Switching Horses Mid-Race: How to know when riding different betas creates alpha

When you think about it, the idea of an investment benchmark is somewhat philosophical.  Although many hedge fund managers profess to deliver uncorrelated “absolute” returns – not relative returns – a large portion of them compare their results to equity markets.  Why?  Not necessarily because they manage to that bogey, but because hedge fund allocations are often made by diverting capital from equity allocations.  In other words, equities are the most likely alternative.   Today’s guest contributor says that the choice of benchmark can make or break a “horse-switching” manager.

Measuring Performance in Active Allocation Amongst Multiple Benchmarks

Exclusive to AllAboutAlpha.com by: Eric Stanhope Hirschberg, Orion Investment Management (with programming assistance from Hamaad Shah)

“Process is the one aspect of investing that I can control. Yet all too often I focus on outcomes rather than process. Yet ironically, the best way of getting good outcomes is to follow a sound process. The research shows that holding people accountable for outcomes tends to lead to suboptimal performance, generally because they spend all their time worrying about the things they can’t control. I’d advise a far better approach to assess people on the criteria of adherence to process.” — James Montier

Benchmarks: What are they good for?

While it’s tempting to say “absolutely nothing”, benchmarks have a meaningful role in the allocation process. How relevant a benchmark is to the investor really comes down to the context in which the benchmark is derived.

My personal preference is to view a benchmark as a tool to understand the alternative to an allocation. In this way the “goodness” of a choice can be measured against the alternative of not choosing, or for that matter making any alternative choice. This is a local perspective, in so much as one’s choices and risk tolerances are not uniform to an entire investment population. For that, I need to accept the nearest correlating passive and investable asset as the benchmark. And while this approach is fine for a strategy that maintains a significant correlation to a passive investable benchmark, how should I view the myriad of strategies in which the Manager / Asset Allocator tells us his expertise is to opportunistically invest?

Consider the manager who is actively making choices to allocate or de-allocate between a number of possible strategies or benchmarks, based on some criterion which he holds as his proprietary domain. I shall demonstrate a methodology that allows one to break down this Manager / Asset Allocator’s strategy into a series of choice components, with a corresponding framework for valuing the various choices made over the investment horizon.

Measurement as a Function of Rational Choice

So let us construct a way to measure this class of active managers.

Firstly, since I cannot know what goes on inside the head of our Manager / Asset Allocator, let us derive a set of assumptions and move on from there.

Assumption 1: The Manager / Asset Allocator chooses the best risk adjusted return he can get us.

While he could just roll the dice on the riskiest assets, his return is maximized by staying in the game, and as the risk adjusted return accounts for the risk of ruin and other shades of undesirable outcomes, it maximizes the chance of success. His success is a function of his ability to forecast his risk adjusted return. His failure to do this does not warrant our discarding the assumption that he is trying.

Assumption 2: The Manager / Asset Allocator actively chooses between assets as opposed to assets held mutating from one exposure to another.

There have certainly been times in the history of finance where a Manager / Asset Allocator maintaining a “static” strategy, experiences a huge style or exposure shift as a result of an exogenous factor he wasn’t aware he was choosing. A short portfolio of Canadian junior mining assets, overnight became a portfolio of internet start ups, as bankers realized buying worthless shells and converting them was cheaper than the opportunity cost of waiting for the public issuance backlog in 2000. I will limit our discussion to the normal case of active allocation by the Manager / Asset Allocator.

Assumption 3: The Manager / Asset Allocator can choose to invest in the passive benchmark or actively replicate it.

The Manager / Asset Allocator uses assumption 1 to choose portfolio A’ with respect to A. If he can’t find anything better than a, he will simply invest in A (provided that he believes the risk adjusted return of A is greater than that of B). This is often the case with CTA’s, many of whom simply choose the ETF or Futures contract over some optimized portfolio.

Assumption 4: The Manager / Asset Allocator’s portfolio constituents and their respective weights are transparent to us.

To calculate the Manager / Asset Allocator’s tracking error (A’-A) I need to know the constituents of the Manager / Asset Allocator’s portfolio.

Analysis of the Two Benchmark Case

For the purposes of this analysis, I consider:

  • A strategy that allocates between 2 available benchmarks (A, B). To be clear, while it is easier to know which benchmarks should be used a priori, it is not a necessity, as there are effective ways in which I can discover the benchmarks through data analysis.
  • The Manager / Asset Allocator invests in or replicate a benchmark A at some particular time, using strategy portfolio A‘. I will assume some unknown difference, but A’=A would still work.

