Saturday, January 22, 2011

Betting Size: Kelly Criterion and Portfolio Variance - Ari Paul

In this issue:
1) Kelly Criterion
2) Portfolio Variance
3) Rules of Thumb
4) Don't Diversify with Dutch Tulip Bulbs

Dear Friends, Colleagues, and Investors,

After identifying a variety of attractive investments, how much of our capital should we risk? Should we use leverage or hold cash in reserve? Should we invest all our money in our best idea or diversify broadly, even if that means devoting money to less attractive investments?


1) The Kelly Criterion
The scientist John Kelly created a simple formula that describes the optimal strategy for non-correlated bets. Before delving into the formula, let's look at a simple example. Let's say that you're going to make a series of 100 bets on a coin flip. The coin is weighted and thus lands on heads 60% of the time. You have $100. How much should you bet each time? It’s tempting to think that you should bet your entire bankroll every time to maximize your expected returns, but this is incorrect. While betting your bankroll does indeed maximize your expected value for any individual bet, it ignores the fact that you have a limited bankroll, and once you lose it, you also lose the opportunity to make future profitable bets. If you bet your entire bankroll each time, you will likely go bankrupt by the fourth bet, so you'll miss out on future advantageous bets.

Here's the Kelly Criterion formula:

where:

  • f* is the fraction of the current bankroll to wager;
  • b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
  • p is the probability of winning;
  • q is the probability of losing, which is 1 − p.

The most intuitive way to think about Kelly’s approach is to consider the outcome of a full “set” of bets. Let’s imagine that we bet our entire bankroll on that 60% weighted coin. If we bet 10 times, we can expect to double our portfolio 6 times, but then we will go bankrupt. If instead we only bet 20% of our portfolio, our bankroll will grow by a factor of 1.2 when we win, and shrink by a factor of 0.8 when we lose. The growth looks like this: 1.2*1.2*1.2*1.2*1.2*1.2*0.8*0.8*0.8*0.8 for a total return of about 22%. If we increased our bet size to 30% of our portfolio, the total return would fall to 16%.

While the Kelly Criterion is a good starting point, it has a lot of problems. First, it assumes that bets are uncorrelated. If I have 20 equities in my portfolio, their returns are certainly very correlated; a global depression will send all equities tanking. Even if we use Kelly for sequential bets, the correlation problem remains. In the real world, corporate earnings, GDP growth, and even stock market returns are not independent events each quarter; the current period impacts the next. Finally, the Kelly Criterion requires that we know the “edge” of any bet. We can never be sure of “edge”, and we are very likely to estimate it poorly. As with any formula, garbage in = garbage out.

The are numerous way to improve on Kelly for investors. First, we can adjust the Kelly weighting by the beta of the security to the rest of our portfolio; this will better take advantage of the benefits of diversification. For example, if Kelly recommends a weighting of 10% of our bankroll but the security has a negative beta to our portfolio, we will probably be happy with a higher weighting, since the security reduces our portfolio volatility. Second, we can recognize that the Kelly allocations will be too high because they fail to account for correlation and we can systematically reduce the allocations. Several prominent hedge fund managers routinely use the Kelly Criterion but divide the result by 2. Finally, we can use Kelly as a rough starting point for deciding how much capital to risk for a given amount of expected reward.

The greatest value of the Kelly Criterion is that it demonstrates that even when we’re sure we have found a great investment, we only want to commit a small portion of our portfolio.

2) Portfolio Variance
A complimentary approach to portfolio construction looks at the relative attractiveness of securities. Many fund managers feel more comfortable investing from the bottom up than trying to time the market. Mutual fund managers and pension managers are in the same boat because they generally prefer to be 100% long the market. To what extent should these managers concentrate their capital in their best idea? Even if you are a market timer, you must consider how to divide your capital between your best and second best investment ideas.

Imagine we’re considering a simple portfolio with two assets, A and B. We believe that asset A will return 20% a year and asset B will return 15% a year, and they are uncorrelated. How should we distribute our funds? To maximize our expected value we would put all our money in Asset A, but we can earn almost as much with much lower risk through diversification. If we split our money evenly between the two assets we can expect a 17.5% return but we cut our portfolio’s variance in half! In the real world, very few assets are uncorrelated and managers face a more subtle decision. Calculating portfolio variance for a portfolio of 50 assets is hard by hand but easy with the proper software. The problem is that, as with any formula, garbage in = garbage out. If we make a poor estimation of the covariance of assets, we may end up with much more risk than we expected.

