Defining Pure Active Return and Risk What Should Managers Be Hired to Deliver

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We agree on the separation of market or policy risk and pure active risk, in principle. Let's go one step further and apply it in practice, separating these "gross" components of market risk (and return) and pure active risk (and return) in a real world context. The returns that usually pass for alphas - the simple differences between the benchmark return and the manager's return - are properly known just as "active returns" (without the "pure"). But these "simple" active returns might have more or less exposure to market risk than the amount implicit in the investor's benchmark.

Market risk can be measured by a CAPM single factor beta, or, more usefully, in a multifactor manner. The most intuitive of these multifactor approaches measures market risk in terms of style factors - the familiar large-capitalization, small capitalization, value, and growth categories.10

10 We aren't really sure who was first to classify investments or managers based on regression factors, but certainly Barr Rosenberg and his successors at BARRA have been the most complete at making this practice into a science. See www.barra.com/reseatrli/barrapub/risk-modeIs.asp for a good introductory level description. Sharpe (1988, 1992) saw that style and size factors could be productively and intuitively used in such regressions (called returns-based style analysis), and did much work in the area involving special types of regressions designed to make the output more intuitive to lay audiences. Sharpe's approach is certainly the most commonly used in practice. For determining which styles or common risk factors are most relevant for use in returns-based style analysis, one authoritative source is Fama and French (1993), drawing on the much earlier observation by Banz (1981) and Reinganum (1981) that small-cap stocks had historically outperformed larger-cap stocks, and the observation by Basu (1977, 1983), among others, that "value" stocks

Other, often more complex, factor models have been identified that more completely explain market risks, but some of these are hard to describe in plain English so we'll stick with style factors in this discussion. Style factors also have the benefit of convenient investability through low-cost style index funds. The manager's returns, then, can be explained in terms of the exposures, or betas, to an intuitive series of style factors that express the manager's return from market risks, plus a pure alpha, or the return generated over and above market returns. Simple active return and pure active return are only the same if the manager's factor exposures are weighted the same as they are in the benchmark. Since the benchmarks flow from the SAA decision, managers can and should be chosen with factor weights such that, at least in sum across the managers, they are consistent with the benchmark's factor weights.

So once a set of market factor weights - a custom benchmark such as "large-cap growth" or "80% large-cap value and 20% large-cap growth" - is set for a given manager, the proper objective of the manager is simply to beat that customized benchmark.

That's why you hire active managers. You hire them to give you the levels of market risk exposure that you expect from them and which you assigned to them through their customized benchmark, and to beat that benchmark. If a manager does anything else, with or without the knowledge of the investor, it is stealthily changing an important aspect of the investor's SAA policy. The common term for this is "misfit" risk, but it's really the risk that the particular mix of benchmarks representing one's SAA policy is not being delivered. In other words, the valuable and important return added by a manager isn't the total return that he or she delivers, but only that part of the return that is beyond what could be delivered through a set of index funds reflecting the manager's persistent style biases, its market risk exposures. This unique contribution of the manager to the return is what we're calling pure active return, or pure alpha. We know we're being redundant to say "pure" alpha, but we're trying to call attention to this precise definition, one that doesn't contain a market (or style) risk component, for the reasons stated in the prior section. It is also what Sharpe meant by his term "selection return" in his work on style analysis, and what those familiar with the very detailed factor models of BARRA know as "specific return."

Realized pure alpha is easily separated from market risk factors and is measured by regression analysis.11 The regression determines the effective style weights of the manager, or mix of style benchmarks that has the "best fit" to the manager's actual

(having low price/book, price/earnings, or other valuation ratios suggesting that the stocks were cheap) had outperformed "growth" stocks (having expensive valuations).

