Optimal Asset Allocation with the Kelly Criterion

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. Roughly… if you can find an exploitable, repeatable edge… Kelly’s system tells the maximum you should bet based upon that criteria.

Extending it a bit further (as done by folks like Ed Thorp, author of two math bibles for the investor/bettor Beat the Dealer and Beat the Market) we can do a bit of hand-waving and make it work for the stock market.  While some derivations of “Stock Market Kelly” involve using back-looking numbers like beta or something else to approximate the continuous (read: they can go to any price) returns of securities, we’re going to do it in a discrete way – assuming discrete numbers for wins and losses.  If you’re itching for something stronger, I’ll see what I can do about another article in this series.

Using the Kelly Criterion For Asset Allocation

On the Worst of the Free Financial Advisor podcast (The predecessor to my current podcast, Stacking Benjamins!), I already discussed how to calculate the proper allocation of capital into a single stock when you have a decent guess as to your edge and your odds. However, what if you’re not into the whole individual stock thing and you are just trying to set up categories of assets for your retirement? Kelly can also guide you here!

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 course of action in allocating those funds between stocks and treasury bonds. Well, then all we have to calculate is your ‘edge’ and your ‘odds’.

Like I said on the show, 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 – you know past performance is no guarantee of future results. As they say, history doesn’t repeat itself but it rhymes.

According to T. Rowe Price, up until 2009 (closest I can find) the S&P 500 beat Treasury Bonds 85% of the time over rolling 10 year periods starting in 1926. We’ll use .85 for our odds of stock out-performance. As for edge? That’s a tough one – but for arguments sake, let’s use the earnings yieldof the S&P 500 and subtract the 10 year treasury yield to come up with (roughly) 7% – 2% = 5% (lots of studies have used an edge of 5% for stock vs. bond returns so this is reasonable, or we’re at least wrong with everyone else). On the downside, let’s say the worst you’ll do in stock over 10 years is 0%, a loss/opportunity cost of 2% (I know, I know. Stock has actually lost money over a ten year period before, namely 1928-1938.).  Remember, normal Kelly is

Picture of John Kelly

John Kelly (wikipedia)

(probability*(1+odds offered)-1)/(odds offered)

It needs to change a bit to take into effect the fact that generally a security won’t ‘go to zero’ – so even a losing ‘bet’ has some value.  Therefore, we simplify to

(Expected Value)/(odds offered)

Here’s the math:

Expected value = .85*.05 (5%) – .15*.02 (-2%) = 0.0395 divide by odds offered (winning bet – so .05) = 79%.

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

Ready to try it out?  Here’s a calculator which applies the concepts above to come up with an allocation.

Kelly Criterion Investment Allocator
Input Data Values
Probability Asset Class or Stock Outperforms (Percent)
Expected Gain When It Outperforms (Percent)
Expected Loss When It Underperforms (Percent)
Bankroll (Dollars)
Type of Kelly – ‘1’ for full, ‘.5′ for Half, ‘.25′ for Quarter… etc.

Kelly Says…
Calculated Results Values
Kelly Number (Percent of Bankroll)
Adjusted Kelly Number for Kelly Type (Percent of Bankroll)
Dollars to Use in Position

Code created with assistance from Political Calculations

Hope this helps… keep an eye out for a full, hardcore calculator! Comments?

Comments

    • says

      Jephiter,

      Pretty awesome that you’re able to use Kelly for Real Estate! Have you found any disadvantages to the system? Any surprises you didn’t account for?

    • says

      Thanks!

      Calculating expected returns is kind of hand wavey, but if you can somehow get a good guess, Kelly is perfect for your asset allocation. I’ll try to get a more advanced Kelly calculator at some point!

  1. says

    As of 7-29-15, the Kelly Criterion Calculator seems to be frozen. I enjoy using this from time to time, and thank you for making it available. Will you please trouble-shoot and let me know if there is a problem or if it is “user error.”

    Thanks in advance!

    • PK says

      My apologies! It was the worst type of error, operator error.

