**By Peter Hecht**

*Vice President, Senior Investment Strategist*

*Evanston Capital Management*

As a former finance professor, I experience heartburn every time I hear the investment community criticize hedge funds for not keeping up with the S&P 500. It is the same type of heartburn I felt when investors thought hedge funds demonstrated skill when they outperformed the S&P 500 in 2008. Both comments reflect a fundamental misunderstanding on how to evaluate hedge funds…or any new investment.

Starting with Markowitz in the 1950's, “mean-variance” modern portfolio theory (MV-MPT) has taught us that investments need to be evaluated in the context of the investor's entire portfolio. In a nutshell, investors want to know if having access to a new investment improves their entire portfolio's expected return for a given level of risk, i.e. improves the efficient frontier.

In practice, it is common for asset allocation specialists to discuss efficient frontiers with institutional investors. Separately, asset managers and other investment professionals report a full array of performance and risk statistics on various asset classes, strategies and specific investment products. However, it is typically not clear how the various performance and risk statistics relate to the investor’s ultimate goal – improving the efficient frontier. In fact, the long list of “jargon filled” statistics is overwhelming and confusing.

If improving the efficient frontier is the ultimate goal, is it easy for the investor to simultaneously account for and evaluate an investment’s expected return, risk and diversification attributes? Fortunately, the answer is yes. The little known “Appraisal Ratio” (AR), defined as the investment’s expected alpha-to-alpha volatility ratio – an alpha consistency measure – does just this. Within the set of new investments being considered, it can be shown that the investment with the highest AR improves the efficient frontier the most.

In order to calculate the AR (to be discussed later), an investment’s beta is necessary, which is why it is so crucial to beta-adjust returns when evaluating performance – whether it be for a hedge fund or any investment product.

**The Important, Yet Underutilized, Appraisal Ratio**

If maximizing the total portfolio’s Sharpe Ratio is the ultimate investor goal under MV-MPT, then the Appraisal Ratio is the relevant risk-adjusted performance statistic when evaluating new investments. In order to clearly define the AR, consider a simple linear regression of the new investment’s excess return (in excess of Tbills) on the existing total portfolio’s excess return.

The alpha is the intercept from the regression while beta is the slope. Epsilon is the regression residual – the part of the new investment’s excess return that can’t be explained by the regression model. Within this framework, the AR is defined as the alpha, **α** , divided by the residual volatility, **σ(ε)**.

When alpha is nonzero, the investor is not on the efficient frontier. In the case of positive alpha, more capital needs to be allocated to the positive alpha asset.

Intuitively, why isn’t it enough to just focus on the alpha when evaluating new investments? In order to answer this question, think of the new investment’s return as containing two parts: a unique part and one that is common to the existing portfolio.

The “common” part of the new investment’s return is irrelevant for improving the efficient frontier. It is already available to the investor via the existing portfolio. On the other hand, the unique part is *unrelated* to anything available in the existing portfolio and, thus, has potential to improve the efficient frontier. The unique part, however, contains more than just the alpha term. It also includes epsilon, the random regression residual. The epsilon makes the return’s unique part uncertain or risky, which is undesirable, all else equal. This is why the AR is divided by the residual volatility. The alpha represents the unique part’s expected excess return while the residual volatility represents the unique part’s risk. Based on this representation, I think of the return’s unique part as the *realized* alpha, where **α** is the expected alpha and **σ(ε) ** is the alpha volatility. In other words, the AR takes on a “return per unit risk” form, which is similar to a Sharpe Ratio but with a beta adjustment.

Mathematically, it can be shown that the maximum attainable total portfolio Sharpe Ratio is directly related to the new investment’s AR. For the purpose of this discussion, I will not go into the level of detail that fully proves this. However, this relation is why the AR is crucial for evaluating investments within MV-MPT. A higher AR translates to a higher total portfolio Sharpe Ratio, i.e. a more favorable efficient frontier.

At this point, it should be clear why beta-adjusting returns is critical to performance evaluation. In order to calculate the AR, the key performance measure in MV-MPT, returns need to be beta-adjusted. It really is that simple.