11 Realized pure alpha may be determined by an actual multivariate regression or by a constrained optimization technique that mimics a regression such as Sharpe's style analysis method. The purpose of the optimization technique is to allow for a no-shorting constraint, that is, to require all factor betas to be between zero and one. We generally prefer ordinary regression, for its greater ability to accurately describe "deep" value and growth managers. Pure active return, of course, is properly thought of after fees. One should also incorporate manager transition costs into the pure active return; these must be amortized over the time period for which the manager is likely to be returns. The pure alpha is then the residual, the manager's actual return in excess of the return on this amount of market risk. On a forward-looking basis, we assign to a manager his or her customized benchmark or "normal portfolio," capturing the style and other market risk exposures that will best describe that manager's neutral, or "home," position. It's easy for index funds or risk-controlled active managers.

For others, this customized benchmark might be informed by the historic regression and by any other information that is useful to characterizing the normal style biases of the manager. Even a TAA manager or a style rotator has such a "home" position. The view of market risk that we've been describing, by the way, is continuous and scalar - that is, a manager can have any amount of exposure to a single benchmark or to multiple style (or any other factor) benchmarks. The market exposure or style weight, at its essence, is just a beta, after all. And betas are a good way to determine or describe the level of exposure to any market risk.12

One additional idea: The investor may or may not be able to collect a portfolio of managers whose normal portfolios, in the aggregate, look like the benchmark. The misfit risk of a single manager goes away if it is cancelled by misfit risk of an opposing character from another manager (a growth manager is offset by a value manager, a large-cap manager with some small-cap exposure is offset by a small-cap manager with some large cap exposure, etc.). But such perfect offsets of style and other factors are not often the case. So when we're optimizing a manager structure, as we do below, we'll be optimizing pure active return against total active risk -and on the risk side, we'll control not only the pure active risk added up from each manager but also the net misfit risk taken across all managers, calculated properly using scalar values for the managers' exposures to all the market risk factors we are tracking. This net misfit risk is a part of the active risk investors actually face.

The standard deviation of the period-by-period pure alphas may be thought of as the pure active risk, representing the tracking error to the manager's customized benchmark.

These two parameters,

• Pure active return, or pure alpha, a

• Pure active risk, q (known as the Greek letter "omega"), can be combined to arrive at a single measure of manager achievement (either historical or expected), the:

• Pure information ratio, IR = a/a representing the amount of pure active return delivered (or expected) per unit of pure active risk taken (or expected) by an individual manager, relative to its customized benchmark.13

held, so that the cost (which is paid only when the manager is hired or fired) is properly converted into annualized return form.

12 The CAPM uses just a single factor to capture the market. We are simply trying to control risk relative to the asset allocation policy better, by dividing market risk into more granular subcomponents. Either way, regression is a useful model for sorting out the market and idiosyncratic components of risk and return.

13 In our experience, some investors don't find that the term "information ratio" conveys much intuition about its meaning. So, observing that IR measures the consistency with which the active

Across the portfolio of managers held by the investor, the denominator would be the aggregation of the q terms plus any net misfit risk remaining across the group of managers. We indicate this "simple" active risk as jA. One wishes for the misfit component to be zero across all managers, but in practice, it is difficult to make every last bit of misfit risk go away.

Now that we've defined pure active return and risk, we can use these measures (and particularly the ratio of return to risk, the pure information ratio) as well as misfit risk, to compare any manager with any other - across asset classes, styles, and risk levels - creating a level playing field for all managers. Even more importantly, we can use these measures to properly separate investment results that are the investor's responsibility from those that are created by the manager. The returns delivered by the capital markets on the particular mix of styles that constitute the manager's custom benchmark are the responsibility of the investor who selected the manager, if only because the investor is the only party in a position to control the market risk exposures across his or her whole portfolio of managers. Too often, performance evaluation practices confuse the benchmark return and the pure alpha, apportioning credit and blame incorrectly. Even the smartest and most well-intentioned investors are sorely tempted to blame the active manager, rather than themselves, when the manager's asset class delivers a poor policy return (no matter what pure alpha the manager achieved). With the pure active return and risk clearly defined and calculated, these errors need no longer occur. As a common example, think of the value managers that boast of beating the S&P even when they fail to beat the value benchmark. Which one should they really be held against? If managers persistently choose to exercise their expertise in one domain of market risk such as "deep value," isn't that the domain against which their value-added should be measured by clear-eyed investors?

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