Systematic Trading research and development, with a flavour of Trend Following

## Geometric Information Ratio

#### April 26th, 2010 · 25 Comments · Fund Review

Following the previous post where we ranked CTA performance using the Geometric Information Ratio, a couple of readers have requested clarification/more information on it. So I thought a reference post dedicated to the Geometric Information Ratio might be a good idea.

This post contains a spreadsheet used to calculate the Geometric Information Ratio, and expanded information on how the classic Information Ratio fails to accurately do its job. We’ll also reiterate the rationale behind the new Geometric Information Ratio and the formula used to derive it.

### Where the Information Ratio fails

The main issue with the classic calculation of the Information Ratio is its use of arithmetic returns and lack of leverage adjustment. This is best illustrated with a simplified example:

Consider 3 hypothetical managers:

• Steady Sam: very reliable, she produces 0.3% month after month.
• Volatile Viv, she aims for big moves, which she gets regularly in both directions by alternating 30% up months with 25% down months
• Cautious Carla, Viv’s twin sister, uses the same strategy as her sister but knows she should play it safer with much less leverage: 6% up months followed by 5% down months

Meanwhile, the benchmark regularly alternates 0.5% and -0.3% months.

The formula for the Information Ratio being:

We can calculate the average excess return (alpha) for each manager: Sam gets two months of 0.3% return while the benchmark returns 0.5% and -0.3% giving excess returns of -0.2% followed by +0.6%, foran average of 0.2%. With the same calculation, Viv scores 2.4% per month with Carla scoring 0.4%. The standard deviation figures for the excess returns are respectively 0.4%, 27.1% and 5.1%, which gives information ratios of:

• Steady Sam IR = 0.5
• Volatile Viv IR = 0.09
• Cautious Carla = 0.08

Carla seems to underperform her twin sister, whereas Sam clearly outscores her 2 colleagues. However, looking at a compounded equity curve from the 3 managers give a different picture with Viv’s volatility pushing her into negative territory.

The Information Ratio calculation penalizes Carla for her lack of leverage, while falsely indicating that Viv produces excess return to the benchmark.
Applying the right amount of leverage to a strategy is part of the manager’s skill and should be part of the Information Ratio calculation.

### The Geometric Information Ratio Rationale

The main departure from the classic Information Ratio calculation is the use of a measure of return which takes leverage into account. The geometric average return is such measure: excess leverage applied to a positive strategy will result in a negative geometric average return (unlike the arithmetic return).

We can imagine a strategy where the investor desires exposure to the manager’s alpha with no beta risk. To implement such strategy, the investor would take a hedged position in the manager’s fund. The hedge would consist of a short position in the benchmark, adjusted for the beta coefficient between the manager and the benchmark (if the fund has a beta of 0.5, the short position in the benchmark would be half the size of the long position in the fund).

After one period of such hedged position, the investor would collect both alpha and beta returns from the manager minus the return from the benchmark. Because of the beta adjustment, the beta return and benchmark return would cancel each other out. Rebalancing the positions to repeat the operation for each period would allow to gradually harvest and compound the manager’s alpha only.

The performance of such strategy is a good measure of the manager’s skill and the calculation of the Geometric Information Ratio will follow the same logic.

### The Geometric Information Ratio Formula

1. Calculate the manager’s beta to the benchmark:
2. Normalise both sets of monthly returns using beta as a multiplier
3. Calculate monthly alphas as the difference between normalised returns:
4. Compound alpha and calculate its geometric mean (i.e. compounded growth rate):
5. Calculate the geometric standard deviation of the monthly alpha (variability of alpha, i.e. omega):
6. Divide alpha by omega to obtain the Geometric Information Ratio

Showing my geeky deviance, I enjoyed playing with phpMathPublisher, an online formula tool to express the full formula in simplified form – Rm and Rb being the respective manager and benchmark returns over n periods:

and in fully expanded form (also on the phpMathPublisher tool):

More useful to you is probably this spreadsheet which implements the calculation for an arbitrary CTA (TransTrend) and a benchmark (Conquest Capital Group MFS fund).

