The Kelly Formula: Growth-Optimized Money Management Not for the Faint of Heart

Feb. 4.11 | About: PetMed Express, (PETS)

A law of the theory of betting is that the optimal procedure is to bet proportionally to one's advantage, adjusted by variance. This is the well-known "Kelly Formula" (aka 'Kelly Criterion'), discovered by John Kelly in the 1950’s. It results in the maximum expected rate of bankroll growth, and is mathematically the optimal strategy for money management in betting games.

The Kelly Formula was popularized by Ed Thorp in his 1962 book “Beat the Dealer” This book inspired millions of gamblers and stock investors alike. The concept is simple – when the odds are good, you bet a higher percentage of your bankroll. All great investors use this formula, either implicitly or explicitly. When Warren Buffett managed much smaller sums of money back in the ‘60’s, he is known to have placed up to 40% of the portfolio in a single stock.

The Kelly Formula is: Kelly % = W - (1-W)/R where:

  • Kelly % = percentage of capital to be put into a single trade.
  • W = Historical winning percentage of a trading system.
  • R = Historical Average Win/Loss ratio.

So why doesn’t every money manager follow this simple formula? It’s because betting the standard Kelly % entails wild swings, which are not for the faint of heart. Betting the straight Kelly % results in a 1/3 chance of halving a portfolio before it doubles. Many investors will not hold on to a stock during such extreme drawdowns. A more risk-averse strategy used by some is to scale things back and bet a fraction (such as ½ or 1/3) of the Kelly bet. This is done commonly by blackjack and poker players. The risk of a 50% drawdown of your bankroll is significantly reduced with fractional Kelly betting (less than 1% at ¼ Kelly).

The Kelly formula (and fractional Kelly betting) easily applies to simpler betting games such as Poker, where the gambler can calculate the exact odds, has an immediate payback, and has only one investing opportunity per unit of time (a single hand).

The stock market requires a special application of the Kelly formula because the Odds received cannot be known precisely, the stock investor must wait up to several years for his payback, and the stock market offers thousands of investing opportunities each day. While many investors may walk away from Kelly betting due to these difficulties, I do not think they are insurmountable. Ed Thorp himself ran a hugely successful hedge fund in the 1970’s and ‘80’s using Kelly Formula principles.

I’ve input the Kelly Formula into Excel, and created a spreadsheet with adjustments for stock market investing. Once a favorable stock investment opportunity is identified, I’m going to use this spreadsheet to decide how much of my portfolio to invest. Again, the idea is that you want to find that fraction which maximizes the amount of money you expect to win over a lifetime of investing.

Below is a photo of the spreadsheet with some standard inputs (you can purchase this spreadsheet on the ‘Research Offers’ page of my website). I can easily change the assumptions to generate new Kelly fractional bets. Cells colored Blue require User input, those in yellow show several versions of Kelly fractions. Column definitions with further explanations follow:

The Kelly Formula for Stock Investing

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The Kelly Formula for Stock Investing (Click on Image or press Ctrl+ for larger)Click to enlarge

Column Definitions

(1) This is the discount of a stock from its fair value. For example, you calculate a stock’s value as $100, and the stock is selling for $75 in the market, giving a 25% discount. My own minimum discount is 25%. I have the spreadsheet set up to automatically increase the discount by 5% each successive row but this can easily be changed . Side note, for more information about calculating fair value, see my post here.

(2) One of my favorite Warren Buffett quotes is: “The price you pay determines your rate of return.” column 2, an input into the Kelly Formula, is calculated automatically and displays the rate of return for each given discount in column 1. For example, that $100 stock selling for $75 presents a 100/75 = 1.33, or 33% return opportunity.

(3) and (4) Column 3 is the probability of a winning trade, and column 4 the probability of a Losing trade. Examine your trading history to determine this (ie. 12 winning trades out of 20 total gives 60% Winners) These columns have the greatest impact on the Kelly equation. Be conservative here, thinking of both bull and bear markets. Column 4 is calculated for you as (1-column 3).

(5) The Standard Kelly %, only for the brave of heart, or for use at your Friday night Poker game where you don’t mind some wild swings.

Columns 6 through 11 provide adjustments due to the realities of stock investing.

