Though basic, this illustration draws out two crucial points for investors of all stripes:
• An intelligent investor needs an edge (a view different than that of the market); and
• An investor needs to properly allocate capital to maximize value when an investment idea
does appear.
The Kelly formula contributes to a larger concept known as the Kelly Criterion, or Kelly system.
Based on information theory, the Kelly Criterion says an investor should choose the investment(s) with the highest geometric mean return. This strategy is distinct from those based on mean/variance efficiency. Importantly, however, you can calculate geometric mean using the same arithmetic mean and variance from mean/variance models.
Mathematician and investor Ed Thorp is probably the Kelly Criterion’s most visible advocate and successful practitioner. In the early 1960s, Thorp developed a system of card counting to improve a player’s odds in the card game blackjack and complemented it with the Kelly system to optimize wealth building. 11 Thorp went on to co-found Princeton-Newport Partners, delivering 20 percent annual compounded returns, with a 6 percent standard deviation, over a 20-year span via various investment strategies.
In his book, The Mathematics of Gambling, Thorp explains the Kelly system’s attractive
Features:
1. The chance of ruin is “small.” Because the Kelly system is based on proportional bets, losing all of your capital is theoretically impossible (assuming money is infinitely divisible). Even so, a small chance of a significant drawdown remains.
2. The Kelly system is highly likely to grow a bankroll faster than other systems. Provided comparably attractive opportunities continue to appear, there is a high probability the system will generate a bankroll that exceeds other systems by a determinable multiple.
3. You tend to reach a specified level of winnings in the least average time. If you have a financial end goal in mind and continuous opportunities, the Kelly system will likely allow you to achieve the objective in a shorter time than other systems.
In short, the Kelly system has proven to be both theoretically sound and useful for practitioners.
Still, the most enthusiastic supporters for the approach (information theorists, mathematicians, gamblers, and traders) do not include mainstream economists. We now turn to some of the more practical constraints with the Kelly system, and we contrast the Kelly system with mean/variance efficiency.
Practical Considerations with the Kelly Criterion and Mean/Variance
Under ideal conditions the Kelly Criterion is clearly a powerful concept. Using the Kelly formula’s optimal betting strategy in our coin-toss example is unquestionably valuable. The real world, however, presents a great deal more complexity than a coin toss or blackjack table. In the stock market an investor faces many more outcomes than a gambler in a casino. That said, the Kelly Criterion works well when you parlay your bets, face repeated opportunities, and know what the underlying distribution looks like. We now take a look at these conditions, using the opportunity to compare the Kelly Criterion to mean/variance efficiency.
Parlaying bets. You can approach financial opportunities with one of two betting strategies: bet the same amount each time or reinvest your winnings. As it turns out, what you look for will be very different based on which strategy you select.
Kelly recognized this, writing: “suppose the gambler’s wife allowed him to bet one dollar each week but not to reinvest his winnings. He should then maximize his expectation (expected value of capital) on each bet.” In other words, if you employ the first strategy, you should focus on average payout calculated with the arithmetic mean. In this case, the mean/variance approach is the way to go.
In contrast, the Kelly Criterion assumes you parlay your bets, and says you should choose the opportunities with the highest geometric means.
As an illustration of the difference between arithmetic and geometric returns, consider the following stock price changes (this may be reminiscent of the late 1990s and early 2000s):
T0/100$ T1/$200 T2/$20
What is the arithmetic average return from T0 to T2? The answer is simply the sum of the changes (100 percent + -90 percent = 10 percent) divided by the number of periods (2). The
arithmetic average is 5 percent (10 percent/2).
In contrast, the geometric average is the product of the changes (2.0 x .1) to the Nth root (2) minus 1.
sqrt (2.0 X0.1)-1 = 55.3%
In this case, the arithmetic average shows 5 percent while the geometric average is negative 55
percent. Notably, the geometric mean is always less than or equal to the arithmetic mean. The greater the variance, the larger the difference between the arithmetic and geometric mean. Additionally, if a series contains a single payoff of zero, the geometric mean is always zero. Play a game with a zero payoff long enough and you are assured ruin.
Exhibit 3 reproduces three series of payoffs with varying arithmetic returns, variances, and geometric returns that Poundstone uses in Fortune’s Formula:
Source: Poundstone, 198. If you bet the same amount every time, like Kelly’s once-a-week gambler, you should focus on the arithmetic means. Mean/variance doesn’t determine the best series because individuals may have different preferences. Both the risk and returns rise for these series as you move from left to right. Determine your risk preference and you can settle on the best strategy for you. Clearly, though, the highest expected payoff is with series C.
In contrast, the parlay bettor using the Kelly Criterion will always choose series A. According to Poundstone’s calculation, starting with $1 and reinvesting profits each week for a year leads to an expected fortune of over $67 million. The same strategy with series C amasses an expected value of just under $38,000.
Series B has a favorable arithmetic mean, but the geometric mean is zero. This happens because one of the payoffs is zero, which means you will lose all of your money with this strategy given enough trials.
Leaving aside the technical details of the Kelly Criterion, the central message for investors is that standard mean/variance analysis does not deal with the compounding of investments. If you seek to compound your wealth, then maximizing geometric returns should be front and center in your thinking.
Repeated trials. Both the Kelly Criterion and mean/variance approaches assume lots of trials, or financial propositions. The probabilistic nature of most market-based financial propositions means you need a substantial number of observations to reasonably assure you capture the system’s signal, versus short-term noise.
Know the distribution. Long-term stock market investing differs from casino games, or even trading, because outcomes vary much more than a simple model suggests. Any practical money management system faces the challenge of correcting for more complicated real-world distributions.
Substantial empirical evidence shows that stock price changes do not fall along a normal distribution. Actual distributions contain many more small change observations and many more large moves than the simple distribution predicts. These tails play a meaningful role in shaping total returns for assets, and can be a cause of substantial financial pain for investors who do not anticipate them.
As a result, mean and variance insufficiently express the distribution and mean/variance can at
best crudely approximate market results. Notwithstanding this, practitioners assess risk and reward using a majority of analytical tools based on faulty mean/variance metrics. |
