Estimating the Covariance Matrix for Portfolio Optimization Methodology

For evaluating the performance of the estimators included in our "horse race," one can compare the estimators computed over a certain sample period (the in-sample period) with the covariance matrix realized over a subsequent period (the out-of-sample period).

However, our main interest is to assess how the performance of the estimators translates into the performance of the optimal portfolios obtained from the MV optimization process. Therefore, we find it more useful to conduct an empirical performance contest that focuses on the out-of-sample performance of the respective optimal portfolios. We follow Chan et al. (1999), Bengtsson and Hoist (2002), Jagannathan and Ma (2003) and Ledoit and Wolf (2003) and use the ex-post GMVP as our betterment criterion. We benefit from the fact that finding the GMVP does not require the estimation of the expected stock returns vector, which is out of the scope of our paper.'

In our performance contest we mimic an investor having the following investment strategy

1. The investor is only interested in stocks traded on the NYSE.

2. He only cares about minimizing the risk of his investment and therefore invests in the GMVP.

3. He chooses a method to estimate the covariance matrix based on historical data of stock returns. He chooses the length of the backward period, in which he collects monthly return data on stocks traded on the NYSE (the in-sample-period). Based on this data he computes the estimator of the covariance matrix, finds the GMVP and invests in this portfolio.

4. He has a certain investment horizon. Namely, he keeps his portfolio unchanged for a certain period (the out-of-sample period). When this period is over, he liquidates the portfolio. Then, he starts the whole process of estimating the covariance matrix, constructing the GMVP and holding it until liquidation all over again.

To illustrate the way we conduct our performance contest, let us assume that the first time ,ur investor wishes to invest is January 1974, and that he chooses an in-sample period of 120 nonths and an out-of-sample period of 12 months. Therefore:

1. We collect monthly return data of stocks traded on the NYSE from 1/64 till 1 2 73. We choose an estimator of the covariance matrix, which is computed based on this data.

2. We construct the GMVP from the estimator computed in phase 2.

3. We record the monthly returns on the GMVP From 1/74 till 12/74.

4. We start the whole process all over again. Namely, we collect monthly return data of stocks traded on the NYSE from 1/65 till 12/74, based on this data we compute the same estimator used in phase 1, construct the GMVP and record its monthly returns from 1/75 till 12/75 and so on...

5. We repeat the process of computing the covariance matrix, constructing the GMVP and recording its monthly returns in the out-of-sample period 30 times (the last monthly return recorded is from 12/2003). As a result, all together, we collect 360 monthly returns (from 1/74 till 12/2003) on the GMVP.

6. We compute the standard deviation of the collected 360 monthly returns. This standard deviation represents the risk our investor was exposed to in the 30 years he was running his investment strategy. Given the chosen in-sample and out-of¬sample periods, we can refer to the computed standard deviation as a proxy of the performance of the specific estimator used for estimating the covariance matrix. We conduct our test for seven different estimators. Since the motivation of our investor is to minimize the risk of his investment, the smaller the standard deviation of the collected 360 monthly returns, the better the respective estimator of the covariance matrix.

We run our "horse race" six times, each time changing the length of the in-sample period or the length of the out-of-sample period. We use in-sample periods of 120 months (also used in Ledoit and Wolf [2003]) and 60 months (also used in Chan et al. [1999] and Jagannathan and Ma [2003]).' We use out-of-sample periods of 12 months (also used in Chan et al. [1999], Jagannathan and Ma [2003] and Ledoit and Wolf [2003]), 24 months and 36 months. We chose these three out-of-sample periods, since we believe they correspond to realistic investment horizons (see also Chan et al. [1999]). As an aside, we always construct the first GMVP on 1/74 13 Jobson and Korkie (1981) mention rules of thumb regarding the length of the in-sample period of 4 to 7 years and 8 to 10 years and record the last return data on the last GMVP on 12:'03. Thus, in each of the six runs for every one of the seven estimators participating in our contest, we have a set of 360 monthly returns used to compute the respective standard deviation.

It is also worth mentioning that each time we construct a GMVP, we construct it only out of NYSE stocks whose returns cover the entire in-sample and out-of-sample periods used. For example, in the case of in-sample period of 120 months and out-of sample period of 12 months, for constructing the GMVP of 1/74, we only use NYSE stocks with monthly return data for all the 132 months from 1/64 till 12/74. For constructing the GMVP of 1/75, we only use NYSE stocks with monthly return data for all the 132 months from 1/65 till 12/75 and so on. Therefore, the resulting number of stocks used for constructing the GMVP varies across the years and the runs of the contest (see also Bengtsson and Holst [2002]).'

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