I know, but I'm curious to see what markowitz says for comparison.
I think I see now -- markowitz doesn't say how much of your BR to invest. You can achieve the same maximum sharpe ratio regardless of how much you're investing in total. So markowitz only tells you how to allocate your investment (among a set of picks) once you'd already decided how much to invest total.
For the example I gave, I wrote an R script to brute-force check the integer solutions for the highest Sharpe ratio.
Code:
outer.sum = function(v1,v2) outer(v1,v2,"+")
outer.prod = function(u1,u2) outer(u1,u2,"*")
BR <- 100
p <- .1
q <- .5
r <- 1/3
g1 <- c(15, -1)
g2 <- c(1.5, -1)
g3 <- c(3, -1)
p1 <- c(p, 1-p)
p2 <- c(q, 1-q)
p3 <- c(r, 1-r)
lst.p = list(p1,p2,p3)
prob <- Reduce(outer.prod,lst.p)
max <- 0
for(x in 0:BR) for(y in 0:BR-x){
z <- BR-x-y
lst.g = list(x*g1,y*g2,z*g3)
gain <- Reduce(outer.sum, lst.g)
EV <- sum(prob*gain)
sdev <- sqrt(sum((gain*prob-EV)^2))
if(sdev==0) rat <- 0
else rat <- EV/sdev
if(rat>max){
max <- rat
alloc <- c(x,y,z)
}
}
max
alloc
Answer: if you know you're investing $100 in total, then it says to invest
$36 in A
$38 in B
$26 in C
for a Sharpe ratio of .37
Whereas Kelly would allocate a significantly smaller amount to A since it's a longshot.
If I'm wrong about anything I said then someone correct me.
Edit -- oh, I guess IRL, you also have riskless assets available and so you'd invest your entire BR and allocation would be the only question. I'll add a riskless asset D to the example and see what happens.
Edit: asset D = 100% chance of gaining 3% return
Answer: if your BR is $50, you want it to be (1, 0, 0, 49)
sharpe ratio of .2655 (I'm surprised it's so low)
Kelly would probably subtract some from D and put some in B and C.
Last edited by heehaww; 05-01-2014 at 08:45 AM.