At T=0, the Manager / Asset Allocator Chooses To invest in A (and not B) and does so by construction of replicant A’.

Piece-wise Analysis: Period 1

At T=100 I consider five pieces of information:

  1. Corr (A’, A). I want to understand the portfolios active versus passive allocation component. In other words, was there any value created in the replication process?
  2. Ret (A’) – Ret (A). I want to understand the value of active versus passive allocation. What was the value of replication?
  3. Ret (A)-Ret (B) .I want to understand the value of switching.
  4. Information Ratio* A’/A= Ret (A’-A)/ Stdev (A’-A). I want a measure of the risk adjusted return of the active allocation.
  5. Information Ratio* A/B = Ret (A-B)/Stdev (A-B). I want a measure of the risk adjusted return of switching.

* The information ratio used is based upon geometric returns and adjusted for leverage effects.At T=100, the Manager / Asset Allocator Switches from Benchmark A to Benchmark

Piece-wise Analysis: Period 2

At T=200 I again consider five pieces of information:

  1. Corr(B’,B)
  2. Ret(B’-B)
  3. Ret(B-A)
  4. Information Ratio B’/B = Ret(B’-B)/Stdev(B’-B)
  5. Information Ratio B/A = Ret(B-A)/ Stdev(B-A)

At T=200, the Manager / Asset Allocator Switches from Benchmark B back to Benchmark A

Piece-wise Analysis: Period 3

At T=300 I again consider five pieces of information:

  1. Corr(A’,A)
  2. Ret(A’-A)
  3. Ret(A-B)
  4. Information Ratio A’/A = Ret(A’-A)/Stdev(A’-A)
  5. Information Ratio A/B = Ret(A-B)/ Stdev(A-B)

Adding it all up

Taking the prior example, a tactical allocator made 3 allocation decisions in a 300 day period. The total return is as follows:

tRet’ = Ret (A’)0 to 100 + Ret(B’)100 to 200 + Ret(A’)200 to 300

At this point I should be asking three questions:

  • First, did the Manager / Asset Allocator add any value over the piecewise benchmarks using replicants, i.e. was tRet’ > tRet ? What was the weighted sum of the information ratios?
  • Secondly, did the Manager / Asset Allocator add any value over a simulated tactical switching model with the same benchmark universe and an equal number of randomly timed switches?
  • Thirdly, how did the Manager / Asset Allocator’s allocation compare to all convex combinations of the two benchmark assets held passively over the same period?

Example: Three Allocation Decisions

To illustrate an example, I use SP500 index and US Treasury Index Data for a two and one half year interval (2005-2007).

I create a fictitious Manager / Asset Allocator who makes three allocation decisions over this interval. For simplicity’s sake, I have chosen a Manager / Asset Allocator who is limited to investment in the benchmarks themselves (A’ equal to A). The identical analysis can be performed on the portfolios where, the Manager / Asset Allocator chooses as a replacement for the simple benchmark (A’ not equal to A).

  • No transaction costs are considered as there are a total of six transactions (in liquid securities) over the entire period.
  • I do not consider corr(A’,A) as it remains constant 1 in this example due to Manager / Asset Allocator investment limitations.
  • No leverage is used in this example. That said, one can easily adjust for the effects of leverage by normalizing using the beta coefficients (M) of A’= MA +b and B’=MB + c for each interval chosen.

In practice, this normalization should always be undertaken when considering quality of returns.

The Return Distribution for the Three Allocation Simulation

By plotting the Manager / Asset Allocator’s return as a function of switching versus the boot-strapped distribution of outcomes, I can ascertain a probability of skill for the Manager / Asset Allocator’s ability to time benchmark exposure. By comparing the replicant returns as a function of deviation, I can formulate a statement about the Manager / Asset Allocator’s ability to create additional value given a benchmark exposure. The following chart shows the outcome from randomly sampling 2000 sets of three switches ABA or BAB from the previously mentioned data. The red dot marks our Manager / Asset Allocator’s return in this distribution.

