All trading strategies suffer occasional losses, technically known as drawdowns. The drawdowns may last a few minutes or a few years. To profit from a quantitative trading business, it is essential to manage your risks in a way that limits your drawdowns to a tolerable level and yet be positioned to use optimal leverage of your equity to achieve maximum possible growth of your wealth. Furthermore, if you have more than one strategy, you will also need to find a way to optimally allocate capital among them so as to maximize overall risk-adjusted return.

The optimal allocation of capital and the optimal leverage to use so as to strike the right balance between risk management and maximum growth is the focus of this chapter, and the central tool we use is called the Kelly formula.

OPTIMAL CAPITAL ALLOCATION AND LEVERAGE

Suppose you plan to trade several strategies, each with their own expected returns and standard deviations. How should you allocate capital among them in an optimal way? Furthermore, what should be the overall leverage (ratio of the size of your portfolio to your account equity)? Dr. Edward Thorp, whom I mentioned in the preface, has written an excellent expository article on this subject in one of his papers (Thorp, 1997), and I shall follow his discussion closely in this chapter. (Dr. Thorp's discussion is centered on a portfolio of securities, and mine is constructed around a portfolio of strategies. However, the mathematics are almost identical.)

Every optimization problem begins with an objective. Our objective here is to maximize our long-term wealth—an objective that I believe is not controversial for the individual investor. Maximizing long-term wealth is equivalent to maximizing the long-term compounded growth rate g of your portfolio. Note that this objective implicitly means that ruin (i.e., equity's going to zero or less because of a loss) must be avoided. This is because if ruin can be reached with nonzero probability at some point, the long-term wealth is surely zero, as is the long-term growth rate.

(In all of the discussions, I assume that we reinvest all trading profits, and therefore it is the levered, compounded growth rate that is of ultimate importance.)

One approximation that I will make is that the probability distribution of the returns of each of the trading strategy i is Gaussian, with a fixed mean m_i and standard deviation s_i. (The returns should be net of all financing costs; that is, they should be excess returns.) This is a common approximation in finance, but it can be quite inaccurate. Certain big losses in the financial markets occur with far higher frequencies (or viewed alternatively, at far higher magnitudes) than Gaussian probability distributions will allow. However, every scientific or engineering endeavor starts with the simplest model with the crudest approximation, and finance is no exception. I will discuss the remedies to such inaccuracies later in this chapter.

Let's denote the optimal fractions of your equity that you should allocate to each of your n strategies by a column vector F^* = (  f^*₁, f^*₂, …, f^*_n)^T. Here, T means transpose.

Given our optimization objective and the Gaussian assumption, Dr. Thorp has shown that the optimal allocation is given by

Here, C is the covariance matrix such that matrix element C_ij is the covariance of the returns of the i^th and j^th strategies, –1 indicates matrix inverse, and M = (m₁, m₂, …, m_n)^T is the column vector of mean returns of the strategies. Note that these returns are one-period, simple (uncompounded), unlevered returns. For example, if the strategy is long $1 of stock A and short $1 of stock B and made $0.10 profit in a period, m is 0.05, no matter what the equity in the account is.

If we assume that the strategies are all statistically independent, the covariance matrix becomes a diagonal matrix, with the diagonal elements equal to the variance of the individual strategies. This leads to an especially simple formula:

This is the famous Kelly formula (for the many interesting stories surrounding this formula, see, for example, Poundstone, 2005) as applied to continuous finance as opposed to gambling with discrete outcomes, and it gives the optimal leverage one should employ for a particular trading strategy.

Interested readers can look up a simple derivation of the Kelly formula at the end of this chapter in the simple one-strategy case.

Often, because of uncertainties in parameter estimations, and also because return distributions are not really Gaussian, traders prefer to cut this recommended leverage in half for safety. This is called half-Kelly betting.

If you have a retail trading account, your maximum overall leverage l will be restricted to either 2 or 4, depending on whether you hold the positions overnight or just intraday. In this situation, you would have to reduce each f_i by the same factor l/(|f₁|+|f₂|+…+|f_n|), where |f₁|+|f₂|+…+|f_n| is the total unrestricted leverage of the portfolio. Here, we ignore the possibility that some of your individual strategies may hold positions that offset each other (such as a long and a short position each balanced with short and long T-bills, respectively), which may allow you to hold a higher leverage than this formula suggests.