### More Performance Metrics and CTA Ranking Information

The Geometric Information Ratio is one way to look at the value added by a manager, improving the concept of the classic Information Ratio. There are, of course, other ways to evaluate a manager or investment’s performance.

A large amount of litterature has actually been written on performance measurement and how best to evaluate and rank managers. For those readers wanting to investigate this topic more in depth, here is what I could find on my bookshelf, related to this topic:

I read half of Schwager’s book, which is an interesting discussion on various performance ratios, but have not done more than leafing through the 2 other ones (more maths-oriented) yet.

Also these 2 papers discussing the more ubiquitous Sharpe ratio are an interesting read:

### 25 Comments so far ↓

• Stephane

Hi Jez,

great Post, very good explanation of how the GIR improves the classic Alpha calculation. Thank you for the excel file.

Stephane

• Hi Jez,

Your two posts on this topic plus the Conquest Capital paper have got me thinking a bunch on this topic. Thank you.

I think you are 100% right that the logic of taking arithmetic means of monthly returns is flawed and you clearly demonstrate the flaw with your examples.

However, I am not sure if your solution is perfect yet. A couple of comments:

1) If one works with continuously compounded returns, I think life is easier – your geometric means and variances are built in as they add linearly (a 1% month continuously compounded followed by a 3% month results in a mean of 2% per month and a std dev of 1%) – no need for all those ugly capital PI’s!

2) Is it possible that “geometric mean” is a mis-nomer? You are calculating the ((geometric mean of (1+monthly return))-1) and ((geometric Std Dev of (1+monthly return))-1). Perhaps “Compound Information Ratio”? It’s just semantics I guess, but I started off trying to reproduce your results using monthly return instead of (1+monthly return). Once again, using CC returns avoids all that unpleasantness!

3) Should the risk-free rate figure in here somewhere? I am thinking that Alpha(i) = MgrRet(i) – Beta x BRet(i) – (Beta-1) x Rf
where i denotes the month, MgrRet is the manager’s return, Beta is the Cov/Var of the manager’s return to the benchmark, BRet is the benchmark return and Rf is the risk-free rate.

4) Finally, I have issues with the whole idea of creating benchmarks like this to measure alpha of a CTA. First, the benchmark has an active trading strategy embedded in it. Compare this to stockmarket indices which simply embed a passive indexing strategy. Surely the equivalent for CTA’s would be a portfolio passively holding a basket of futures. If not, then one can argue endlessly about which strategy to embed in the index.

Secondly, it seems to me that applying the concept of beta to ad hoc indices is not well founded. Commodities are not capital assets, they are inputs to production, they do not generate cash-flows like bonds or stocks. They are not on the Capital Asset Pricing Line!

• Hi Eventhorizon – thanks, I am glad that my posts are thought-provoking.
Thanks for the comments. I agree with you re: points 1 and 2 Continuously Compounded returns would have made the calculation a bit easier (but with the same end result). I think geometric return is a fairly standard definition but I can see how it could be slightly confusing. Hopefully the Excel spreadsheet clears that up..
Re: points 3 and 4, I think the Conquest Capital paper makes a pretty good argument on why they believe their active strategy can be used as a benchmark, which is supported by their calculations of correlation with other CTAs. As such I dont think a passive index of Futures is a good representation because most CTAs use a long/short Trend Following active strategy. As for the risk-free rate, the comparison is with the benchmark so i dont think it should be taken into account (unless you want to calculate something like the Sharpe ratio of the manager’s alpha?).
Finally, I see beta mostly as a gross “leverage” coefficient and not really in the context of the CAPM

• Thanks for the good reply, Jez.

I noticed an error in my original comment, for the record I would like to correct it …

Alpha(i) = MgrRet(i) – Beta x BRet(i) – (Beta-1) x Rf

Alpha(i) = MgrRet(i) – Beta x BRet(i) – (1-Beta) x Rf

Personally, I think a good argument can be made that the risk-free rate should be in there, after all, we are trying to identify excess returns: those of the manager vs the benchmark and the benchmark vs the risk-free rate.

