Maximizing Compounded Rate of Return

A simple formula that few traders utilize

Here is a little puzzle that may stymie many a professional trader. Suppose a certain stock exhibits a true (geometric) random walk, by which I mean there is a 50-50 chance that the stock is going up 1% or down 1% every minute. If you buy this stock, are you most likely, in the long run, to make money, lose money, or be flat?

Most traders will blurt out the answer “Flat!”, and that is wrong. The correct answer is you will lose money, at the rate of 0.5% every minute! That is because for a geometric random walk, the average compounded rate of return is not the short-term (or one-period) return m (1% here), but is m – s²/2, where s (also 1% here) is the standard deviation of the short-term return. This is consistent with the fact that the geometric mean of a set of numbers is always smaller than the arithmetic mean (unless the numbers are identical, in which case the two means are the same). When we assume, as I did, that the arithmetic mean of the returns is zero, the geometric mean, which gives the average compounded rate of return, must be negative.

This quantity m – s²/2 holds the key to selecting a maximum growth strategy. In a previous article (“How much leverage should you use?”), I described a scheme to maximize the long-run growth rate of a given investment strategy (i.e., a strategy with a fixed m and s) by leveraging. However, often we are faced with a choice of different strategies with different expected returns and risk. How do we choose between them? Many traders think that we should pick the one with the highest Sharpe ratio. This is reasonable if a trader fix each of his or her bet to have a constant size. But if you are a trader interested in maximizing long-run wealth (like the Kelly investor I mentioned in the previous article), the bet size should always be proportional to the compounded return. Maximizing Sharpe ratio does not guarantee maximal growth for multi-period returns. Maximizing m – s²/2 does.

For further reading:

Miller, Stephen J. The Arithmetic and Geometric Mean Inequality. ArithMeanGeoMean.pdf

Sharpe, William. Multi-period Returns. http://www.stanford.edu/~wfsharpe/mia/rr/mia_rr3.htm

Poundstone, William. (2005). Fortune’s Formula. New York: Hill and Wang.

How much leverage should you use?

Maximizing growth without risking bankruptcy

Many hedge fund disasters come not from making the wrong bets – that happen to the best of us – but from making too big a bet by overleveraging. On the other hand, without using leverage (i.e. borrowing on margin to buy stocks), we often cannot realize the full growth potential of our investment strategy. So how much leverage should you use?

Surprisingly, the answer is well-known, but little practiced. It is called the Kelly criterion, named after a mathematician at Bell Labs. The leverage f is defined as the ratio of the size of your portfolio to your equity. Kelly criterion says: f should equal the expected excess return of the strategy divided by the expected variance of the excess return, or

f = (m-r)/s²

(The excess return being the return m minus the risk-free rate r.)

This quantity f looks like the familiar Sharpe ratio, but it is not, since the denominator is s², not s as in the Sharpe ratio. However, if you can estimate the Sharpe ratio, say, from some backtest results of a strategy, you can also estimate f just as easily. Suppose I have a strategy with expected return of 12% over a period with risk-free rate being 4%. Also, let’s say the expected Sharpe ratio is 1. It is easy to calculate f, which comes out to be 12.5.

This is a shocking number. This is telling you that for this strategy, you should be leveraging your equity 12.5 times! If you have $100,000 in cash to invest, and if you really believe the expected values of your returns and Sharpe ratio, you should borrow money to trade a $1.2 million portfolio!

Of course, estimates of expected returns and Sharpe ratio are notoriously over-optimistic, what with the inevitable data-snooping bias and other usual pitfalls in backtesting strategies. The common recommendation is that you should halve your expected returns estimated from backtests when calculating f. This is often called the half-Kelly criterion. Still, in our example, the recommended leverage comes to 6.25 after halving the expected returns.

