# How to evaluate minimum-variance strategies against perfect information mvp

The minimum variance portfolio should minimize the standard deviation (variance) of the portfolio at time $t+1$. The covariance matrix $\Sigma_{t+1}$ needs to be estimated in order to form the mvp.

I have a list of forecast methods of the covariance matrix and I would like to evaluate their performance. My current strategy consists of estimating the covariance matrix at the end of each week using the past 2 year daily returns, forming the MVP and rebalancing it at the end of each week.

Between-method's evaluation is trivial since i just compare performance statistics. However, I would also like to know what the performance statistics are under PERFECT information. How would i go about calculating the covariance matrices for each time period and forming the true (proxied) optimal portfolio.

Would I simply be forming a covariance matrix using daily returns of the entire week or would I be taking the single week return vector and forming the covariance based on that? e.g $\Sigma_{t+1} = (r_{t+1} - \mu)(r_{t+1} - \mu)'$ where $r_{t+1}$ is a vector of the week's return (under out-of-sample, it is unknown, for perfect information, I know this). That brings to the question of $\mu$, how would i calculate that under perfect information? Should I be using another measure such as the realized variance by Merton?

Just to save you some time, there is a non-existence proof for this class of problems. The models assume perfect information, what has been missed is that there are no estimators that converge to the population parameter under incomplete information.

Consider the static model equation $\tilde{w}=R\bar{w}+\epsilon,R>1$. The maximum likelihood estimator for $R$ for any distribution of $\epsilon$ centered on zero with finite, positive variance is the least squared estimator. The test distribution for $\hat{R}-R$ is the Cauchy distribution. The least squared estimator is a variant of the mean. The Cauchy distribution has no population mean. Only the zeroeth moment is defined.

See

Mann, H. and Wald, A. (1943) On the Statistical Treatment of Linear Stochastic Difference Equations. Econometrica, 11, 173-200.

and

White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197.

For an extended discussion, you can see

Harris, D.E. (2017) The Distribution of Returns. Journal of Mathematical Finance, 7, 769-804.

There is a Bayesian solution, but it doesn't create a covariance matrix. The distributions involved lack a covariance matrix, even in log form. I believe White's proof was missed because a non-existence proof generates no literature.

Here is a simple way of solving the above:

1. Take future returns (use the daily return matrix of all the assets for the future investment horizon), calculate covariance matrix.
2. Calculate weights using future perfect data.
3. Repeat for all time periods.

Pick whichever method of estimating the covariance matrix you prefer the most and plug it into 1), likewise for optimizer choice.

• if my investment horizon is one week, should my covar be calculated from daily returns of that week or should i calculate it from a week return (1xN vector of return) Aug 13 '15 at 17:36
• Daily, or the highest frequency you have to best estimate the covariance. You could also use the weekly returns but mathematically speaking you will run into issues due to the lack of observation. Aug 13 '15 at 17:51
• Would I be using some form of realized covariance where the mean is set to zero? Aug 13 '15 at 18:54
• Depends on what you're trying to capture and your actual research question. IMO you would do as you would normally except use the future data. Aug 14 '15 at 4:16
• Imho the term "minimum variance portfolio under perfect information" needs clarifying. If perfect information means knowing all future returns, than all random variables become deterministic and all portfolios have a variance of zero. If perfect information means knowing the true mean and variance of your random variables than you don't really have that and it doesn't seem clear to me, that using the next weeks return to estimate them is better than any other estimate. Aug 7 '16 at 20:36