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A while ago, we adressed the following question: A portfolio manager knows that his strategy can, on average, outperform the benchmark index by 3% annually. His portfolio has an annual volatility (standard deviation) of 25% against the index's 15%. Assuming that the correlation between the returns of the portfolio and the returns of the index is 0.9, how many years would it take to outperform the index with 90% probability?

The correct answer is a whopping 300 years! (apply the Itô-Döblin formula)

Today somebody asked me if I could run a couple of simulations to get a better understanding of the result. What I did was plot 20 simulations of log(Portfolio/Index) for varying correlations. For ρ = 0.9 2 out of 20 (10%) are--as expected--below zero:
skillluckparadox
anonymous (guest) meinte am 16. May, 15:19:
Doesn't make sense
If for rho = 0.9, 2 out 20 are below 0, should it not be the case that for rho = 0.5, 10 out of 20 are below 0? Yet I do not see 10 out of 20 below 0. Explain, please. 
Mahalanobis antwortete am 16. May, 18:12:
Huh?
Why should (1- rho) be the probability that the series is below zero after 300 days? Check the original story and keep in mind that
Wr,t = zstandard normal*sqrt(t)

u.p <- 0.08
u.i <- 0.05
sd.p <- 0.25
sd.i <- 0.15
t <- 300
rho <- c(0,0.5,0.9)
u.r <- u.p - u.i + sd.i^2 - rho*sd.i*sd.p
sd.r <- sqrt(sd.p^2 + sd.i^2 - 2*rho*sd.p*sd.i)
pnorm((u.r - 0.5*sd.r^2)*sqrt(t)/sd.r)

Out: 0.7237735 0.7866116 0.9047849

So for rho = 0 (0.5) there is a 0.27622645 (0.21338837) chance that the series is below zero. 
stxx meinte am 16. May, 15:26:
Wile it is tempting to use sophisticated quantitative analysis to derive conclusions about market behaviour I still have to remind myself to think of how I can improve my forecasts through real world actions (e. g. call the portfolio manager and ask whether good risk-adjusted performance has an impact on his salary ;)