11-14-2024 - Portfolio Optimization

Background

Basic probability Basic calculus

Random variable Expectation Covariance Variance

Method of Lagrange Multipliers (constrained optimization)

Main point

I have $r$ random variables (outcomes of tossing a dice, but can have real values) $R_1,\dots ,R_r$. $R_i$ represents the value of stock $i$. This value oscillates - it’s random.

I have to make some decisions about which stock to purchase - need to decide how much of each stock to include in portfolio.

e.g. I want to aim for a constant income, or I want to minimize risk, etc.

How to convert this into linear equations?

Possible if our model is simple enough.

Notation

$$\begin{array}{r}E(R)\\ (x_1,\dots ,x_r)\in {\mathbb{R}}_{+}^r\\ R=x_1{R}_1+\cdots +x_r{R}_r\text{ (portfolio)}\end{array}$$

$x_i$ represents the quantity (= percentage of stock in portfolio) of stock $i$. $x_1+\cdots +x_r=1$.

Problem 1

Let us minimize the risk, without any concern for the gains. I don’t want them to go up or down a lot - not vary too much with respect to one another.

Covariance Matrix

Takes $R_1,\dots ,R_r$ and gives a symmetric matrix that has positive eigenvalues

$$\mathbf{\Sigma }=\left[\begin{array}{ccc}Var(R_1)& Cov(R_i,R_j)& \dots \\ ⋮& \ddots & ⋮\\ \dots & \dots & Var(R_r)\end{array}\right]$$

Say $R_1$ and $R_2$ are random variables which takes values in $\{1,2,3\}$ When they are measured together, only the outcomes $(1,2),(2,3),(3,3)$ are possible. Positive covariance because if one grows, the other one also grows. If $(1,3),(2,1),(3,1)$, then the covariance of these values would be negative.

$$Var(R_i)=Cov(R_i,R_i)$$

The variance is always a positive number, which is why $\mathbf{\Sigma }$ has positive eigenvalues.

Risk

Risk is $Var(R)=E(R-E(R){)}^2$. Minimize this.

$$Var(R)=Var\left(\sum _i{x}_i{r}_i\right)={\mathbf{x}}^T\sum \mathbf{x}$$ $${\nabla }_x\left({\mathbf{x}}^T\sum \mathbf{x}\right)=\text{linear in x!}=2\sum \mathbf{x}=0$$

(linear because $x^{T}x=x\cdot x\approx x^2$) This is not actually what we want - we want a constraint

$$F(x)=x^T\sum x+\lambda (x_1,\dots ,x_r)$$

is what we want. (Lagrange Multipliers)

$$\begin{array}{r}{\nabla }_{x}F=0\\ {\nabla }_{\lambda }F=0\end{array}$$ $$2\sum x+\left[\begin{array}{c}\lambda \\ \lambda \\ \lambda \\ ⋮\\ \lambda \\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ 0\\ 0\end{array}\right]$$ $$x_1+\cdots +x_r=1$$ $$\underset{\text{this matrix}}{\underbrace{\left[\begin{array}{cc}2\sum x& 1\\ & ⋮\\ & 1\\ & \\ 1\dots 1& 0\end{array}\right]}}\left[\begin{array}{c}{x}_1\\ ⋮\\ x_r\\ \lambda \end{array}\right]=\left[\begin{array}{c}0\\ ⋮\\ 0\\ 1\end{array}\right]$$

Theorem: this is invertible. $(r+1)\times (r+1)$ matrix.

Problem 2

Do not allow value to drop below a line, without oscillating too much. Minimize $Var(R)$ while $x_1,\dots ,x_r=1$ and $E(R)={\mu }_0$. Need to know $E(R)=?$

If $E(R_i)={\mu }_i$ given, then $E(R)={\mu }_0=x_1{\mu }_1+\cdots +x_r{\mu }_r=\vec{\mathbf{x}}\cdot \vec{\mathbf{\mu }}$.

In this case everything is added one dimension because now we have

$$F(x)=x^T\sum x+{\lambda }_1(x_1,\dots ,x_r)+{\lambda }_2(x_1{\mu }_1+\cdots +x_r{\mu }_r)$$

and

$$\begin{array}{r}{\nabla }_{x}F=0\\ {\nabla }_{{\lambda }_1}F=0\\ {\nabla }_{{\lambda }_2}F=0\end{array}$$

Constraints:

$$\begin{array}{r}{x}_1+\cdots +x_r=1\\ x_1{\mu }_1+\cdots +x_r{\mu }_r={\mu }_0\end{array}$$

Then:

$$2\sum x+\left[\begin{array}{c}{\lambda }_1\\ {\lambda }_1\\ {\lambda }_1\\ ⋮\\ {\lambda }_1\\ 0\end{array}\right]+\left[\begin{array}{c}{\lambda }_2{\mu }_1\\ {\lambda }_2{\mu }_2\\ {\lambda }_2{\mu }_3\\ ⋮\\ {\lambda }_2\mu r\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ 0\\ 0\end{array}\right]$$

So:

$$\underset{(r+2)\times (r+2)}{\underbrace{\left[\begin{array}{ccc}2\sum \mathbf{x}& \vec{1}& \vec{\mathbf{\mu }}\\ {\vec{1}}^T& 0& 0\\ {\vec{\mathbf{\mu }}}^T& 0& 0\end{array}\right]}}\left[\begin{array}{c}{x}_1\\ ⋮\\ x_r\\ {\lambda }_1\\ {\lambda }_2\end{array}\right]=\left[\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ {\mu }_0\end{array}\right]$$

This is invertible under some natural assumption.