Fixing the leverage of a portfolio is not as easy or intuitive as it sounds. Back to our $100,000 example. Say you followed the (half-) Kelly criterion and bought a portfolio worth $625,000 with some borrowed money. The next day, disaster struck, and you lost 5%, or $31,250, of the value of your portfolio. So now your portfolio is worth only $593,750, and your equity is now only $68,750. What should you do? Most people I know will just stick to their guns and do nothing, hoping that the strategy will “recover”. But that’s not what the Kelly criterion would prescribe. Kelly says, if you want to avoid eventual bankruptcy (i.e. your equity going to zero or negative), you should immediately further reduce the size of your portfolio to $429,688. Why? Because the recommended leverage, 6.25, times your current equity, $68,750, is about $429,688.

Thus Kelly criterion requires you to sell into a loss (assuming you have a long-only portfolio here), and buys into a profit – something that requires steely discipline to achieve. It also runs counter to the usual mean-reversion expectation. But even if you strongly believe in mean-reversion, as no doubt many of the ruined hedge funds did, you need to consider protecting you and your investors from the possibility of bankruptcy before the market reverts.

Besides helping you to avoid bankruptcy, the Kelly criterion has another important mathematically proven property: it is a “growth-optimal” strategy. I.e. if your goal is to maximize your wealth (which equals your initial equity times the maximum growth rate possible using your strategy), Kelly criterion is the way.

Notice this goal is not the same as many hedge managers’ or their investors’ goal. They often want to maximize their Sharpe ratio, not growth rate, for the reason that their investors want to be able to redeem their shares at any time and be reasonably sure that they will redeem at a profit. Kelly criterion is not for such investors. If you adopt the Kelly criterion, there may be long periods of drawdown, highly volatile returns, low Sharpe ratio, and so forth. The only thing that Kelly guarantees (to an exponentially high degree of certainty), is that you will maximize the growth potential of your strategy in the long run, and you will not be bankrupt in the interim because of the inevitable short-term market fluctuations.

For further reading:

Poundstone, William. (2005). Fortune’s Formula. New York: Hill and Wang.

Thorp, Edward O. (1997; revised 1998). The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market. www.bjmath.com/bjmath/thorp/paper.htm

An arbitrage trade between energy stocks and futures

With a lesson costing 6 billion dollars, Amaranth has taught us an, albeit disastrous, arbitrage trading technique in energy futures: buying the March-over-April spread in natural gas futures, and betting that it will increase in value. Unfortunately for Amaranth and its head trader Brian Hunter, the spread decreased rather than increased in September, resulting in a $6 billion drop in value. Many commentators breathed a sign of relief that this has caused no widespread disruptions in the financial market, reasoning that this is too obscure a corner in arbitrage trading to matter. However, as we learned in the Long Term Capital Management debacle, spreads in the financial markets often move in tandem, especially during times of market stress. Since mid-August, I have been following another obscure spread heading towards a 3-year low, and it may present a profitable trading opportunity.

As oil prices go to historic highs during this past year, energy stocks have followed a similar course. For example, front-month light sweet crude oil E-mini contract QM reached a historic intraday high of $78.30 on July 14, while the energy sector exchange-traded fund XLE reaches its historic intraday high of $60.15 on May 11. Since energy companies typically own rights to oil either above or under ground, it is reasonable that their stock prices follow the price of oil. In technical terms, we say that energy stock price “cointegrates” with the crude oil price, a concept pioneered by the Nobel laureates Robert Engle and Clive Granger. To prove that they do in fact cointegrate, I ran a Matlab cointegration package developed at University of Toledo in Ohio on the closing prices of QM (using a perpetual futures series) and XLE for the last 3 years. The program determined that they cointegrate with a 95% probabilty. Now, what this does not mean is that QM and XLE prices will always move up or down in a similar percentage everyday. This also doesn’t mean that there won’t be periods of time when the spreads between QM and XLE will go way out of sync, just as the gas futures spread did for Amaranth. What this tells us is that with high probability, the spread will eventually goes back to their historic average, and then probably goes in the opposite direction for a while.