I stated that adopting this capital allocation and leverage will allow us to maximize the long-term compounded growth rate of your equity. So what is this maximum compounded growth rate? It turns out to be

where the S is none other than the Sharpe ratio of your portfolio! As I mentioned in Chapter 2, the higher the Sharpe ratio of your portfolio (or strategy), the higher the maximum growth rate of your equity (or wealth), provided you use the optimal leverage recommended by the Kelly formula. Here is the simple mathematical embodiment of this fact.

Note that following the Kelly formula requires you to continuously adjust your capital allocation as your equity changes so that it remains optimal. Based on the SPY example (Example 6.2), let's say you followed the Kelly formula and bought a portfolio worth $252,800. The next day, disaster struck, and you lost 10 percent on SPY. So now your portfolio is worth only $227,520, and your equity is now only $74,720. What should you do now? Kelly's criterion will dictate that you immediately reduce the size of your portfolio to $188,892. Why? Because the optimal leverage of 2.528 times the current equity of $74,720 is $188,892.

As a practical procedure, this continuous updating of the capital allocation should occur at least once at the end of each trading day. In addition to updating the capital allocation, one should also periodically update F^* itself by recalculating the most recent trailing mean return and standard deviation. What should the lookback period be, and how often do you need to update these inputs to the Kelly formula? These depend on the average holding period of your strategy. If you hold your positions for only one day or so, then as a rule of thumb, I would advise using a lookback period of six months. Using a relatively short lookback period has the advantage of allowing you to gradually reduce your exposure to strategies that have been losing their performance. As for the frequency of update, it should not be a burden to update F^* daily once you have written a program to do so.

One last point: Some strategies generate a variable number of trading signals each day, which may result in a variable number of positions and thus total capital each day. How should the Kelly formula be used to determine the capital in this case when we don't know what it will be beforehand? One can still use the Kelly formula to determine the maximum number of positions and thus the maximum capital allowed. It is always safer to have a leverage below what the Kelly formula recommends.

RISK MANAGEMENT

We saw in the previous section that the Kelly formula is not only useful for the optimal allocation of capital and for the determination of the optimal leverage, but also for risk management. In fact, the SPY example (Example 6.2) illustrated that the Kelly formula would advise you to reduce the portfolio size in the face of trading losses. This selling at a loss is the frequent result of risk management, whether or not the risk management scheme is based on the Kelly formula.

Risk management always dictates that you should reduce your position size whenever there is a loss, even when it means realizing those losses. (The other face of the coin is that optimal leverage dictates that you should increase your position size when your strategy generates profits.) This kind of selling is believed by some analysts to be the cause of “financial contagion” affecting many large hedge funds simultaneously when one faces a large loss.

An example of this is the summer 2007 meltdown, described in the previously cited article “What Happened to the Quants in August 2007?” by Amir Khandani and Andrew Lo (Khandani and Lo, 2007). During August 2007, under the ominous cloud of a housing and mortgage default crisis, a number of well-known hedge funds experienced unprecedented losses, with Goldman Sachs's Global Alpha fund falling 22.5 percent. Several billion dollars evaporated within one week. Even Renaissance Technologies Corporation, arguably the most successful quantitative hedge fund of all time, lost 8.7 percent in the first half of August, though it later recovered most of it. Not only is the magnitude of the loss astounding, but the widespread nature of it was causing great concern in the financial community. Strangest of all, few of these funds hold any mortgage-backed securities at all, ostensibly the root cause of the panic. It therefore became a classic study of financial contagion as propagated by hedge funds.

Another example is the January 2021 GameStop short squeeze Wang (2021). As this stock was surging due to traders’ promotion on Reddit's r/wallstreetbets forum and the subsequent coordinated buying of the stock and especially its call options, large hedge funds suffered enormous losses as they bought cover for their short positions. Renaissance Institutional Equities Fund fell 9.5 percent, while Melvin Capital lost a whopping 53 percent.

