### Your Alpha is My Beta

A couple of weeks ago, I had the pleasure of a short correspondence with Lars Kestner, a well known quant and derivatives trader, and creator of the thoughtful K-ratio as a measure of risk adjusted performance. We connected on the definition of alpha, and how the term has been so abused in media and marketing as to become almost meaningless. To help make his point, Lars quoted a passage from his recent whitepaper, “My Top 8 Pet Peeves“, which I’ve taken the liberty of copying below:

Incorrect casual use of the term alpha

This complaint may stem from the statistician in me, but the casual use of the term alpha irritates me quite a bit. Returning to very basic regression techniques, the term alpha has a very specific meaning.rp = α + β1 r1 + β2 r2 + β3 r3 + … + εAlpha is just one of the estimated statistics of a return attribution model. The validity of the regression outputs, whether parameter estimates such as alpha or various betas, or risk estimates such as standard errors, depend on the model used to specify the return stream. Independent variables should be chosen such that the resulting error residuals cannot be meaningfully explained further by adding independent variables to the regression. In the most prevalent return attribution model, the typical one factor CAPM model, returns are explained by one independent variable – broad market returns.Defining an appropriate return attribution model is necessary to estimate a manager’s alpha. I find it ironic that the use of the term alpha is most frequently applied to a subset of asset managers called hedge funds where defining the return attribution model is often the hardest. Long-short equity managers can display non-constant beta as their net exposures change. Fixed income arbitrage managers typically display very non-normal return distribution patterns. Managed futures traders can capture negative coskewness versus equity markets that provide additional benefits beyond their standard return and risk profile. Calculating these managers’ alpha is a difficult task if for no other reason that specifying the “correct” return attribution model is problematic.Consider the specific example of a hedge fund manager whose net exposure is not constant. In this case, a one factor market model is not necessarily optimal and other factors such as the square of market returns might need to be added to account for time varying beta. If a manager makes significant use of options, the task of specifying a proper model becomes even harder. Also, consider a manager whose product specialty is volatility arbitrage and an appropriate market benchmark may not be available. How then to estimate alpha?I prefer using the term “value-add” to be a generic catch-all for strategies that increment a portfolio’s value. Whether that incremental value is generated though true alpha, time varying beta, short beta strategies with low return drag, cheap optionality, negative coskewness to equity markets, or something else that is not able to be estimated directly from a return attribution model, it saves me from having to misuse the term alpha.

Lars raises great questions about the relevance of alpha derived from a linear attribution model with Gaussian assumptions when a strategy may exhibit non-linear and/or non-Gaussian risk or payoff profiles. Unfortunately, this describes many classes of hedge funds. While this is true, his comments took me in a different direction altogether.

It’s interesting to contextualize alpha not just in terms of the factors that an experienced expert might consider, but rather in terms of what a

*specific**target*investor for a product might have knowledge of, and be able to access elsewhere at less cost. In this way, a less experienced investor might perceive a product which harnesses certain non-traditional beta exposures to have delivered ‘alpha’, or more broadly ‘value added’, where an experienced institutional quant with access to inexpensive non-traditional betas would assert that the product delivers little or no alpha whatsoever.Let’s start with the simplest example: imagine a typical retail investor who invests through his bank branch. A non-specialist at the bank branch recommends a single-manager balanced domestic mutual fund, where the manager is active with the equity sleeve, exerting a value bias on the portfolio. The bond sleeve tracks the domestic bond aggregate. The fund charges a 1.5% fee.

Subsequently, the investor meets a more sophisticated Advisor and they briefly discuss his portfolio. The Advisor consults his firm’s software and determines the fund’s returns arecompletely explained by the bond aggregate index returns, domestic equity returns, and the Fama French (FF) value factor. In fact, after accounting for these factors, the mutual fund delivers -2% annualized alpha.

The Advisor suggests that the client move his money into his care, where he will preserve his exact asset allocation vis-a-vis stocks and bonds, but invest the bond component via a broad domestic bond ETF, and use a low-cost value-biased equity ETF for the equity sleeve. The Expense Ratio (ER) of the ETF portfolio is 0.1% per year, and the Advisor proposes to charge the client 0.9% per year on top, for a total of 1% per year in expenses. The Advisor, by identifying the underlying exposures of the client’s first fund, and engineering a solution to replicate those factors with lower cost, has generated 1% per year in alpha (1.5% mutual fund fee – 1% all-in Advisor fee + 0.5% by eliminating the negative mutual fund alpha).

At the client’s next annual review, the Advisor recommends that the client diversify half of his equities into international stocks, at a fee of 0.14%. An unbiased estimate of non-domestic equity returns would be similar to domestic returns, minus the 0.6*0.5*(0.14-0.1) = 0.012% increase in total portfolio fees. However, currency and geographic diversification are expected to lower portfolio volatility by 0.5% per year, so the result is similar returns with lower risk = higher risk adjusted returns = higher value added = higher alpha.

After another year or so, the new Advisor discusses adding a second risk factor to the equity sleeve to compliment the existing value tilt: a domestic momentum ETF with a fee of 0.15%. Based on the relatively low correlation between value and momentum tilts (keeping in mind they are all long domestic equity portfolios), the Advisor believes the new portfolio will deliver the same returns over the long-run, but diversification between value and momentum tilts will slightly reduce the portfolio volatility, by another 0.2%. Same returns with less risk = higher alpha.

At each stage, the incremental increase in returns and reduction in portfolio ‘beta’ (vis-a-vis the original fund) results in a higher ‘alpha’ for the client. Now obviously the actions that the Advisor took are not traditional sources of alpha – that is, they are not the result of traditional active bets – but they nevertheless add meaningful value to the client.

