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Developing a Mean-Variance Efficient Stock Portfolio

The art of professional portfolio management experienced a revolution after the concepts in Harry Markowitz’s seminal paper explained how investors view risk and return. In his work, he derived formulas used to calculate volatility and expected return of a diversified portfolio with the ultimate goal of his to find the best portfolio allocation to maximize return for a given level of risk.


His work is based on three assumptions: 1) Investors are generally risk-averse, 2) investors base their portfolio decisions on risk and expected return only, and 3) investors measure risk as the variance (or standard deviation) of expected returns. The third assumption led Markowitz to conceive the idea of a portfolio being “mean-variance” efficient when no other portfolio offers a higher expected return at a given level of risk, or lower level of risk for a given expected return. Creating portfolios that have higher mean-variance efficiency depends on certain statistical characteristics of the returns of the assets held in a portfolio.

First, an arithmetic mean of historical returns is used to calculate the expected value of portfolio returns. This mean can be used to provide a reasonable forward-looking expected return for the next time period of the same magnitude. As Markowitz mentions, investors will seek out the highest returns. However, based on Markowitz’s work higher expected return is accompanied by a commensurate increase in the level of total risk.  

Using Markowitz’s framework, the total risk of a portfolio equals its volatility (i.e. standard deviation) of past returns. Even though standard deviation assumes a normal distribution of data which is not an accurate measurement of financial time series, the risk quantified describes how far outcomes are likely to be from the distribution’s mean, or expected value in this case. In other words, the calculation of standard deviation (i.e. square root of the variance of returns) allows for investors to see how standard deviation measures the average deviation from the series expected value and can be used to interpret the expectation for future volatility in return streams.

After identifying that total risk can be measured with the standard deviation of past returns, investors need to address the two types of portfolio volatility. The total risk referenced above is comprised of market and firm-specific risk, also known as undiversifiable and diversifiable risk, respectively. In other words, firm-specific or unsystematic risk can be diversified away in a portfolio but market or systematic risk cannot be removed entirely.

From a statistical perspective, both macroeconomic and firm-specific risk can be defined using two calculations. As mentioned above, standard deviation of a component’s returns will estimate the range of returns for a security over a specified period of time. This calculation is the first portfolio risk metric.

Second, investors need to calculate the systematic risk that reflects how individual stock returns in a portfolio move together and respond simultaneously to macroeconomic news. The statistical term used for this calculation is called covariance. By contrasting variance (i.e. the value for which the standard deviation calculates a square root) and covariance we can better understand the fundamental differences between these two components of portfolio risk.

For a two stock portfolio, variance and covariance formulas differ in that when calculating covariance, instead of multiplying the first stock’s deviation from its mean in each period by itself (i.e. squaring the quantity) we multiply it by the second stock’s deviation from its mean. In other words, variance measures how one stock varies around its own mean and covariance assesses how two stocks vary around their respective means.

Readers who have a background in statistics may have already foreseen how based on the calculations mentioned above diversification reduces only firm-specific risk. That is, as the number of holdings in a portfolio increases, the individual firm-specific variance terms will increase linearly, but the covariance terms will increase exponentially. With more holdings in a portfolio the more macroeconomic risk dominates the portfolio volatility.

Studies have consistently shown that in most cases once a portfolio contains 15-30 stocks only the effect of covariance matters. However, the optimal number of holdings will depend on the correlations and covariance between each pair of stocks in the portfolio. In other words, the lower the correlation and covariance across holdings the greater the opportunity to reduce portfolio volatility even when holding fewer positions.

Key Takeaways:

  • Basic statistics (e.g. arithmetic mean, standard deviation, covariance, and correlation) provide the foundation for understanding portfolio management.
  • In modern finance, total risk (i.e. volatility) is explained with the standard deviation of returns
  • This total risk is comprised of macroeconomic and firm-specific risk
  • Only firm-specific risk can be reduced through diversification
  • As a portfolio increases its number of holdings, the covariance (macro risk) effects eventually dominate the portfolio volatility calculation.
  • Creating a portfolio diversified among stocks with low return correlations improves the portfolio’s mean-variance efficiency.
Summary adapted from Robert Weigand’s Applied Equity Analysis and Portfolio Management.

As always, please feel free to contact me with any comments or questions. Thanks for reading.


John
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