Afterall, we can get the risk-free rate at zero cost, so when we buy the index we are buying (BRet-Rf), not BRet.

Based on his leverage, we expect an average manager (whose alpha=0) in the CTA space to deliver R0 = Beta x (BRet-Rf) + Rf.

So the alpha delivered by a manager is Rm-R0.

When Rf is close to zero, this argument is academic, but when Rf gets back up to higher values we have seem in the not so distant past, it becomes significant. Indeed beta IS a gross leverage coefficient, but it is gross leverage to EXCESS returns not absolute!

Keep up the great work – I have taken a few tentative steps on my own blog and I am discovering how much work it is!!

• Peter Andersen

Hi Jez,
Im not a stat-wizz, just an ordinary asset manager. I copied in data from my global equity portfolio from june-04 and onwards together with the MSCI The World monthly performance data in USD. The Geometric Information ratio is shown as 0,4 – how do I interpret that number correctly?
Best,
Peter

• Peter,
The Geometric Information Ratio calculates a score taking into account excess return and the variability of that excess return.
0.4 sounds like a fairly good score (check the previous post for the figures of the Trend Following Wizards: http://www.automated-trading-system.com/cta-alpha-calculate/ ). The first point is that your portfolio adds alpha (GIR > 0). The 0.4 figure points to a fairly high monthly excess return compared to its variability. You could have the same excess return but with greater variability, which would lower the Generic Information Ratio.

You might want to run some comparisons with your peer fund managers GIR to get an idea of how you rank comapred to them

Cheers,
Jez

• Izz

I did the same thing Peter Anderson did and obtained a negative information ratio. How would I interpret this?

I used daily returns of my portfolio against the daily changes in the benchmark. Does this affect the quality of results?

• Izz – the basic interpretation is that your beta-adjusted returns do not produce alpha to the benchmark. I havent done any comparison between monthly vs. daily return but I suspect the result would still be the same – the main difference would probably be with volatility (ie impacting the denominator, which is irrelevant with negative GIR).

• Jim

Jez: I really enjoyed reading your analysis. I have a question on the model however. In column E where you calculate the Beta-normalized excess return shouldn’t the formula be (B3-\$C3)*D\$3 instead of B3-\$C3*D\$3?

You get a much different answer between the 2 variations.

Thanks again,
Jim

• Hi Jim – thanks for the comment.
I double-checked the spreadsheet and it still seems correct to me:
1. We calculate the beta (in cell \$D\$3)
2. We beta-adjust the return from the benchmark (C3*D\$3) so that it can be compared to the fund’s return (B3)
3. The comparison/excess return is therefore B3 – C3*\$D\$3

Let me know if you’re seeing it differently

• This mixture of geometric and arithmetic calculations
should not be called a geometric information ratio. To begin, it
seems it would be more consistent to define alpha geometrically by:
α = (1+ Rm)/(1 + β*Rb). But even this is problematic since β
incorporates arithmetic comparisons. A better approach is to redo
from scratch the whole standard arithmetic calculation, which you
introduced in your first equation, using the natural log of 1 + R
in place of R, always taking anti logs before taking any
ratios.

• Thanks for the suggestion Andre – I might look into them although I feel the method described in the post ties in with a real but theoretical investment strategy between the benchmark and the strategy (which is the driver for the way alpha is calculated). As such it fills the main objective of taking compounding into account – as the geometric average does…

• Andrew

For the benefit of an outsider, could you explain how the GIR accounts for leverage?

Thks.

• @Andrew – the returns are beta-adjusted for normalisation, which would take care of (i.e. remove) the differences that would arise from different levels of leverage.

• Andrew Chalk

Thanks for the response. So the ‘geometric’ part of the GIR accounts for compounding. The use of beta, to account for leverage, could be done to the standard IR (arithmetic). I.e. There are really two, independent, improvements being made here.

• Correct Andrew!

• Bevan

Jez,

Could you provide a simple explanation on how to convert the GEO spread-sheet into an ARITH example or perhaps create an addtional sheet to compare the differences.
Cheers

• Bev, the “arithmetic” Information Ratio is the standard way of calculating it. I think you should be able to find good examples on the web of how to use it such as this:
http://www.investopedia.com/terms/i/informationratio.asp
I would personally not try to convert from one to the other but rather treat them as two different calculations.

