25. With a $5 mark-to-market loss, you decide to sell anothercontract at $25, knowing that your system has excellent historical performance.The market continues to trade higher—rallying up to $30. At first you become disappointed because you have a $10 loss on the first sale and a $5 loss on the second sale. Your account has lost $15 of the original $45. At the same time, youcannot believe your good fortune. Able to make sales at $30, you feel that you aresure to win big when the mean reversion occurs. You sell one more contract at $30.As soon as the sale hits the tape, buying pressure fuels the market to $35. At thispoint your account is down $30; that is, $15 on the first sale, $10 on the secondsale, and $5 on the last sale.You would love to sell more, but because of the leverage and margin you’reusing on the trade, you can’t. In fact, you’ll be forced to close your positions if themarket trades much higher, due to a lack of funds. Of course, that is exactly whathappens. The market rallies to a never before seen value of $40. Your broker callsto tell you that he is buying your three contracts at $40 and that your account hasnow lost $45. Your capital is wiped out at a time when the best sales in the historyof the market are possible.In the weeks after closing your position, slowly but surely the market tradesback down. Not too long after you were forced to cover at the high of $40, pricesfall to $25. Eventually, levels of $15 are seen—just as your model predicted.This story may sound farfetched to some. Trust me, this type of poor moneymanagement occurs every day. Most frequently, this type of trading occurs withsmall undercapitalized traders taking positions that are too large. Other times itcan happen on a much larger scale and even shake world financial markets.
THE RELATIONSHIP BETWEEN LEVERAGE AND RETURNSWe’ve been focusing throughout the book on maximizing the performance of tradingstrategies through the use of reward-to-risk measurements such as the Sharpe ratioand K-ratio. It’s important to remember that our goal as traders is to maximize reward
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for a given unit of risk and then leverage up our risk to achieve the desired returns.Leverage involves borrowing money and using that money to fund trading decisions.These days, leverage is often easy to come by. In futures trading, it isobtained by placing a performance bond, called margin, as collateral for thenotional value of the trades. For example, in order to trade the 10-year Treasurynote on the Chicago Board of Trade, I need only deposit $2700 to gain long orshort exposure to $100,000 face value of Treasury notes. In the stock market, 100percent leverage is obtainable in margin accounts. For every $1 that an investor hasin cash, he or she can purchase $2 of stocks. The $1 that creates this leverage isborrowed from the broker in the form of a margin loan.Expected returns scale proportionally to risk. If I expect a 10 percent returnusing no leverage, then by borrowing $1 for every dollar I have (ignoring interestcosts for now) and applying the $2 to trading, I now have an expected return of 20percent. Using 3:1 leverage—borrowing $2 for every $1 in capital and subsequently using the $3 for trading—my expected return increases to 30 percent. Thistactic is used by many hedge fund managers to scale unleveraged returns toachieve a desired return. For example, if ABC Hedge Fund’s unleveraged expectedreturn is 10 percent and the manager wishes to make 30 percent per year, then thatmanager would use 200 percent leverage. He or she would borrow $2 for everydollar invested in the fund, using that total amount for trading.But something happened on the way to leverage. When we invest over multiple periods, each single period return affects the starting capital of the nextperiod. If returns are risky, leverage leads to an often bumpy road and creates awide variation in the resultant possibilities, which we’ll call “ending wealth”—theamount of money in possession at the end of some length of time. If a trader beginswith $100 and ends with $500 after five years of trading, his profit would be$400—ending capital minus beginning capital—and his ending wealth wouldbe $500. Although two strategies with identical average returns will yield an equivalent average ending wealth, the distribution of ending wealth varies dramaticallywhen the volatility in returns differs.To see how volatility can affect the distribution of ending wealth, considerthe example of two strategies. Strategy #1 returns 5 percent per year with annualvolatility of 20 percent. Strategy #2 returns 15 percent per year with annual volatility of 60 percent. Both have the same reward-to risk characteristics, but Strategy#2 is leveraged 3:1. Both yield equal measurement of reward divided by risk:5%/20% = 0.25, 15%/60% = 0.25. The only difference between strategies is theamount of leverage employed in each style of trading. Strategy #2 has much higherreturns, but also higher volatility.After 10 years, $100 invested in the lower return/lower risk Strategy #1 will10have an average ending wealth of $100 ⭈ (1 + 5%) = $162. One hundred dollarsinvested in the higher return/higher risk Strategy #2 will have an average ending10wealth of $100 ⭈ (1 + 15%) = $404 (Figure 11.1).
