \frac{IR}{2\sigma_{\alpha}} $$
$$ Utility = \alpha_p - \lambda_A {\sigma_{\alpha}}^2 $$
#### Transaction costs
$$ MCVA_n = \alpha_n - 2 \lambda_A \sigma_{\alpha} MCAR_n $$
* No trade region
$$ -SC_n \le MCVA_n \le PC_n $$
#### Portofolio VaR measures
$$ VaR=V Z_c \sigma $$
$$VaR_i = Z_c \sigma_i V_i = Z_c \sigma_i w_i V_p $$
$$ MVaR_i = \frac{\beta_i VaR_p}{V_p} $$
$$ VaR_{c1} = VaR_{c2} \frac{Z_{c1}}{Z_{c2}} $$
$$ VaR_p = Z_c \sigma_p V_p = \sqrt{VaR^2_1 + VaR^2_2 + 2 VaR_1 VaR_2 \rho_{12}} $$
#### VaR tools
##### Marginal VaR
$$ MVaR_i = Z_c \frac{Cov(r_i, r_p)}{\sigma_p} = Z_c \sigma_p \beta_i = \frac{VaR_p \beta_i}{V_p} $$
##### Incremental VaR
$$ IVaR_a = VaR_{p+a} - VaR_p $$
##### Component VaR
$$ CVaR_i = MVaR_i V_i = MVaR_i w_i V_p = VaR_p \beta_i w_i $$
$$ CVaR_i \% = \ frac{CVaR_i)}{VaR_p} = w_i \beta_i $$
#### Risk Management
##### Global Minimal
* Lower a portfolio VaR
$$\forall i, j MVaR_i = MVaR_j $$
$$\forall i, j \beta_i = \beta_j $$
##### Optimal portfolio
* Maximize the Sharpe ratio with VaR of the portfolio
$$ \frac{r_p - r_f}{VaR_p} $$
$$ \frac{r_i - r_f}{MVaR_i} = \frac{r_j - r_f}{MVaR_j} $$
$$ \frac{r_i - r_f}{\beta_i} = \frac{r_j - r_f}{\beta_j} $$
#### Surplus
$$ Surplus = Assets - Liabilities $$
$$ \Delta Surplus = \Delta Assets - \Delta Liabilities = r_a A - r_l L $$
$$ r_s = r_a - \frac{L}{A}r_l $$
$$ \Delta P \% = -D \Delta y $$
$$ \Delta P = D \Delta y P_0 $$
$$ SaR = |E(S) - Z_c \sigma_s | $$
$$ E(S_t) = S_t \pm Z_c \sigma_s $$
$$ \sigma_{surplus} = \sqrt{A^2 {\sigma_A}^2 + L^2 {\sigma_L}^2 - 2AL \sigma_A \sigma_L \rho } $$
#### Risk budgeting
$$w_i = \frac{IR_i/TE_i}{IR_p/TE_p} = \frac{\alpha_i/{TE_i}^2}{\alpha_p/{TE_p}^2}$$
$$ \sum{w_i} \le 1 $$
* The residual weight is allocated to the index
### Risk monitoring and performance measurement
#### Liquidity Duration
$$ LD_i = \frac{Q_i}{0.15 V_i} $$
$$ \alpha = r_p - r_b $$
#### Return calculation
##### Time-weighted returns (geometric average)
$$ 1+r_g = ({\prod{1+r_i}})^{\frac{1}{n}} $$
##### Dollar-weighted returns: IRR
$$ PV(out_flow) = PV(in_flow) $$
$$ \sum_{i=0}^n{\frac{CF_i}{(1+r_{dw})^i}} = 0 $$
#### Risk-adjusted performance measures
* Sharpe Ratio
$$ \frac{r_p- r_f}{\sigma_p} $$
* Treynor ratio
$$ \frac{r_p-r_f}{\beta_p} $$
* Information Ratio
$$ \frac{\alpha_p}{\sigma(\alpha_p)} $$
#### Market timing
* No market timing
$$ r_p - r_f = a+b(r_m - r_f) +e_p $$
* Treynor and Mazuy
$$ r_p - r_f = a+b(r_m - r_f) + c(r_m - r_f)^2 + e_p $$
* Henriksson and Merton
$$ r_p - r_f = a+b(r_m - r_f) + c(r_m - r_f)D + e_p $$
#### Performance attribution
$$ (W_{Pi}-W_{Bi})r_{Bi} + W_{Pi}(r_{Pi} - r_{Bi}) = W_{Pi} r_{Pi} - W_{Bi}r_{Bi} $$
* Contribution from asset allocation
$$ (W_{Pi}-W_{Bi})r_{Bi} $$
* Contributio from security selection
$$ W_{Pi}(r_{Pi} - r_{Bi}) $$