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Optimal Asset Allocation with the Kelly Criterion

Written by:
PK

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.

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