Mean Variance Optimization Part I
Measuring Aggregate Risk
In the 1950s, Professor Harry Markowitz of the City University of New York developed an approach to investment analysis that has become known as Modern Portfolio Theory (MPT). Instead of traditional asset management using just fundamental or technical analysis, his system looks at the performance of a portfolio of assets based on the combination of its components' risk and return. His hypothesis and subsequent work were so revolutionary that Markowitz was a joint Nobel Laureate for economics in 1990. Markowitz’ “Efficiency Frontier” - and the resulting Capital Asset Pricing Model (CAPM) - made it possible to determine whether or not a portfolio is optimal in terms of its risk/reward characteristics. Markowitz referred to this determination as Mean Variance Optimization (MVO).
Covariance and MVO
The objective for MVO is clear, however the method of measuring MVO is yet undeveloped. The purpose of this three-part Risk Manager series on MVO is to offer dialogue about how portfolio-level (“aggregate”) variance can be managed, optimized, and ultimately measured. First, it is essential to understand the meaning of aggregate as: the entirety of a household or an institution’s investable asset base. Second, the role of covariance among asset classes (e.g., emerging market stocks, domestic high-quality bonds, U.S. small cap stocks) in any potential portfolio must be a first priority so the investor can better achieve lower covariance between holdings – that is, hold low or negatively-correlated asset classes that “offset” the variance (i.e., total risk) in other asset classes. Whatever the method of determining the desired covariance among asset classes, including this step is essential prior to beginning portfolio construction.
Portfolio Construction and MVO
If the overriding goal of portfolio construction is to carry out the proper allocation to meet the risk/reward expectations of the investor, and the proper asset classes have been predetermined, then only 2 questions remain: 1) should the investor generally index, or pursue an actively-managed approach to each asset class?, and 2) if actively managed, who should manage those assets? These questions will receive added attention in the months ahead. For now, we know anecdotally that a well-constructed portfolio (with relatively low covariance among asset classes), should render a risk-adjusted result that is preferable to the alternative: an asset base that is not built with optimization in mind. A primary example of this simple truth is visible in a review of a portfolio’s aggregate Sharpe Ratio. Developed by William Sharpe in 1966, a low (or negative) Sharpe Ratio – net of fees - indicates that a portfolio is performing below expectations on a risk-adjusted basis, while a positive Sharpe Ratio – net of fees – indicates outperformance.
Recognizing the value of aggregate Sharpe Ratio as the first step to developing an MVO measurement model. A Sharpe Ratio does not compare to an index and therefore does not risk irrelevance when comparing two sets of holdings. It simply answers the question: did the investor receive any excess reward for the risk taken? To develop a MVO scoring mechanism from Sharpe Ratio analysis involves two points: 1) recognizing that William Sharpe did not create his formula for portfolio-level analysis, but for security-level analysis, and 2) that Sharpe Ratios can serve to measure whether an investor may perform better or worse than the relevant marketplace(s) involved. This “market return” can be measured according to the Capital Market Line (CML) as follows:
And how would the possibility of risk-adjusted outperformance according to Sharpe Ratios be determined? By comparing the Sharpe Ratio of the portfolio to that of the market!
Measuring Aggregate MVO
It follows from a portfolio-to-market Sharpe Ratio comparison, that any formula for measuring Aggregate MVO should quantify the success or failure of achieving this risk-adjusted (net of fees) outperformance. Further, the MVO measurement should attribute any outperformance to the asset classes involved. Finally, the measurement should attribute any outperformance to the holdings within each asset class. This final point is the crux of long-standing debate regarding active vs. passive (index) investing and the subject of the April edition of The Risk Manager.
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