What are the difficulties of implementing the Markowitz mean-variance framework in constructing portfolios?

Question

What will be an ideal response?

Answer 1

The implementation for constructing portfolios requires the estimation of the inputs (mean, variance, and covariance) for all of the securities that are candidates for inclusion in the portfolio. These inputs are not easily estimated and this presents a variety of difficulties. For example, the number of inputs that must be calculated isenormous. For example, if there are N securities that can be included in a portfolio, there are N variances and (N2– N)/2 covariances to estimate. Hence, for a portfolio of just 50 securities that could be included in a portfolio, there are 1,224 covariances that must be calculated. For 100 securities, there are 4,950 covariances.

Holding aside estimation risk, the enormity of the estimations that must be made was clear to Markowitz. It was clear to Markowitz that some kind of model of covariance structure was needed for the practical implementation of the theory to large portfolios. He did little more than point out the problem and suggest some possible models of covariance.

The use of portfolio variance as a risk measure presents additional difficulties. First, we have to make an assumption about the return distribution. If we assume normal distribution of returns, then the variance is the appropriate measure of risk. However, empirical and theoretical evidence suggests that stock returns and bond returns are not normally distribution. As a result, extensions of the Markowitz optimization framework have been suggested that include other risk measures such skewness and kurtosis. Second, we use of portfolio variance is questioned in regards to the objective of portfolio managers: outperforming a benchmark. The measure used with this objective is a portfolio's tracking error. This measure is the standard deviation or variance of the difference between the portfolio return and the benchmark return. The key point is that in constructing a portfolio where there is a benchmark, the relevant risk measure is not the portfolio variance but the portfolio tracking error.

Finally, consider the notion of the decomposition of portfolio total risk (i.e., portfolio variance) into systematic risk and idiosyncratic risk. Studies of the stock market indicate that it does not take more than 25 or so randomly selected stocks to remove most of the idiosyncratic risk of a portfolio. That is, a randomly selected portfolio of stocks is mostly exposed to systematic risk. However, when risk is measured in terms of tracking error, it takes a considerably larger number of stocks to remove idiosyncratic risk. Typically, this is not the case when dealing with bonds where the benchmark is one of the standard bond market indexes.