This kind of contagion occurs because a large loss by one hedge fund causes it to sell off some large positions that it holds (whether or not these are the positions that cause the loss in the first place). This selling causes the prices of the securities to drop (or rise in the case of short positions). If other hedge funds are holding similar positions, they will then suffer large losses also, causing their own risk management system to sell off their own positions, and on and on. For example, in the summer of 2007, one large hedge fund might have been holding subprime mortgage-backed securities and suffered a large loss in that sector. Risk management then required that it sell off liquid stock positions in their portfolio that might, up to that point, be unaffected by the subprime debacle. Because of the selling of such stock positions, other statistical arbitrage hedge funds that hold no mortgage-backed securities might now have suffered big losses, and have proceeded to sell their stocks as well. Hence, a sell-off in the mortgage-backed securities market suddenly became a sell-off in the stock market—a nice demonstration of the meaning of contagion. Similarly, in January 2021, the losses in short GameStop positions in some long–short funds caused them to buy cover other unrelated short positions while simultaneously selling unrelated long positions due to overall portfolio deleveraging, leading to losses in other funds that hold similar short and long positions. They were also forced to buy and sell the same stocks due to their own deleveraging, leading to a contagion.

Given the necessity of realizing losses as well as the scale and frequency of trading required to constantly rebalance the portfolio in order to closely follow the Kelly formula, it is understandable that most traders prefer to trade at half-Kelly leverage. A lower leverage implies a smaller size of the selling required for risk management.

Sometimes, even taking the conservative half-Kelly formula may be too aggressive, and traders may want to limit their portfolio size further by additional constraints. This is because, as I pointed out previously, the application of the Kelly formula to continuous finance is premised on the assumption that return distribution is Gaussian. (Finance is continuous in the sense that the outcomes of making bets in the financial market fall on a continuum of profits or losses, as opposed to a game of cards where the outcomes fall into discrete cases.) But, of course, the returns are not really Gaussian: large losses occur at far higher frequencies than would be predicted by a nice bell-shaped curve. Some people refer to the true distributions of returns as having fat tails. What this means is that the probability of an event far, far away from the mean is much higher than allowed by the Gaussian bell curve. These highly improbable events have been called black swan events by the author Nassim Taleb (see Taleb, 2007).

To handle extreme events that fall outside the Gaussian distribution, we can use our simple backtest technique to roughly estimate what the maximum one-period loss was historically. (The period may be one week, one day, or one hour. The only criterion to use is that you should be ready to rebalance your portfolio according to the Kelly formula at the end of every period.) You should also have in mind what is the maximum one-period drawdown on your equity that you are willing to suffer. Dividing the maximum tolerable one-period drawdown on equity by the maximum historical loss will tell you whether even half-Kelly leverage is too large for your comfort. The leverage to use is always the smaller of the half-Kelly leverage and the maximum leverage obtained using the worst historical loss. In the S&P 500 index example in the previous section, the maximum historical one-day loss is about 20.47 percent, which occurred on October 19, 1987—“Black Monday.” If you can tolerate only a 20 percent one-day drawdown on equity, then the maximum leverage you can apply is about 1. Meanwhile, the leverage recommended by half-Kelly is 1.26. Hence, in this case, even half-Kelly leverage would not be conservative enough to survive Black Monday.

The truly scary scenario in risk management is the one that has not occurred in history before. Echoing the philosopher Ludwig Wittgenstein, “Whereof one cannot speak, thereof one must be silent”—on such unknowables, theoretical models are appropriately silent.

Beyond position risk (which is comprised of both market risk and specific risk), there are other forms of risks to consider: model risk, software risk, and natural disaster risk, in decreasing order of likelihood.

PSYCHOLOGICAL PREPAREDNESS

It may seem strange that a book on quantitative trading would include a section on psychological preparedness. After all, isn't quantitative trading supposed to liberate us from our emotions and let the computer make all the trading decisions in a disciplined manner? If only it were this easy: human traders who are not psychologically prepared will often override their automated trading systems' decisions, especially when there is a position or day with abnormal profit or loss. Hence, it is critical even if we trade using quantitative strategies to understand some of our own psychological weaknesses.

Fortunately, there is a field of financial research called behavioral finance (Thaler, 1994) that studies irrational financial decision-making. I will try to highlight a few of the common irrational behaviors that affect trading.

The first behavioral bias is known variously as the endowment effect, status quo bias, or loss aversion. The first two effects cause some traders to hold on to a losing position for too long, because traders (and people in general) give too much preference to the status quo (the status quo bias), or because they demand much more to give up the stock than what they would pay to acquire it (the endowment effect). As I argued in the risk management section, there are rational reasons to hold on to a losing position (e.g., when you expect mean-reverting behavior); however, these behavioral biases cause traders to hold on to losing positions even when there is no rational reason (e.g., when you expect trending behavior, and the trend is such that your positions will lose even more). At the same time, the loss aversion bias causes some traders to exit their profitable positions too soon, even if holding longer will lead to a larger profit on average. Why do they exit the profitable positions so soon? Because the pain from possibly losing some of the current profits outweighs the pleasure from gaining higher profits.

