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AHA


Total Posts: 1
Joined: Jul 2017
 
Posted: 2017-07-19 01:21
Hi I'm new to the forum and I have some questions about how you would go about applying Kelly sizing to an equity portfolio.

I understand that in the limit of a small edge and if µ is small relative to σ the trading fraction according to the Kelly criterion is approximately f=return over risk free rate/variance.

Most value PMs come up with some form of ex ante expected return and variance. Would you simply use these estimates for sizing?

The position size would then just be f in percent terms? i.e if f = 5 for Stock A and f=2 for Stock B with a 100k account, one would invest .05*100k = 5k in stock A and .02*95k = 1.9k in Stock. Or would it be 5k in stock A and 2k in Stock B.

I'm skeptical to see that that the order of investments can change the dollar amounts so dramatically. If we use the former then we will never invest all of our capital. The latter may imply lots of leverage or dry powder depending on how many potential investments we can identify.

Most of my f's come along the .5-3.0 range and their sum is approximately 50, ideally I would like to deploy all of my capital or even use some leverage.

Any thoughts or papers would be much appreciated.

AIC


Total Posts: 167
Joined: Apr 2008
 
Posted: 2017-08-04 15:48
For multiple bets, Markowtiz mean variance may be a better way to handle it.

Train yourself to let go of everything you fear to lose- Master Yoda

finanzmaster


Total Posts: 159
Joined: Feb 2011
 
Posted: 2017-11-11 18:58
>For multiple bets, Markowtiz mean variance may be a better way to handle it.
Though Markowitz and Kelly are based on different assumptions, numerically they often deliver very similar results:
Max Dama, Kelly Criterion Position Sizing == Mean-Variance Optimization

In either case, neither Kelly nor Markowitz can be considered as "do as I say" approach.
The reason is, inter alia, extreme sensitivity of both methods to the parameter estimation errors.

Here is an R-script to make a simple Monte Carlo experiment:
#####set the "genuine" market parameters#######################
sigma1 = 0.4 #vola of the 1st asset
sigma2 = 0.3 #..of the 2nd asset
rho = 0.7 #correlation coefficient, must be in [-1, 1]
mu1 = 0.12 #expected return of the 1st asset
mu2 = 0.09 #..of the 2nd asset
rfr = 0.01 #risk-free return
N_SIM = 1000 #number of simulations

######## theoretical solution with "genuine" market parameters ########
library("mvtnorm") #multivariate Normal distribution
library("MASS") #[generalized] matrix inverse
covMatrix <- matrix(c(sigma1*sigma1, sigma1*sigma2*rho, sigma1*sigma2*rho, sigma2*sigma2), 2,2)
vectorOfMeanReturns = c(mu1, mu2)
H=chol(covMatrix)
print( "Optimal portfolio with TRUE market parameters" )
ginv(t(H)%*%(H)) %*% (vectorOfMeanReturns - rfr)


################################################################################
######## solution with market parameters, estimated from "historical" data #####
################################################################################
#### three trials.
### NB! All trials have identical input data but pretty different output! ###
for(trial in 1:3)
{
historicalData = mvrnorm(n=N_SIM, vectorOfMeanReturns, covMatrix)
empSigma1 = sd(historicalData[,1])
empSigma2 = sd(historicalData[,2])
empRho = cov(historicalData[,1], historicalData[,2]) / empSigma1 / empSigma2
empiricalCovMatrix = matrix(c(empSigma1*empSigma1, empSigma1*empSigma2*empRho,
empSigma1*empSigma2*empRho, empSigma2*empSigma2), 2,2)
empricialVectorOfMeanReturns = c(mean(historicalData[,1]), mean(historicalData[,2]))
H=chol(empiricalCovMatrix)
print( paste("Optimal portfolio with EMPIRICAL market parameters - TRIAL ", trial) )
print( ginv(t(H)%*%(H)) %*% (empricialVectorOfMeanReturns - rfr) )
}


If you want to dig deeply have a look at : my paper on Kelly Criterion

www.yetanotherquant.com - Knowledge rather than Hope: A Book for Retail Investors and Mathematical Finance Students
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