Although the Appraisal Ratio is critical to MV-MPT investment evaluation, it is not well known and, thus, not widely used. To be fair, I only know about it because George Constantinides, a Finance professor at the University of Chicago, included it in a problem set for his Asset Pricing Theory course – rather practical for a “theory” class. Clearly, the AR could benefit from a more effective marketing campaign.

In addition to being practical, the AR is easy to calculate. Since the inputs come from widely used linear regression techniques, it can be computed in many software packages, including Microsoft Excel, within seconds. No fancy optimization is needed. Given this, there is no reason to avoid using the AR as one relevant performance evaluation metric.

Before leaving this section, it is important to note that I don’t believe the world of applied portfolio theory is always about mean and variance. “Skewness” and “fat tails” can be important, especially for short volatility strategies. However, MV-MPT is a reasonable starting point in addition to providing many valuable, timeless portfolio management lessons.

**Common Performance Evaluation Mistakes & Misperceptions**

In order to complement a discussion dedicated to advocating one relevant performance evaluation metric, it feels appropriate to address a few of the commonly observed “don’t do’s”.

**1) Do not forget to deduct the risk free rate when estimating alphas.** It is common for me to come across analysis - even from vended performance software packages - that forgets to work with excess returns.

**2) Unless the investment has zero beta and/or the investor plans to allocate 100% to one investment, do not focus on the individual investment’s Sharpe Ratio.** The Sharpe Ratio is a total portfolio performance metric. Unlike the AR, it does not correctly control for the individual investment’s beta.

**3) Do not be seduced by high average alphas.** Similar to total portfolio returns, alphas need risk-adjusting too. The AR is a risk-adjusted alpha.

**4) Do not necessarily avoid investments with a high correlation to your existing portfolio.** A high (low) correlation translates to a low (high) residual volatility, which is a good (bad) attribute if the investment has a positive alpha. Remember, the AR is defined as the alpha divided by the residual volatility.

**5) Do not focus too much on individual expected returns.** Returns are risky, even over the long run. The expected return represents the “base case” from a distribution with many outcomes below and above. In order to construct a portfolio with the most favorable prospective return distribution, returns need to be appropriately risk-adjusted in the context of the entire portfolio. The AR does just this.

**6) Lastly, do not forget to beta-adjust returns when performing individual investment performance evaluation.** The AR is a key performance metric, and its components, require beta-adjusting.

Portfolio construction and performance evaluation are some of the key functions performed by an individual or institution responsible for an investment portfolio. Yet, these important tasks are rarely performed in a manner that is consistent with achieving the highest total portfolio expected return for a given level of total portfolio risk – the investor’s ultimate goal.

Focusing on individual returns does not correct for risk. Focusing on individual Sharpe Ratios does not correct for beta. Focusing on alpha controls for beta but doesn’t account for the alpha’s volatility (risk). Only the Appraisal Ratio, defined as *the expected alpha-to-alpha (residual) volatility ratio*, correctly accounts for an individual investment’s return attributes. In sum, the new investment with the highest AR delivers the highest total portfolio Sharpe Ratio. This is why the AR is the most relevant performance metric in a MV-MPT world.

**Peter Hecht** is a Vice President and Senior Investment Strategist at Evanston Capital Management, LLC. Prior to joining Evanston Capital Management, Mr. Hecht served in various portfolio manager and strategy roles for Allstate Corporation’s $35 billion property & casualty insurance portfolio and $4 billion pension plan. He also had the opportunity to chair Allstate’s Investment Strategy Committee, Global Strategy Team, and Performance Measurement Authority. Mr. Hecht also served as an Assistant Professor of Finance at Harvard Business School. His research and publications cover a variety of areas within finance, including behavioral and rational theories of asset pricing, liquidity, capital market efficiency, complex security valuation, credit risk, and asset allocation. Mr. Hecht previously served at investment banks J.P. Morgan and Hambrecht & Quist, and as a consultant for State Street Global Markets. Mr. Hecht has a bachelor’s degree in Economics and Engineering Sciences from Dartmouth College and an MBA and Ph.D. in Finance from the University of Chicago’s Booth School of Business.