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Risk and Return

Amateur traders first look at profits while professionals first look at risk. It is far more interesting to search for a profitable system than to concentrate on losses, but an understanding of risk is essential to every aspect of trading. Otherwise, it is only a matter of time before a price shock forces you out of the market, or a series of losses become enough of a worry to stop your trading. In this book, you will find that some trading methods are considered better than others. This chapter will explain how those choices were made. The preferred methods are based on sound rules that incorporate both risk and return. These techniques can be applied to any trading method.

Figure 4-1 shows two equity plots for the same time period. The left one (a) shows much higher profits, but more risk. The price fluctuations above and below the straight line (the rate of return) are clearly larger than in the right plot (b). Which is better, the one with higher profits, or the returns with lower risk? Is it investor "risk preference" that decides the answer?

The "best" performance is not based on whether the investor is con­ servative or risk-seeking. It is entirely a function of which performance gives the highest returns for the lowest risk, which in turn will translate back into higher profits. In Figure 4-1, it is difficult to decide which is better without being given the exact risk and return for each chart.

Risk Preference

Risk preference is the investor's willingness to accept more or less risk in exchange for profits. When two systems have the same returns, rational investors will choose the system with the lowest risk. Similarly, when

(a) Currency Portfolio
(b) Bond Portfolio

two systems have the same risk, investors would choose the one with the highest returns. Among programs with the best combination of risk and reward, it is necessary to accept proportionately higher risk to increase returns by a small amount.

Standardizing Risk and Return

Before comparing the currency and bond portfolios in Figure 4-1, the returns and risk must be represented in a standard, convenient form. This allows different test periods to be compared equally with one another. The most popular approach is used extensively by equities and financial analysts:

•  Return is the annualized, compounded rate of return.

•  Risk is the annualized standard deviation of the equity changes.

Compounded Rate of Return

A compounded rate of return means that the accumulated funds are rein­ vested. In the case of a simple interest-bearing savings account, "daily compounding" means that the interest earned each day is added to the total savings and earns interest starting the next day.

Using annualized compounding, you may earn, for example, 5 percent on an investment, leaving the interest to accumulate. Because the inter­ est is posted only once each year, a $1,000 investment is worth $1,050 at the end of the first year. At the end of the second year, the $1,050 has grown by another 5 percent to $1,102.50, and after the third year to $1,157,625. The same result can be found by multiplying the initial investment of $1,000 by the rate of interest raised to the number of years that the funds were invested.

This technique is useful for system evaluation when you have the starting and ending value of an investment and you want to find the compounded rate of return that would give those values. The calcula­ tion can simply be reversed, as shown in Box 4-1 .

Annualized Standard Deviation

The annualized standard deviation measures risk. It is simply the stan­ dard deviation of the annual increase or decrease in equity (compounded returns). Therefore, if the annualized return is only one value each of 20 years, the annualized standard deviation uses only those 20 values. You may use monthly changes in equity to get more accuracy. The next few sections will compare risk and return evaluated over different time periods, and show that they are not easily interchanged.

Using a Standard Deviation to Determine the Chance of Loss

The standard deviation is the classic measurement of distribution. It shows how data is clustered around the average value. It assumes that the pattern of results is symmetrical. For example, a chart of trading performance is shown in Figure 4-2(a). The rate of return is the solid angled line drawn upward through the center of the equity. The stan­ dard deviation of the monthly profits and losses (subtracting last mon­th's equity, or accumulated profits and losses, from this month's value) will show how the returns are clustered above and below the rate-of- return line.

It is easier to see the distribution of equity changes in Figure 4-2(b). By subtracting the equity value of each month from the previous month, the equity has been detrended. Fluctuations are centered above and below a zero line, rather than an upward regression line, as in Figure 4- 2(a). The standard deviation is useful because it puts a value on the chance of loss. In Figure 4-2(b) bands are drawn parallel to the straight

Most market analysts have standardized their way of representing both risk and return. This book will adopt the same notation.

Returns are the annualized, compounded rate of return. This is given as

CROR = (Ending_Equity/lnvestment) A (1/Period) - 1

where CROR is the compounded rate of return Ending_Equity is the last account value Investment is the starting account value Period is the number of years (for annualized returns)

The Period, or number of years, is expressed as a decimal fraction (e.g., 8V 2 years is 8.5). Therefore, if your $1,000 investment is worth $1,300 after 4 1 / 2 years, you have earned a compounded rate of return equal to 6.00 percent:

CROR = (1300/1000) A (1/4.5) - 1 = 1.0600-1.00 = 6.00% per annum

For a monthly compounded rate of return, the period would be 54.0 months, or .487 percent per month.

