# Optimize Your Portfolio with the Kelly Formula

Suppose there were a simple and elegant formula that helped you to maximize your long-term investment returns while minimizing the risk of total ruin. Sounds like something you'd like to use for your portfolio, doesn't it? In his recent book, Fortune's Formula, author William Poundstone details the development, use, and criticism of an equation that purports to do just what I have described. It is known as the Kelly Formula.

In the 1950s, Bell Labs scientist Claude Shannon's work on the way information is transmitted gave birth to the field of information theory. Shannon's key insight was that information is tied to chance. As Poundstone says of Shannon's concept on Page 56 of Fortune's Formula, "Information exists only when the sender is saying something that the recipient doesn't already know and can't predict. Because true information is unpredictable, it is essentially a series of random events like spins of a roulette wheel or rolls of a dice."

To illustrate this concept, Poundstone used the example of a phone company ad in which a rancher orders 200 oxen. Because of poor phone quality the rancher was misunderstood and instead received 200 dachshunds. While most people would think a rancher ordering 200 dachshunds is a low-probability event, the person taking the order did not have more information, so this improbable request seemed entirely reasonable.

At this point, you're probably wondering how all of this relates to the investment field. Well, around the same time of Shannon's work, another Bell Labs scientist--John Kelly--took Shannon's proposition that information was probabilistic one step further. He recognized that having information that was not already widely dispersed to the "market" provides an advantage. Thus, he developed an equation--the Kelly Criterion--to indicate the optimal percentage of one's bankroll that should be bet if you possess superior information. Kelly applied this to betting on horse racing or sports; others have applied it to investing in the stock market.

Even though the mathematics underlying the Kelly Criterion are complex, the simplified edge/odds approximation of this equation--also known as the Kelly Formula--is very practical. The "edge" in this estimate can be thought of as your expected winnings given your information advantage, and "odds" can be thought of as the consensus expectations of your system or market. As Poundstone writes in Fortune's Formula, the power of this formula is that in the long run it "�offers the highest compound return consistent with no risk of going broke."

It is the "no risk of going broke" proposition that has infuriated many economists, most prominent among them Nobel laureate Paul Samuelson. These economists have argued that the long run might be so long that many individual investors will never realize the high compound returns that Kelly's formula promises. What's more, while the Kelly Formula may help investors avoid going broke, as Poundstone says on Page 225 of Fortune's Formula, there "�is an inadequate guarantee of safety." For example, if an investor only has a \$100,000 portfolio, and loses 90% of his wealth, he may not be left with nothing, but the remaining \$10,000 is insufficient to provide for his needs.

Morningstar's Approach
While I'll be the first to acknowledge some of the drawbacks of the Kelly Formula and will also admit that it's not the complete answer to portfolio management, I do think that anything which can help investors get a feeling for rank ordering their position sizes is worth exploring.

Thus, I've come up with an approximation of Kelly position sizes using metrics from Morningstar's research. I've labeled our edge under the Kelly Formula as the expected excess annual rate of return for a 5-star stock over the next three years, which is based on its percentage discount to our fair value estimate (or margin of safety) and our cost of equity assumption for the firm. I've called the odds under our system the likelihood of Morningstar's 5-star stocks outperforming the market since we started rating stocks.

A few weeks ago, I wrote an article about the holdings of my favorite value managers, and in the table below I've listed the five most common holdings of these managers, along with my approximation of the Kelly allocation for each of the five stocks based on the Morningstar Rating for stocks as of Sept. 25.

 Managers' Common Holdings and Kelly Allocations Fair Value Estimate Cost-of-Equity Assumption 3-Year Annual Expected Return 5-Star Odds of Outperformance Full Kelly Alloc. Half Kelly Alloc. Tyco TYC \$36 10.50% 21.19% 1.78 6.00% 3.00% \$58 9.01% 15.45% 1.78 3.62% 1.81% Dell DELL \$39 9.00% 31.60% 1.78 12.70% 6.35% \$4,303 10.00% 22.40% 1.78 6.97% 3.48% Time Warner TWX \$20 9.50% 13.14% 1.78 2.05% 1.02% * Fair value estimates as of 09-25-06

Let's take  Tyco TYC as our example. As of Sept. 25, our fair value estimate for Tyco was \$36 per share. Given our 10.5% cost-of-equity assumption for Tyco and the stock's Sept. 25 stock price of \$27.29, we'd expect Tyco's annual return to approximate 21.19% per year over the next three years. The difference between the expected annual return and the cost-of-equity assumption can be thought of as our expected excess return per year, which in the case of Tyco is 10.69%. This is our edge. For the odds portion of our equation, I've calculated Morningstar's success rate in selecting 5-star stocks that outperform a cap-weighted S&P 500 benchmark. Based on my calculations, our success rate since inception is approximately 56%. Thought of another way, one out of every 1.78 5-star stocks will outperform the market. I will note, however, that this is sensitive to the benchmark selected. Had I chosen a different benchmark, this percentage could have been modestly lower. Nevertheless, dividing 10.69% by 1.78 yields our full position size for Tyco of 6%.

Now, for most investors, having as much as 6% of their assets invested in one stock may seem too aggressive. In addition, by following the full Kelly allocation, one may end up having most of his or her assets invested in only a few stocks. For some, a portfolio with this type of concentration may also seem too risky. Thus, taking a fraction of our estimated full Kelly allocation--in our case I've shown a half Kelly allotment--will yield more-conservative position sizes.

Given that the Kelly Formula is only an approximation of the Kelly Criterion, and that the metrics I've used in this article are only estimates of a myriad of potential edge/odds combinations, it should be noted that these percentages are not hard and fast rules for portfolio management. Rather, I think they should be thought of as a framework that an investor can use in order to rank the attractiveness of potential investments when allocating capital among a group of stocks. The key concept to remember, though, is that the larger your edge--think margin of safety--the bigger the investment you should make.

Justin Fuller, CFA has a position in the following securities mentioned above: DELL. Find out about Morningstar's editorial policies.