Statistical Arbitrage Week4

Portfolio Optimization

Structure of Optimizer

• Risk Model
  • Shrinkage estimator of the covariance matrix of stock returns
• Transaction Cost Model
  • $1 \text{bp} + \frac12 \text{bid-ask spread}$
• Alpha
  • Weighted blend of various standardized, winsorized alphas

Input

• position as of close of business on day $t-1$
• alphas using data observed up to day $t-1$
• transaction costs using data observed up to day $t-1$
• risk model using data observed up to day $t-1$
• constraints using data observed up to day $t-1$

Objectives & Constraints

• Minimize risk: $x’ \Sigma x$
• Maximize exposure to alpha: $\alpha’ x$
• Neutralize exposure to beta: $\beta’ x = 0$
• Minimize transaction costs: $\tau’ \lvert x-w \rvert$
• Other constraints:
  • maximum trade size
  • maximum position size
  • maximum industry and country exposure
• Notations

Variable	Dimension	Definition
$\mathbf{x}$	$n \times 1$	desired portfolio weights
$\mathbf{w}$	$n \times 1$	initial portfolio weights
$\boldsymbol{\Sigma}$	$n \times n$	covariance matrix of stock returns
$\boldsymbol{\alpha}$	$n \times 1$	aggregate alphas
$\boldsymbol{\beta}$	$n \times 1$	historical betas
$\boldsymbol{\tau}$	$n \times 1$	transaction costs

The parametric problems formed as:

$$max_{\mathbf{x}} \boldsymbol{\alpha}' \mathbf{x} - \lambda \boldsymbol{\tau}' \lvert \mathbf{x}-\mathbf{w} \rvert - \mu \mathbf{x}' \boldsymbol{\Sigma} \mathbf{x}$$

subject to:

• beta neutrality: $\boldsymbol{\beta}’ \mathbf{x} = 0$
• max trade and position: $\boldsymbol{\gamma} \leq \mathbf{x} \leq \boldsymbol{\delta}$
  • Max trade size for $i^{th}$ stock: $\theta_i$
    • $\Rightarrow w_i - \theta_i \leq x_i \leq w_i + \theta_i$
  • Max position size for $i^{th}$ stock: \pi_i
    • $\Rightarrow -\pi_i \leq x_i \leq \pi_i$
• industry constraint: $-r^* \cdot \mathbf{1} \leq \mathbf{R}’ \mathbf{x} \leq r^* \cdot \mathbf{1}$
  • Sectors are a factor of risk
  • Difficult to time sector performance
  • Constrain industry exposure
  • But not to zero (will incur transaction cost)
  • $r^* = $ 300,000 \text{ limit for }$ 50 \times 50M \text{ book size}$
  • Industry Dummy
    • $\rho$ industries
    • Boolean matrix $\mathbf{R}$ of dimension $n \times \rho$
    • $R(i, j) = 1$ if $i^{th}$ stock belongs to $j^{th}$ industry else 0
    • Every row of matrix $\mathbf{R}$ has exactly one entry equal to 1; all other entries are equal to 0
• country constraint: $-f^* \cdot \mathbf{1} \leq \mathbf{F}’ \mathbf{x} \leq f^* \cdot \mathbf{1}$
  • Countries are a factor of risk
  • Difficult to time country performance
  • Constrain country exposure
  • But not to zero (will incur too much transaction cost)
  • $f^* = $ 100,000 \text{ limit for } $ 50 \times 50M \text{ book size}$
  • Industry Dummy
    • $\varphi$ industries
    • Boolean matrix $\mathbf{F}$ of dimension $n \times \varphi$
    • $F(i, j) = 1$ if $i^{th}$ stock belongs to $j^{th}$ industry else 0
    • Every row of matrix $\mathbf{F}$ has exactly one entry equal to 1; all other entries are equal to 0

Output

trade to be executed on day $t$

$$\text{Final position}(t+1) = \text{Final position}(t) + \text{trade}(t)$$

Backtest Process

1. Load all necessary data into memory
2. Create the alphas
3. Start from portfolio with zero dollar invested
4. Loop over all days in backtest period
   • Every day: call optimizer to find optimal rebalancing trade given initial position
   • End-of-day position becomes initial position of next day
5. Compute P&L