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Ending WealthAverage
Median
Strategy #1
$162
$136
Strategy #2
404
74
FIGURE
11.1
Ending Wealth for Strategy #1 and Strategy #2. The average and median ending wealth paints a differentpicture for each strategy.
We simulate 5000 random outcomes each with 120 monthly returns usingthe previously mentioned assumptions for Strategy #1 and Strategy #2 to analyze therange of outcomes in a 10-year investment period. While it’s true that the averageending wealth of Strategy #2 is much higher than that of Strategy #1 ($404 vs. $162),we find that the median ending wealth for Strategy #1 is actually higher than that ofStrategy #2. In fact, approximately 60 percent of the time the higher return/higherrisk Strategy #2 produces ending wealth less than the starting amount of $100. Thisdecrease in wealth occurs despite annual average returns of 15 percent. Obviously, itis important to control the risk of trading strategy returns.We have two statistics telling us very different stories. Strategy #1’s mean ending wealth is much lower than the mean ending wealth of Strategy #2, but its medianwealth is much higher. The median of a series is the middle value when values arearranged in numerical order. For example, in the series 23 … 32 … 35 … 47 … 57,the average is (23 + 32 + 35 + 47 + 57)/5 = 38.8. The median is the middle of the fivevalues when they are sorted from high to low; in this case it’s 35.The mean and median can be very different values. In the case of comparingending wealth, we should focus on median wealth. While the mean is the averageof all ending wealth possibilities, the median is the middle of a distribution wherehalf the scores are below and the other half are above. The median is less sensitiveto extreme scores, which makes it a better measure for skewed distributions.As we see in Figures 11.2 and 11.3, the leverage we employ in our trading canaffect our outcome dramatically. While leverage might increase average returns,there’s a point we cross where adding extra volatility hurts the likelihood of futureprofits. Despite Strategy #1’s returns of only 5 percent compared to Strategy #2’sreturns of 15 percent, Strategy #1 makes money after the 10-year period roughly 67percent of the time, while Strategy #2 only makes money 43 percent of the time.While the chances of doubling or tripling our money is higher with Strategy #2, it’sprobably not worth the risk, considering how often we lose money at the end of the10-year period.The reason the excess leverage in Strategy #2 harms ending wealth isbecause of the asymmetrical property of leverage. If I start with $100 and lose 10percent, I require an 11 percent return on my remaining $90 investment to breakeven. If I start with $100 and lose 25 percent, I need a 33 percent return on my
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Distribution of Ending Wealth: Unleveraged Strategy3.0%
Occurrence
2.5%2.0%1.5%1.0%0.5%
955
1000
910
865
820
775
730
685
640
595
550
505
460
420
380
340
295
255
215
175
95
135
55
15
0.0%Portfolio Value
FIGURE
11.2
Distribution of Ending Wealth: Unleveraged Strategy. The bulk of ending wealth values fall between 50 and 300.
Distribution of Ending Wealth: Leveraged Strategy18%16%
Occurrence
14%12%10%8%6%4%2%1000
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
0
50
0%
Portfolio ValueFIGURE
11.3
Distribution of Ending Wealth: Leveraged Strategy. The bulk of ending wealth values fall between zero and 600.
