Is outperformance versus an index a signal of managerial talent or the result of market randomness? The question is more than academic considering the size of the money management industry today. In Financial Derivatives: Pricing, Applications, and Mathematics Jamil Baz and George Chacko give a stylized answer to this question.
Consider a money manager whose portfolio value P and the index he is trying to beat (I) follow geometric Brownian motion
where dWP and dWI are Wiener increments with correlation ρ. With F(P,I) = P/I = R denoting relative performance, one can build a new process Rt (by using Itô's lemma for the two-dimensional geometric Brownian motion (P,I)):
The value of the process at time t is:
By setting σP = 25%, σI = 15%, ρ = 90% and μP - μI = 3% the following table is obtained:
If a portfolio manager outperforms the index by 3% per year on average and under the volatility and correlation conditions described above, it would take 300 years for this manager to outperform the index with 90% probability...
Consider a money manager whose portfolio value P and the index he is trying to beat (I) follow geometric Brownian motion
where dWP and dWI are Wiener increments with correlation ρ. With F(P,I) = P/I = R denoting relative performance, one can build a new process Rt (by using Itô's lemma for the two-dimensional geometric Brownian motion (P,I)):
The value of the process at time t is:
By setting σP = 25%, σI = 15%, ρ = 90% and μP - μI = 3% the following table is obtained:
If a portfolio manager outperforms the index by 3% per year on average and under the volatility and correlation conditions described above, it would take 300 years for this manager to outperform the index with 90% probability...
Mahalanobis - am 2005-05-02 05:12 - Rubrik: Finance