To illustrate this point, let’s take a look at a plot of the spreads between QM and XLE over the last 3 years. (Click on the graph twice to enlarge it.) Suppose we are long a front-month QM contract (rolling over the contract every month), and are simultaneously short 640 shares of XLE. The number of shares is determined by the Matlab package mentioned above. The y-axis shows the dollar value of this pair of positions. We can see that in the past 3 years, the value went as high as $5,550 on October 14^th, 2004, and as low as -$4,152 on February 16^th, 2006. The average is $57, which is almost zero. As of this writing (at the close of September 28, 2006), the value is -$2,584. While this is not near the 3-year low yet, it is getting there. Those who have a strong stomach will buy this spread now, and hope that the value will move back up to it long-run average of near zero. (Click on the graph twice to enlarge it.)

Some people may feel uneasy about trading oil futures because they have to keep rolling over to the next nearby contract every month, or maybe their brokerage doesn’t allow futures trading at all. There is now a convenient alternative: an exchange-traded fund called USO. This fund trades like a stock on the American Stock Exchange, just like XLE. USO closely reflects the value of the nearby contracts in crude oil (with a small percentage that reflects the value of other energy futures such as natural gas or heating oil). And yes, I have checked that it cointegrates equally well, if not better, with XLE.

Some thoughtful readers may wonder whether there are any fundamental reason energy stocks have dropped much less in value since the summer than energy futures prices. Now energy companies are valued much like any other companies: roughly speaking, their stock is worth the present value of their anticipated future cash-flow plus their current net asset value. The current net asset value certainly should follow the front-month crude oil contracts very closely, in fact, more closely than their stock price. However, their anticipated future cash-flow reflects the expected price of oil in the years to come, not the current cash price of oil. (For those readers who enjoy a bit of exercise, they can look up the oil contracts that expire in 2007, 2008 and beyond to see if they in fact has higher prices than the front contract.) At this time, the stock (and futures) market is telling us this: oil price will go back up in the future.

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A “Highly Improbable” Event?

A historical analysis of the natural gas spread trade that bought down Amaranth

Nick Maouis, the founder of Amaranth, claims that the 6-billion dollar loss that his fund suffers is due to a “highly improbable” event in the natural gas market. Some analysts have thrown doubts on this claim. To see how improbable this loss is, let’s take a quick look at the historical performance of this trade since 2000. This is not only of forensic (and perhaps legal) interest: if Mr. Maouis’ claim were true, it would have furnished us a glimpse of a potentially highly profitable trading strategy.

The bet that Amaranth and its head trader Brian Hunter made is that the March-over-April spread in natural gas futures will increase in value throughout the year prior to the contract expiration. Unfortunately for their investors, the spread decreases rather than increases in September, resulting in a $6 billion drop in value. We don’t know the exact time when Amaranth bought this spread. However, it is likely that they have started buying in April of this year. April is the time when the nation’s natural gas storage inventory coming out of the winter is known and thus provides a foundation to bet on next winter’s natural gas sufficiency. I plotted below the profit-or-loss of buying this spread (long one March contract of the following year, and short one April contract) in April and exiting the position at the end of September every year since 2000. (Click on the graph twice to make it bigger.)

To my surprise, this trade loses money 3 out of 6 previous years. The one year that this trade was very profitable is 2005: it made more than $16,000 profit per pair of contracts. This is consistent with a Wall Street Journal report that Mr. Hunter made $1 billion for Amaranth in 2005. That was indeed due to an improbable event last year: Hurricane Katrina.

Note also from the 2006 graph that, consistent with news reports, the trade was actually quite profitable up till the beginning of September. This paper profit may not be easily realized by Mr. Hunter though, since a lot of it may be due to his aggressively increasing his position and driving up the market.

Now there can be several objections to my analysis. You might think that if we hold on to this spread position longer, say till December, it would have been more profitable historically. My research shows otherwise. Holding till December would have resulted in losing 4 out of the previous 5 years, losing even in 2005. You might also argue that this is an extremely simplistic version of Mr. Hunter’s strategy. No doubt Mr. Hunter used various complex options strategies, continuously adjusted with various fundamental factors such as weather prediction and natural gas inventory reports as inputs. However, from a risk management point of view, the portfolio that Mr. Hunter owns seems highly correlated to a plain vanilla spread position that I described. The fact that this plain vanilla position loses money half the time historically would not have been reassuring.

In a future article, I will describe some calendar spread trades in energy futures that do have a much better profit consistency.