This behavioral bias manifests itself most clearly and most disastrously when one has entered a position by mistake (because of either a software bug, an operational error, or a data problem) and has incurred a big loss. The rational step to take is to exit the position immediately upon discovery of the error. However, traders are often tempted to wait for mean reversion such that the loss is smaller before they exit. Unless you have a model for mean reversion that suggests now is a good time to enter into this position, this wait for mean reversion may very well lead to bigger losses instead.

While loss aversion leads to suboptimal trading in such a situation, there are other times when loss aversion is wise and is not a behavioral bias. Most economic arguments against loss aversion assumes that we have a large number of “gamblers” (read: traders) playing a risky game, and as long as the average return is positive, the economist suggests that this risky game is worthwhile, even if some of these gamblers may be ruined. However, when you are the trader, it is rational to avoid ruin at all cost, no matter what return the “average” trader enjoys. This contrast between the ensemble average (across different traders) and the time series average (over a long time horizon for a single trader) is a profound mathematical observation by the physicists Ole-Peters and Nobel laureate Murray Gell-Mann. See Box 6.1 for more details.

Another common bias that I have personally experienced is the representativeness bias—people tend to put too much weight on recent experience and underweight long-term average (Ritter, 2003). (This reference has a good introduction to various biases studied by behavioral finance.) After a big loss, traders—even quantitative traders—tend to immediately modify certain parameters of their strategies so that they would have avoided the big loss if they were to trade this modified system. But, of course, this is unwise because this modification may invite some other big loss that is yet to happen, or it may have eliminated many profit opportunities that existed. We must remember that we are operating in a probabilistic regime: No system can avoid all the market vagaries that can result in losses.

If you feel that your system really is deficient and want to tweak it, you should always backtest the modified version to make sure that it does outperform the old system over a sufficiently long backtest period, not just over the last few weeks.

There are two major psychological weaknesses that are more well known to the traders than to economists: despair and greed.

Despair occurs when a trading model is in a major, prolonged drawdown. Many traders (and their managers, investors, etc.) will be under great pressure under this circumstance to shut down the model completely. Other overly self-confident traders with a reckless bent will do the opposite: They will double their bets on their losing models, hoping to recoup their losses eventually, if and when the models rebound. Neither behavior is rational: if you have been managing your capital allocation and leverage by the Kelly formula, you would lower the capital allocation for the losing model gradually.

Greed is the more usual emotion when the model is having a good run and is generating a lot of profits. The temptation now is to increase its leverage quickly in order to get rich quickly. Once again, a well-disciplined quantitative trader will keep the leverage below the dictates of the Kelly formula as well as the caution imposed by the possibility of fat-tail events.

Both despair and greed can lead to overleveraging (i.e., trading an overly large portfolio): In despair, one tries to recoup the losses by adding fresh capital; in greed, one adds capital too quickly after initial successes with a strategy. Therefore, the one golden rule in risk management is to keep the size of your portfolio under control at all times. This is, however, easier said than done. Large, well-known funds have succumbed to the temptation to overleverage and failed: Long-Term Capital Management in 2000 (Lowenstein, 2000) and Amaranth Advisors in 2006 (Chan, 2006a). In the Amaranth Advisors case, the leverage employed on one single strategy (natural gas calendar spread trade) due to one single trader (Brian Hunter) is so large that a $6 billion loss was incurred, comfortably wiping out the fund's equity—a textbook case of risk mismanagement.

I have experienced this pressure myself both in an institutional setting and in a personal setting, and the unfortunate result both times was to succumb prematurely. When I was with a money management firm, I lost over $1 million for the fund's investors because, in a fit of greed, I added over $100 million to a portfolio based on a strategy that had been traded for barely six months. (That was before I learned of the Kelly criterion and other stress testing methodologies.) As if this is not enough lesson, I repeated the same mistake again when I started trading independently. It concerns a mean-reverting spread strategy involving XLE, an energy exchange-traded fund (ETF) and the crude oil future (CL). When the spread refused to mean revert over time, I stubbornly increased the size of the spread to almost $500,000. Finally, despair set in, and I exited the spread with close to a six-figure loss. Naturally, the spread started to revert afterward when I wasn't around to benefit. (Fortunately, several of my other strategies performed well in that first year of my independent trading, so the fiscal year ended with only a small overall loss.)

