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10
11 Managed Portfolio Portfolio Actual Portfolio
12 Component Weight Return Return
13 Equity 0.75 6.5000% 4.8750%
14 Bonds 0.12 1.2500% 0.1500%
15 Cash 0.13 0.4800% 0.0624%
16
17 Return on Managed 5.0874%
18
19 Excess Return 1.1184%
20
21
22 Contribution of Asset Allocation
23 Actual Weight Benchmark Excess Market Performance
24 Market in Portfolio Weight Weight Return Contribution
25
26 Equity 0.75 0.6 0.15 5.8100% 0.8715%
27 Fixed Income 0.12 0.3 -0.18 1.4500% -0.2610%
28 Cash 0.13 0.1 0.03 0.4800% 0.0144%
29 Contribution of
30 Asset Allocation 0.6249%




margin; and (3) some anomalies in realized returns”such as the turn-of-the-year effects”
have been sufficiently persistent to suggest that managers who identified them, and acted on
them in a timely fashion, could have beaten the passive strategy over prolonged periods.
These observations are enough to convince us that there is a role for active portfolio man-
agement. Active management offers an inevitable lure, even if investors agree that security
markets are nearly efficient.
At the extreme, suppose capital markets are perfectly efficient, an easily accessible market
index portfolio is available, and this portfolio is the efficient risky portfolio. In this case, se-
curity selection would be futile. You would do best following a passive strategy of allocating
funds to a money market fund (the safe asset) and the market index portfolio. Under these sim-
plifying assumptions, the optimal investment strategy seems to require no effort or know-how.
But this is too hasty a conclusion. To allocate investment funds to the risk-free and risky
portfolios requires some analysis. You need to decide the fraction, y, to be invested in the risky
market portfolio, M, so you must know the reward-to-variability ratio
E(rM) rf
SM
M

700
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20 Performance Evaluation and Active Portfolio Management


where E(rM) rf is the risk premium on M, and M is the standard deviation of M. To make a
rational allocation of funds requires an estimate of M and E(rM), so even a passive investor
needs to do some forecasting.
Forecasting E(rM) and M is complicated further because security classes are affected
by different environment factors. Long-term bond returns, for example, are driven largely by
changes in the term structure of interest rates, while returns on equity depend also on changes
in the broader economic environment, including macroeconomic factors besides interest rates.
Once you begin considering how economic conditions influence separate sorts of investments,
you might as well use a sophisticated asset allocation program to determine the proper mix for
the portfolio. It is easy to see how investors get lured away from a purely passive strategy.
Even the definition of a “pure” passive strategy is not very clear-cut, as simple strategies
involving only the market index portfolio and risk-free assets now seem to call for market
analysis. Our strict definition of a pure passive strategy is one that invests only in index funds
and weights those funds by fixed proportions that do not change in response to market condi-
tions: a portfolio strategy that always places 60% in a stock market index fund, 30% in a bond
index fund, and 10% in a money market fund, regardless of expectations.
Active management is attractive because the potential profit is enormous, even though
competition among managers is bound to drive market prices to near-efficient levels. For
prices to remain efficient to some degree, decent profits to diligent analysts must be the rule
rather than the exception, although large profits may be difficult to earn. Absence of profits
would drive people out of the investment management industry, resulting in prices moving
away from informationally efficient levels.



Objectives of Active Portfolios
What does an investor expect from a professional portfolio manager, and how do these ex-
pectations affect the manager™s response? If all clients were risk neutral (indifferent to risk),
the answer would be straightforward: The investment manager should construct a portfolio
with the highest possible expected rate of return, and the manager should then be judged by
the realized average rate of return.
When the client is risk averse, the answer is more difficult. Lacking standards to proceed
by, the manager would have to consult with each client before making any portfolio decision
in order to ascertain that the prospective reward (average return) matched the client™s attitude
toward risk. Massive, continuous client input would be needed, and the economic value of
professional management would be questionable.
Fortunately, the theory of mean-variance efficiency allows us to separate the “product deci-
sion,” which is how to construct a mean-variance efficient risky portfolio, from the “con-
sumption decision,” which describes the investor™s allocation of funds between the efficient
risky portfolio and the safe asset. You have learned already that construction of the optimal
risky portfolio is purely a technical problem and that there is a single optimal risky portfolio
appropriate for all investors. Investors differ only in how they apportion investment between
that risky portfolio and the safe asset.
The mean-variance theory also speaks to performance in offering a criterion for judging
managers on their choice of risky portfolios. In Chapter 6, we established that the optimal
risky portfolio is the one that maximizes the reward-to-variability ratio, that is, the expected
excess return divided by the standard deviation. A manager who maximizes this ratio will sat-
isfy all clients regardless of risk aversion.
Clients can evaluate managers using statistical methods to draw inferences from realized
rates of return about prospective, or ex ante, reward-to-variability ratios. The Sharpe measure,
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702 Part SIX Active Investment Management


or the equivalent M 2, is now a widely accepted way to track performance of professionally
managed portfolios:
E(rP) rf
SP
P

The most able manager will be the one who consistently obtains the highest Sharpe meas-
ure, implying that the manager has real forecasting ability. A client™s judgment of a manager™s
ability will affect the fraction of investment funds allocated to this manager; the client can in-
vest the remainder with competing managers and in a safe fund.
If managers™ Sharpe measures were reasonably constant over time, and clients could reli-
ably estimate them, allocating funds to managers would be an easy decision.
Actually, the use of the Sharpe measure as the prime measure of a manager™s ability re-
quires some qualification. We know from the discussion of performance evaluation earlier in
this chapter that the Sharpe ratio is the appropriate measure of performance only when the
client™s entire wealth is managed by the professional investor. Moreover, clients may impose
additional restrictions on portfolio choice that further complicate the performance evaluation
problem.


