Clarifying the Information Ratio and Sharpe Ratio

With the interest in hedge funds and other alternative investment mechanisms soaring, here is an attempt to provide an intuitive explanation for understanding and interpreting the Sharpe Ratio and the Information Ratio. They are both indicators of risk adjusted returns, and are ratios of mean returns to standard deviations of some flavor, but different from each other.

The Sharpe Ratio:

The Sharpe Ratio reflects the ratio of all excess returns over the risk free rate to the total risk (or standard deviation) of the return stream. In other words, we strip out the risk free rate from the earned returns, and divide that by the total standard deviation of the returns.

The Information Ratio:

The Information Ratio, on the other hand, is the ratio of the alpha component of total returns to the standard deviation of these excess alpha returns. The alpha component is the return that is attributable to the manager’s skill (or luck ;-), and is the residual after taking out the risk free return and the beta components from the total returns. Also note the difference in the denominator – while the Sharpe ratio considers the standard deviation of the total returns, the Information Ratio considers the variability of only the alpha component of the return (which also forms the numerator).

Conceptually, think of this like this: if total returns can be considered the sum of the risk free rate, the beta (i.e the compensation for taking on systematic risk, or market risk), and the alpha, then the Sharpe Ratio looks to express the ‘excess’ returns over the risk free return (i.e, beta plus alpha) per unit of total risk undertaken. That is intuitive because the risk free rate, by definition, has no risk anyway and all risk arises from the beta and alpha components.

The Information Ratio only looks to compute the return per unit of risk undertaken for the alpha component. This is important because alpha is always in a very risky category – its mean for the market as a whole is zero (in practice slightly less than zero because of transaction and other costs) and therefore it is easy to lose money on alpha that will bite into the beta returns.

Here is a graphic that explains the whole reasoning:

Interpreting the Information Rat, or, Why is the Information Ratio Important?

The Information Ratio is very useful to understand how risky is dabbling with the alpha in question. If we were to assume that alpha returns will be normally distributed, then the Information Ratio allows us to model the alpha as being a distribution with mean = IR and standard deviation = 1. This is intuitive because IR = (mean alpha return/standard deviation of alpha returns). A ratio of say, 0.4 can be interpreted to imply a normal distribution with mean equal to 0.4 and a standard deviation of one. From this point, everything is easy because we can now estimate the probability of losing money, or the probability of meeting a benchmark.

Note that just simply putting the formula =normsdist(-IR) gives us the probability of losing money in one year.

We can extend the analysis to multiple years – for example, consider a manager with an alpha of say, 3%, and standard deviation of say 10% (IR = 0.3). The probability of him losing money over a one year period is 38%. Now think of a three year horizon. The mean returns over a three year period will be 9%, and the standard deviation will be (3^1/2)*10%, or 17.3%, and therefore a possibility of losing money over a three year period to be about 30%.

Comments

[User 148992]

DISCLAIMER: PAST PERFORMANCE IS NO INDICATION OF FUTURE RESULTS...

That should be somewhere in your article whenever you bring up statistical principals and say that because a managers historical alpha is .4, in one year there is a 38% chance that he will not meet his benchmark, in three there is a 30% chance. What happens if the manager has an awful year (i.e. alpha of -.4). Do you then revise your historical alpha to find that it is only, lets say, .2 and that actually over the next two years (one year is already a negative) and now there is a 45% chance of that manager underperforming over the duration of your three-year prediction period. Try explaining that to a client!

Statistics like this are useful for gaining insight, but real investment decisions CANNOT be predicated on them alone. Please read a few studies on mean reversion...you say that alpha is zero-sum (technically, negative-sum) long-term, so how can you expect this same manager to continually produce an alpha of .4 over any long-term time horizon.

When posting like this, please try to use statistics in a resposible manner and make sure that you don't go misleading people into thinking that their portfolios are safe and sound because statistics says so. Thats what causes people to lose their life savings in the market.

[Mukul]

Thank you for taking the time to read and respond to my post. If you are reading this article as investment advice, well then I am a bit speechless at your criticism. I have only tried to explain what these ratios are, and what they are expected to measure, that is all. If you believe in the distinction between alpha and beta, than I would assume you believe in the CAPM, and you could challenge that too on the same grounds. Since the CAPM, EMH, MPT etc with all their faults are the basis of how the investment world frames discussions, I see little need to state the obvious (eg, that no one knows the future, the assumption of alpha or beta being normally distributed is a very strong assumption, you need to look on both sides before you cross the road, I could go on, but this isn't the subject of the post).

[User 148992]

Thank you for your response, and the article does good job of clarifying the IR and Sharpe ratio. My issue comes in the latter part when you describe the practical use. This piece of the article was written to show people how to utilize the IR. I don't mean to be the statistics police (and I do realize I am going a little overboard here), I just think that when you are teaching predictive statistics you have to make a point to ensure things like "consider a manager with an alpha of say, 3%, and standard deviation of say 10%" really means that "you believe the manager will continue to produce an information ratio of .3 with a standard deviation of 10%". What you don't explain here is that the probability of a manager actually achieving an IR of .3 or greater over the following year (given the assumption that total IR of the market is normally distributed with mean 0 and std dev 1) is only 38%. This might make people a little more wary of using .3 or .4 or whatever arbitrary number they decide to use for their predicted IR.

The point I am trying to make is that this article is written as an introduction/clarifica... of the ratios aimed at an audience that my not understand some of these concenpts fully. While I understand that you might be speechless if you think I might be reading this article for investment advice, I bet at least one someone used it that way. When writing an explanatory piece like this for the masses, it needs to be wary of the traps people can fall into when using historical analysis. All I'm saying is please try not to assume too much, and make sure it is understood that people need to use their brains and are personally responsible for the inputs that they use and cannot attribute losses (or hopefully gains) to historical data.