The Information Coefficient Distribution for the Three Allocation Simulation

As the Manager / Asset Allocator is claiming his skill comes from his ability to time switches, our analysis now allows us to make an objective assessment of his claim. Our Manager / Asset Allocator’s resultant IC is shown as a red point on the chart below. In this case, it is clear that he should be compensated beyond the cost of active replication as replacing him with a random switching algorithm would likely produce an inferior result. Had the Manager / Asset Allocator’s IC been less than the mean of the distribution, the expected value from employing a switching algorith would have been greater, and it would be economically beneficial to replace him.

Mapping the Three Allocation Simulation Over All Convex Combinations of the Two Benchmark Buy and Hold

Now let us compare the Manager / Asset Allocator’s total return and deviation with both simulated switches and all possible passive benchmark mixes. These mixes take the form Total Return (cA+dB) and standard deviation of returns (cA+dB) where c=0 to 1 and d=1-c. The lower left endpoint of the line represents a 100% allocation to the Bond Index while the upper right endpoint represents a 100% allocation to the equity index.

Note that our Manager / Asset Allocator has performed quite well. He beats a significant majority of our simulated switches, and his active return could not be passively replicated without a significant increase in risk. In this case the Manager / Asset Allocator’s compensation for actively switching is well deserved.


I have endeavoured to formulate a framework for evaluating a Manager / Asset Allocator who actively moves between two investments. A simple example of equities and bonds was illustrated for an initial selection plus two additional switches. I could have as easily used two styles (growth and value, or large cap and small cap) as benchmark assets. Under this regime, the Manager / Asset Allocator can switch between benchmarks tactically, opportunistically or even unknowingly, it really does not affect our ability to measure the Manager / Asset Allocator’s utility in doing what he/she does.

The methodology presented in this paper forms the basis for evaluating a Manager / Asset Allocator that moves, in part or in whole, between multiple benchmarks. My work on this expanded problem will presented at a later date.

I note that there is a school of thought that believes that one way to measure a Manager / Asset Allocator’s addition of active value is to take his initial portfolio and hold it passively as the benchmark “inertia benchmark”. This notion, while creative, is flawed in a number of respects.

Firstly, as open funds can receive allocations from investors somewhat continuously (depending on addition/redemption terms); the benchmark could vary dramatically by date.

Secondly, just because an investor starts off in a particular portfolio by means of subscription, that does not imply that the portfolio would have been purchased by the manager had he been required to hold it for some fixed period. Based on my experience transacting institutional portfolio rebalances for a number of years, I would say that individual component durations are far from uniform.

Lastly, one utility of a benchmark is that it allows the investor access to an alternative allocation. If the investor has to rely upon the Manager to present the initial portfolio, it is unlikely that the investor would be able to construct the aforementioned portfolio without the assistance of the Manager, thereby invalidating the concept of the benchmark as a means of alternative allocation.

Authors Note: If you would like to learn more about implementing my risk and performance measurements for your organization, feel free to contact me via Eric@KapitalMarks.com or contact me via LinkedIn. Any comments or criticisms are most appreciated.

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  1. Jez Liberty
    August 11, 2010 at 8:42 am

    Very interesting and thorough post – thanks. I completely agree with the concept of reducing the manager compensation based on their Information Coefficient. And this concept gives useful leads to evaluate the merits of dynamic asset allocations.

    One question regarding your calculation of the Information Ratio, as you mention it is based upon geometric returns, do you have an exact formula for the ratio calculation. Or the underlying concept of how it is calculated.

    Not wanting to use arithmetic returns myself, I have devised a formula for a (beta-adjusted) “Geometric Information Ratio”, mostly using common sense and would be interested in checking how it compares with your methodology.

    The formula for “my” Geometric Information Ratio can be found here:

    hope you find it interesting – and thanks in advance for any feedback.

  2. Patrick Burns
    September 16, 2010 at 3:43 am

    The methodology presented here is similar to using random portfolios for performance measurement. That’s explained, for instance, at:

    I suspect that the “inertia benchmark” is a better performance measure than what is almost always done now, but I agree that it has problems. There is a blog post on that at

  3. Eric Hirschberg
    September 24, 2010 at 7:45 am

    There are some similarities. One difference is certainly the underlying benchmark distribution. This methodology encompasses an active benchmark as opposed to a passive and this may be quite different and would require further work to verify. I believe methodology becomes a random portfolio methodology in the limit case of continuous switching with degrees of freedom = all combinatorials of assets. In that sense your random portfolio methodology is the unconstrained special case of my methodology.

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