Important Mathematical Measures Of Investment Performance

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by: Tiger Technologies

Whether you invest in a stock portfolio managed by yourself or by some other manager, you want to measure the investment performance not by looking at just the returns, but by looking at the risk adjusted returns.

This is crucial, because it helps you determine:

1. Whether to pay alpha type fees or beta fees for the performance

2. If you can replicate the performance by taking lower risk and having more transparency

3. If you can sleep at night for generating those returns

Here are 3 important mathematical measures we use for measuring an investment's risk adjusted performance.

1. Fama-French Three-Factor Model1

Eugene Fama, Nobel Laureate and Professor of Finance at the University of Chicago Booth School of Business and Kenneth R. French, Professor of Finance at the Tuck School of Business at Dartmouth College attempted to explain the outperformance of certain stock market sub-sectors through the use of statistical methods.

They found that value stocks outperform growth stocks; similarly, small cap stocks tend to outperform large cap stocks. They arrived at a three factor model to explain this outperformance. But the model can also be used to measure the alpha produced by any stock portfolio.

Please see the complete article for the math behind the model.

Forgetting the math for a second, the important thing to remember about the Fama-French three factor model is that it tells you if you can replicate, for example, a long/short stock portfolio being offered by a hedge fund manager with 3 beta portfolios. If the hedge fund manager is not producing any alpha after fees compared to the Fama French portfolio, then you are paying for beta and not alpha. This test can be done for any stock portfolio to test for beta replication.

2. Volatility Profiling

Once you as an investor have determined that a stock portfolio does produce alpha, then comes the question on how to rank the various managers offering these portfolios. Ranking portfolios bases on 1 month or 1 year returns is fool hardy. It tells you nothing about the robustness of the portfolio. Robustness is important as it will tell you if you can sleep at night knowing that your money is being managed safely regardless of the market environment we are in. Whether it is the calm times of 2013 or the rough ride of 2008, you should be able to rest at ease.

Volatility profiling breaks down the S&P 500 returns monthly by market environment. At a fundamental level, there are three types of market environments:

1. Trending Market

This market is defined as one that is made up of a series of smaller periods with a relatively constant rate of return, such that the instantaneous volatility of returns as measured per smaller time slice is close to zero. For our discussion, we have also defined a trending market as a market in a bull phase.

2. Shock Market

This market is one that is made up of a series of smaller periods with high volatility and low volatility. For our discussion, we also define a shock market as a market in a bear phase.

3. Noise Market

This market is one that is made up of a series of smaller periods with a relatively constant volatility. A noise market bears no trend.

Here is an example of how we classified the S&P 500 months by volatility profile.

Month&Year Volatility Period

Source: MA Capital Management

Now you as an investor can take your stock portfolio's returns and break out its performance by volatility profile to see how it performed in the 3 different market environments. This should give you a much better idea of the portfolio's robustness.

As an example, this is how the various asset classes do under the 3 volatility environments:

Source: Barclay Hedge, HFR, Yahoo! Finance, MA Capital Management

S&P is the S&P 500 Index, BONDS are 30 year US Treasuries

3. BTOP 50, Barclay Hedge CTA Index, as a proxy for managed futures

4. HFR Global Hedge Fund Index and the Equity Hedge Index as a proxy for hedge fund returns. (Note: data prior to 1998 for the HFR Global Hedge Fund Index was not available).

3. MAR Ratio Versus Sharpe/Sortino Ratios

Smart investors look at risk adjusted returns and not just returns. To measure risk adjusted returns, investors generally use Sharpe or Sortino ratios. We prefer the MAR ratio. Here is what these 3 different ratios measure:

1. Sharpe Ratio: Sharpe ratio is a measure of a portfolio's return less the risk free rate divided by its volatility.

2. Sortino Ratio: Sortino ratio is a measure of a portfolio's return less the risk free rate divided by its downside volatility. The main difference between the Sharpe and Sortino ratios is that the Sortino ratio does not penalize the portfolio for outsized positive returns as it only takes the volatility of its negative returns.

3. MAR Ratio: MAR ratio is a measure of a portfolio's return divided by its worst loss.

The reason why we like the MAR ratio is because it mimics human psychology the best. Investors care less about volatility as it is measured for the purpose of Sharpe or Sortino ratios and more about loss. Loss is real pain even if it is just a paper loss and not a realized loss. Loss tests an investor's ability to stay in the investment, therefore as investment managers we ensure that the loss is minimized, even if it takes away from the returns.

Here is an example that compares MAR ratio with Sharpe or Sortino ratios to show its effectiveness. Consider 3 portfolios, A, B and C over a period of 24 months as shown below. All 3 portfolios start at an initial value of \$100 and have different returns and volatilities.

Portfolio A has average monthly returns of 0.34%, annualized volatility of 0.50% and a Sharpe ratio of 68% (assuming a risk free rate of 0).

Portfolio B has average monthly returns of 0.80%, annualized volatility of 1.36% and a Sharpe ratio of 59%.

Portfolio C has average monthly returns of 1.18%, annualized volatility of 2.74% and a Sharpe ratio of 43%.

Sharpe ratio will tell you that portfolio A is the best and portfolio C is the worst. But one look at the profiles will tell me that we would rather be in portfolio C than portfolio A.

Next, we also gave all 3 portfolios a small loss period over 2 months that totaled 0.20%. The MAR ratio tells us exactly the opposite story. It tells us that with a MAR ratio of 71, portfolio C is the best and with a MAR ratio of 21, portfolio A is the worst.

There are several other measures of risk adjusted returns. When looking at any alpha portfolio like a hedge fund or a CTA that trades in esoteric markets, the risk adjusted returns become even more important. The investor has to account for currency risk, foreign market beta, correlations between the betas, illiquidity premiums, etc. All of these factors add to the portfolio risk and it becomes that much more important to test the viability of alpha in those strategies.

[1] The Journal of Finance, Vol. 25. No. 2, May 1970, Efficient Capital Markets, Eugene F. Fama

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.