Risk is the annualized standard deviation of the equity, which we will take as the standard deviation of the annual returns. Then

ASD = @SUM(Yearly_Returns*2)/Period

where ASD is the annualized standard deviation

@SUM is the program function that adds a list of values Yearly_Returns is a list of yearly changes in returns Period is the number of years (for annualized returns)

•  Spreadsheet. This calculation is readily available on spreadsheet
programs as a built-in function. In Quattro, it is simply specified
as @STD(B4..B20) in order to find one standard deviation of the
values in column B, rows 4 through 20.

•  TeleTrac and System Writer. Both testing software programs have sim­
ilar built-in standard deviation functions, Std_Dv(series,period) in
TeleTrac and @StdDev(series,period) for System Writer. Each calculates
and returns the value of one standard deviation.

Figure 4-2. Standard deviation of equity and equity changes, (a) The

standard deviation of the equity shows the chance of a total profit or loss swing with respect to the straight line approximation of the rate of return, (b) The standard deviation of monthly equity changes shows the likelihood of profit or loss during any one month.

line, detrended rate of return (0). These lines show the grouping of 1, 2, and 3 standard deviations, as follows:

68% of all data falls between ± standard deviation (the band from A to -A)

95% of all data falls between ± 2 standard deviation (the band from B to -B)

Because we are only interested in the risk of loss, we need to know the chance of equity falling below —A, -B, and -C. If 68 percent of the data falls between +A and -A then 32 percent falls outside. One half of the 32 percent will be above + A and the other half below -A. Therefore, there is only a 16 percent chance that losses will exceed 1 standard deviation, a 2.5 percent chance they will exceed 2 standard deviations, and a .5 percent chance they will exceed 3 standard deviations.

Choosing Between the Currency and Bond Portfolios

Once the returns and risk are in uniform notation, the comparison between the currency and bond portfolios becomes simple. The portfo­ lios are risk-adjusted by dividing the larger risk (20 percent in the cur­ rency portfolio) by the smaller risk of the bond portfolio, giving 20/2 = 10. If the currency risk is reduced by a factor of 10, then the currency profit of 50 percent must also be divided by 10. The result is that the cur­ rency portfolio returns only 5 percent, compared with 8 percent for bonds, when they are both at the 2 percent risk level. The bond portfo­ lio (Figure 4-lb) is 60 percent better.

Leverage

The risk-adjusted return is a better measurement of performance because it is the simplest way of getting the most portfolio diversifica­tion, especially when futures or options programs allow leverage. The bond portfolio, with 8 percent returns, could have been leveraged as high as 20 times using the futures markets, returning 160 percent with a 40 percent risk.

Options and other derivatives allow the investor considerable flexibility in varying leverage. Portfolios that deleverage by holding cash reserves also have latitude for varying the investments based on risk and reward. Only the fully invested account (which can assume more risk than either the currency or bond portfolios) may choose the highest profits.

Actual System Test for Finding the Best Choice

A trend-following system was tested on two years of the Bombay stock index, the SENSEX. Table 4-1 shows nine tests progressing from short-term, faster trends at the top to slower ones at the bottom. The best prof­its were 3681 points for test 8 and the lowest risk was 814 points for test 4. However, the best performance choice was test 2, with an adjusted, compounded rate of return of 34.5 percent. Profits and risk in columns 2 and 3 were both converted to percentages based on the current SEN- SEX price of 2700.

A fast way of arriving at the best choice is simply to divide the total profits (column 2) by the total risk (column 3). The greatest return/risk ratio will be the best choice, as long as standard measurements were used and both returns and risk were calculated over the same time period within the same test. Even if test 4 was evaluated over a slightly different length interval than test 9, the ratio would give a valid com­ parison.

Table 4-1. Risk-Adjusting Results of a Trending System

Finding the best risk-adjusted returns for the Bombay SENSEX trend-following program can be done by combining a spreadsheet and system tests. In this case, profits and risk (columns 2 and 3) were found using CompuTrac SNAP for 2 years of data. Risk was calculated as 1 standard deviation of the equity. Percentage returns (column 4) were calculated using 100, the initial investment, plus the gross returns divided by 2700, the current SENSEX price. The compounded rate of return for the 2- year period is the return (column 4) less the initial investment of 100, raised to the 1 II power (see calculation for compounded rate of return). The percentage risk (col­ umn 6) is the risk in points (column 3) divided by the current SENSEX price of 2700. The risk-adjustment factor is the percentage risk (column 6) divided by the lowest percentage risk in column 6 (30 in test 4). The risk-adjusted compound rate of return (column 8) is the compound rate of return (column 5) divided by the adjust­ ment factor (column 7). The profit/risk ratio (column 9) is the profit (column 2) divided by the risk (column 3).

Columns 8 and 9 both show that the best choice for trend speed was test 2, which had neither the highest profits nor the lowest risk. Although the ratio does not tell the relative rate of return, using the highest value is a fast way of identifying the best test.

Risk to Time

The standard deviation using the quarterly change in equity must be bigger than the standard deviation of monthly changes because longer time intervals allow larger fluctuations. Even though a quarter is 3.0 times longer than a month, the standard deviation is 4.74 times larger. As the time interval becomes even larger, however, the variation does not increase at the same rate. Most price or equity data will tend toward a maximum variance, regardless of the time interval. A typical risk-to- time curve is shown in Figure 4-3.

Time Periods Tell Different Stories

Although equity changes get larger when annual data is used instead of monthly or daily returns, longer periods can also hide risk. During October 1987, the S&P 500 plunged 44 percent but recovered to show a smaller loss of 21 percent by the end of the month, and actually closed out the year up 2 percent.

Using monthly data, the chances are only 1 in 21 that the returns for the month will be the low of that month, therefore the performance is smoothed. The risk of loss is there, but you cannot see it from the num­bers.

Drawdown

To assess risk it is always necessary to know the maximum drawdown, the largest peak-to-valley equity drop. The maximum drawdown is likely to be much larger than any monthly equity change, as seen in the S&P example. Even though you may not see a loss of that magnitude soon, it is inevitable that a swing as big or greater will occur sometime in the future. It is both naive and unrealistic to assume that a faster, larger rally or bigger plunge will never occur.

Monthly Quarterly Semi-Annually

Interval of Calculation

Fignre 4-3. S&P risk to time relationship. The standard deviation of the S&P price changes from 1983-1992 shows that price change approaches a limit as the measurement time period gets larger.

The 5O-Tear Rule

Environmental planning uses a 10-year rule for water relief. Ditches are built along country roads to control the maximum runoff measured over the past 10 years. If the runoff overflows the ditch, there is unfor­tunate erosion (although no obvious harm to people). To protect farmers settling on the banks of the Mississippi River , levies were built to a height that satisfies the "50-year rule." What does that mean to a family living in a home along the river? It says that water will rise above the levy, but not very often. It may have overflowed the levy 75 years ago, but not during the past 50 years; therefore, flooding could happen once in the lifetime of each family member.

The same situation exists in trading. Professional traders will be faced with extreme moves and extraordinary price shocks during their careers. Those who plan to use the markets for only a short time may never see these moves and could take the chance that these extremes will not occur during their short trading stay. It is a classic risk-and- reward decision.

Trading safety relates directly to capitalization. Larger, more diversi­fied investments, with proportionately greater reserves, are safer. If you are fortunate enough to profit from a small amount of capital, then decrease the relative risk by increasing the percentage of reserves. Traders who continue to leverage all their profits increase their chance of a complete loss.

Specifying Acceptable Risk

Standardizing the risk measurement is important because it allows you to specify the amount of risk you are willing to accept for any trading method. For example, as a foreign exchange trader, you are given US$10 million to trade. You are told that you should not risk more than 10 per­cent. Having traded for some time, you have a good idea about the fluc­ tuations of your performance, and have a track record that can be mea­ sured. Or, as an experienced analyst, you expect to devise a method based on previously successful ideas.

Using annual figures, your performance shows an admirable 40 percent return with a 10 percent standard deviation. You know that 1 standard deviation means that there is an 84 percent chance that the equity won't drop more than 10 percent, and 3 standard deviations gives a 99.5 percent chance that equity won't drop more than 30 percent. Looking at the results in reverse, a risk of 10 percent means that there is a 16 percent chance that you will lose more than 10 percent during a year.

You cannot have such a large risk of losing 10 percent; therefore, you bring this down to a safe level by dividing by 4. This really means that you will use only one-fourth of the capital in the account to trade. The account is "deleveraged." The original 10 percent standard deviation becomes 2.5 percent and there is a

16.0% chance of losing more than 2.5% (1 standard deviation) 2.5% chance of losing more than 5.0% (2 standard deviations) .5% chance of losing more than 7.5% (3 standard deviations)

which is about right for your risk limits. Because you divided the risk by 4, the returns must also be divided by 4, giving an adjusted annualized return of 10 percent—still pretty good. To implement this plan, you need to trade only one-fourth of the funds allocated to you, or $2.5 mil­lion instead of $10 million, holding the rest as reserve in the event of an undesirable (but possible) run of bad trades. You are experienced enough to know that a 0.5 percent chance means that you will lose more than 7.5 percent, but not often. Since this risk measurement used month­ ly values, a daily maximum drawdown is still needed to decide a safe level of capitalization.

Foreign Exchange Trader's Dilemma

Even if everything works as planned, producing a 10 percent return with a 5 percent risk during the next year, there is still a problem. The trading manager calls you aside and asks, "Why aren't you using all the money? You returned 10 percent using only $2.5 million. If you had used all the money, you could have returned 40 percent. If you're not going to use it all, I'll have to give it to someone else."

That's the dilemma. You needed $7.5 million in reserve for bad times, but you didn't use it. You look brilliant, but too conservative. How do you explain that you were using the money? There was a 16 percent chance that you would need more than $2.5 million and a 2.5 percent chance of using more than $5 million. It is difficult to explain, but it needs to be done. If you trade a larger percentage, you will eventually be shut down by losing more than 10 percent. This same problem will be seen in Chapter 7, when we look at the effects of price shocks.

Graphing Risk and Reward

Graphing the risk and reward leads to important observations. This approach is often used by asset allocators, to show the trade-offs between systematic combinations of profits and losses. The returns and risk (in points) given in Table 4-1 have been plotted in Figure 4-4 and marked 1 through 9.

Results that are plotted higher and to the left have more profit and less risk. Those that are lower and to the right are worse choices because they have less profit and more risk. A rational investor would always choose a point directly above another one because it has a higher return for the same level of risk (given a choice of points 2 or 3, point 2 is the better one).

Efficient Frontier

A curve can be drawn through the points that return the highest profit at each level of risk. Note that the curve, seen as a broken line in Figure 4-4, flattens out as it goes to the right. This means that the investor or trader must assume a proportionately higher risk to gain a small increase in profits.

This curve is similar in concept to an efficient frontier, used in asset allo­ cation. However, if you use the performance of a single trading system, one combination may show exceptionally good returns by chance. A curve drawn through the best cluster of points, rather than the absolute highest combinations, will give a more realistic appraisal of expectations.

The Best Choice

Investors are always free to choose the combination of risk and reward that satisfies their objectives. A rational choice must be on the efficient frontier. A more conservative investor will be more comfortable with a selection further to the left, while a risk seeker will prefer the higher profits even at the cost of increased risk.

What is normally considered the "best" choice is found by drawing a straight line from the risk-free rate of return (on the left scale), tangent to the efficient frontier curve, as seen in Figure 4-4. The risk-free rate of return assumes 5 percent over 2 years applied to the amount of money needed to buy the stock index at 2700 points. Keeping the calculation in SENSEX points,

(investment in points) (yeari) (year 2)

The straight line drawn from 277 touches the efficient frontier at exactly test 2, the best choice.

Diversification and Risk Reduction

Diversification is the best form of risk reduction. Investing in more than one asset, each with a good return, will reduce risk by benefiting from equity variation that occurs at different times. Investment managers are constantly searching for ways to place funds that yield better-than-aver- age returns and modest risk. Combining a number of medium-risk, diverse investments often nets a low-risk portfolio. Diversification can also be gained by using different trading strategies in the same market.

Applying Asset Allocation Techniques

Asset allocation is the process of distributing investment funds into one or more markets or vehicles to create an investment profile with the most desirable return/risk ratio. In its simplest form, asset allocation would use only one active investment, such as a stock portfolio, placing the remain­ ing funds into risk-free, short-term government bonds. In its most compli­ cated form, many investment vehicles are combined on a dynamic basis. Assets may vary from a passive gold portfolio to active international stocks and discretionary trading of foreign exchange markets.

Simple Risk Reduction

The stock market produces healthy long-term returns of about 10 per­ cent per year (16.4 percent compounded rate of return from 1983 to 1992), with a return/risk ratio of approximately 1:1. If you were 100 per­ cent invested in stocks, there would be a 16 percent chance of a loss greater than the annualized return sometime during that year. To reduce the risk, you simply trade a smaller amount of stocks and put the bal­ ance in a money market account or Treasury bills. Figure 4-5 shows that the risk and returns are reduced linearly by holding varying amounts in reserve.

Classic Stock and Bond Portfolio

A more popular alternative is to allocate a portion of the portfolio to bonds. Although risk-free when held to maturity, a passive bond portfo­ lio will fluctuate in value in the same way as equity holdings. Therefore, the bond investor is subject to fluctuation in the underlying interest rates, profiting when interest rates drop and losing when they rise.

Figure 4-6 compares the return/risk ratios of a stock and bond port­ folio for varying percentage combinations of those two items. The monthly returns were combined using a spreadsheet, and the com­ pounded rate of return and monthly standard deviations were recalcu­ lated on the new portfolio. The result is a classic relationship, where a small addition of a higher risk asset (in this case the stock market) adds more return than risk. When more than 20 percent equities are added to the portfolio, the incremental risk becomes larger than the return. In both cases, the final portfolio has lower risk than stocks alone, achiev­ ing the goal of the asset manager.

Adding Other Assets

To see whether a new trading method or alternate investment would improve the returns of a stock and bond portfolio, we can use the same spreadsheet technique that was applied in Figure 4-6. Because most fund managers use a portfolio of 60 percent stocks and 40 percent bonds as a benchmark, the following calculations assume that the new trading

11

1.5 2 2.5 3 3.5 4 4.5 5

Risk (1 Standard Deviation of Monthly Equity Changes)

Figure 4-6. Combining stocks and bonds. Because stocks and bonds have different risk and return profiles, and provide portfolio diversification, plotting combinations of these assets results in a curve, rather than a straight line. The best choice favors a smaller allocation of stocks.

