Variance with complete covariance information
Bounding variance with incomplete covariance information
Consider (see here for more background) a portfolio of assets, which is described by a vector , with the amount (in, say, US dollars) invested in the -th asset. We are interested in defining, and quantifying the risk associated with holding the position (as described by ) over a certain time period into the future.
We can define the vector , with the rate of return of the -th asset at the end of the period considered. The return of the portfolio is .
Of course, is unknown. Let us model this uncertainty by assuming that is a random variable. Then, the return is also a random variable. We define the risk of the portfolio , and denote by , the variance of its return . That is:
where is the expected value of the portfolio's return, and denotes the expectation operator with respect to the distribution of the random variable .
Evaluating the risk is useful to compare different portfolios. Some might have a higher expected return than others; this usually comes at the expense of higher values of risk. The risk defined above suffers from several practical limitations, and we explore one of them here.
With , and not random (we just take a position at the beginning of the investment period, and hold it until the end), we have
where is the covariance matrix of the vector of returns.
If this matrix is known, the risk (as we defined it) is easily computed via the matrix-vector product. However, in practice, the covariance matrix is hard to estimate precisely.
Assume now that the covariance matrix is only partially specified. For example, in a case with assets, we might describe our uncertainty as
where the symbol denotes that we believe that the corresponding assets (in this case, assets and ) are positively correlated, while denotes negative correlation. Here, the symbol “” denotes complete uncertainty on the sign of the correlation. The above covariance information is only partial, with only the diagonal elements (variance of each asset) given hard numbers. Denote by the set of symmetric matrices that satisfy the above pattern. This set is a polytope, which we will denote , in the space of symmetric matrices, since it is defined as the following ordinary inequalities:
Of course, any symmetric matrix in the set is not necessarily a covariance matrix. In order for it to be a covariance matrix, it has to be positive semi-definite.
We define the worst-case risk as the largest variance of the portfolio that can be attained by some covariance matrix:
The above is an SDP in (matrix) variable . It is implementable in CVX as follows.CVX syntax
vars = [0.1; 0.03; 0.6]; % given asset variances cvx_begin variable S(n,n) symmetric maximize( x'*S*x ) subject to diag(S) == vars; S(1,2) >= 0; S(1,3) <= 0; S == semidefinite(n); cvx_end
Consider for example the case when the portfolio is given by the vector . The maximum portfolio variance is . We can compare this number with the best (that is, smallest) portfolio achievable. Replacing the maximization by minimization (which is permitted in SDP, as the objective is linear), we obtain the most optimistic value of . Thus, under our uncertainty model for the covariance, the portfolio variance can change in relative amount by the staggering amount of