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
- Calculate the manager’s beta to the benchmark:
- Normalise both sets of monthly returns using beta as a multiplier
- Calculate monthly alphas as the difference between normalised returns:
- Compound alpha and calculate its geometric mean (i.e. compounded growth rate):
- Calculate the geometric standard deviation of the monthly alpha (variability of alpha, i.e. omega):
- 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: