Balancing an investment portfolio
For some time, I have been fretting over whether carrying 100% of a retirement portfolio in stocks or keeping some mix of treasuries or bonds was prudent. Recently, I came across the Kelly Criterion, as a general topic applicable to both betting and investments. After some searching of the literature, I found the specific application which I wanted.
The Kelly criterion tells you how much to bet. In investing, this equals the amount which is invested in stock, assuming the remainder is kept in a purportedly safe instrument, high grade bonds or treasury notes. The Kelly criterion strikes a balance between betting too much, in which case the variablity of expected returns could result in ruin and betting too little, in which case, the amount won is less than optimum. The Kelly Criterion requires there be some probability that there is a positive payoff. As the probability of a positive payoff approaches 0.5, the betting fraction, the amount that should be wagered in an incremental bet, approaches zero.
With faith, either by relying on others to critique this, or by your trusting my math (a dubious bet in itself,) the following two plots are sufficient to apply the Kelly criterion to an investment portfolio.
These plots show the Optimal Betting Fraction, i. e. the amount of a portfolio which should be put at risk, as a function of the annual total return of the equities into which the investment is planned. The several lines are for different volatilities, which could apply. As an example, investing in a S&P 500 Index fund, for which a 20% volatility might apply and for which an 8% long tern annual total return would be reasonable, use of a betting fraction of 0.75 will yield the optimal portfolio growth.
These plots, now show the rate of portfolio growth that is predicted, given the same market growth and volatility assumptions that were used to determine the betting fraction. For the assumptions which resulted in a 0.75 betting fraction in the example we started in the previous paragraph, a portfolio growth of only a tiny bit more than the riskless return results! We never said this was a get-rich-quick scheme.
At this stage, we now must face several issues. Is the math right? This is the primary reason I am making this available to a few people who can critique it. Are the parameter assumptions right? They may be conservative, but they were not intentionally that conservative. Finally, are we interpreting the results properly? The Kelly Criterion is said to be an objective confirmation of the wisdom of a somewhat conservative utility function. The theory is intended to produce a result, a growth rate of a portfolio, which cannot be bettered by more aggressive investing assuming the input assumptions are true. Remember: the criterion says, if volatility is 20% and the rate of return in greater than 4% more than the risk free interest rate, borrow money to invest! That's not conservative. On the other hand, for rates of return which are less than 4% above risk free rates, hold some capital back. Else, the theory says, you will loose in the swings.
Appendix--Derivation of equations used to create the plots
Kelly Criterion methodology applied to the portfolio balancing problem.
We start with a single market tick, in which a stock is equally likely to go up by an amount (s + m) or go down by an amount (s - m), where s corresponds to the stock's volatility and m corresponds to the stocks average annual growth or drift. For this situation E(X) = m, Var(X) =
s2. If the investor starts with initial capital V0, investing a fraction f, the return from this incremental investment is:
where r is the rate of return on the remaining capital invested in risk free bonds.
We want to determine a return g as a function of
f, the betting fraction. Thus:
We now carry this discrete incremental step over into a continuous process by dividing the time interval into successively smaller steps while retaining the same drift, m and variance,
s2. This gives us
For which
If we take E(log(.)) of both sides , we obtain g(f). Expanding g(f) in a power series gives
which, as n goes to infinity becomes
These results apply to any random variable X with E(X) = m and variance Var(X) = s2. Thus it applies to the lognormal distribution used to represent stock price in, for example, option pricing calculations. This application of the above results leads to
Acknowledgements:
This derivation in outline and the actual equations as graphics are taken from
THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET
by Edward O. Thorp ©1997 Edward O. Thorp and Associates, Newport Beach, CA 92660
Paper presented at: The 10th International Conference on Gambling and Risk Taking
Montreal, June 1997
Chapter 7, pages 23-27
http://www.bjmath.com/bjmath/thorp/paper.htm
http://www.bjmath.com/bjmath/thorp/ch7.pdf