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 \mathbb{R}^n the vector containing the rates of return of each asset. A portfolio corresponds to a vector x \in \mathbb{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, 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.

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