Analytical portfolio optimization for VaR under multivariate normality

Question

Given a set of assets with returns following a multivariate normal distribution with a known mean vector and a known covariance matrix, $$ r \sim N(\mu,\Sigma), $$ I want to find optimal portfolio weights $w^*$ that maximize value at risk (VaR), subject to weights summing up to 1 and being nonnegative: $$ w^*:=\arg \max_{w} F^{-1}(q) \\ \text{subject to} \\ \quad w^{\top}e=1 \quad \text{and} \quad w>0 $$ where $F(\cdot)$ is the CDF of the porfolio return, $q$ is the quantile of interest, $e$ is a vector of ones and $>$ holds for each element of $w$. $F$ is a normal CDF with expectation $w^{\top}\mu$ and variance $w^{\top} \Sigma w$, the parameters being functions of $w$.

Does this problem have an analytical solution?

The elements of the mean vector are not necessarily equal; otherwise a solution is readily available and equals the well-known solution of minimization of portfolio variance.

Note: I do not care about the plausibility of multivariate normality for asset returns; I need this for testing and benchmarking a numerical optimization routine for VaR.

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Edit: updated to include formulas and to restrict the question to the case of nonnegative weights (in the original question the latter restriction was optional).

Answer 1

If you define VaR as a quantile of the portfolio's returns distribution about the mean of those returns, then the minimum-variance portfolio is what you look for: it minimises the volatility around the mean, whereever this mean is. When returns are jointly normal, maximising VaR as just defined does the same.

Regarding my second comment: numerical procedures for VaR typically directly maximise a given order statistic on a sample of portfolio returns. But then there is no guarantee that in the sample the optimal VaR-portfolio will be the same as the minimum-variance portfolio. Thus, for testing a numerical procedure, knowing the true distribution and generating data from it may not be enough. Example (using R):

I create 2000 scenarios for 10 assets; each may have a weight between 0 and 25%. The returns use ridiculously high mean returns, so that the distribution is shifted away from zero.

library("NMOF")        ## https://github.com/enricoschumann/NMOF
library("neighbours")  ## https://github.com/enricoschumann/neighbours

set.seed(32923)

## create random data
na <- 10              ## number of assets
ns <- 2000            ## number of scenarios
k <- ceiling(ns*0.1)  ## order statistic to maximise
R <- randomReturns(na = na, ns = ns,
                   mean = runif(na, 200/100/255, 300/100/255),
                   sd = runif(na, 0.005, 0.02),
                   rho = 0.6)


## minimum-variance solution
x.qp <- minvar(cov(R), wmin = 0, wmax = 0.25)

The VaR optimisation. The algorithm I use minimises, so I put a minus in front of the VaR.

### 1) objective function
of_var <- function(x, R, k, ...)
    -(sort(R %*% x, partial = k)[k] - sum(R %*% x)/nrow(R))

### 2) optimisation with Threshold Accepting
x.ta <- TAopt(
    OF = of_var,                    ## SETTINGS
    list(nI = 20000,                ### number of iterations
         neighbour = neighbourfun(  ### neighbourhood function
             min = 0,
             max = 0.25,
             stepsize = 1/100),
         x0 = rep(1/na, na)         ### initial solution: equal weights
         ),
    R = R, k = k)$xbest

We can compare the resulting portfolios and their returns distributions: the portfolios are very similar.

## compare weights
data.frame(MV  = round(100*x.qp, 2),
           VaR = round(100*x.ta, 2))

## compare returns distributions under given sample
plot(ecdf(R %*% rep(1/na, na)),
     main = "distribution of portfolio returns")
lines(ecdf(R %*% x.qp), col = "blue")
lines(ecdf(R %*% x.ta), col = "darkgreen")

Finally, I evaluate the VaR-objective function at the minimum-variance and at the minimum-VaR portfolio.

of_var(rep(1/na, na), R, k)  ## equal weight
## [1] 0.01316127
of_var(x.qp, R, k)           ## minimum variance
## [1] 0.008222392
of_var(x.ta, R, k)           ## minimum VaR
## [1] 0.008178637

As you can see, Threshold Accepting found a portfolio that provides a tiny advantage over the minimum-variance portfolio in this particular sample. (This could probably be improved, but it only serves to make the point here.) Under the true distribution, both portfolios should be the same.

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You may also define VaR as a quantile of the returns distribution, without the centering. You cannot generally maximise any quantile of the returns distribution. Just think of the median: if you can use leverage and the portfolio has a positive return, you can increase the median (which is the same as the mean under the Gaussian distribution) without bound. So you had better define VaR as a lower quantile. Maximising it is, in a Gaussian world, equivalent to maximising

$$\mu'x - \lambda \sqrt{x' \Sigma x}$$

in which $\lambda$ is an appropriate multiplier for the standard deviation, such as 1.645 for the 5% VaR:

> qnorm(0.05)
## [1] -1.644854

This model should not pose much difficulty to a numeric solver. (For the median, $\lambda$ would be zero.)