Elo rating and funds’ performance

We want to thank Prof. Dr. Roberto Frota Decourt from Unisinos for visiting us at TU Dortmund and presenting one of his working papers, which inspired us for this little blog post.
 
The elo rating system is a by Aprad Elo created system for calculating relative skill levels in games such as chess or video games. Although this system couldn’t establish its implementation in many other forms of sport, there are several websites publishing these elo rankings (e.g. Word Football Elo Ratings).
The elo rating number is based on pairwise comparisons. Players‘ ratings are not measured absolutely, but rather depend on their own rating, the rating of their opponents and the results of the game. If Player A has a elo rating of R_A and Player B a rating of R_B, the expected score for Player A is given by
 

    \[E_A=\frac{1}{1+10^{(R_A-R_B)/400}}.\]

 
Analogously, Player B‘s expected score can be determined. We note E_A + E_B = 1. Obviously, if a player with a higher elo rating wins against a player with a lower elo rating, he will gain less points then he would lose if he lost the game. The magnitude of this score adjustment can be set by a factor k, as the updated elo rating of Player A is given by
 

    \[R_A^n = R_A + K * (S_A-E_A).\]

 
where S_A are the actually scored points. The factor K is usually set to values between 10 and 40. Recent studies suggest a value of 24.
 
Ok, let‘s now apply this rating system to financial markets. We regard 41 actively managed fonds, each of them wants to beat its competitors. We take weekly performances as measured by returns into account and let each fond play against each other each month. One could think of several ways to set S_A. One possibility is to set S_A=1, if the fond generated a higher return than its competitor for the respective month and S_A=0 otherwise. However, one could also think of setting S_A as follows:
 

    \[S_A= \begin{cases} 1, \textrm{ if the monthly return is the highest of all,}\\ 0,\textrm{ if the weekly return is the lowest of all,}\\ \in (0,1),\textrm{ linear interpolation between highest and lowest monthly return.} \end{cases}\]

 
By this, we do not only capture if a fond beats its competitor or not, but also quantify by how much the fond over-/underperformed. Now, we can simulate these monthly competitions. For five years of competition, here is the distribution of the elo scores obtained at the end of the period:
 

 
The evolution of the elos looks like this:
 

 
One can interpret the final elo score as a performance measure. An often used performance measure is e.g. the sharpe ratio (for actively managed fonds probably rather Jensen’s Alpha), which is defined as the excess return of a fond divided by its volatility. We can compare the top ten fonds of each performance measure:
 

RankFondEloSharpe RatioRank (Sharpe Ratio)
1Prudential Jennison Financial Services Fund Z1554.3311.91%1
2Prudential Jennison Financial Services Fund A 1551.0111.74%2
3Prudential Jennison Financial Services Fund R 1548.7011.31%3
4John Hancock Financial Industries Fund A 1547.190.90%14
5Prudential Jennison Financial Services Fund C 1546.3311.22%5
6Prudential Jennison Financial Services Fund B 1546.2011.23%4
7John Hancock Financial Industries Fund C 1543.000.76%15
8John Hancock Financial Industries Fund B1542.940.74%16
9Fidelity Select Consumer Finance Mutual Fund 1536.08-12.68%41
10Diamond Hill Financial Long-Short Fund A1525.950.31%19
 
Interestingly, the worst fond as measured by sharpe ratio is in the top ten as measured by elo. The spearman rank correlation amounts to 63\%. For shorter periods the correlation lies between 70\% and 95\%.
 
All in all, this procedure is just intended as a little gimmick. However, by this we are able to regard the competitiveness between fonds and their aim to beat each other (or the market, different elo game) from a different perspective. How applicable and meaningful this “shift” is – or if it is really just a gimmick – might be an interesting question for a variety of research topics.

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