The search for alpha is an often heard refrain in investment circles. Unfortunately, many of those seeking alpha—or attempting to deliver alpha—have only a nebulous idea of what alpha is. First, then, let’s attempt to define exactly what this performance measure means. Then we’ll take a stab at how alpha is best used to evaluate particular investors, investment producets, or investment managers.

Wikipedia’s entry is a good place to start and defines alpha as “… a risk-adjusted measure of the so-called active return on an investment… Often the return of a benchmark is substracted in order to consider relative performance, which yields Jensen’s alpha.” When people talk about alpha, they usually mean performance relative to a benchmark,* i.e.*, Jensen’s alpha, and that’s the approach we’ll take here.

#### The Mathematics of Alpha

At time *t* for an investment with return against a benchmark with return the estimate of alpha is based on the equation:

The term is called the exposure of investment to benchmark , and is a mean-zero noise or error term that is idiosyncratic to .

It is important to adjust the returns by subtracting the risk-free rate. There are technical reasons why this is necessary, but the intuitive idea is that the asset’s return over the risk-free rate gives us a picture of the additional return we receive from taking the risks associated with the investment. Failing to subtract the risk-free rate as shown causes significant distortions in and is the most common mistake made when estimating it.

Thus, the alpha we get is dependent on the benchmark we choose. It is a relative measure that can be estimated statistically but is *never* observed directly.

#### More Complex Models for Alpha

In the cases above, we modeled the returns of an investment as a linear equation based on a particular benchmark. A natural question is how appropriate is a given benchmark as the basis for building the return model for a given investment? One response is to use multiple benchmarks, or factors, to achieve a more complete model.

For such a multi-factor model the equation is

where is the return of factor and is the exposure of investment to factor .

#### The Challenges of Model Development and Estimation

There are significant challenges to building a multi-factor model. Selecting the factors may be done by collecting a series of representative indices or other financial data; however, fitting such factor models must deal with the inevitable correlations which exist across benchmarks. Uncorrelated factors can be derived by projecting the returns of a collection of representative investments onto a lower dimensional subspace; however, this means that the factors are also not observable and are estimated simultaneously with the parameters.

The parameters of the model are almost certainly time-dependent. Thus, the factor model becomes:

Financial returns are heteroskedastic. This demands that we also estimate time-dependent variances for the model data. (See “All Correlations Tend to One…”, http://wp.me/p3uMW7-2v)

These technical challenges can be dealt with, but the task isn’t easy. Certainly, the naïve approach of collecting a few indices and throwing the data into an off-the-shelf regression program isn’t going to cut it. Unfortunately, it is all too often exactly what is done.

#### Alpha, the “Average” Investor, and Market Efficiency

Consider the case where we are using some broad, capital-weighted market index as our benchmark and the target investment is one whose potential components are the same as those that constitute the benchmark. The capital-weighted gross returns of the portfolios of all such investors *is* the benchmark. Therefore, the capital-weighted mean of all alphas will be zero and of all betas will be one.

If, however, we look at the net returns, *i.e.*, returns net of costs such as fees, then the capital-weighted mean alpha *must be below zero*, because net performance includes costs that are absent from the benchmark or factors. Remember, alpha is a *relative* measure. If the investment management industry magically experienced a universal rise in competence and allocated capital more efficiently, then the mean alpha would still be slightly below zero.

Arguing that the investment management industry is not doing its job because of that fact is like arguing that American education is failing because half of the students are below average. There certainly may be problems in American education, but any argument for or against that conclusion that is based on that sort of observation is vacuous.

Similarly, pointing out that alpha is approximately zero says nothing about whether markets are efficient or not. Market efficiency means something completely different. Efficient or not efficient, alpha has nothing to say about it.

#### Forecasting Alpha

Forecasting what alpha we can expect from an investment going forward introduces further complications. The obvious but naïve choice of using the “best fit” estimates as a forecast is not a good idea. Again, there are technical issues why this is so, but we’ll rely on a simple intuitive argument despite the fact that it may make a few of my mathematical colleagues cringe.

Consider the observed alpha of an investment. We can think of this observation as being roughly composed of two parts: skill (model) and luck (noise). Even under the strong assumption that our model is in some fashion complete, the forecast return of the investment should only include the skill component because the future expected value of the luck component is zero.

The luck, although unobserved, has an expected value of zero, so it may seem that the best forecast is still just the model fit. However, if we examine the cross section of alphas across investments, then it is likely that further from the average an estimate is, the greater the likely impact of luck on that realization.