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remaining $75 to break even. If I’m really risky and lose 50 percent, it would takea 100 percent gain on my remaining $50 just to return to my starting capital point.The larger my loss, the larger the proportional gain required to break even (seeFigure 11.4).Strategies with very high reward-to-risk characteristics are able to takelarger risks, since large losses can be overcome due to superior profitability.Strategies with lower reward-to-risk characteristics are unable to overcome largelosses. Here lies the mantra of optimal leverage: Higher reward to risk strategiescan be traded at higher leverage than lower reward-to-risk strategies due to theability to recover from large losses. Determining the optimal leverage becomesvery important, since the volatility of our strategy will affect our median ending wealth.1Required gain to regain loss = ᎏᎏ − 11 − %LossThe fact that volatility affects the distribution of ending wealth explains theconcept of risk aversion in human judgment. If individuals were risk neutral, thenwe would play any game with positive expected value, regardless of the riskinvolved. In any positive expectancy trading strategy, we would leverage returns as
Gain Required to Offset Loss120%
Gain Required
100%80%60%40%20%
–5%
–10%
–15%
–20%
–25%
–30%
–35%
–40%
–45%
–50%
0%
Loss
FIGURE
11.4
Gain Required to Offset Loss. A 10 percent loss requires an 11 percent gain to break even. A 50 percentloss requires a 100 percent gain to break even.
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far as possible because it would maximize our expected profit. For example, a riskneutral individual would view both bets below as equally good:Bet 1:Bet 2:
50 percent chance of winning $250 percent chance of losing $150 percent chance of winning $10,00250 percent chance of losing $10,001
While both bets have an expected value of + $0.50, very few people wouldview these two options as equally good. We do care about the risks we take withour money. Larger risks command higher returns. If we assume that individualsare risk averse, it would make sense to adjust profits by the risk taken to achievethem. By maximizing the median ending wealth instead of mean ending wealth,we penalize risk and maximize our most likely ending wealth.
THE DANGER OF LEVERAGEWhy do so many traders lose, even those who have an edge in trading the markets?More times than not, losses can be explained by one factor: excessive leverage. The problem is most commonly found in the futures markets, where traderscan utilize anywhere between 5:1 and 20:1 leverage. Two examples illustrate thishazard of excess leverage. One example involves a profitable trader employing toomuch leverage, while the other involves a trader with no edge employing varyingdegrees of leverage.Leverage and the Trader with an EdgeLet’s first consider a trader with a methodology that produces 55 percent successful trades where each winning and losing trade is of equal dollar amount and equalto the bet size. This trader begins with $1 and must decide how much capital to beton each trade. When betting too small, the trader will not take full advantage ofthe mathematical advantage of his strategy. When betting too large, losing tradeswill wreck successive winning streaks.Numerous studies have been conducted on the subject of optimal betting strategies. There is a unique optimal bet size to maximize the median ending wealth. As wewill study later in this chapter, the optimal bet size in this case above is 10 percent ofthe trader’s wealth on each trade. If we simulate 100 trades and assume the traderbegins with $1 and uses varying bet sizes, we see why this 10 percent bet size is optimal. The median ending wealth increases consistently up to the 10 percent bet size.After this is exceeded, the median ending wealth actually decreases (Figure 11.5).It’s interesting to note that if we increase our bet size past the optimal point,we can actually decrease return while increasing risk. If we take our bet size toofar past the optimal point, then we can actually lose money more times than we
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Median Ending Wealth$1.80$1.60$1.40
Wealth
$1.20$1.00$0.80$0.60$0.40$0.20$0.000%
5%
10%
15%
20%
25%
30%
Bet Size
FIGURE
11.5
Median Ending Wealth. Median ending wealth is maximized by betting 10 percent of assets on each bet.
make money. In the example above, the median wealth declines below the startingwealth of $1 when we risk more than 20% of our capital on each trade. Theseresults certainly explain why money management can be so crucial. Imagine a situation where we bet our entire capital on every bet. Even if we’re extremely luckyand win 50 bets in a row, our first loss will wipe out all our capital. An understanding of this principle is of great importance to traders, and it’s certainly onereason why most traders lose—even those with expected edge.Leverage and the Trader with No EdgeIn our second example of why traders lose, consider an untalented trader who hasno edge in the markets. Each trade has a 50 percent chance of success, with winning and losing trades equal to the bet size. If we start with $100, make 10 trades,and always bet 20 percent of our bankroll on each trade, the ending profit and lossdistribution is broken down in Figure 11.6.Note that in this example our trader ends with negative returns roughly 62percent of the time at the end of 10 trades. Why does the trader lose despite having a break-even trading strategy? Although the expected value of this game is stillzero, the ending wealth follows a skewed distribution. In some rare instances,everything falls in our trader’s favor. But in most cases, the trader finishes with
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Ending Wealth
Occurrence
11
0%
16
1%
2436
4%12%
54
21%
82122
25%21%
183
12%
275413
4%1%
619
0%
FIGURE
313
11.6
Table of Ending Wealth. This table details the skew in ending wealth values.
