860
P A R T V I I
Applied Portfolio Management
The Value of Imperfect Forecasting
A weather forecaster in Tucson, Arizona, who always predicts no rain may be right 90%
of the time. But a high success rate for a “stopped-clock” strategy is not evidence of fore-
casting ability. Similarly, the appropriate measure of market forecasting ability is not the
overall proportion of correct forecasts. If the market is up 2 days out of 3 and a forecaster
always predicts market advance, the two-thirds success rate is not a measure of forecasting
ability. We need to examine the proportion of bull markets ( r
M
, r
f
) correctly forecast and
the proportion of bear markets ( r
M
. r
f
) correctly forecast.
If we call P
1
the proportion of the correct forecasts of bull markets and P
2
the propor-
tion for bear markets, then P
1
1 P
2
2 1 is the correct measure of timing ability. For exam-
ple, a forecaster who always guesses correctly will have P
1
5 P
2
5 1, and will show ability
of P
1
1 P
2
2 1 5 1 (100%). An analyst who always bets on a bear market will mispredict
all bull markets ( P
1
5 0), will correctly “predict” all bear markets ( P
2
5 1), and will end
up with timing ability of P
1
1 P
2
2 1 5 0.
What is the market timing score of someone who flips a fair coin to predict the market?
CONCEPT CHECK
24.4
When timing is imperfect, Merton shows that if we measure overall accuracy by the
statistic P
1
1 P
2
2 1, the market value of the services of an imperfect timer is simply
MV(Imperfect timer)
5 (P
1
1 P
2
2 1) 3 C 5 (P
1
1 P
2
2 1) 32N(½ s
M
"T) 2 14 (24.7)
The last column in Table 24.4 provides an assessment of the imperfect market-timer. To
simulate the performance of an imperfect timer, we drew random numbers to capture the
possibility that the timer will sometimes issue an incorrect forecast (we assumed here both
P
1
and P
2
5 .7) and compiled results for the 86 years of history.
26
The statistics of this
exercise resulted in an average terminal value for the imperfect timer of “only” $8,859,
compared with the perfect timer’s $352,796, but still considerably superior to the $2,562
for the all-equity investments.
27
A further variation on the valuation of market timing is a case in which the timer does
not shift fully from one asset to the other. In particular, if the timer knows her forecasts are
imperfect, one would not expect her to shift fully between markets. She presumably would
moderate her positions. Suppose that she shifts a fraction v of the portfolio between T-bills
and equities. In that case, Equation 24.7 can be generalized as follows:
MV(Imperfect timer)
5 v(P
1
1 P
2
2 1)32N(s
M
"T) 2 14
If the shift is v 5 .50 (50% of the portfolio), the timer’s value will be one-half of the value
we would obtain for full shifting, for which v 5 1.0.
26
In each year, we started with the correct forecast, but then used a random number generator to occasionally
change the timer’s forecast to an incorrect prediction. We set the probability that the timer’s forecast would be
correct equal to .70 for both up and down markets.
27
Notice that Equation 24.7 implies that an investor with a value of P 5 0 who attempts to time the market
would add zero value. The shifts across markets would be no better than a random decision concerning asset
allocation.
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C H A P T E R
2 4
Portfolio Performance Evaluation
861
24.5
Style Analysis
Style analysis was introduced by Nobel laureate William Sharpe.
28
The popularity of the
concept was aided by a well-known study
29
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’s idea was to regress fund returns on indexes representing a range of asset
classes. The regression coefficient on each index would then measure the fund’s implicit
allocation to that “style.” Because funds are barred from short positions, the regression
coefficients are constrained to be either zero or positive and to sum to 100%, so as to rep-
resent a complete asset allocation. The R -square of the regression would then measure the
percentage of return variability attributable to style or asset allocation, while the remainder
of return variability would be attributable either to security selection or to market timing
by periodic changes in the asset-class weights.
To illustrate Sharpe’s approach, we use monthly returns on Fidelity Magellan’s Fund
during the famous manager Peter Lynch’s tenure between October 1986 and September
1991, with results shown in Table 24.5 . While seven asset classes are included in this
analysis (of which six are represented by stock indexes and one is the T-bill alternative),
the regression coefficients are positive for only three, namely, large capitalization stocks,
medium cap stocks, and high P/E (growth) stocks. These portfolios alone explain 97.5%
of the variance of Magellan’s returns. In other words, a tracking portfolio made up of the
three style portfolios, with weights as given in Table 24.5 , would explain the vast majority
of Magellan’s variation in monthly performance. We conclude that the fund returns are
well represented by three style portfolios.
The proportion of return variability not explained by asset allocation can be attrib-
uted to security selection within asset classes, as well as timing that shows up as periodic
changes in allocation. For Magellan, residual variability was 100 2 97.5 5 2.5%. This sort
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