Now let’s extend the logic into a more traditional institutional discussion. The institution is generally applying attribution analysis for one or both of the following purposes. The two applications are obviously linked in process, but have substantially different objectives.

- To discover how well systematic risk factors explain portfolio returns over a sample period. For example, we might determine that a long-short equity manager derives some returns from idiosyncratic equity selection, some from the Fama French value factor, and some returns from time-varying beta. If we hired the manager for exposure to these factors, this would confirm our judgement. Otherwise it might prompt some questions for the manager about ‘style drift’ or some other such nonsense.
- To determine if a manager has delivered “value added”, or alpha. For example, perhaps the manager delivered excess returns, but we discover that the excess returns can be explained away by adding traditional Fama French equity factors to the regression. Since it is a simple and inexpensive matter to replicate these risk factor exposures through ‘passive’ allocations to these factors (using ETFs or DFA funds for example), it might be reasonable to discount this source of ‘value added’ for most investors, and trim the alpha estimate accordingly.

This should be pretty straightforward so far. Using a long-short equity mandate as our sandbox, we discussed how a manager’s returns might result from exposure to the FF factors, time-varying exposure to the market, and an idiosyncratic component called alpha. But now let’s get our hands dirty with some nuance.

Let’s assume the long-short manager has been laying on a derivative strategy with non-linear positive payoffs. Imagine as well that a wily quant suspects he knows the method that the manager is using, can replicate the return series from the derivative strategy, and includes this factor in his attribution model. Once this factor is added, the manager’s alpha is stripped away. While the quant may feel that there is no ‘value add’ in the derivative strategy because he can replicate it for cost, surely an average investor would have no way to gain exposure to such an exotic beta. As such, the average investor might perceive the strategy as ‘value added’, or ‘alpha’ while the quant would not.

Ok, let’s back out the derivative strategy, and assume our long/short manager exhibits positive and significant alpha after standard FF regression. In other words, the manager’s excess returns are not exclusively due to systematic (positive) exposure to market beta or standard equity factors, such as value, size, or momentum. Of course, since it is a ‘long/short’ strategy, the manager can theoretically add value by varying the portfolio’s aggregate exposure to to the market itself. When he is net long, the strategy should exhibit positive beta risk, and when he is net short it should manifest negative beta risk. How might we determine if this time-varying beta risk explains portfolio returns?

Engel (1989) demonstrated how regressing portfolio returns on squared CAPM returns will tease out time-varying beta effects. So let’s assume that adding a squared CAPM beta return series to the attribution model explains away a majority of this ‘alpha’ source. Therefore, including this factor in the model increases the explanatory power (R2) of the model, and reduces the alpha estimate. But is this fair or relevant in the context of ‘value added’? After all, while we can say that the manager is adding value by varying CAPM beta exposure, we have not demonstrated how an investor might generate these excess returns in practice. I have yet to see a product that delivers the squared absolute returns of CAPM beta, but please let me know if I’ve missed something.

I submit that it’s useful to identify the time-varying beta decisions for attribution purposes. This source of returns may represent true “value add” or (dare I say alpha), because it can not (presumably) be inexpensively and passively replicated by the investor. To the extent an investor is experienced enough, and/or sophisticated enough to identify factors which can inexpensively replicate the time-varying beta decisions (such as via bottom-up security selection, or top-down timing models), then, and only then, might the investor discount this source of ‘value added’.

So far we’ve discussed hypothetical examples, but a recent lively debate on APViewpoint is a great real-life case study. Larry Swedroe at Buckingham has long militated against traditional active management in favour of DFA style low-cost factor investing. It took many by surprise, then, when Larry wrote a compelling argument for including a small allocation to AQR’s new risk premia fund (QSPIX) in traditional portfolios. After all, at first glance this fund is a major departure from Larry’s usual philosophy, with high fees, and leveraged long and short exposures to a wide variety of more exotic instruments. Thus ensued 100 short dissertations from a host of respected and thoughtful Advisors and managers on APViewpoint’s forum about why the fund’s leverage introduces LTCM style risk; why the factor premia the fund purports to harvest can not exist in the presence of efficient markets, and; why the fund’s high fees present an insurmountable performance drag.

Notwithstanding these potentially legitimate issues, I’m uniquely interested in how one might view this fund in terms of alpha and beta. The fund’s strategy involves making pure risk-neutral bets on well documented factors, such as value, momentum, carry, and low beta, across a variety of liquid asset classes. In fact, AQR published a paper describing the strategy in great detail. Presumably even a low-level analyst with access to long-term return histories from the factors the fund has exposure to could explain away all of the fund’s returns. From this perspective then, the fund would deliver zero alpha. However, it is far easier to gather the return streams from these more ‘exotic’ factors than it is to operationalize a product to effectively harvest them. So for most investors, this product represents a strong potential source of ‘value add’.

The goal of this missive was to demonstrate that, when it comes to alpha, where you stand depends profoundly on where you sit. Different investors with varying levels of knowledge, experience, access, and operational expertise will interpret different products and strategies as delivering different magnitudes of value added. At each point, an investor may be theoretically ‘better off’ from adding even simple strategies to the mix, perhaps at lower fees, and even

*after*a guiding Advisor extracts a reasonable fee on top. More experienced investors may be able to harness a broader array of risk premia directly, and thus be willing to pay for a smaller set of more exotic risk premia.It turns out that ‘alpha’ is a remarkably personal statistic after all.