Optimal Asset Allocation with the Kelly Criterion

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The Kelly Criterion is the brilliant summation of a betting strategy first discovered by Information Theorist John Kelly. Kelly came up with a betting system which optimizes bankroll growth based upon known odds and a definite payout. If you can find an exploitable, repeatable edge, Kelly's system tells the maximum you should bet based upon that criteria.

Kelly Criterion Optimal Asset Allocation Calculator

Here's a calculator which applies the concepts in this post to come up with an allocation:

Using the Kelly Criterion with Your Portfolio

Extending Kelly a bit further (like Ed Thorp, author of two math bibles for the investor/bettor Beat the Dealer and Beat the Market, has done) we can do a bit of hand-waving and make it work for the stock market. 

Some derivations of "Stock Market Kelly" involve using back-looking numbers such beta to approximate the continuous returns of securities. We're going to do it in a discrete way, and use discrete numbers for wins and losses.

The Kelly Criterion For Asset Allocation

Let's say that you're investing with a 10 year time-frame – you want to buy a house or retire, for example. You have an extra $100,000 and are trying to determine the best allocating between stocks and treasury bonds.

Let's try to calculate is your 'edge' and your 'odds'.

It's true: garbage in, garbage out. All we can do is take an educated guess and hope that it is close enough to reality to guide our choices. (See: past performance is no guarantee of future results.)

As they say, history doesn't repeat itself but it often rhymes.

Odds: The S&P 500 beats 10 Year Treasuries roughly 85% of the time over rolling 10 year periods. We'll then enter .85 for our odds of stock out-performance.

Edge: Edge is tough, but for arguments sake, let's use 5%.

Historically 5% is a decent choice; sometimes authors will take average earnings yield and subtract Treasury yield. Change it as you desire.

Using Odds and Edge to Optimize Asset Allocation

'Normal' or 'Full' Kelly is

\frac{probability*(1+odds\ offered)-1}{odds\ offered}

We need to modify the Kelly Criterion a bit to take into effect the fact that generally a security won't 'go to zero'. (Even a losing 'bet' almost always has some value in the stock market).

We simplify the equation to

\frac{expected\ value}{odds\ offered}

Here's the math using the assumptions in the previous section:

\frac{expected\ value}{odds\ offered} = \\~\\
\frac{.85*.05*(5\%) - .15*.02*(-2\%)}{79\%} = \\~\\
\frac{0.0395}{5\%} = 79\%

So, in this theoretical portfolio with your historic estimate of odds and edge, aim for 79% stocks and 21% bonds. The standard disclaimer applies: these numbers are guesses, so adjust your expectations accordingly.

For traditional Kelly applications, also try the Kelly Calculator for bet sizing.



PK started DQYDJ in 2009 to research and discuss finance and investing and help answer financial questions. He's expanded DQYDJ to build visualizations, calculators, and interactive tools.

PK lives in New Hampshire with his wife, kids, and dog.

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