20.4 MARKET TIMING
Consider the results of three different investment strategies, as gleaned from Table 5.3:
1. Investor X, who put $1 in 30 day T-bills (or their predecessors) on January 1, 1926, and
always rolled over all proceeds into 30-day T-bills, would have ended on December 31,
2001, 76 years later, with $16.98.
2. Investor Y, who put $1 in large stocks (the S&P 500 portfolio) on January 1, 1926,
market timing and reinvested all dividends in that portfolio, would have ended on December 31, 2001,
with $1,987.01.
Asset allocation in
which the investment 3. Suppose we define perfect market timing as the ability to tell with certainty at the
in the market is
beginning of each year whether stocks will outperform bills. Investor Z, the perfect timer,
increased if one
shifts all funds at the beginning of each year into either bills or stocks, whichever is
forecasts that
going to do better. Beginning at the same date, how much would Investor Z have ended
the market will
up with 76 years later? Answer: $115,233.89!
outperform bills.



>
3. What are the annually compounded rates of return for the X, Y, and perfect-timing
Concept
strategies over the period 1926“2001?
CHECK
These results have some lessons for us. The first has to do with the power of compounding.
Its effect is particularly important as more and more of the funds under management represent
pension savings. The horizons of pension investments may not be as long as 76 years, but they
are measured in decades, making compounding a significant factor.
The second is a huge difference between the end value of the all-safe asset strategy
($16.98) and of the all-equity strategy ($1,987.01). Why would anyone invest in safe assets
given this historical record? If you have absorbed all the lessons of this book, you know the
reason: risk. The averages of the annual rates of return and the standard deviations on the all-
bills and all-equity strategies were

Arithmetic Mean Standard Deviation
Bills 3.85% 3.25%
Equities 12.49 20.30
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20 Performance Evaluation and Active Portfolio Management


The significantly higher standard deviation of the rate of return on the equity portfolio is
commensurate with its significantly higher average return. The higher average return reflects
the risk premium.
Is the return premium on the perfect-timing strategy a risk premium? Because the perfect
timer never does worse than either bills or the market, the extra return cannot be compensa-
tion for the possibility of poor returns; instead it is attributable to superior analysis. The value
of superior information is reflected in the tremendous ending value of the portfolio. This value
does not reflect compensation for risk.
To see why, consider how you might choose between two hypothetical strategies. Strat-
egy 1 offers a sure rate of return of 5%; strategy 2 offers an uncertain return that is given by
5% plus a random number that is zero with a probability of 0.5 and 5% with a probability of
0.5. The results for each strategy are

Strategy 1 (%) Strategy 2 (%)
Expected return 5 7.5
Standard deviation 0 2.5
Highest return 5 10
Lowest return 5 5

Clearly, strategy 2 dominates strategy 1, as its rate of return is at least equal to that of strat-
egy 1 and sometimes greater. No matter how risk averse you are, you will always prefer strat-
egy 2 to strategy 1, even though strategy 2 has a significant standard deviation. Compared to
strategy 1, strategy 2 provides only good surprises, so the standard deviation in this case can-
not be a measure of risk.
You can look at these strategies as analogous to the case of the perfect timer compared with
either an all-equity or all-bills strategy. In every period, the perfect timer obtains at least as
good a return, in some cases better. Therefore, the timer™s standard deviation is a misleading
measure of risk when you compare perfect timing to an all-equity or all-bills strategy.

Valuing Market Timing as an Option
Merton (1981) shows that the key to analyzing the pattern of returns of a perfect market timer
is to compare the returns of a perfect foresight investor with those of another investor who
holds a call option on the equity portfolio. Investing 100% in bills plus holding a call option
on the equity portfolio will yield returns identical to those of the portfolio of the perfect timer
who invests 100% in either the safe asset or the equity portfolio, whichever will yield the
higher return. The perfect timer™s return is shown in Figure 20.5. The rate of return is bounded
from below by the risk-free rate, rf.
To see how the value of information can be treated as an option, suppose the market index
currently is at S0 and a call option on the index has exercise price of X S0(1 rf). If the
market outperforms bills over the coming period, ST will exceed X; it will be less than X other-
wise. Now look at the payoff to a portfolio consisting of this option and S0 dollars invested
in bills.

Payoff to Portfolio
Outcome: ST X ST X
S0(1 rf ) S0(1 rf )
Bills
ST X
Option 0
S0(1 rf ) ST
Total
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704 Part SIX Active Investment Management




F I G U R E 20.5 Rate of return
Rate of return of a
perfect market time




rf




rM
rf




The portfolio returns the risk-free rate when the market is bearish (that is, when the market
return is less than the risk-free rate) and pays the market return when the market is bullish and
beats bills. This represents perfect market timing. Consequently, the value of perfect timing

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