Previous blog post on the Kelly Criterion

As pointed out in a previous blog post, the Kelly Criterion is an interesting option to decide on position sizing in portfolio selection. While the previous post looked at single stocks, I will today show how to optimize position sizes for a portfolio with multiple stocks.

The core function

At the core of my portfolio optimization is this function:

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opt_portfolio <-function(shares, dpf, maxshare) {
  
  # calculate portfolios return vector
  exp_returns <- dpf%*%shares
  
  obj = -sum(log(1+exp_returns))
  weight.penalty = 1000*(1-sum(shares))^2
  
  # max share penalty:
  maxpen <- sum(shares[shares>maxshare])*1000
  return(obj + weight.penalty + maxpen)
}

The function has three parameters. shares is the vector of shares (position size for each stock) that is going to be estimated by optimization. dpf is a matrix where each stock is a column and each line contains the daily stock price movements like e.g.: diff(stock)/lag(stock). The maxshare option can be used to restrict the position size of a single stock to a maximum.

exp_returns <- dpf%*%shares calculates the daily (or weekly) portfolio returns. obj is the Kelly Criterion. The higher the volatility, the larger values obj will take. We are going to minimize the function so low values, i.e. low volatility is preferred. The next line is a trick to restrict the optimizer to values that sum to 1 (100%). If the sum of all position sizes is 1, weight.penalty is 0 (the minimum possible value). Would the sum deviate from 1, the weight.penalty would quickly increase. The second penalty term is similar. If a position is sized above the maximum allowed share, the penalty scores. Finally the function returns a value which the optimizer will try to minimize.

The wrapper function

Now to feed stock data into this function I wrote a wrapper function that prepares the matrix of returns and calls optim().

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library(quantmod)
library(dplyr)
opt_portfolio_wrapper <- function(stocks,r=rep(0.03,length(stocks)),maxshare=1,daily=FALSE,short=FALSE) {
  
  # test which stocks are already in workspace, else download data
  for(stock in stocks) {
    if(!exists(stock)) getSymbols(stock,from="1970-01-01",env=parent.frame(2))
  }
  
  # transform return vector from yearly to daily or weekly
  if(daily==TRUE) {
    r <- (1+r)^(1/250)-1
  } else {
    r <- (1+r)^(1/52)-1
  }
  
  portfolio <- NULL
  for(stock in stocks) {
    if(daily) assign(stock,get(stock)[,6])
    if(!daily) assign(stock,to.weekly(get(stock))[,6])
  }
  # merge all stocks together
  portfolio <- Reduce(function(...) merge(..., all=TRUE), mget(stocks))
  
  # build returns by diff()/lag()
  d.portfolio <- data.frame(na.omit(diff(portfolio)/stats::lag(portfolio)))
  
  # center around the mean to eliminate past performance as an information for portfolio selection
  d.portfolio.future <- scale(d.portfolio, scale=F)
  
  # add the (personally) expected return for each stock
  for(i in 1:ncol(d.portfolio.future)) {
    d.portfolio.future[,i] <- d.portfolio.future[,i] + r[i]
  }
  
  # define lower und upper bounds for the parameters. 0 to 1 or -1 to 1 if short positions are allowed
  lower <- rep(ifelse(short==TRUE,-1,0),length(stocks))
  upper <- rep(1,length(stocks))
  
  # starting values for optimizer
  start <- rep(1/length(stocks),length(stocks))
  
  res <- optim(start, opt_portfolio, dpf=d.portfolio.future, maxshare=maxshare, method="Nelder-Mead")
  
  Portfolio <- data.frame("Share"=res$par,"Stock"=stocks) %>% arrange(desc(Share))
  return(Portfolio)
}

Most of the function is commented between the lines. I introduced two options: daily and short. The default values are, that short positions are not allowed and the returns are calculated on a weekly, not daily basis. One thing thats important for me is, that I dont use the stock returns as they are but I center them around the mean return that I expect. If I would not do this, the algorithm would always pick the historically best performing stock(s). However past performance is not a good predictor of future performance and one has to do his homework and build own (and realistic) expectations.

Testing

To test the function I select some random stocks (not a recommendation to trade these stocks).

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rstocks <- c("AAPL","GOOG","AMZN","GIS")
opt_portfolio_wrapper(rstocks)
#        Share Stock
# 1 0.80851815   GIS
# 2 0.09979312  AAPL
# 3 0.06343397  GOOG
# 4 0.02829082  AMZN

This tells the following story: Assuming all stocks have the same expected yearly return of 3%, the long term growth rate of wealth (Kelly Criterion) is achieved by investing in the stock with a lowest volatility (here GIS). At the same time this is a hint, that the assumption that all returns are to be expected as equal is too strong. One could now play with the function to find out how high the expected return of a risky stock has to be, to be included into an existing portfolio.

Summary

I personally use this tool as a rough orientation, how stocks from my watchlist would contribute to the risk-return-profile of my portfolio. I usually use the maxshare option and rather “flat” and moderate return expectations, because the Kelly Criterion highly favors stocks with better return expectations. E.g. if I add 1% additional expected return for GOOG, the numbers change drastically:

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opt_portfolio_wrapper(rstocks, r=c(0.03,0.04,0.03,0.03))
#          Share Stock
# 1  0.749115293   GIS
# 2  0.200639803  GOOG
# 3  0.051410711  AAPL
# 4 -0.001125542  AMZN

If find the tool to be useful because it keeps me from overbetting on very volatile stocks and gives hints, if and when risky stocks are worth to bet on.

Other

There are a few additional points worth mentioning:

• while opt_portfolio_wrapper() gives the shares to achieve the Kelly-optimal portfolio selection, it does not tell, how much one should bet on the whole portfolio. Approximately this be calculated by dividing the expected portfolio return by the annualized variance. One could also adapt the kelly() function from the previous blog post to get a number that incorporates the fact, that the return distribution is fat-tailed and non-normal.
• I excluded some of the options that my function originally had for the sake of readability. Options that I found interesting are:
  • transaction costs
    
  • “fixed shares” (e.g. my pension account is invested in index funds and bonds. When picking stocks, I might want to incorporate the fact, that I am already exposed to these securities)
    
  • a from option for getSymbols() if the historic returns should be restricted to a certain range