1 Topic 10 (Ch. 24) Portfolio Performance Evaluation Measuring
investment returns The conventional theory of performance
evaluation Market timing Performance attribution procedures
Slide 2
2 Measuring Investment Returns One period: Find the rate of
return (r) that equates the present value of all cash flows from
the investment with the initial outlay.
Slide 3
3 Example: Consider a stock paying a dividend of $2 annually
that currently sells for $50. You purchase the stock today and
collect the $2 dividend, and then you sell the stock for $53 at
year- end.
Slide 4
4 Multiperiod: Arithmetic versus geometric averages: Arithmetic
averages:
Slide 5
5 Geometric averages: The compound average growth rate, r G, is
calculated as the solution to the following equation: In general:
where r t is the return in each time period.
Slide 6
6 Geometric averages never exceed arithmetic ones: Consider a
stock that doubles in price in period 1 (r 1 = 100%) and halves in
price in period 2 (r 2 = -50%). The arithmetic average is: r A =
[100 + (-50)]/2 = 25% The geometric average is: r G = [(1 + 1)(1 -
0.5)] 1/2 1 = 0
Slide 7
7 The effect of the -50% return in period 2 fully offsets the
100% return in period 1 in the calculation of the geometric
average, resulting in an average return of zero. This is not true
of the arithmetic average. In general, the bad returns have a
greater influence on the averaging process in the geometric
technique. Therefore, geometric averages are lower.
Slide 8
8 Generally, the geometric average is preferable for
calculation of historical returns (i.e. measure of past
performance), whereas the arithmetic average is more appropriate
for forecasting future returns: Example 1: Consider a stock that
will either double in value (r = 100%) with probability of 0.5, or
halve in value (r = -50%) with probability 0.5.
Slide 9
9 Suppose that the stocks performance over a 2-year period is
characteristic of the probability distribution, doubling in one
year and halving in the other. The stocks price ends up exactly
where it started, and the geometric average annual return is zero:
which confirms that a zero year-by-year return would have
replicated the total return earned on the stock.
Slide 10
10 However, the expected annual future rate of return on the
stock is not zero. It is the arithmetic average of 100% and -50%:
(100 - 50)/2 = 25%. There are two equally likely outcomes per
dollar invested: either a gain of $1 (when r = 100%) or a loss of
$0.50 (when r = -50%). The expected profit is ($1 - $0.50)/2 =
$0.25, for a 25% expected rate of return. The profit in the good
year more than offsets the loss in the bad year, despite the fact
that the geometric return is zero. The arithmetic average return
thus provides the best guide to expected future returns.
Slide 11
11 Example 2: Consider all the possible outcomes over a
two-year period:
Slide 12
12 The expected final value of each dollar invested is: (4 + 1
+ 1 + 0.25)/4 = $1.5625 for two years, again indicating an average
rate of return of 25% per year, equal to the arithmetic average.
Note that an investment yielding 25% per year with certainty will
yield the same final compounded value as the expected final value
of this investment: (1 + 0.25) 2 = 1.5625.
Slide 13
13 The arithmetic average return on the stock is: [300 + 0 + 0
+ (-75)]/4 = 56.25% per two years, for an effective annual return
of 25% since: (1 + 25%)(1 + 25%) 1 = 56.25%. In contrast, the
geometric mean return is zero since: [(1 + 3)(1 + 0)(1 + 0)(1
0.75)] 1/4 = 1.0 Again, the arithmetic average is the better guide
to future performance.
Slide 14
14 Dollar-weighted returns versus time-weighted returns:
Example:
Slide 15
15 Dollar-weighted returns: Using the discounted cash flow
(DCF) approach, we can solve for the average return over the two
years by equating the present values of the cash inflows and
outflows:
Slide 16
16 This value is called the internal rate of return, or the
dollar-weighted rate of return on the investment. It is dollar
weighted because the stocks performance in the second year, when
two shares of stock are held, has a greater influence on the
average overall return than the first-year return, when only one
share is held.
Slide 17
17 Time-weighted returns: Ignore the number of shares of stock
held in each period. The stock return in the 1 st year: The stock
return in the 2 nd year:
Slide 18
18 The time-weighted (geometric average) return is: This
average return considers only the period- by-period returns without
regard to the amounts invested in the stock in each period. Note
that the dollar-weighted average is less than the time-weighted
average in this example because the return in the second year, when
more money is invested, is lower.
Slide 19
19 Note: For an investor that has control over contributions to
the investment portfolio, the dollar-weighted return is more
comprehensive measure. Time-weighted returns are more likely
appropriate to judge the performance of an investor that does not
control the timing or the amount of contributions.
Slide 20
20 Several risk-adjusted performance measures: Sharpes measure:
Sharpes measure divides average portfolio excess return over the
sample period by the standard deviation of returns over that
period. It measures the reward to (total) volatility trade-off.