Hope this helps.
Jez

• Bevan

Thanks Jez – if I isolate 2 year return from Jun04-May06 I get product 11.278% Transtrend and 14.088% Conquest & STDEV of 2 yrs alphas of 2.645%. to get IR of -1.0625. =+(11.278-14.088)/2.645

So is the -ve IR is reflecting the impact of large negative alphas that are skewing the mean of the denominator. Hope I’m not completely on another planet here!
Cheers

• Bevan,
The sign of the IR always reflect whether the asset under measurement produces alpha to the benchmark. This is the first/main info.
The denominator then helps quantify the amount of (positive) alpha in light of volatility (in case of positive alpha, a less volatile asset will have a greater IR). In case of negative alpha, the actual IR value is not very relevant.
Cheers,
Jez

• Hi Jez,

I have a comment about this, particularly “Applying the right amount of leverage to a strategy is part of the manager’s skill and should be part of the Information Ratio calculation.”

This statement makes sense if you have the rule that you will only give 100% or 0% of your money to a manager to invest. Otherwise it does not make sense.

If you allow yourself the flexibility of assigning some fraction, f, of your money to the best manager, and putting the rest in the benchmark, then all that matters is their original IR. Their geometric IR is neither necessary nor sufficient information to make an investment decision. You can invest with Viv above, and have your wealth grow over time if you use an appropriate f<1, and rebalance each period.

Since assigning all or none seems like a rather artificial constraint, I don't see the utility of this geometric IR.

• Winslow,
I see your point and t’s true that “You can invest with Viv above, and have your wealth grow over time if you use an appropriate f<1, and rebalance each period." but compared to a _beta-adjusted_ investment in the benchmark this would under-perform (hence the negative GIR).
Actually, it seems to me that the beta-adjustment included in the GIR calculation is a way to implicitly allow for various values of f.

• Hi Jez,
Thanks for the reply. Yes, I am with you on the wisdom of beta-adjusting. In Grinold and Kahn, they only define IR in what we are referring to as the beta-adjusted sense. After all, it makes no sense to ascribe performance to a manager based on their level of leverage in the benchmark (unless they are benchmark timing). I don’t understand why anyone would not beta-adjust (other than laziness).

But given beta-adjustment to put it on equal footing with GIR, then the traditional IR has merit. It’s a mathematical result (it’s in G&K I think) that IR totally orders mutually exclusive, scalable investment opportunities for all mean-variance investors (with the benchmark as their numeraire). That is, these investors always prefer the higher IR investment, no matter what other statistical information is made available to them about the distribution of returns. This is not true of GIR.

In continuous time, the result is much stronger than that even, essentially true for all utility functions.

So I think that if you are unable or unwilling to rebalance, then GIR is what you want, but otherwise (beta-adjusted) IR is more applicable.

• Thanks for the note back Winslow.
You’re right, there are two aspects to this GIR and I “kind of” mixed the two together.

Thanks for the pointer to Grinold and Kahn, I’ll have to check it out to understand that point better.

Based on your last point, GIR might have some use for a portfolio of CTAs, where rebalancing is not practical/possible on a short (e.g. monthly) timeframe.

• Hi again Jez,

Re: rebalancing – it is important whether or not one rebalances, but it’s not terribly important that one does so frequently (once a year is actually pretty good).

This paper has a nice analysis of the effect on growth rate of rebalancing frequency

http://www.tandfonline.com/doi/abs/10.1080/14697680802629400

I couldn’t find a free version of that one, sorry, but the abstract should tell you something.

As for dealing with a portfolio choice problem over CTAs, then properly you should care about the highest IR or GIR portfolio that can be achieved, which is not calculable from either merely the individual IRs nor GIRs (you need a correlation matrix also).

Since IR is independent of leverage, “the” maximal IR portfolio is actually a family of portfolios, parametrized by leverage, and the highest GIR one is a member of that family – the member with the “optimal” leverage for asymptotic growth.