Evaluating Minimum-Variance Portfolios with Future Return Data
Summary
The document asks how to benchmark covariance forecasting methods used to build minimum-variance portfolios. The strategy described estimates covariance from historical daily returns, forms portfolio weights, and rebalances weekly. For a perfect-information comparison, one answer proposes using the daily returns observed during each future investment horizon to compute a covariance matrix, then calculating weights and repeating for each period. Forecast methods and optimizer choices can then be compared against this benchmark.
A second answer cautions that perfect-information benchmarks may rest on problematic statistical assumptions, invoking results about explosive stochastic models and estimators that lack conventional limiting moments. It argues that a Bayesian approach does not restore a covariance matrix in that setting. These answers do not reconcile their assumptions, and the warning concerns a particular class of models rather than establishing that ordinary realized-covariance benchmarks are universally invalid. The choice of return horizon and the benchmark’s interpretation therefore require care.
Key ideas
- A proposed perfect-information benchmark estimates covariance from daily returns realized over each future investment horizon.
- Portfolio weights can be formed from that future covariance estimate and recomputed for each period.
- Historical covariance forecasts and optimizer choices can be compared against the benchmark.
- The document also presents a model-specific warning that some statistical settings lack conventional estimators or covariance matrices.
- The answers rely on different assumptions and do not resolve how broadly the warning applies.
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Full text
# How to evaluate minimum-variance strategies against perfect information mvp
# 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?
## Answer by Dave Harris (score 1)
https://quant.stackexchange.com/a/39082
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.
## Answer by Kyle Balkissoon (score 0)
https://quant.stackexchange.com/a/19286
Here is a simple way of solving the above:
- Take future returns (use the daily return matrix of all the assets for the future investment horizon), calculate covariance matrix.
- Calculate weights using future perfect data.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.