4-1 shows nine tests progressing from short-
47
Risk and Return
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 test4. However, the best performance choice was test 2, with an adjusted,compounded rate of return of 34.5 percent. Profits and risk in columns2 and 3 were both converted to percentages based on the current SENSEX price of 2700.A fast way of arriving at the best choice is simply to divide the totalprofits (column 2) by the total risk (column 3). The greatest return/riskratio will be the best choice, as long as standard measurements wereused and both returns and risk were calculated over the same timeperiod within the same test. Even if test 4 was evaluated over a slightlydifferent length interval than test 9, the ratio would give a valid comparison.
Table 4 - 1 . Risk-Adjusting Results of a Trending SystemTest(1)Fasti2345678Slow 9
ProfitRisk(in SENSEX pts)(2)(3)
Return
CROR
Risk
(%)
(%)
(%)(6)
AdjustFactor(7)
Riskadj Prof/RiskCRORRatio(8)(9)
313330462808203220692136322036813311
216213204175177179219236223
47.045.928.332.333.033.848.053.649.3
454041303133515854
1.501.331.371.001.031.101.701.931.80
31.334.520.632.332.030.728.227.827.4
120810931113814827905139115681451
(4)
(5)
2.592.792.522.502.502.362.312.352.28
Finding the best risk-adjusted returns for the Bombay SENSEX trend-followingprogram 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 yearsof 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 returnsdivided by 2700, the current SENSEX price. The compounded rate of return for the 2year period is the return (column 4) less the initial investment of 100, raised to the1 /2 power (see calculation for compounded rate of return). The percentage risk (column 6) is the risk in points (column 3) divided by the current SENSEX price of2700. The risk-adjustment factor is the percentage risk (column 6) divided by thelowest percentage risk in column 6 (30 in test 4). The risk-adjusted compound rate ofreturn (column 8) is the compound rate of return (column 5) divided by the adjustment 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, whichhad neither the highest profits nor the lowest risk. Although the ratio does not tellthe relative rate of return, using the highest value is a fast way of identifying thebest test.
48
How Changing Markets and Technology Affect Results
Risk to TimeThe standard deviation using the quarterly change in equity must bebigger than the standard deviation of monthly changes because longertime intervals allow larger fluctuations. Even though a quarter is 3.0times longer than a month, the standard deviation is 4.74 times larger.As the time interval becomes even larger, however, the variation doesnot increase at the same rate. Most price or equity data will tend towarda maximum variance, regardless of the time interval. A typical risk-totime curve is shown in Figure 4-3.
Time Periods TellDifferent StoriesAlthough equity changes get larger when annual data is used instead ofmonthly or daily returns, longer periods can also hide risk. DuringOctober 1987, the S&P 500 plunged 44 percent but recovered to show asmaller loss of 21 percent by the end of the month, and actually closedout the year up 2 percent.Dec 87 S&P Futures
Sep 30,30,19871987
321.69 (closing price)
Oct 20,1987
181.00 (low)
down 44%
Oct 31,1987
255.74 (closing price)
downdown 21%21 %
Mar 87 S&P Futures
Dec 31,1986
242.17 (closing price)
Mar 88 S&P FuturesFutures
Dec 31,1987
247.09 (closing price)
up 2%
Using monthly data, the chances are only 1 in 21 that the returns for themonth will be the low of that month, therefore the performance issmoothed. The risk of loss is there, but you cannot see it from the numbers.Maximum DrawdownTo assess risk it is always necessary to know the maximum drawdown, thelargest peak-to-valley equity drop. The maximum drawdown is likelyto be much larger than any monthly equity change, as seen in the S&Pexample. Even though you may not see a loss of that magnitude soon, itis inevitable that a swing as big or greater will occur sometime in thefuture. It is both naive and unrealistic to assume that a faster, largerrally or bigger plunge will never occur.
49
Risk and Return
Monthly
Quarterly
Seml-Annually
Annually
Interval of Calculation
Fignre 4 - 3 . S&P risk to time relationship. The standard deviation of theS&P price changes from 1983-1992 shows that price change approaches alimit as the measurement time period gets larger.
The SO-Tear RuleEnvironmental planning uses a 10-year rule for water relief. Ditches arebuilt along country roads to control the maximum runoff measuredover 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 aheight that satisfies the "50-year rule." What does that mean to a familyliving in a home along the river? It says that water will rise above thelevy, but not very often. It may have overflowed the levy 75 years ago,but not during the past 50 years; therefore, flooding could happen oncein the lifetime of each family member.The same situation exists in trading. Professional traders will be facedwith extreme moves and extraordinary price shocks during theircareers. Those who plan to use the markets for only a short time maynever see these moves and could take the chance that these extremeswill not occur during their short trading stay. It is a classic risk-andreward decision.Trading safety relates directly to capitalization. Larger, more diversified investments, with proportionately greater reserves, are safer. If youare fortunate enough to profit from a small amount of capital, then
50
How Changing Markets and Technology Affect Results
decrease the relative risk by increasing the percentage of reserves.Traders who continue to leverage all their profits increase their chanceof a complete loss.
Specifying Acceptable RiskStandardizing the risk measurement is important because it allows youto specify the amount of risk you are willing to accept for any tradingmethod. For example, as a foreign exchange trader, you are given US$10million 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 fluctuations of your performance, and have a track record that can be measured. Or, as an experienced analyst, you expect to devise a methodbased on previously successful ideas.Using annual figures, your performance shows an admirable 40 percentreturn with a 10 percent standard deviation. You know that 1 standarddeviation means that there is an 84 percent chance that the equity won'tdrop more than 10 percent, and 3 standard deviations gives a 99.5 percentchance that equity won't drop more than 30 percent. Looking at the resultsin reverse, a risk of 10 percent means that there is a 16 percent chance thatyou will lose more than 10 percent during a year.You cannot have such a large risk of losing 10 percent; therefore, youbring this down to a safe level by dividing by 4. This really means thatyou will use only one-fourth of the capital in the account to trade. Theaccount is "deleveraged." The original 10 percent standard deviationbecomes 2.5 percent and there is a16.0% 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 by4, the returns must also be divided by 4, giving an adjusted annualizedreturn of 10 percent—still pretty good. To implement this plan, youneed 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 anundesirable (but possible) run of bad trades. You are experiencedenough to know that a 0.5 percent chance means that you will lose morethan 7.5 percent, but not often. Since this risk measurement used monthly values, a daily maximum drawdown is still needed to decide a safelevel of capitalization.
Risk and Return
51
Foreign ExchangeTrader's DilemmaEven if everything works as planned, producing a 10 percent returnwith a 5 percent risk during the next year, there is still a problem. Thetrading manager calls you aside and asks, "Why aren't you using all themoney? You returned 10 percent using only $2.5 million. If you hadused all the money, you could have returned 40 percent. If you're notgoing 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 doyou explain that you were using the money? There was a 16 percent chancethat you would need more than $2.5 million and a 2.5 percent chance ofusing more than $5 million. It is difficult to explain, but it needs to bedone. If you trade a larger percentage, you will eventually be shut downby losing more than 10 percent. This same problem will be seen inChapter 7, when we look at the effects of price shocks.
Graphing Risk and RewardGraphing the risk and reward leads to important observations. Thisapproach is often used by asset allocators, to show the trade-offsbetween systematic combinations of profits and losses. The returns andrisk (in points) given in Table 4-1 have been plotted in Figure 4-4 andmarked 1 through 9.Results that are plotted higher and to the left have more profit and lessrisk. Those that are lower and to the right are worse choices because theyhave less profit and more risk. A rational investor would always choose apoint directly above another one because it has a higher return for the samelevel of risk (given a choice of points 2 or 3, point 2 is the better one).Efficient FrontierA curve can be drawn through the points that return the highest profitat each level of risk. Note that the curve, seen as a broken line in Figure4-4, flattens out as it goes to the right. This means that the investor ortrader must assume a proportionately higher risk to gain a smallincrease in profits.This curve is similar in concept to an efficient frontier, used in asset allocation. However, if you use the performance of a single trading system,one combination may show exceptionally good returns by chance. A curvedrawn through the best cluster of points, rather than the absolute highestcombinations, will give a more realistic appraisal of expectations.
52
How Changing Markets and Technology Affect Results
Risk (in SENSEX Points)Figure 4 - 4 . Plot of risk and returns. Plotting the risk andreturn of each test provides a clear picture of investment choices.Those points higher and to the left are better; those lower and tothe right are worse. The standard selection is found at the pointwhere a straight line, anchored at the risk-free rate-of-return onthe left scale, touches the curve. The curve is called an efficientfrontier.
The Best ChoiceInvestors are always free to choose the combination of risk and rewardthat satisfies their objectives. A rational choice must be on the efficientfrontier. A more conservative investor will be more comfortable with aselection further to the left, while a risk seeker will prefer the higherprofits even at the cost of increased risk.What is normally considered the "best" choice is found by drawing astraight line from the risk-free rate of return (on the left scale), tangentto the efficient frontier curve, as seen in Figure 4-4. The risk-free rate ofreturn assumes 5 percent over 2 years applied to the amount of moneyneeded to buy the stock index at 2700 points. Keeping the calculation inSENSEX points,2700X.05(investment in points)(yearl)
X
.05(year 2)
277(interest in points)
The straight line drawn from 277 touches the efficient frontier at exactlytest 2, the best choice.
Risk and Return
53
Diversification andRisk ReductionDiversification is the best form of risk reduction. Investing in more thanone asset, each with a good return, will reduce risk by benefiting fromequity variation that occurs at different times. Investment managers areconstantly searching for ways to place funds that yield better-than-average returns and modest risk. Combining a number of medium-risk,diverse investments often nets a low-risk portfolio. Diversification canalso be gained by using different trading strategies in the same market.Applying AssetAllocation TechniquesAsset allocation is the process of distributing investment funds into one ormore markets or vehicles to create an investment profile with the mostdesirable return/risk ratio. In its simplest form, asset allocation would useonly one active investment, such as a stock portfolio, placing the remaining funds into risk-free, short-term government bonds. In its most complicated form, many investment vehicles are combined on a dynamic basis.Assets may vary from a passive gold portfolio to active internationalstocks and discretionary trading of foreign exchange markets.Simple Risk ReductionThe stock market produces healthy long-term returns of about 10 percent per year (16.4 percent compounded rate of return from 1983 to1992), with a return/risk ratio of approximately 1:1. If you were 100 percent invested in stocks, there would be a 16 percent chance of a lossgreater than the annualized return sometime during that year. To reducethe risk, you simply trade a smaller amount of stocks and put the balance in a money market account or Treasury bills. Figure 4-5 shows thatthe risk and returns are reduced linearly by holding varying amounts inreserve.Classic Stock and Bond PortfolioA more popular alternative is to allocate a portion of the portfolio tobonds. Although risk-free when held to maturity, a passive bond portfolio 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.
54
Stocks 100%Reserve0%
How Changing Markets and Technology Affect Results
75%25%
50%50%
25%75%
0%100%
StocksReserve
Figure 4-5. Combining stocks and cash reserves. As cash reservesIncrease, both risk and return decrease at the same rate.
Figure 4-6 compares the return/risk ratios of a stock and bond portfolio for varying percentage combinations of those t w o items. Themonthly returns were combined using a spreadsheet, and the compounded rate of return and monthly standard deviations were recalculated on the new portfolio. The result is a classic relationship, where asmall addition of a higher risk asset (in this case the stock market) addsmore return than risk. When more than 20 percent equities are added tothe portfolio, the incremental risk becomes larger than the return. Inboth cases, the final portfolio has lower risk than stocks alone, achieving the goal of the asset manager.A d d i n g Other A s s e t sTo see whether a new trading method or alternate investment wouldimprove the returns of a stock and bond portfolio, we can use the samespreadsheet technique that was applied in Figure 4-6. Because mostfund managers use a portfolio of 60 percent stocks and 40 percent bondsas a benchmark, the following calculations assume that the new trading
55
Risk and Return
2
2.533.544.5Risk (1 Standard Deviation of Monthly Equity Changes)
Figure 4 - 6 . Combining stocks and bonds. Because stocks and bondshave different risk and return profiles, and provide portfolio diversification,plotting combinations of these assets results In a curve, rather than astraight line. The best choice favors a smaller allocation of stocks.
program or asset replaces a percentage of that standard portfolio. Eachspreadsheet row, representing one month, becomesNew 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 usedReturn is the compounded (annual or period) rate of returnRisk is the (annualize or period) standard deviationDiversifying with DerivativesIf 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 assetwill be a better investment than the stock and bond portfolio. But that israrely the case. The new asset will usually have a higher profit and ahigher risk or a lower profit at a lower risk. By diversifying the originalportfolio with a small allocation of the new asset, the portfolio returnswill be improved at some investment levels.Derivatives have become a closely watched asset for portfolio diversification. The most popular index for measuring performance ofCommodity Trading Advisors is the MAR Dollar-Weighted CTA Index(compiled by Managed Account Reports). For the period 1983 through1992, the Index showed a compounded rate of return of 12.84 percent
56
How Changing Markets and Technology Affect Results
with a monthly standard deviation of 5.62 percent. For the same period,the performance profile of stocks, bonds, and the CTA Index was:Monthly Returns, 1983-1992
Monthly standard deviationCompounded rate of return
S&P
LB Bonds
$W CTA
4.5216.40
1.5611.45
5.6212.84
MAR changed the Dollar-Weighted Index retroactively as ofJanuary 1,1993.
We would not expect the CTA performance to improve the returns ofa stock and bond portfolio because the risk was higher and the returnslower than stocks. The unique monthly patterns, however, prove thatdiversification can improve performance even though the cumulativestatistics make it seem unlikely. Figure 4-7 shows the risk and return ofa portfolio where different percentages of the CTA Index are combinedwith a 60 percent to 40 percent stock and bond portfolio. As allocationsincrease to a 20 percent use of the CTA Index, risk decreases whilereturns improve marginally. At the 20 percent level, risk has droppedmore than 9 percent while returns are unchanged. As allocations becomelarger, the unfavorable profile begins to show and the total performancedeteriorates. In this case, diversification alone improves results.
H.3-I2.7
1
12.8
1
12.9
1
|3
1
13.1
1
13.2
13.3
Figure 4-7. CTA Index withstocksandDeviation)bonds. Adding derivatives to aRisk(1 Standardstock and bond portfolio Improves results when less than up to 20% isallocated. Risk declines due to diversification, although returns remainnearly unchanged.
Risk and Return
57
U s i n g Correlations t oSelect AssetsRisk is reduced when the assets in the portfolio are diversified using different strategies or unrelated markets. But diversification is not alwaysobvious. Most investors would consider a portfolio of stocks and bonds tobe diversified, offering some risk reduction because stocks might post aprofit on the same day that bonds lose. That is true sometimes.When an unfavorable economic report is released by the federal government, stock prices may react with a sharp drop. Traders expect theFed to lower interest rates to offset this decline, therefore they b u ybonds, moving prices higher (and rates lower). If you have a portfolio ofstocks and bonds, this represents risk protection. The loss in stocks ispartially offset by a profit in bonds.But under most economic conditions, profits from stocks and bondshave similar movements. That does not mean that the stock market riseson one out of every two days that bond prices rise. When interest ratesmove slowly lower, as seen in the prolonged recession of the early 1990s,the stock market moves slowly higher. This may be a combination of higher earnings (more productivity, lower cost of money) and anticipation ofeconomic stimulation. The result is that the price movements of stocks andbonds are fundamentally related and do not offer as much diversificationas an entirely unrelated investment (see Figure 4-8).
- 1 012345Lehman Bros. Bond Index Monthly Returns (in Percent)
Figure 4-8. Scatter diagram of monthly stock and bond returns.The elongated pattern shows returns of passive portfolios of stocks andbonds. This similarity limits the potential diversification.
58
How Changing Markets and Technology Affect Results
Correlations and Risk ProtectionThe similarity of two price series can be measured by calculating the correlation coefficient, r, which compares how two corresponding sets ofnumbers in a times series vary with respect to one another. The resultsof the relationship are expressed on a scale from +1 (perfect positivecorrelation, where the two series move exactly together) to 0 (no relationship between the data movements) to —1 (perfect negative correlation, where the two items move exactly in opposite directions).Correlation coefficients are a good indication of how much diversification you will get by combining markets or strategies in a portfolio. As
Box 4-2. CALCULATION AND INTERPRETATIONOF CORRELATIONSThe correlation coefficient measures the variation in the corresponding values of two data series. Formally, it is the ratio of the unexplained deviationto the total deviation of each value from the average or trend. It can beexpressed for two equity series in the following general notation:R = @SUM(Equity1_deviations*Equity2_deviations,N)/(@SQRT(@SUM(Equity1_deviationsA2,N)* @ SQRT( @ SUM(Equity2_deviationsA2,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 valuesentered. Each row below is copied down from 2 through 100.CellsA1-A100B1-B100C1-C100D1E1F1G1H1
FormulasDate of the EntryEquitylEquity2B1 2 @AVG($B$1..$B$100)C1 2 @AVG($C$1..$C$100)D1*E1D1A2D2*2
DescriptionDaily, Monthly, or Yearly1st Equity values2nd Equity valuesDeviations of Equity 1Deviations of Equity 2Products of DeviationsSquare of Equity 1 DeviationsSquare of Equity 2 Deviations
The following is calculated once after rows 1-100 are complete:R = @SUM(F1..F100)/(@SQRT(@SUM(G1..G100)))*(@SQRT(@SUM(H1..H100)))
Risk and Return
59
the positive correlations become stronger, the potential diversificationdeclines. When a correlation is +1.00, there is no diversification. Butwhen the correlation is zero (where there is no predictable relationshipbetween the two items), it is still not clear how much risk reduction is tobe gained (see Box 4-2).A comparison of the average of the standard deviation and rate ofreturn with the same values found using a spreadsheet portfolio, showsthe real effect of diversification for three passive portfolios (see Table 42). Using stocks, bonds, and a simple foreign exchange basket, more riskreduction is gained when the correlation is .00 for the FX index withstocks 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 movesfrom 0 to 1.r=0There 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= —1A perfect negative (opposite) correlation exists. For every move in onedata series, there is an opposite, equivalent move in the other series.
Note that results are more meaningful when the series is detrended bytaking the first differences of the prices.ExampleThe scatter diagram (Figure 4-8) shows an elongated pattern, indicating a modest relationship between monthly stock and bond pricemovements. When bond prices rise during one month, there is a reasonable chance that stock prices will rise. The result of the calculationis r = .34; there is a 34 percent positive correlation between annualstock and bond price movement.Built-in Spreadsheet FunctionsThe correlation coefficient can be found for most spreadsheet programs, under the "Tools" menu. The result is instantaneous. Just selectTOOLS/ADVANCED MATH/REGRESSION in Quattro, or the equivalent in other programs. Most spreadsheets give R Squared, (r2) ratherthan r; therefore the correlation value ranges from 0 to +1, rather than-1 to +1. If you have r2, you must compare the slopes of the individual series. If they are the same, then r is positive; if they are moving inopposite directions, r is negative.
How Changing Markets and Technology Affect Results
60
Table 4-2. Risk Reduction Associated with Correlationsa. S&P vs. Bonds with a .34 Correlation (Modest Pattern)S&P500
Bonds
Avg Return16.7811.05St Dev10.654.89b. S&P vs. FX with a .00 Correlation (No Pattern)S&P500FXAvg ReturnSt Dev
16.7810.65
2.9014.52c. Bonds vs. FX with a .00 Correlation (No Pattern)
Avg ReturnSt Dev
Bonds
FX
11.054.89
2.9014.52
Average13.927.77Average9.8412.58Average6.989.70
Equal Allocation13.927.04 (-9.4%)Equal Allocation9.848.92 (-29.1%)Equal Allocation6.987.80 (-19.6%)
bonds. It is still necessary to create a portfolio using a spreadsheet tofind the expected level of risk reduction.The 9.4 percent risk reduction for the S&P bond portfolio is less than eitherof the two FX portfolios. That is expected because the correlation of .34 ishigher than the other portfolios. However, the Stocks-FX and Bonds-FX portfolios had very different amounts of risk reduction, even though they bothhad zero correlations. Correlations are helpful, but not complete.Fast-Netting Method forAsset AllocationComputers allow us to shortcut the mathematics and go directly to theanswer. The purpose of standard deviations and correlation coefficientsis to find out which combination produced the greatest return for thelowest risk. Although the traditional method is correct, it still leaves alarge degree of uncertainty.Spreadsheets can find more complete, understandable answersquickly. How did we know that the two FX portfolios in Table 4-2reduced risk by 29 percent and 19 percent? By putting the annualreturns in spreadsheet columns, it is a simple matter to create anothercolumn by adding half of the Stocks return plus half the FX returns.Then calculate the standard deviation of the new column. That is asdefinitive as you can get. The same procedure can be followed formonthly and daily returns. The computer may only take another 3 seconds to calculate a portfolio of daily, rather than monthly returns.Computers now allow us to perform operations directly, rather thanestimate the results. Some of us need time to adjust to this way of oper-
Risk and Return
61
ating, of using the power available in the computer. The spreadsheetsolution is much simpler, but requires such a large number of calculations that we would never have considered doing this manually. Thecorrelation 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 riskthan the future; one standard deviation represents the probability of 68percent occurrence; and using 10 years of annual data gives a sampleerror of 1 /SQRT(N-l), or 32 percent.
Adding Common Senseto StatisticsProbability and statistics are not a substitute for common sense. Thescatter diagram, Figure 4-8, shows one isolated point far away from thetypical pattern, a loss of 22 percent. That outlier was one month out of120, 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 "lessthan a 1 percent chance of losing."The outlier in Figure 4-8 was the stock market plunge of 1987. Stocksplummeted and traders ran to buy bonds. There was also the invasionof Kuwait and Gorbachev's abduction a few years later. All of these areprice "shocks" that can cause extreme, highly correlated moves whenstatistics have ruled them out for all practical purposes. These infrequent but important events are discussed in Chapter 7.Business RiskCommon sense allows us to create a worst-case scenario. It is a necessary 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 firmhas 10,000 clients trading stocks, options, bonds, foreign exchange, anda variety of commodity markets. That would seem highly diversifiedand very safe for the brokerage firm, which must use its own capital tocover underfunded trading losses while it tries to collect money fromcustomers. But Kuwait is invaded. Investors run to the U.S. dollar, bondprices soar, commodity prices rise, and the stock market moves up on acombination of economic stimulation and lower interest rates.Newspapers cover the financial stories, showing individuals who haveprofited, and more follow. Soon the 10,000 individual investors are all
62
How Changing Markets and Technology Affect Results
holding positions that would benefit from the continuing war. The brokerage firm is faced with the situation