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ability is equivalent to the value of the call option, for a call enables the investor to earn the
market return only when it exceeds rf.
Valuation of the call option embedded in market timing is relatively straightforward using
the Black-Scholes formula. Set S $1 (to find the value of the call per dollar invested in the
market), use an exercise price of X (1 rf) (the current risk-free rate is about 3%), and
a volatility of .203 (the historical volatility of the S&P 500). For a once-a-year timer,
T 1 year. According to the Black-Scholes formula, the call option conveyed by market tim-
ing ability is worth about 8.1% of assets, and this is the annual fee one could presumably
charge for such services. More frequent timing would be worth more. If one could time
the market on a monthly basis, then T 1„12 and the value of perfect timing would be 2.3%
per month.


The Value of Imperfect Forecasting
But managers are not perfect forecasters. While managers who are right most of the time
presumably do very well, “right most of the time” does not mean merely the percentage of the
time a manager is right. For example, a Tucson, Arizona, weather forecaster who always
predicts “no rain” may be right 90% of the time, but this “stopped clock” strategy does not
require any forecasting ability.
Neither is the overall proportion of correct forecasts an appropriate measure of market
forecasting ability. If the market is up two days out of three, and a forecaster always predicts
a market advance, the two-thirds success rate is not a measure of forecasting ability. We need
to examine the proportion of bull markets (rM rf) correctly forecast and the proportion of
bear markets (rM rf) correctly forecast.
If we call P1 the proportion of the correct forecasts of bull markets and P2 the proportion
for bear markets, then P1 P2 1 is the correct measure of timing ability. For example, a
forecaster who always guesses correctly will have P1 P2 1 and will show ability of 1
(100%). An analyst who always bets on a bear market will mispredict all bull markets
(P1 0), will correctly “predict” all bear markets (P2 1), and will end up with timing
ability of P1 P2 1 0. If C denotes the (call option) value of a perfect market timer, then
(P1 P2 1)C measures the value of imperfect forecasting ability.
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F I G U R E 20.6
rP “ rf
Characteristic lines
Slope = 0.6 A: No market timing,
beta is constant
B: Market timing,
beta increases with
expected market
rM “ rf excess return



A. No Market Timing, Beta Is Constant


rP “ rf Steadily
increasing
slope




rM “ rf



B. Market Timing, Beta Increases with Expected Market
Excess Return




The incredible potential payoff to accurate timing versus the relative scarcity of billionaires
should suggest to you that market timing is far from a trivial exercise and that very imperfect
timing is the most that we can hope for.


<
4. What is the market timing score of someone who flips a fair coin to predict the Concept
market?
CHECK

Measurement of Market Timing Performance
In its pure form, market timing involves shifting funds between a market index portfolio and
a safe asset, such as T-bills or a money market fund, depending on whether the market as a
whole is expected to outperform the safe asset. In practice, most managers do not shift fully
between bills and the market. How might we measure partial shifts into the market when it is
expected to perform well?
To simplify, suppose the investor holds only the market index portfolio and T-bills. If the
weight on the market were constant, say 0.6, then the portfolio beta would also be constant,
and the portfolio characteristic line would plot as a straight line with a slope 0.6, as in Figure
20.6A. If, however, the investor could correctly time the market and shift funds into it in peri-
ods when the market does well, the characteristic line would plot as in Figure 20.6B. The idea
is that if the timer can predict bull and bear markets, more will be shifted into the market when
the market is about to go up. The portfolio beta and the slope of the characteristic line will be
higher when rM is higher, resulting in the curved line that appears in 20.6B.
Treynor and Mazuy (1966) tested to see whether portfolio betas did in fact increase prior
to market advances, but they found little evidence of timing ability. A similar test was imple-
mented by Henriksson (1984). His examination of market timing ability for 116 funds in
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706 Part SIX Active Investment Management


Regression
TA B L E 20.7 Coefficient*
Sharpe™s style
Bills 0
portfolios for the
Magellan fund Intermediate bonds 0
Long-term bonds 0
Corporate bonds 0
Mortgages 0
Value stocks 0
Growth stocks 47
Medium-cap stocks 31
Small stocks 18
Foreign stocks 0
European stocks 4
Japanese stocks 0
Total 100.00
R-squared 97.3

*Regressions are constrained to have nonnegative coefficients and to have coefficients
that sum to 100%.
Source: William F. Sharpe, “Asset Allocation: Management Style and Performance
Evaluation,” Journal of Portfolio Management, Winter 1992, pp. 7“19.