Note: The risk-free rate may not be constant over the measurement
period, so we are taking a sample average, just as we do for r P.
The Conventional Theory of Performance Evaluation
Slide 21
21 Treynors measure: Like Sharpes, Treynors measure gives
excess return per unit of risk, but it uses systematic risk instead
of total risk. Jensens measure: Jensens measure is the average
return on the portfolio over and above that predicted by the CAPM,
given the portfolios beta and the average market return. Jensens
measure is the portfolios alpha value.
Slide 22
22 Information ratio: The information ratio divides the alpha
of the portfolio by the nonsystematic risk of the portfolio. It
measures abnormal return per unit of risk that in principle could
be diversified away by holding a market index portfolio. Note: Each
measure has some appeal. But each does not necessarily provide
consistent assessments of performance, since the risk measures used
to adjust returns differ substantially.
Slide 23
23 Example: Consider the following data for a particular sample
period: The T-bill rate during the period was 6%. Portfolio PMarket
M Average return35%28% Beta1.201.00 Standard deviation42%30%
Nonsystematic risk, (e)18%0
Slide 24
24 Sharpes measure: Treynors measure:
Slide 25
25 Jensens measure: Information ratio:
Slide 26
26 While the Sharpe ratio can be used to rank portfolio
performance, its numerical value is not easy to interpret. We have
found that S P = 0.69 and S M = 0.73. This suggests that portfolio
P under-performed the market index. But is a difference of 0.04 in
the Sharpe ratio economically meaningful? We often compare rates of
return, but these ratios are difficult to interpret. The M 2
measure of performance
Slide 27
27 To compute the M 2 measure, we imagine that a managed
portfolio, P, is mixed with a position in T- bills so that the
complete, or adjusted, portfolio (P*) matches the volatility of a
market index (such as the S&P500). Because the market index and
portfolio P* have the same standard deviation, we may compare their
performance simply by comparing returns. This is the M 2
measure:
Slide 28
28 Example: P has a standard deviation of 42% versus a market
standard deviation of 30%. The adjusted portfolio P* would be
formed by mixing portfolio P and T-bills and : weight in P: 30/42 =
0.714 weight in T-bills: (1 - 0.714) = 0.286. The return on this
portfolio P* would be: (0.286 6%) + (0.714 35%) = 26.7% Thus,
portfolio P has an M 2 measure: 26.7 28 = -1.3%.
Slide 29
29
Slide 30
30 We move down the capital allocation line corresponding to
portfolio P (by mixing P with T- bills) until we reduce the
standard deviation of the adjusted portfolio to match that of the
market index. The M 2 measure is then the vertical distance (i.e.,
the difference in expected returns) between portfolios P* and M. P
will have a negative M 2 measure when its capital allocation line
is less steep than the capital market line (i.e., when its Sharpe
ratio is less than that of the market index).
Slide 31
31 Suppose that Jane constructs a portfolio (P) and holds it
for a considerable period of time. She makes no changes in
portfolio composition during the period. In addition, suppose that
the daily rates of return on all securities have constant means,
variances, and covariances. This assures that the portfolio rate of
return also has a constant mean and variance. We want to evaluate
the performance of Janes portfolio. Appropriate performance
measures in 3 scenarios
Slide 32
32 Jane's portfolio P represents her entire risky investment
fund: We need to ascertain only whether Janes portfolio has the
highest Sharpe measure. We can proceed in 3 steps: Assume that past
security performance is representative of expected performance,
meaning that realized security returns over Janes holding period
exhibit averages and covariances similar to those that Jane had
anticipated.
Slide 33
33 Determine the benchmark (alternative) portfolio that Jane
would have held if she had chosen a passive strategy, such as the
S&P 500. Compare Janes Sharpe measure to that of the best
portfolio. In sum: When Janes portfolio represents her entire
investment fund, the benchmark is the market index or another
specific portfolio. The performance criterion is the Sharpe measure
of the actual portfolio versus the benchmark.
Slide 34
34 Janes portfolio P is an active portfolio and is mixed with
the market-index portfolio M: When the two portfolios are mixed
optimally, the square of the Sharpe measure of the complete
portfolio, C, is given by: where P is the abnormal return of the
active portfolio relative to the market-index, and (e P ) is the
diversifiable risk.
Slide 35
35 The ratio P / (e P ) is thus the correct performance measure
for P in this case, since it gives the improvement in the Sharpe
measure of the overall portfolio. To see this result intuitively,
recall the single-index model: If P is fairly priced, then P = 0,
and e P is just diversifiable risk that can be avoided.
Slide 36
36 However, if P is mispriced, P no longer equals zero.
Instead, it represents the expected abnormal return. Holding P in
addition to the market portfolio thus brings a reward of P against
the nonsystematic risk voluntarily incurred, (e P ). Therefore, the
ratio of P / (e P ) is the natural benefit-to-cost ratio for
portfolio P. This performance measurement is the information
ratio.