New Portfolio Return = (1 - % New Asset)

x (.60 x Stock Return + .40 x Bond Return) + % New Asset x New Asset Return

where % New Asset is the percentage of the new asset used

Return is the compounded (annual or period) rate of return Risk is the (annualize or period) standard deviation

Diversifying with Derivatives

If the new asset has better returns at lower risk than the old portfolio, then it will raise the efficient frontier everywhere. In fact, the new asset will be a better investment than the stock and bond portfolio. But that is rarely the case. The new asset will usually have a higher profit and a higher risk or a lower profit at a lower risk. By diversifying the original portfolio with a small allocation of the new asset, the portfolio returns will be improved at some investment levels.

Derivatives have become a closely watched asset for portfolio diver­ sification. The most popular index for measuring performance of Commodity Trading Advisors is the MAR Dollar-Weighted CTA Index (compiled by Managed Account Reports). For the period 1983 through 1992, the Index showed a compounded rate of return of 12.84 percent

5with a monthly standard deviation of 5.62 percent. For the same period, the performance profile of stocks, bonds, and the CTA Index was:

MAR changed the Dollar-Weighted Index retroactively as of January 1,1993 .

We would not expect the CTA performance to improve the returns of a stock and bond portfolio because the risk was higher and the returns lower than stocks. The unique monthly patterns, however, prove that diversification can improve performance even though the cumulative statistics make it seem unlikely. Figure 4-7 shows the risk and return of a portfolio where different percentages of the CTA Index are combined with a 60 percent to 40 percent stock and bond portfolio. As allocations increase to a 20 percent use of the CTA Index, risk decreases while returns improve marginally. At the 20 percent level, risk has dropped more than 9 percent while returns are unchanged. As allocations become larger, the unfavorable profile begins to show and the total performance deteriorates. In this case, diversification alone improves results.

2.7

2.9 3 3.1

Risk (1 Standard Deviation)

Figure 4-7. CTA Index with stocks and bonds. Adding derivatives to a stock and bond portfolio Improves results when less than up to 20% is allocated. Risk declines due to diversification, although returns remain nearly unchanged.

Using Correlations to Select Assets

Risk is reduced when the assets in the portfolio are diversified using dif­ ferent strategies or unrelated markets. But diversification is not always obvious. Most investors would consider a portfolio of stocks and bonds to be diversified, offering some risk reduction because stocks might post a profit on the same day that bonds lose. That is true sometimes.

When an unfavorable economic report is released by the federal gov­ ernment, stock prices may react with a sharp drop. Traders expect the Fed to lower interest rates to offset this decline, therefore they buy bonds, moving prices higher (and rates lower). If you have a portfolio of stocks and bonds, this represents risk protection. The loss in stocks is partially offset by a profit in bonds.

But under most economic conditions, profits from stocks and bonds have similar movements. That does not mean that the stock market rises on one out of every two days that bond prices rise. When interest rates move slowly lower, as seen in the prolonged recession of the early 1990s, the stock market moves slowly higher. This may be a combination of high­ er earnings (more productivity, lower cost of money) and anticipation of economic stimulation. The result is that the price movements of stocks and bonds are fundamentally related and do not offer as much diversification as an entirely unrelated investment (see Figure 4-8).

Correlations and Risk Protection

The similarity of two price series can be measured by calculating the cor­ relation coefficient, r, which compares how two corresponding sets of numbers in a times series vary with respect to one another. The results of the relationship are expressed on a scale from +1 (perfect positive correlation, where the two series move exactly together) to 0 (no rela­ tionship between the data movements) to —1 (perfect negative correla­ tion, where the two items move exactly in opposite directions).

Correlation coefficients are a good indication of how much diversifi­ cation you will get by combining markets or strategies in a portfolio. As

The correlation coefficient measures the variation in the corresponding values of two data series. Formally, it is the ratio of the unexplained deviation to the total deviation of each value from the average or trend. It can be expressed for two equity series in the following general notation:

R = @SUM(Equity1_deviations*Equity2_deviations,N)/ (@SQRT(@SUM(Equity1_deviations A 2,N) * @ SQRT( @ SUM(Equity2_ deviations A 2,N)))

For a spreadsheet, it is necessary to create a new column with the dif­ference between each equity value and the average equity:

Spreadsheet solution. Assume there are 100 rows of equity values entered. Each row below is copied down from 2 through 100.

Cells Formulas Description

The following is calculated once after rows 1-100 are complete:

R = @SUM(F1..F100)/(@SQRT(@SUM(G1..G100))) *(@SQRT(@SUM(H1..H100)))

A comparison of the average of the standard deviation and rate of return with the same values found using a spreadsheet portfolio, shows the real effect of diversification for three passive portfolios (see Table 4-2). Using stocks, bonds, and a simple foreign exchange basket, more risk reduction is gained when the correlation is .00 for the FX index with stocks or bonds, compared with a higher .34 correlation for stocks and

Box 4-2 . (Continued)

The results r are interpreted as follows:

r = +1 A perfect positive correlation exists. For every move in one data series,

there is an equivalent move in the other series. +1 > r > 0 The similarity of price movement increases as the value of r moves

from 0 to 1.

r = 0 There is no relationship between the two sets of points.