less money than at the start of trading. How is this skewed distribution formed? Ifour trader has one winning trade followed by one losing trade, the ending equityis actually less than where it began. Consider the four possibilities occurring at theend of two trades where the trader begins with $100 and bets 20 percent of hisassets on each trade (Figure 11.7):Our trader loses money 75 percent of the time despite the dollar profit of theone winner equaling the dollar loss in the three losers combined. As our bet sizeincreases, this skewed profit and loss distribution becomes more pronounced. Itmight explain why 95 percent of futures traders lose money while the other 5 percent are making a killing, whether due to luck or skill.
LEVERAGE IN THE REAL WORLDAs illustrated in the above examples, volatility and leverage are hazardous. Theseexamples do not exaggerate. Just consider a real-world example of the perils ofleverage, one of the most popular trends in the equity market runup of the late1990s: indexing.As noted earlier in the book, indexing is accomplished by structuring portfolios to mimic the performance of popular benchmarks such as the S&P 500 orNasdaq 100. Some money managers believed that if market returns were good,then leveraging these returns would be better. During the late 1990s, leveragedindex funds began to be marketed to individual investors. These funds borrowedmoney or used futures markets to achieve as much as 2:1 leverage.One group of funds offered both a leveraged long and leveraged short indexfund designed to track the Nasdaq 100. The “bull” fund was designed to provide
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Trade 1
Trade 2
Win
Win
WinLoss
LossWin
Loss
Loss
FIGURE
11.7
Ending Wealth$100 ⭈ 1.2 ⭈ 1.2 = $144
$100 ⭈ 1.2 ⭈ 0.8 = $96$100 ⭈ 0.8 ⭈ 1.2 = $96
$100 ⭈ 0.8 ⭈ 0.8 = $64
Probability25%25%25%25%
Example of Simple Game. Seventy-five percent of the time, our ending wealth is below our starting capitalof $100.
twice the performance of the Nasdaq 100, while the “bear” fund was designed toprovide twice the performance of being short the Nasdaq 100. Figure 11.8 detailsthe performance of investing $100 on December 31, 2000, into one leveraged longNasdaq 100 index fund and simultaneously purchasing $100 of one leveraged shortNasdaq 100 index fund. Both funds are in the same mutual fund family, and reallife net asset values (NAVs) were used to compute daily portfolio value statistics.While the Nasdaq 100 finished down 32 percent in 2001, both the “bull” andthe “bear” fund actually lost money. A $100 investment in the bull fund becameroughly $31 at the end of 2001, while a $100 investment in the bear fund became $95.How can this be if one fund is long the Nasdaq 100 and the other is short?Similar to the trade example above, the funds lose due to the leverage employed.As the market rose, both funds were required to buy assets to maintain the properleverage ratio. When the market fell, both funds sold these assets, again to adjustto the perfect ratio. This behavior is completely consistent with each funds’ mission of maintaining 2:1 leverage.The problem is, volatile sideway price action will eat into both funds’ assetvalues. When the market runs in one direction or another, the leveraging effect willbenefit the fund’s customers. But in a market that trades in a volatile sidewaysmanner, the extra leverage seriously hampers performance.I cannot overemphasize the importance of managing risk in a trading portfolio. It is a natural habit for traders to feel the need to push the risk envelope andtake full advantage of their expected edge by leveraging assets. I know this needbecause I’ve been there.At age 15, I developed a countertrend system to pick tops and bottoms in theTreasury bond futures market. While paper-trading the system for three fullmonths, I correctly picked eight short-term tops and bottoms in a row. Armed withmy new system, I opened a trading account and began trading the T-bond contractat the CBOT.I could not wait for the money to come rolling in from trading. I still remember sitting in high school class calculating my projected income from pyramidingand constantly scaling up the size of the trading program. After the first year of
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Value of Nasdaq 100 Leveraged Mutual Fund250Value of $100 investment
Leveraged Short FundLeveraged Long Fund
200
150
100
50
9/01/2
9/0112
/211
9/01/2
/0110
299/
/01
/01
298/
29
/017/
/01
296/
/01
295/
/01
294/
/01
293/
2/
28
/01291/
12
/2
9/00
0
DateFIGURE
11.8
Value of Nasdaq 100 Leveraged Mutual Fund. Despite the Nasdaq 100’s precipitous decline in 2001,both leverage long and leveraged short funds lost money.