(6) Column 6 addresses the issue of not knowing a stock’s true fair value precisely. Some stocks you may have 90% or more confidence in your calculation of its fair value. Others may be far less. For example, I can more confidently estimate the value of a steady grower like Johnson & Johnson (NYSE:JNJ) or Coca-Cola (NYSE:KO) than a small-cap stock with erratic earnings. I leave this value as a judgment call. If I believe I can estimate a stock’s true value within 10%, I’ll use a 90% confidence. Greater range of value will lead to less confidence.

(7) Column 7 is simply column 6 multiplied by column 2.

(8) Column 8 introduces the ‘Time Value of Money’. At the base of the column, the user enters the expected number of years for a discounted stock to reach fair value (3 years is often given in literature), and the return on cash as an alternate investment opportunity.

(9) Column 9 shows the impact of columns 6 and 8 on the Kelly Formula. As seen by comparing column 9 to column 5, a high confidence level and current low level of return on cash result in a somewhat minimized impact on the optimal Kelly fraction. But don’t be fooled, changing those assumptions has very interesting results. I call this the ‘Non-Diversified’ Kelly %, as it does not yet account for a situation such as the stock market, with many simultaneous investment opportunities.

10) Column 10 is the ‘magic formula’ which provides diversification, found in an Ed Thorp paper written in 1997 describing (among other things) how to adjust the Kelly approach when multiple investing opportunities - such as encountered in the stock market – are offered. Thorp first used this formula for sports betting in Las Vegas, where he could find many decent bet opportunities each day. The formula is fairly simple, and is always a ratio less than the difference between winning and losing percentage (column 3 – column 4).

11) Column 11 presents the final modified Kelly fraction, which accounts for all prior adjustments as well as the diversity found in stock investing. Naturally, the investor places a greater fraction of his portfolio in stocks which provide a greater opportunity of return, and this column shows how.

Undervalued Stock Application
Petmed Express (NASDAQ:PETS), better known as 1-800-PETMEDS, is a highly successful pet pharmacy specializing in online and direct-to-consumer marketing of pet medications and health products. This small-cap stock has had consistent earnings growth for the last 10 years. PETS has no debt, holds over 20% of its market value in cash, offers an attractive 3.3% dividend yield, and management has been shareholder-friendly. The PETS stock is at $14.72 per share, and I calculate fair value at $22.60, giving us a 35% discount from fair value.

Given its steady earnings growth, I'm at a high (90%) confidence on fair value. Utilizing the Kelly Formula, I should place 5% of my portfolio in PETS stock (column 11) and hold until the market recognizes this value. If I could find 19 more stocks offering this set of discount/confidence levels, I would construct a 20 stock portfolio of undervalued stocks.

Further Notes on the Spreadsheet Results:

a) Using the results shown in column 11, an investor would end up with a 10 to 25 stock portfolio (ie, 10 stocks offered at an 80% discount each, up to 25 stocks offered at a 25% discount each). I’ve found that many value investing managers have said that range of number of stocks was an ideal portfolio. Personally, I’d be willing to hold just 10 stocks if I can purchase each at an 80% discount to fair value.

b) Joel Greenblatt, hedge fund manager and creator of the Magic Formula stock investing method, used a 25 stock portfolio to generate 30% annual returns for almost 20 years. I note that the ‘Diversified’ Kelly fraction of 4% (25 stocks) correlates well with an expected return of 33% (column 2).

c) In my own experience, I can now see where I should have been using a method such as this to place a higher percentage of my portfolio in stocks with a greater discount to fair value. The best two stocks that I’m currently holding (HOS and GOK - average return of 84% in 6 months - were deeply discounted to fair value, around 70%. Yet, I only bought 3% of each, when I should have invested 9% each (18% total) – see column 11 using current assumptions. The resulting gains in my portfolio would have been significantly more.

d) Warren Buffett has said investors should invest as if they will only purchase 20 stocks in their entire lives. If you want to play Warren Buffett, and have high confidence in your abilities – then use column 9, the ‘Non-Diversified’ fractional Kelly %. But you can expect significant volatility (hopefully, of course, only on the upside).

Disclosure: I am long PETS.