Using the Sharpe ratio for football - Introduction

The Sharpe ratio is a measurement commonly used in finance that was developed by Nobel Prize winner William Sharpe. It is a combination statistic that measures an investment's return over its risk, or equivalent its expected value over its standard deviation.

Lost? Bear with me.

In football we constantly use averages, which becomes a proxy for what we "expect." If a play has averaged 4.5 yards a carry in a given situation over time we expect it to result in about 4.5 yards the next time we use it. Also, we constantly use averages to determine everything from yards per pass, tackles per game, 3rd down percentage, and various success and failure rates throughout.

Coaches at every level including youth usually keep a log of what plays were run or what defenses were called, what the situation was, and what the net result of the play was (or at least they all can and should). From this decisions can be made about a given play or situation's effectiveness, or at minimum the stats are compiled into the final game statistics. They may even be compiled in yearly statistics, etc.

The point of this article is to expand upon the use of only simple averages into a more dynamic statistic that can help us all make decisions, and can be computed by anyone with a computer. Most of the coaches who view my site and email me are not NFL or big time D-1 coaches with lots of money and computing power. Moreover, I'd actually say most of the coaches who look at my site are youth coaches--theres lots more of them than there are high school coaches and they don't have the benefit of ANY budget to spend on football or computers.

Sharpe Ratio

The Sharpe Ratio is defined as the ratio of the difference between the expected return of some strategy minus the expected return of some riskless benchmark and the standard deviation of the strategy. This can be written as such:


Standard deviation is a measurement of the volatility of a set of values. You can read more about standard deviation here. For example, if a given pass play was run four times, and the results (in yards) of the play was 10, 10, 10, and 10, it has a standard deviation of 0. However, if it was run four times and the results were 0, 0, 40, and 0, it has the same average gain (10 yards) but its standard deviation would be 20. We would prefer pass play one to pass play two. It has the same expected gain, but it is less risky than the second.

Bill Walsh talked about this explicitly in his book Finding the Winning Edge. This is the framework we will go from when using the Sharpe ratio: that volatility or risk (as measured by standard deviation) would not be accepted unless it is appropriately rewarded.

Thus by doing so we will have moved beyond the generic coach and as we compile our gameplans, make personnel decisions, or analyze tendencies and are not constricted to only the averages, which is not at all the whole story.

This is not to say that using this will restrict you to only non-risky strategies; you come closer to eliminate unrewarded risks, and you can more clearly evaluate the risks and rewards and you may discover you are overestimating your risk. I will talk more about this later or in future posts.

Calculating the Sharpe Ratio

To begin we will discuss evaluating a given play's performance over an entire season. We will make up a hypothetical run play, say, the counter trey. Let's assume you ran the play 50 times over 10 games, or what averaged out to 5 times a game. For now we will ignore down and distance and situation (when doing this you can maybe only look times it was run on first down, or non goal-line situations, etc).

Below is a link to a spreadsheet that I made using made up results. The yardage gain of the play is in the left hand column. Just for interest, let's say that four times we fumbled the exchange to the other team, so I gave that a value of -20 to indicate the loss of field position etc. This number is made up.

Spreadsheet

In the second column is a benchmark number, or, in the Sharpe ratio terminology, a risk free strategy. In this case I assumed that at any time you could run a QB sneak and get 2 yards. This is actually very important. This provides a benchmark, because at any point you call the counter trey you are essentially deciding not to call some other play. One of them being a QB sneak, for a guarenteed 2 yards.

(I'm limiting this to middle of the field/1st and 2nd down, I know a guarenteed 2 yards is unrealistic for a short yardage or goal line situation. The exact value of this number is not totally important.)

The gist is that you are bearing more risk by calling some other play, in this example the counter trey, when you could have the 2 yards (which obviously wouldn't get you too many 1st downs). As we all know, the game of football becomes about calculated risks.

In the third column is the difference between the observed gains and losses and the benchmark QB sneak. This is called the excess gain or loss or the differential return.

Lastly, in the fourth is the calculated Sharpe ratio, which is the average of the differential returns divided by their standard deviations. In Microsoft excel this can be done very simply, where the values are in C2 to C51:

=AVERAGE(C2:C51)/STDEV(C2:C52)



This will result in a Sharpe ratio for this play. In this case it is:

If you are comparing similar statistics (i.e. if you did this same method to other plays) you can compare the Sharpe ratios. The higher Sharpe ratio should be preferable because it is providing a higher return compared to its risk or volatility.

In the next part I will show more specific applications of the Sharpe ratio as well as some other recommendations and some caveats when using it.