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712 Part SIX Active Investment Management

The mathematics of the efficient frontier reveal that the Sharpe measure of the risky port-
folio is
[S2(M) 2 2
S(P) A/ (20.3)

Thus, the critical variable in determining the success of the active portfolio is its ratio of
alpha to nonsystematic risk, A/ (eA).
The intuition here is straightforward. You mix the active portfolio with the index for the
benefit of diversification. The position to take in the active portfolio relative to the market
portfolio depends on the ratio of the active portfolio™s abnormal return, A, relative to its
weakness given by its diversifiable risk, (eA). This ratio is sometimes referred to as the
appraisal ratio.
The contribution of individual securities (say, k) to the active portfolio (A) is analogous to
that of the active portfolio to the risky portfolio (P). It is measured by the appraisal ratio,
k / (ek).

SUMMARY • The appropriate performance measure depends on the investment context. The Sharpe
measure is most appropriate when the portfolio represents the entire investment fund.
The Treynor measure or Jensen measure is appropriate when the portfolio is to be mixed
with several other assets, allowing for diversification of firm-specific risk outside of each
• The shifting mean and variance of actively managed portfolios make it harder to assess
performance. A typical example is the attempt of portfolio managers to time the market,
resulting in ever-changing portfolio betas and standard deviations.
• Common attribution procedures partition performance improvements to asset allocation,
sector selection, and security selection. Performance is assessed by calculating departures
of portfolio composition from a benchmark or neutral portfolio.
• Active portfolio managers attempt to construct a risky portfolio that improves on the
reward-to-variability (Sharpe) ratio of a passive strategy.
• Active management has two components: market timing (or, more generally, asset
allocation) and security analysis.
• The value of perfect market-timing ability is enormous. The rate of return to a perfect
market timer will be uncertain, but the risk cannot be measured by standard deviation,
because perfect timing dominates a passive strategy, providing only “good” surprises.
• Perfect-timing ability is equivalent to having a call option on the market portfolio. The
value of that option can be determined using valuation techniques such as the Black-
Scholes formula.
• The value of imperfect market timing depends on the sum of the probabilities of the true

outcome conditional on the forecast: P1 P2 1. If perfect timing is equivalent to call
option C, then imperfect timing can be valued by: (P1 P2 1)C.
• The Treynor-Black model is based on an index model that takes market-timing forecasts
as given. The investment manager uses security analysis to construct an active portfolio.
The active portfolio is mixed with the index portfolio to maximize the Sharpe measure
of the optimal risky portfolio.
• In the Treynor-Black model, the weight of each analyzed security is proportional to
the ratio of its alpha to its residual variance.
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management

20 Performance Evaluation and Active Portfolio Management

active portfolio, 711 Jensen measure, 686 Treynor-Black model, 710
bogey, 694 market timing, 702 Treynor measure, 686
comparison universe, 684 Sharpe measure, 686

Questions 1“3 appeared in past CFA examinations.
1. A plan sponsor with a portfolio manager who invests in small-capitalization, high-growth
stocks should have the plan sponsor™s performance measured against which one of the
a. S&P 500 index.
b. Wilshire 5000 index.
c. Dow Jones Industrial Average.
d. Russell 2000 index.
2. Assume you purchased a rental property for $50,000 and sold it one year later for
$55,000 (there was no mortgage on the property). At the time of the sale, you paid $2,000
in commissions and $600 in taxes. If you received $6,000 in rental income (all of it
received at the end of the year), what annual rate of return did you earn?
a. 15.3%
b. 15.9%
c. 16.8%
d. 17.1%
3. A two-year investment of $2,000 results in a return of $150 at the end of the first year
and a return of $150 at the end of the second year, in addition to the return of the original
investment. The internal rate of return on the investment is:
a. 6.4%
b. 7.5%
c. 15.0%
d. None of the above
4. Based on current dividend yields and expected capital gains, the expected rates of return
on portfolios A and B are 11% and 14%, respectively. The beta of A is 0.8 while that of B
is 1.5. The T-bill rate is currently 6%, while the expected rate of return of the S&P 500
index is 12%. The standard deviation of portfolio A is 10% annually, while that of B is
31%, and that of the index is 20%.
a. If you currently hold a market index portfolio, would you choose to add either of these
portfolios to your holdings? Explain.
b. If instead you could invest only in bills and one of these portfolios, which would you
5. Evaluate the timing and selection abilities of four managers whose performances are
plotted in the following four scatter diagrams.
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management

714 Part SIX Active Investment Management

rP “ rf

rP “ rf

rM “ rf

rM “ rf

rP “ rf rP “ rf

rM “ rf rM “ rf

6. Consider the following information regarding the performance of a money manager
in a recent month. The table presents the actual return of each sector of the manager™s
portfolio in column (1), the fraction of the portfolio allocated to each sector in
column (2), the benchmark or neutral sector allocations in column (3), and
the returns of sector indexes in column (4).