trading, I estimated my net worth would be approximately $1 million, give or takea couple hundred thousand dollars. Ready to begin my journey, I placed the entryfor my first trade. But something strange happened shortly after the trade wasfilled—the market went against me. Because the possibility of losing money neverentered my mind, I had no contingency plan to adjust my trading. The market wentagainst me further, and I became paralyzed by fear. Fifteen hundred of my $5000was gone. I was confused, frightened, and unprepared. Eventually I closed theposition with a $1500 loss.Did the strategy eventually make money? I have no idea. I was so shellshocked from the first trade that I never traded that strategy again.
THE KELLY CRITERIAThe first major work in the area of optimal leverage was researched by J. L. Kelly,Jr., of Bell Labs in 1956. Kelly studied the rate of transmission over noisy communication channels. Although his work was not particularly useful for telecommunications, Kelly noted that it would be very relevant to the gambling profession.In essence, Kelly discovered that gamblers should determine their bet size suchthat the logarithm of capital is maximized. Maximizing the logarithm of wealth is
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similar to maximizing the median of ending wealth if sufficient time periods arepresent.Kelly found that there is an optimal amount of a bettor’s bankroll to bet oneach trial in order to maximize the median ending wealth. For simple games wherea winning payout is equal to losing payouts, the formula is very simple:Optimal bet size as a percent of capital = 2p – 1where p is the probability of winningIf I have a 55 percent change of winning, then I should bet (2 ⭈ 55%) – 1 =10% of my bankroll on each bet. For games where I have a 70 percent chance ofwinning, I should bet (2 ⭈ 70%) – 1 = 40% on each bet. Logically, the maximumbet size is obtained if there is a 100 percent chance of winning. At that point I bet(2 ⭈ 100%) – 1 = 100% of my bankroll on each bet. If I can never lose, it makessense that I would always bet my entire stake on each trial.The above formula works for bets where winning payouts are equal to losingpayouts; that is, if I collect $1 during wins and lose $1 during losses. The formulacan be adjusted for games that do not have this fixed win/loss payout. Here’s amore general version of the Kelly formula:Optimal bet size as a percent of capital = p – (1 – p)/rwhere p is the probability of winning and r is the ratio of the winning payout tothe losing payout
VINCE’S OPTIMAL FRalph Vince was the first to apply Kelly’s work to trading (1990). Vince calculatesthe optimal leverage point using trade-by-trade returns. His version of optimal f issolved by maximizing the ending wealth of a series of trades by betting a fixedpercentage of the largest losing trade. The idea is that the largest losing trade willoccur at some point, so we risk some fraction of the largest loss, which maximizesour ending wealth.Although Vince’s method should yield beneficial results, using daily return datafrom a strategy will provide a tighter, better measure of optimal leverage due to theincreased number of data points. Vince’s method suffers if the number of trades issmall enough that the ending wealth curve becomes jagged. When the ending wealthcurve is not smooth, it becomes difficult to determine the true optimal leverage point.