-1 < r < 0 The negative similarity increases as the value of r moves from 0 to -1. r = — 1 A perfect negative (opposite) correlation exists. For every move in one

data series, there is an opposite, equivalent move in the other series.

Note that results are more meaningful when the series is detrended by taking the first differences of the prices.

Example

The scatter diagram (Figure 4-8) shows an elongated pattern, indicat­ ing a modest relationship between monthly stock and bond price movements. When bond prices rise during one month, there is a rea­ sonable chance that stock prices will rise. The result of the calculation is r = .34; there is a 34 percent positive correlation between annual stock and bond price movement.

Built-in Spreadsheet Functions

The correlation coefficient can be found for most spreadsheet pro­ grams, under the "Tools" menu. The result is instantaneous. Just select TOOLS/ADVANCED MATH/REGRESSION in Quattro, or the equiva­ lent in other programs. Most spreadsheets give R Squared, (r 2 ) rather than r; therefore the correlation value ranges from 0 to +1, rather than —1 to +1. If you have r 2 , you must compare the slopes of the individ­ual series. If they are the same, then r is positive; if they are moving in opposite directions, r is negative.

Table 4-2. Risk Reduction Associated with Correlations

bonds. It is still necessary to create a portfolio using a spreadsheet to find the expected level of risk reduction.

The 9.4 percent risk reduction for the S&P bond portfolio is less than either of the two FX portfolios. That is expected because the correlation of .34 is higher than the other portfolios. However, the Stocks-FX and Bonds-FX port­ folios had very different amounts of risk reduction, even though they both had zero correlations. Correlations are helpful, but not complete.

Fast-Netting Method for Asset Allocation

Computers allow us to shortcut the mathematics and go directly to the answer. The purpose of standard deviations and correlation coefficients is to find out which combination produced the greatest return for the lowest risk. Although the traditional method is correct, it still leaves a large degree of uncertainty.

Spreadsheets can find more complete, understandable answers quickly. How did we know that the two FX portfolios in Table 4-2 reduced risk by 29 percent and 19 percent? By putting the annual returns in spreadsheet columns, it is a simple matter to create another column by adding half of the Stocks return plus half the FX returns. Then calculate the standard deviation of the new column. That is as definitive as you can get. The same procedure can be followed for monthly and daily returns. The computer may only take another 3 sec­ onds to calculate a portfolio of daily, rather than monthly returns.

Computers now allow us to perform operations directly, rather than estimate the results. Some of us need time to adjust to this way of operating, of using the power available in the computer. The spreadsheet solution is much simpler, but requires such a large number of calcula­ tions that we would never have considered doing this manually. The correlation coefficient is a smart, uniform way of estimating relation­ships, but not as good as the spreadsheet solution.

There are still error factors and probabilities to be considered in the "fast netting" approach. Past performance is likely to show less risk than the future; one standard deviation represents the probability of 68 percent occurrence; and using 10 years of annual data gives a sample error of 1 /SQRT(N-l), or 32 percent.

Adding Common Sense to Statistics

Probability and statistics are not a substitute for common sense. The scatter diagram, Figure 4-8, shows one isolated point far away from the typical pattern, a loss of 22 percent. That outlier was one month out of 120, less than 1 percent of the data. Statistically, that is small. In reality, it means that on average, after you have traded successfully for 5 years, you will lose all your investment. That does not sound as good as "less than a 1 percent chance of losing."

The outlier in Figure 4-8 was the stock market plunge of 1987. Stocks plummeted and traders ran to buy bonds. There was also the invasion of Kuwait and Gorbachev's abduction a few years later. All of these are price "shocks" that can cause extreme, highly correlated moves when statistics have ruled them out for all practical purposes. These infre­ quent but important events are discussed in Chapter 7.

Business Risk

Common sense allows us to create a worst-case scenario. It is a neces­ sary step for trading survival. It is not just traders that need to do this, brokerage firms have done it for years. For example, a brokerage firm has 10,000 clients trading stocks, options, bonds, foreign exchange, and a variety of commodity markets. That would seem highly diversified and very safe for the brokerage firm, which must use its own capital to cover underfunded trading losses while it tries to collect money from customers. But Kuwait is invaded. Investors run to the U.S. dollar, bond prices soar, commodity prices rise, and the stock market moves up on a combination of economic stimulation and lower interest rates. Newspapers cover the financial stories, showing individuals who have profited, and more follow. Soon the 10,000 individual investors are all holding positions that would benefit from the continuing war. The bro­kerage firm is faced with the situation that its total customer assets will be very volatile. If the war ends, customer losses could be huge, and all at the same time. Completely unrelated investments are liquidated to cover losses in other markets. Those customers trading on margin may not meet the calls for new money, placing the brokerage firm's capital­ization at risk.