1968“1980 found that, on average, portfolio betas actually fell slightly during the market ad-
vances, although in most cases the response of portfolio betas to the market was not statisti-
cally significant. Eleven funds had statistically positive values of market timing, while eight
had significantly negative values. Overall, 62% of the funds had negative point estimates of
timing ability.
In sum, empirical tests to date show little evidence of market timing ability. Perhaps this
should be expected; given the tremendous values to be reaped by a successful market timer, it
would be surprising to uncover clear-cut evidence of such skills in nearly efficient markets.


20.5 STYLE ANALYSIS
Style analysis was introduced by Nobel laureate William Sharpe.3 The popularity of the con-
cept was aided by a well-known study4 concluding that 91.5% of the variation in returns of
82 mutual funds could be explained by the funds™ asset allocation to bills, bonds, and stocks.
Later studies that considered asset allocation across a broader range of asset classes found that
as much as 97% of fund returns can be explained by asset allocation alone.
Sharpe considered 12 asset class (style) portfolios. His idea was to regress fund returns on
indexes representing a range of asset classes. The regression coefficient on each index would
then measure the implicit allocation to that “style.” Because funds are barred from short posi-
tions, the regression coefficients are constrained to be either zero or positive and to sum to
100%, so as to represent a complete asset allocation. The R-square of the regression would
then measure the percentage of return variability attributed to the effects of security selection.
To illustrate the approach, consider Sharpe™s study of the monthly returns on Fidelity™s
Magellan Fund over the period January 1985 through December 1989, shown in Table 20.7.

3
William F. Sharpe, “Asset Allocation: Management Style and Performance Evaluation,” Journal of Portfolio Man-
agement, Winter 1992, pp. 7“19.
4
Gary Brinson, Brian Singer, and Gilbert Beebower, “Determinants of Portfolio Performance,” Financial Analysts
Journal, May/June 1991.
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F I G U R E 20.7
30
Fidelity Magellan
Fund cumulative
25
return difference:
fund versus style
20
benchmark
Source: William F. Sharpe,
15 “Asset Allocation:
Management Style and
Performance Evaluation,”
10
Journal of Portfolio
Management, Winter 1992,
pp. 7“19.
5


0
1986 1987 1988 1989 1990




F I G U R E 20.8
12
Fidelity Magellan
Fund cumulative
10
return difference:
fund versus S&P 500
8
Source: William F. Sharpe,
“Asset Allocation:
6
Management Style and
Performance Evaluation,”
4 Journal of Portfolio
Management, Winter 1992,
pp. 7“19.
2

0

2
1986 1987 1988 1989 1990



While there are 12 asset classes, each one represented by a stock index, the regression
coefficients are positive for only 4 of them. We can conclude that the fund returns are well
explained by only four style portfolios. Moreover, these three style portfolios alone explain
97.3% of returns.
The proportion of return variability not explained by asset allocation can be attributed to
security selection within asset classes. For Magellan, this was 100 97.3 2.7%. To evalu-
ate the average contribution of stock selection to fund performance we track the residuals from
the regression, displayed in Figure 20.7. The figure plots the cumulative effect of these resid-
uals; the steady upward trend confirms Magellan™s success at stock selection in this period.
Notice that the plot in Figure 20.7 is far smoother than the plot in Figure 20.8, which shows
Magellan™s performance compared to a standard benchmark, the S&P 500. This reflects the
fact that the regression-weighted index portfolio tracks Magellan™s overall style much better
than the S&P 500. The performance spread is much noisier using the S&P as the benchmark.
Of course, Magellan™s consistently positive residual returns (reflected in the steadily
increasing plot of cumulative return difference) is hardly common. Figure 20.9 shows the
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Fifth Edition Management




708 Part SIX Active Investment Management




F I G U R E 20.9 90
Average tracking error,
80
636 mutual funds,
1985“1989
70
Source: William F. Sharpe,
“Asset Allocation: 60
Management Style and
Performance Evaluation,” 50

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