AN IMPROVED METHOD FOR CALCULATING OPTIMAL LEVERAGEI believe the best method for determining optimal leverage involves maximizing themedian ending wealth. Harry Markowitz (1959) showed that maximizing medianwealth is equivalent to maximizing the mean logarithmic return, and he developed
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a good approximation for calculating the mean logarithmic return based on thearithmetic mean and variance of returns:Expected log return = expected return – 1⁄2 variance of returnsUsing an amount of leverage, M, the log return equation becomes:Expected leveraged log return = M ⭈ expected return – 1⁄2 M 2 variance of returnsIn order to maximize expected log return, we take the derivative of theexpected leveraged log return with respect to M. Setting the derivative to zero, wesolve for M. It happens that the optimal leverage—the optimal value of M—foruse in a trading strategy is calculated by:µOptimal leverage = return / variance = ᎏ2σUsing the optimal leverage calculated from the above formula, we will bemaximizing our median ending wealth, the benefits of which were seen earlier inthis chapter. If our strategy is expected to return 5 percent per year over the riskfree rate, with annual standard deviation of 20 percent, then our optimal leverage is(0.05)/(0.20) 2 = 125%. We would actually borrow 25 percent of our capital to plowback into our trading strategy.Let’s suppose we begin with $100 and the optimal leverage of our strategy is200 percent. We borrow $100 from our broker and invest a total of $200 into ourtrading strategy by buying two shares of ABC stock, trading at $100. Our strategy iscorrect and ABC rallies from $100 to $150. At this point, we are worth $200: $300from our two shares minus the $100 we owe our broker.Even with $200 in capital, we still need 200% leverage. This means that weshould have $400 invested in the strategy, more than the $300 currently being used.To rebalance, we borrow another $100 from our broker, and invest this new moneyinto our strategy by buying two-thirds of a share of ABC. This rebalance creates theoptimal 200% leverage we desire as our $200 in capital is controlling $400 in trading positions.If ABC declines from $150 back to its original $100, we again need to reevaluate our leverage. Our net worth is now $67: $267 from our 2.67 shares minus the$200 total we owe our broker. With $67 in capital and 200% optimal leverage, weneed to control $134 of stock—more than the $267 currently held. To rebalance,we sell $133 of stock and repay this money to reduce our loan to the broker. Ourremaining position is $134 held in stock minus the remaining $67 we owe to ourbroker—leaving $67 in net worth.Our capital and leverage will change over time as profits and losses accrue. Asseen above when our optimal leverage is greater than one, we will need to trade largerand borrow more as our capital grows from trading gains. As capital declines from trading losses, we will trade smaller and borrow less in order to maintain optimal leverage.
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This is arguably where the principals at Long Term Capital Managementmade a strategic flaw. In their strategy, they employed large amounts of leverage as they believed their optimal leverage was well over 100%. Instead ofdecreasing leverage by closing positions as their capital began to evaporate,the partners either attempted to maintain their position size or in some casesincreased position size—effectively increasing their leverage. This mistakeeventually led to their demise and the corresponding shock to world financialmarkets in October 1998.On the other hand, if our return and risk numbers yield an optimal leverageless than one—meaning we use less than 100 percent of our capital for trading—then a different pattern emerges. In this case, we subtract from positions as we winand add to positions as we lose—precisely opposite the situation studied above.
THE ROLE OF DOLLAR AND PERCENTAGE RETURNSReaders may notice that with most strategies in this book, return and risk havebeen calculated in dollar terms, not in the percentage terms used in this chapterto calculate optimal leverage. We can use the optimal leverage formula above todetermine how large we should trade when using a set amount of capital.Assume that we start trading with $10,000. Suppose we create a profitabletrading method which averages profits of $500 per day and has a daily standarddeviation of $5000 per day. Note that it is imperative to measure average returnand volatility in the same unit of time, whether daily, weekly, monthly, or annualmeasures. Using a little algebra, we find that the optimal dollar volatility is a function of capital, average return, and volatility of return.Optimal dollar volatility = (capital) (average dollar return) / (dollar volatility)The resulting optimal volatility is expressed in the same time frame as theaverage return and volatility measurements. Returning to our example, we findthat the optimal dollar volatility of our $10,000 capital is ($10,000)($500)/($5000)= $1000 per