This situation actually happened in early 1980, when silver and gold rallied to all-time highs. The publicity given to the Hunt's silver posi­ tion was so extensive that the general public was heavily committed to holding long positions.

Brokerage firms will deal with this by monitoring the aggregate cus­ tomer positions. When a percentage of correlated holdings exceed a safe level, they recommend other positions, raise the house margin levels, and generally try whatever is needed to get clients out of current, high- risk positions and add diversification. They save the clients and they serve themselves.

Institutions that hold large house trading positions have the same prob­lem. What appears to be diversified under normal market conditions may not provide the risk protection under extreme cases—and those are the most important. A sharp loss once every three to five years is just as bad as high volatility in daily performance. For many risk situations, there is no statistical evaluation. It is simply necessary for someone to reason out what might happen under varying circumstances.

Correlations and Time

The correlation matrix of monthly returns in Table 4-3 shows how the choice of time periods affects the measurement of similarity between assets or systems. Using only the past 10 years of data, the correlations drop significantly when monthly values replace annual data. Those values already near zero may be slightly higher without contradicting the principle. This phenomenon is not surprising. The annual, long- term relationship between interest rates and Producer Price Index may be predictable, but month-to-month reactions can vary. If interest rates overreact to an anticipated increase in prices, then daily or monthly val­ ues may move somewhat apart, while the economic relationship remains steady. As you reduce the time period, the smaller price changes become more important. Noise becomes a bigger factor in the overall pattern, and the correlation drops.

Using daily data to measure correlation is likely to give you an answer near zero for most combinations. But results over a full year could be very similar, and a poor choice if you are looking for portfolio diversification.

Table 4-3. Correlation Matrices for Annual and Monthly Returns (1983-1992)

Correlation of Monthly Returns

Correlation of Annual Returns

The two correlation matrices in Table 4-3 compare the similarity in price movement of passive portfolios of the S&P 500, Lehman Brothers Bond Index, the Producer Price Index, a Forex group, and the MAR dollar-weighted Commodity Trading Advisor Index. The monthly figures show much smaller correlations than the annual returns. As shorter time periods are used for evaluation, the noise of short-term move­ment interferes with the direction of the long-term trend. This makes the correlations seem lower, yet severe, volatile moves may prove that many of these markets move in the same direction.

Forecasting Correlations

Another essential step in evaluating diversification is to forecast the correlations. Markets change, and more so in these times of globalization. World equity markets are becoming interrelated. A severe drop in the Japanese, German, or U.S. stock markets is likely to cause similar drops in other mar­kets, resulting in correlated performance. This tendency can be seen by cal­ culating individual annual correlations (using monthly data) and looking at the trend of r. If the correlation coefficient is moving toward +1, then you will want to decrease the use of these markets for diversification.

Losses
Average
Profits

Skewed Distributions

The standard deviation assumes that profits and losses fluctuate evenly above and below the equity trend, as seen in Figure 4-2, which shows even­ ly spaced parallel lines. That is not a realistic assumption. Many trend- following systems are skewed to show larger profits than losses, but more frequent smaller losses. Then a typical profit/loss trend-following distribu­tion will show a longer "tail" to the right, indicating larger profits, and the highest point to the left of center and below zero, indicating that the most frequent trades are small losses. Figure 4-9 shows the "normal" distribution curve on the same chart as the skewed trend performance.

Frequency Distribution. A practical way to display trading perfor­ mance is by using a frequency distribution, or "histogram." This looks the same as a spreadsheet bar graph, except that the bars have different widths. For example, Figure 4-10 shows that each of the 10 bars represents

10 percent of the trades, when sorted from greatest loss to largest profit. The wide bar at the right indicates that there is a large variance between the largest profits. The narrowest bar near the left shows that losses are clustered because of cutting losses short.

By looking at the chart, we can say there is a 10 percent chance that profits on any one trade will be greater than 50 percent or less than —10 percent. There is also a 20 percent chance that profits will be greater than 40 percent or less than -6 percent.

Semivariance. A useful way of measuring risk for equity distributions is with semivariance. This simply divides the equity into separate sets of continuous profits and losses. For example, if a trading program had the following sequence of profits and losses:

2.5%, (1.0%), 4.1%, 3.5%, (.6%), (1.1%), (.7%), 1.5%

it would be regrouped to show alternating, cumulative profits and losses: 2.5%, (1.0%), 7.6%, (2.4%), 1.5%

The losses can be evaluated separately, giving the probability of a draw­ down based on sequences of trades, rather than the probability of a loss on a single trade. This is a practical approach to evaluating risk.

