Portfolio Optimization via Linearly Constrained Least-Squares

L. El Ghaoui

Portfolio Optimization via Linearly Constrained Least-Squares

We consider a universe of $n$ financial assets, in which we seek to invest over one time period. We denote by $r \in R^n$ the vector containing the rates of return of each asset. A portfolio corresponds to a vector $x \in R^n$ , where $x_i$ is the amount invested in asset $i$ . In our simple model, we assume that ‘‘shorting’’ (borrowing) is allowed, that is, there are no sign restrictions on .

As explained here, the return of the portfolio is the scalar product $R(x) := r^Tx$ . We do not know the return vector $r$ in advance. We assume that we know a reasonable prediction $\hat{r}$ of $r$ . Of course, we cannot rely only on the vector $\hat{r}$ only to make a decision, since the actual values in $r$ could fluctuate around $\hat{r}$ . We can consider two simple ways to model the uncertainty on $r$ , which result in similar optimization problems.

Mean-variance trade-off

A first approach assumes that $r$ is a random variable, with known mean $\hat{r}$ and covariance matrix $\Sigma$ . If past values $r_1, \ldots, r_N$ of the returns are known, we can use the following estimates

$\hat{r}=\frac{1}{N} \sum_{i=1}^N r_i, \quad \Sigma=\frac{1}{N} \sum_{i=1}^N\left(r_i-\hat{r}\right)\left(r_i-\hat{r}\right)^T .$

Note that, in practice, the above estimates for the mean $\hat{r}$ and covariance matrix $\Sigma$ are very unreliable, and more sophisticated estimates should be used.

Then the mean value of the portfolio’s return $R(x)$ takes the form $\hat{R}(x)=\hat{r}^T x$ , and its variance is

$\sigma(x)^2:=\frac{1}{N} \sum_{i=1}^N\left(r_i^T x-\hat{r}^T x\right)^2=x^T \Sigma x .$

We can strike a trade-off between the ‘‘performance’’ of the portfolio, measured by the mean return, against the ‘‘risk’’, measured by the variance, via the optimization problem

$\min _x x^T \Sigma x: \hat{r}^T x=\mu,$

where $\mu$ is our target for the nominal return. Since $\Sigma$ is positive semi-definite, that is, it can be written as $\Sigma=A^T A$ with $A=\left(r_1-\hat{r}, \ldots, r_N-\hat{r}\right)$ , the above problem is a linearly constrained least-squares.

An ellipsoidal model

To model the uncertainty in $r$ , we can use the following deterministic model. We assume that the true vector $r$ lies in a given ellipsoid $\mathbf{E}$ , but is otherwise unknown. We describe $\mathcal{E}$ by its center hat{r} and a ‘‘shape matrix’’ determined by some invertible matrix $L$ :

$\mathbf{E}:=\left\{r=\hat{r}+L u:\|u\|_2 \leq 1\right\} .$

We observe that if $r \in \mathbf{E}$ , then $r^T x$ will be in an interval $\left[\alpha_{\min }, \alpha_{\max }\right]$ , with

$\alpha_{\min }=\min _{r \in \mathbf{E}} r^T x, \alpha_{\max }=\max _{r \in \mathbf{E}} r^T x .$

Using the Cauchy-Schwartz inequality, as well as the form of $\mathbf{E}$ given above, we obtain that

$\alpha_{\max }=\hat{r}^T x+\max _{u:\|u\|_2 \leq 1} u^T\left(L^T x\right)=\hat{r}^T x+\left\|L^T x\right\|_2 .$

Likewise,

$\alpha_{\min }=\hat{r}^T x-\left\|L^T x\right\|_2 .$

For a given portfolio vector $x$ , the true return $r^T x$ will lie in an interval $\left[\hat{r}^T x-\sigma(x), \hat{r}^T x+\sigma(x)\right]$ , where $\hat{r}^T x$ is our ‘‘nominal’’ return, and $\sigma(x)$ is a measure of the ‘‘risk’’ in the nominal return:

$\sigma(x)=\left\|L^T x\right\|_2 .$

We can formulate the problem of minimizing the risk subject to a constraint on the nominal return:

$\min _x x^T \Sigma x: \hat{r}^T x=\mu,$

where $\mu$ is our target for the nominal return, and $\Sigma:=L L^T$ . This is again a linearly constrained least-squares. Note that we obtain a problem that has exactly the same form as the stochastic model seen before.

Portfolio Optimization via Linearly Constrained Least-Squares

Mean-variance trade-off

An ellipsoidal model

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