      I realize now I never made my ‘hardcore’ calculator… if you’re still using this one, is there anything you’d like to see in a sequel? I’ve built up all sorts of tools in the meantime which might make useful inputs to a Kelly Calculator: http://dqydj.net/calculators-and-visualizations/

      • Irish57 says

        Hi PK:

        Since you asked…

        I have read DQYDJ’s posts on Valuation-based investing based on the Shiller P/E levels. Wade Pfau recently had a couple of articles on this formula method earlier in July in Forbes. (links) I believe it is under-appreciated, and I am using it in my investing program.

        Part 1 of Pfau series: http://www.forbes.com/sites/wadepfau/2015/07/08/is-a-high-cape-cause-for-alarm-part-1-capes-relationship-to-stock-returns/

        Part 2 of Pfau series: http://www.forbes.com/sites/wadepfau/2015/07/09/is-a-high-cape-cause-for-alarm-part-2-valuation-based-asset-allocation/

        A few years ago, there was an article at Business Insider (BI) which I reference from time to time (link). It had a table that showed Credit Suisse’s analysis of historical dispersions of subsequent 3-year stock returns at various Shiller P/E levels.

        http://www.businessinsider.com/chart-shiller-pe-returns-2013-6

        Pfau’s article (Forbes) recommended break points at 2/3 and 4/3 of the Shiller long term mean for allocation decisions (based on Graham and Dodd suggestions). But that basically only boils down to three stock allocations buckets: 25% stocks, 50% stocks, and 75% stocks.

        As of July 2015, with Shiller P/E of 26, a valuation- based investor following the Pfau methodology would already be hunkered down with only 25% stocks. In this era of sub-atomic interest rates on cash and bonds, I think most of us would like to keep our stock allocations as high as possible–within reason–always with eye on valuation and probable outcomes.

        I have been trying to determine the probabilities of outcomes shown in the BI table and have been inputitng them into with the Kelly calculator to determine stock allocations at the various Shiller P/E levels. At Shiller P/E of 26 (where we are today) , the article said this is the level when problems start to crop up, i.e. possible negative subsequent 3- year returns.

        For example, the table in the article implies there were nine (9) historical market experiences starting at the Shiller P/E 26 and higher. Three (3) of those 9 were negative returns in the subsequent 3- year periods. That seems to suggests that the success rate of a positive return is 66% at Shiller P/E of 26. (6 out of 9). At those odds of success, the Kelly calculator suggests a 50% bet. At Shiller P/E of 27, the success rate of a positive outcome drops to about 62%. At those odds, the Kelly output is 47% stocks. And so on. Both are considerably higher than 25% as recommended at this Shiller level in the Pfau article. Pretty big differences.

        Is it a reasonable request that a DQYDJ team member build a calculator in which an investor could plug in the various Shiller P/E levels, and based on the historical outcomes from the Credit Suisse findings and the Kelly calculator criterion running in the back ground, it would output stock allocations ?

        If that is too much to ask, I do have a question now that you understand how I am using the existing calculator. Since the BI article shows the dispersion of outcomes and subsequent 3 year returns (not 10 years as hinted in your original explanation for input) at the various Shiller levels, what are the best inputs I should be using given 3- year returns in the BI table at the current Shiller P/E of 26? (i.e. 66% success rate for a positive return)

        Thanks for your time and consideration.

        Sincerely,
        Irish57

        • PK says

          I think that is a reasonable request, yes – but I can’t guarantee it’ll come quickly (or before any future stock market falls, heh). Let me think on some of the options – it seems that any sort of a Kelly Calculator Reprise should do some sort of automatic odds calculations – perhaps with Shiller PE, or perhaps with something else (or even a blend). Have you seen our recent Shiller PE/CAPE calculator?

          My issue would be with considering something like 3 years ‘long term’. I don’t know what the exact definition of long term should be, but common rules of thumb are usually something like “if you need it in the next 5 years, be safe” or something like that – and we usually average around a decade for a full market cycle. So, I don’t think I can comment on how to adjust it for 3 years – but I’d suggest using a conservative Kelly value, perhaps half or quarter Kelly?

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