Another valuable measurement is risk of ruin. It is the chance of losing so much that you must stop trading. The formula that follows allows us to specify the maximum amount that we are willing to lose, constituting ruin. Before looking at the calculations, consider these premises:

•  In real trading, once profits accumulate, the chance of ruin decreases.
The greatest risk is at the beginning.

•  If we plan to withdraw profits, thereby maintaining the same relative
commitment to the market, then the risk of ruin must be greater than if
we accumulate profits and keep the trading position the same.

The probability of risk of ruin is expressed as: Risk_of_Ruin = ((1 - Edge)/(1 + Edge)) A Units_Capital

where Edge = 2*Probability_of_Win - 1, the "trader's advantage"

Units_Capital allows risk to be given relative to the size of the investment

Example 1. A typical trend-following system will have larger profits than losses and more losing trades than winning ones. If

Probability_of_Win = 40% and Units_Capital = 1 for a $10,000 investment

Risk_of_Ruin = ((1 - .40)/(1 + .40)) A 1 = (.60/1.40) A 1 = .42857 or 42.8%

If the investment is increased to $20,000, or 2 units of capital,

Risk_of_Ruin = ((1 - .40)/(1+.40)) A 2

= (.60/1.40) A 2 = .18367 or 18.3%

By doubling the investment, the risk of ruin decreases by 57 percent.

Profit Goals

Taking profits will reduce the risk of ruin. The closer the profit goal, the less chance of ruin:

Risk and Return 67

Risk_of_Ruin_2 = ((((1 + Edge)/(1 - Edge^^Goal) - 1)/(((1 + Edge)/ (1 - Edge)) A (Units_Capital + Goal) - 1)

where all the terms are the same as the previous calculation, except Goal is the profit objective in units of trading capital (profit objective divided by one unit of trading capital).

Example 2. If you traded the same trend-following system as example 1, with a 2-unit investment, but had a profit goal of half of one capital unit (or $5,000), then

Risk_of_Ruin_2 = ((((1 + .40)/(1 - .40)) A 5) - 1)/(((1 + .40)/(1 - .40))* (2 + .5)-1)

= (((1.40/.60) A .5) - 1)/(((1.40/.60) A 2.5) - 1) = .5275/7.3165 = .0721 or 7.21%

Specifying Wins, Losses, and Risk

To be practical, a risk evaluation must use the performance profile of the system being traded. That includes the size of the profits and losses, as well as the amount of risk the investor is willing to accept. The follow­ ing formula, from Ralph Vince's Portfolio Management Formulas (New York: Wiley, 1990) is a summary of P. Griffin's work, The Theory of Blackjack (Las Vegas: Gamblers Press, 1981) and a "fair approximation" of risk:

Risk of Ruin = ((1 - P)/P) A (MaxRisk/A) where the following terms are defined as

AvgWin is the average winning trade (e.g., $600)

AvgLoss is the average losing trade (e.g., $200)

Investment is the amount invested (e.g., $20,000)

ProbWin is the probability (percentage) of a winning trade (e.g., .40)

ProbLoss is the probability (percentage) of a losing trade (e.g., .60)

MaxRisk is expressed in percent (e.g., 25% = .25)

AvgWin% is @ABS(AvgWin/lnvestment)

AvgLoss% is @ABS(AvgLoss/lnvestment)

Z is ProbWin*AvgWin% - ProbLoss*AvgLoss%

A is (ProbWin*(AvgWin%) A 2 + ProbLoss*(Avgl_oss%) A 2) A ( 1/2)

Pis.5*(1+(Z/A))

Example 3. Using the values given after the definitions, we get

AvgWin% = @ABS(600/20000) = .03 AvgLoss% = @ABS(200/20000) = .01

Z = .40*.03 - .60\01 = .012 - .006 = .006

A = (.40*.03 A 2+.60*].01 A 2) A (1/2) = (.OOOSe+.OOOOe^i^) = .0605

P = .5*(1+.006/.0605)) = .5495

Risk_of_Ruin = ((1 - .5495)/.5495) A (.25/.0605) = .8198*4.132 = .4408 or 44.1%

Summary

Risk is important, but it does not have to be complex. The simplest risk- adjusted measure of trading performance is the return/risk ratio, which allows a fair comparison of values that are determined during different time intervals. It only requires that you use the same measure of risk and reward for all the information being evaluated.

Time periods can be deceptive. All time periods smooth performance, eliminating interim equity fluctuations that could have been large. To offset part of this problem, the maximum drawdown is needed, which puts an absolute value on the expected loss. Over time, even the maximum loss will be replaced by a larger loss. The longer the tested histo­ ry, the less likely it will occur, but inevitably, all programs see larger profits and larger losses than those on record.

Diversification is the best approach to risk reduction. Deciding how much safety is gained by combining assets has be&n a very mathemati­cal process; however, a simple spreadsheet program now allows anyone to combine performance history and evaluate the return/risk ratio of the proposed portfolios.

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