3) Rules of Thumb
We can use a computer program to calculate the risk/return profiles of possible portfolios, but this is so imprecise as to be little better than simple rules of thumb in most cases.

Here's a very simple approach: Let’s start with the world of equities. Compared to holding just 1 stock, a portfolio of 4 stocks will have roughly 65% of the volatility, 12 will have 50%, and a portfolio of 60 stocks will have 40% of the volatility. As you can see, diversifying beyond 12 equities provides little benefit. We need to tweak this assumption when considering stocks that have very low correlation to our portfolio. For example, if you hold a dozen pro-cyclical stocks (like airplane and car makers), adding a couple counter-cyclical stocks (like an outplacement agency or fast food company) to your portfolio would provide significant diversification benefits.

In allocating capital between two stocks with average correlation (roughly 0.6), a useful rule of thumb is to adjust the capital allocation by the square root of the edge differential. This is a lot simpler than it sounds. For example, let’s say I have a portfolio consisting of only Apple and Caterpillar and I believe that Apple will appreciate by 20% next year and Caterpillar by only 10%. Since Apple has twice the edge of Caterpillar, I will devote sqrt(2) as much capital to Apple. If I had $100, I would put $58 into Apple and $42 into Caterpillar. This simplistic approach will generally deliver a happy medium between maximizing your returns and minimizing your portfolio’s risk. If the two stocks are very highly correlated, there is less benefit from diversification, so we might put $80 into stock A and only $20 into B. Alternatively, if the two assets are completely uncorrelated, the diversification benefits are greater so we should consider a 50/50 split.

If we’re building a long equity portfolio, we should aim for 6-20 stocks (depending on the prevalence of opportunities and our time available to analyze them), and overweight the most attractive investments by the square root of the extra edge they provide
.

When we include non-equity assets, the issue of correlation becomes much more difficult. Correlation deserves (and will get) it's own Risk over Reward essay, but I'll offer some simple ideas today. A very conservative approach is to assume that correlations will rise to their historic highs. For example, commodities and equities have at various times had very negative correlations, very positive correlations, and even been uncorrelated. If I have a portfolio that is long commodities and long equities, a very conservative measure of its risk assumes that the assets are very positively correlated. The absolute most conservative assumption is a correlation of 1. If I assumed a very negative correlation, the portfolio would look like it had very little risk; a poor assumption because even if correlations are negative today, they may become positive in the future.

4. Don't Diversify with Dutch Tulip Bulbs
Most investment managers are chronic over-diversifiers. A portfolio of 20 stocks has almost the same risk as a portfolio of 100 stocks, so why dilute your best investment ideas? Warren Buffett wrote: "Many pundits would therefore say [that concentrating your capital] must be riskier than [diversifying like] more conventional investors. We disagree. We believe that a policy of portfolio concentration may well decrease risk if it raises, as it should, both the intensity with which an investor thinks about a business and the comfort-level he must feel with its economic characteristics before buying into it." Another great quote from the oracle, "Diversification is a protection against ignorance. It makes very little sense for those who know what they're doing."

The very best investment managers like Warren Buffett and Seth Klarman usually have 70%+ of their capital in just 5 stocks! Diversification is a crutch used by managers who lack the confidence or will to perform deep due diligence on the companies in which they invest.

The modern twist on this fallacy is diversifying with asset classes. Funds are putting capital in commodities and real estate because those asset classes historically have a low correlation to equities...but that's not a good reason to put your money in an asset. No matter how uncorrelated a bad investment is to your portfolio, it's still a bad investment.

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  2. There's actually a game theory based equilibrium strategy well connected with the Kelly criterion regarding diversifying with multiples asset classes. The concept is you position evenly in all asset classes because capital has to go somewhere and you rebalance to take advantage of capital flow. That positions you to profit over time regardless of how unevenly capital is positioned provided there is movement and either no deflation or proportionally benefiting from inflation and deflation.

    Kelly criterion basically has you allocating proportional to the edge. There's a problem in the book fortune's formula that uses the example of horse betting and talks about risk management with multiple bets and the simple solution is to allocate proportional to the chances of winning... assuming your total edge is based upon having inside knowledge to more accurately handicap the horses then the crowd. In the market that would basically be like investing in the assets based upon probability of risk adjusted outperformance.

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