How should we train ourselves to overcome these psychological weaknesses and learn not to override the models manually and to remedy trading errors correctly and expeditiously? As with most human endeavors, the way to do this is to start with a small portfolio and gradually gain psychological preparedness, discipline, and confidence in your models. As you become emotionally more able to handle the daily swings in profit and loss (P&L) and rein in the primordial urges of the psyche, your portfolio's actual performance will hew to the theoretically expected performance of your strategy.

I have certainly found that to be the case after getting over those aforementioned disastrous trades. My newfound discipline and faith in the Kelly formula has so far prevented similar disasters from happening again.

SUMMARY

Risk management is a crucial discipline in trading. The trading world is littered with numerous examples of giant hedge funds and investment banks laid low by enormous losses due to a single trade or in a very short period of time. Most of these losses are due to overleveraging positions and not to an inherently erroneous model. Typically, traders will not overleverage a model that has not worked very well. It is a hitherto superbly performing model that is at the greatest risk of huge loss due to overconfidence and overleverage. This chapter therefore provides an important tool for risk management: the determination of the optimal leverage using the Kelly formula.

Besides the determination of the optimal leverage, the Kelly formula has a very useful side benefit: It also determines the optimal allocation of capital among different strategies, based on the covariance of their returns.

But no risk management formula or system will prevent disasters if you are not psychologically prepared for the ups and downs of trading and thus deviating from the prescriptions of rational decision making (i.e., your models). The ultimate risk management mind-set is very simple: Do not succumb to either despair or greed. To gain practice in this psychological discipline, one must proceed slowly with small position size, and thoroughly test various aspects of the trading business (model, software, operational procedure, money and risk management) before scaling up according to the Kelly formula.

I have found that in order to proceed slowly and cautiously, it is helpful to have other sources of income or other businesses to help sustain yourself either financially or emotionally (to avoid the boredom associated with slow progress). It is indeed possible that finding a diversion, whether income producing or not, may actually help improve the long-term growth of your wealth.

APPENDIX: A SIMPLE DERIVATION OF THE KELLY FORMULA WHEN RETURN DISTRIBUTION IS GAUSSIAN

If we assume that the return distribution of a strategy (or security) is Gaussian, then the Kelly formula can be derived very easily. We start with the formula for a compounded, levered growth rate applicable to a Gaussian process:

where f is the leverage; r is the risk-free rate; m is the average simple, uncompounded one-period excess return; and s is the standard deviation of those uncompounded returns. This formula for compounded growth rate can itself be derived quite simply, but not as simply as the Kelly formula, so I leave its derivation for the reader to look up in the Thorp article referenced earlier.

To find the optimal f, which maximizes g, simply take its first derivative with respect to f and set the derivative to zero:

Solving this equation for f gives us f = m/s ² , the Kelly formula for one strategy or security under the Gaussian assumption*.

REFERENCES

1. Chan, Ernest. 2006a. “A ‘Highly Improbable’ Event? A Historical Analysis of the Natural Gas Spread Trade That Bought Down Amaranth.” Quantitative Trading blog, October 2, http://epchan.blogspot.com/2006/10/highly-improbable-event.html.
2. Kahneman, Daniel. 2011. Thinking, Fast and Slow. Farrar, Straus and Giroux.
3. Khandani, Amir E., and Andrew Lo. 2007. “What Happened to the Quants in August 2007?” MIT. https://web.mit.edu/Alo/www/Papers/august07.pdf.
4. Lowenstein, Roger. 2000. When Genius Failed: The Rise and Fall of Long-Term Capital Management. Random House.
5. Peters, O., and M. Gell-Mann. 2016. “Evaluating Gambles Using Dyanmics.” Chaos 26, 023103. https://doi.org/10.1063/1.4940236.
6. Poundstone, William. 2005. Fortune's Formula. New York: Hill and Wang.
7. Ritter, Jay. 2003. “Behavioral Finance.” Pacific-Basin Finance Journal 11(4, September): 429–437.
8. Taleb, Nassim. 2007. The Black Swan: The Impact of the Highly Improbable. Random House.
9. Thaler, Richard. 1994. The Winner's Curse. Princeton, NJ: Princeton University Press.
10. Thorp, Edward. 1997. “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market.” Handbook of Asset and Liability Management, Volume I, Zenios and Ziemba (eds.). Elsevier 2006. www.EdwardOThorp.com.