(1) (2) (3) (4)
Actual Actual Benchmark Index
Return Weight Weight Return
Equity 2.0% 0.70 0.60 2.5% (S&P 500)
Bonds 1.0 0.20 0.30 1.2 (Aggregate Bond index)
Cash 0.5 0.10 0.10 0.5

a. What was the manager™s return in the month? What was her over- or
b. What was the contribution of security selection to relative performance?
c. What was the contribution of asset allocation to relative performance? Confirm that
the sum of selection and allocation contributions equals her total “excess” return

relative to the bogey.
7. Conventional wisdom says one should measure a manager™s investment performance
over an entire market cycle. What arguments support this contention? What arguments
contradict it?
8. Does the use of universes of managers with similar investment styles to evaluate relative
investment performance overcome the statistical problems associated with instability of
beta or total variability?
9. During a particular year, the T-bill rate was 6%, the market return was 14%, and a
portfolio manager with beta of 0.5 realized a return of 10%. Evaluate the manager based
on the portfolio alpha.
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Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management

20 Performance Evaluation and Active Portfolio Management

10. The chairman provides you with the following data, covering one year, concerning the
portfolios of two of the fund™s equity managers (manager A and manager B). Although
the portfolios consist primarily of common stocks, cash reserves are included in the
calculation of both portfolio betas and performance. By way of perspective, selected
data for the financial markets are included in the following table.

Total Return Beta
Manager A 24.0% 1.0
Manager B 30.0 1.5
S&P 500 21.0
Lehman Bond Index 31.0
91-day Treasury bills 12.0

a. Calculate and compare the risk-adjusted performance of the two managers relative to
each other and to the S&P 500.
b. Explain two reasons the conclusions drawn from this calculation may be misleading.
11. Carl Karl, a portfolio manager for the Alpine Trust Company, has been responsible since
1990 for the City of Alpine™s Employee Retirement Plan, a municipal pension fund.
Alpine is a growing community, and city services and employee payrolls have expanded
in each of the past 10 years. Contributions to the plan in fiscal 1995 exceeded benefit
payments by a three-to-one ratio.
The plan™s Board of Trustees directed Karl five years ago to invest for total return
over the long term. However, as trustees of this highly visible public fund, they
cautioned him that volatile or erratic results could cause them embarrassment. They also
noted a state statute that mandated that not more than 25% of the plan™s assets (at cost)
be invested in common stocks.
At the annual meeting of the trustees in November 1995, Karl presented the
following portfolio and performance report to the Board.
At Cost At Market
Asset Mix as of 9/30/95 (millions) (millions)
Fixed-income assets:
Short-term securities $ 4.5 11.0% $ 4.5 11.4%
Long-term bonds and mortgages 26.5 64.7 23.5 59.5
Common stocks 10.0 24.3 11.5 29.1
$41.0 100.0% $39.5 100.0%

Annual Rates of
Return for Periods
Ending 9/30/95

5 Years 1 Year
Total Alpine Fund:
Time-weighted 8.2% 5.2%
Dollar-weighted (Internal) 7.7% 4.8%
Assumed actuarial return 6.0% 6.0%
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management

716 Part SIX Active Investment Management

U.S. Treasury bills 7.5% 11.3%
Large sample of pension funds
(average 60% equities, 40% fixed income) 10.1% 14.3%
Common stocks”Alpine Fund 13.3% 14.3%
Average portfolio beta coefficient 0.90 0.89
Standard & Poor™s 500 stock index 13.8% 21.1%
Fixed-income securities”Alpine Fund 6.7% 1.0%
Salomon Brothers™ bond index 4.0% 11.4%

Karl was proud of his performance and was chagrined when a trustee made the
following critical observations:
a. “Our one-year results were terrible, and it™s what you™ve done for us lately that
counts most.”
b. “Our total fund performance was clearly inferior compared to the large sample of
other pension funds for the last five years. What else could this reflect except poor


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