A reasonable approach, then, might be to produce a forecast that is shrunk back towards the mean. For example, we could take a convex combination of the alpha estimated by the model fit, , and the mean alpha, :

The average alpha is near zero. This means that the best forecast is one in which we have shrunk the magnitude of the estimated alpha. Given the other complications we discussed, determining the optimal shrinkage is just one more difficulty we must face.

This says nothing about the fact that things change. Managers are replaced. The competitive landscape evolves. Thus, the relevance of the data we used in our estimation may be in question.

#### Non-Linear Models, Neural Networks, …

You might remark at this point that a linear model has limitations and a more sophisticated approach would provide more insight. No arguments here. More sophisticated models do nothing, however, to address the concerns discussed above and often exacerbate them. For our purposes here, they change nothing.

#### Is Estimating Alpha Worthwhile?

So where are we so far? Alpha is a relative, not absolute, measure. There is not “an” alpha; there are a multitude of alphas dependent upon your modeling choices. Alpha is not directly observable; it can only be estimated from the data we have available to us. Those estimates are not only subject to the usual statisitical uncertainties but are also affected by the fact that the behavior of investments change over time. Even if we get all of that right, the mechanics of actually performing the estimation present formidable challenges. Alpha has definite limitations in what it tells us and many read far too much into it. Finally, even when we have done a solid job of estimating historic alpha, there are additional issues we must face in producing a forecast, which is ultimately what we need to do if we wish to use alpha to help us guide our investment decisions.

So, is it worthwhile estimating alpha? Sure if you can be confident that you have something that is remotely usable. If your approach to alpha is to pick a single benchmark, dump data into a spreadsheet, and hit them with an off-the-shelf OLS routine, then the answer still may be “yes”, but at best that’s a highly qualified “yes”. You must understand what you have done and be certain to take that measure with more than a grain of salt. You must also be certain that those who will use ths estimate to make decisions understand its limitations.

*In practice, however, the answer is probably “no”, except perhaps as a side-effect of an analysis performed with different objectives.*

So if alpha is not the magic number that allows us to evaluate the performance of an investment advisor or manager, then what do we do? More on that in later posts.

Looking forward to more posts on this!

Small correction: all the “m(t)” terms should be “r_m(t)” instead

The “m(t)” as representing the return of a benchmark was an arbitrary choice. It wasn’t a typo. Thanks for reading and commenting.

Robert –

Great post, of course, but I do take issue with this one point:

Arguing that the investment management industry is not doing its job because of that fact is like arguing that American education is failing because half of the students are below average.This doesn’t seem right to me, because not all investors are professional money managers. There are also individual investors. Thus, it is mathematically possible for the alpha of the investment management industry as a whole to be positive.

It seems to me that if the alpha of the money management industry as a whole is not positive after fees, then this has an important implication for individual investors. It means that if you don’t know how good an investor you are, you will be better off investing your money yourself than picking a third-party manager purely at random (well, “purely at random” = probabilities weighted by existing market capitalization).

That strikes me as a non-trivial, useful thing to know.

The trouble people have with that fact is that they view alpha as an absolute rather than relative measure of performance. The education metaphor about half of the students being below average is appropriate: It is a trivially true fact that tells you nothing about the overall performance of the educational system. In the same way mean alpha is going to be approximately zero. That in and of itself tells you nothing about the overall performance of investors, the vast majority of whom in capital-weighted terms are professional investment managers. As for whether you would be better off investing on your own or by using a professional, that question wasn’t addressed at all and wasn’t relevant to the issues I was investigating. Thanks for reading and commenting.

So just out of curiosity, what kind of mistakes might people make as a result of thinking alpha is an absolute instead of a relative measure?

I should mention this to my MBAs this fall…

Perhaps the relevant items you could take from the post:

Alpha is a relative measure, that is not only not observable but is also highly model dependent. Thus, there are alphas, not an alpha.

Alpha can, in some contexts, help rank investors’ performances, but it tells you nothing about overall performance. It also tells you nothing about market efficiency.

The default approach of throwing a few years of returns of an investment and benchmark into a spreadsheet and doing a vanilla ordinary least squares linear regression isn’t going to give you a meaningful estimate of alpha. Perhaps the best you can hope for is some insight about relative volatility and covariance.

Doing alpha the right way is hard, really hard. It demands professional level statisitical modeling skills and tools.

In using alpha the model-dependence, the statistical uncertainty, the non-stationarity, and the need to shrink forecasts all must be kept in mind. Often when you do this and assess what you’re getting from alpha, then you realize it’s not going to be able to give you what you need.

It would be a good exercise to ask the students to prove the “zero mean alpha” result. That is, given a capital weighted benchmark and the set of frictionless investors which collectively equal that benchmark, then the mean capital-weighted alpha is zero. This doesn’t require much beyond high school algebra skills.

Hope that helps.

That does help!