The Capital Asset Pricing Model of financial theory is the root of the Alpha, Beta distinction. This model is shown below:

where R is the return from respectively the portfolio and market, alpha and beta the objects of interest and epsilon a residual error term.

This is a simple linear model that appears in every elementary finance text. The returns from a portfolio can be divided into a market return term, with beta being interpreted as an exposure or weighting to this market return term, and a fixed return, alpha. Mathematically this is the intercept with the returns axis.

In its original form, the sole source of returns was the broad market and an index was used to proxy for this. We can of course develop far more complex versions of this model while still retaining its linear simplicity. We might, for example, choose to use

linear combinations of many sources of return - these sources of return have come to be known as risk factors. The seminal work of this type was that of Fama and French.

Multi-factor models may complicate matters mathematically but the central issue is that we need to estimate the model from a set of portfolio returns and from a set of index or factor returns. The question is, which line (hyper-plane) best fits the cloud of available data.

To estimate this capital market line, we need to add a further condition; it is necessary to make the return on the market, or risk factor, and the residual error term independent of one another. That is the covariance of these should be set to zero:

which now allows us to apply the standard technique of minimising the sum of the squares of the distances of each point from the line to determine it. We can then read off the alpha as the return intercept and beta as the slope of the market line.

We show, in figure 1, the regression line for a long-short UK equity fund (NAV) using monthly data for the fund and the FTSE all-share index. Table 1 shows results. This fund has a beta of 0.207 relative to the FTSE all-share index over this period. Its alpha is 1.279%, (per month). The explanatory power of the regression line is low (Adj. R^2 = 13.5%) as can be seen from observation of the dispersion data in figure 1.

If we move to consider a multi-factor world by adding the FTSE small cap as an explanatory variable, the explanatory power of the regression actually declines and the beta relative to the all share increases. The alpha produced by this fund is significant, much higher than usually observed from fund managers. The portable alpha concept would have us hedge the fund by shorting a contract on the FTSE all-share index in an amount determined by the beta, which would remove the sensitivity to the market return entirely, that is achieve market neutrality. The problem is that the data is not homogenous and the beta value has a standard deviation of 0.08 so our hedge at that level of confidence lies between 0.13 and 0.29, which is rather wide for effective use. There is also a further problem with this simple regression technique. Notice that time does not enter the earlier equation at all. Simply put, the technique cannot capture any market-timing skill that the manager may possess. In fact this will usually show up as a higher beta and lower alpha – penalising the manager.

It is quite common to move to rolling regressions based upon a moving window of data but even this does not satisfactorily resolve this market-timing skill problem.

There are further complications in the real world in that the returns from many of the risk factors are intrinsically non-linear – in many advanced investment strategies, utilising options and the like, this is by design. These linear models are poorly specified in this situation.

The interpretation of beta as a hedge ratio raises a fundamental question for the investor wishing to pursue an alpha strategy. The investor now has a short position in the index, which will lose money if the index rises. The investor also has cash from the short sale in hand – should (s)he repeat the process by awarding a further mandate of the short sale proceeds and hedging again. The production of alpha fully leveraged must exceed the returns from the market to justify following the strategy. In theory this may always be achievable, though clearly margins and market frictions limit the degree of leverage practicable, but perhaps the more important question is to what extent is this leverage desirable? This has been overlooked by some in their pursuit of ‘portable alpha’ strategies – usually those who are driven by the cost advantage of delivering Beta through passive market replicating portfolios.

If we return to the requirement for independence between the market return and the residual error term, we may begin by asking why should we wish this? Surely what we want to achieve is to avoid the downside of the market but capture the upside.

Let us just re-arrange the capital market equation:

with all terms as previously. We now have a distribution for alpha rather than the single intercept value. Again if we interpret the beta as a hedge ratio, by varying this hedge ratio we may optimise the alpha distribution. Our objective now is to maximise the value of this alpha distribution, we wish to achieve the highest mean and greatest upside asymmetry possible. This requires a metric for distributions that can capture these facets, such as the Omega F2 score.

Does it work ? Let us compare the results of a regression based hedge with the results of the Omega hedge. In this instance (figure 2) we show the results from hedging a US fixed income mutual fund with its benchmark index, to which it is very highly correlated. We use three years of monthly data to derive initial beta hedges using both standard regression and Omega regression techniques, we then update the estimates and re-hedge quarterly thereafter. The cash proceeds of the hedging shortsales are left in a non-interesting bearing account.

Notice here that the fund outperforms its benchmark over the period under review. Notice also that the total return from both benchmark and fund are higher than the returns from either alpha portfolio. The regression-based hedge produces a total alpha of 4.84%. The Omega regression produces a total alpha of 9.66%, a materially superior result. With either technique we could leverage the portfolio until we deliver the same variability as the benchmark or fund, and that would outperform both the fund and the benchmark by a comfortable margin, again with the Omega portfolio superior. This of course is precisely the objective of these alpha strategies.

Con Keating is with the Finance Development Centre in London

## No comments yet