Slide 37
37 Janes choice portfolio P is one of many portfolios combined
into a large investment fund: The Treynor measure is the
appropriate criterion. E.g.: Portfolio PPortfolio QMarket
Beta0.901.601.00 Excess return 11%19%10% Alpha*2%3%0
Slide 38
38
Slide 39
39 Note: We plot P and Q in the expected return-beta (rather
than the expected return-standard deviation) plane, because we
assume that P and Q are two of many sub-portfolios in the fund, and
thus that nonsystematic risk will be largely diversified away,
leaving beta as the appropriate risk measure.
Slide 40
40 Suppose portfolio Q can be mixed with T-bills. Specifically,
if we invest w Q in Q and w F = 1 - w Q in T-bills, the resulting
portfolio, Q*, will have alpha and beta values proportional to Qs
alpha and beta scaled down by w Q : Thus, all portfolios Q*
generated from mixing Q with T-bills plot on a straight line from
the origin through Q. We call it the T-line for the Treynor
measure, which is the slope of this line.
Slide 41
41 P has a steeper T-line. Despite its lower alpha, P is a
better portfolio after all. For any given beta, a mixture of P with
T-bills will give a better alpha than a mixture of Q with T-
bills.
Slide 42
42 Suppose that we choose to mix Q with T-bills to create a
portfolio Q* with a beta equal to that of P. We find the necessary
proportion by solving for w Q : Portfolio Q* has an alpha of: which
is less than that of P.
Slide 43
43 In other words, the slope of the T-line is the appropriate
performance criterion for this case. The slope of the T-line for P,
denoted by T P, is: Treynors performance measure is appealing
because when an asset is part of a large investment portfolio, one
should weigh its mean excess return against its systematic risk
rather than against total risk to evaluate contribution to
performance.
Slide 44
44 An example: Excess returns for portfolios P & Q and the
benchmark M over 12 months:
Slide 45
45 Performance statistics:
Slide 46
46 Portfolio Q is more aggressive than P, in the sense that its
beta is significantly higher (1.40 vs. 0.70). On the other hand,
from its residual standard deviation P appears better diversified
(2.02% vs. 9.81%). Both portfolios outperformed the benchmark
market index, as is evident from their larger Sharpe measures (and
thus positive M 2 ) and their positive alphas.
Slide 47
47 Which portfolio is more attractive based on reported
performance? If P or Q represents the entire investment fund, Q
would be preferable on the basis of its higher Sharpe measure (0.49
vs. 0.43) and better M 2 (2.66% vs. 2.16%). As an active portfolio
to be mixed with the market index, P is preferable to Q, as is
evident from its information ratio (0.81 vs. 0.54).
Slide 48
48 When P and Q are competing for a role as one of a number of
subportfolios, Q dominates again because its Treynor measure is
higher (5.38 versus 3.97). Thus, the example illustrates that the
right way to evaluate a portfolio depends in large part how the
portfolio fits into the investors overall wealth.
Slide 49
49 Relationships among the various performance measures The
relation between Treynors measure and Jensens :
Slide 50
50 The relation between Sharpes measure and Jensens :
Slide 51
51
Slide 52
52 Estimating various statistics from a sample period assuming
a constant mean and variance may lead to substantial errors.
Example: Suppose that the Sharpe measure of the market index is
0.4. Over an initial period of 52 weeks, the portfolio manager
executes a low-risk strategy with an annualized mean excess return
of 1% and standard deviation of 2%. Performance measurement with
changing portfolio composition
Slide 53
53 This makes for a Sharpe measure of 0.5, which beats the
passive strategy. Over the next 52-week period this manager finds
that a high-risk strategy is optimal, with an annual mean excess
return of 9% and standard deviation of 18%. Here, again, the Sharpe
measure is 0.5. Over the two-year period our manager maintains a
better-than-passive Sharpe measure.
Slide 54
54 Portfolio returns in last four quarters are more variable
than in the first four:
Slide 55
55 In the first 4 quarters, the excess returns are -1%, 3%,
-1%, and 3%, making for an average of 1% and standard deviation of
2%. In the next 4 quarters the returns are -9%, 27%, -9%, 27%,
making for an average of 9% and standard deviation of 18%. Thus
both years exhibit a Sharpe measure of 0.5. However, over the
8-quarter sequence the mean and standard deviation are 5% and
13.42%, respectively, making for a Sharpe measure of only 0.37,
apparently inferior to the passive strategy.
Slide 56
56 The shift of the mean from the first 4 quarters to the next
was not recognized as a shift in strategy. Instead, the difference
in mean returns in the two years added to the appearance of
volatility in portfolio returns. The active strategy with shifting
means appears riskier than it really is and biases the estimate of
the Sharpe measure downward. We conclude that for actively managed
portfolios it is helpful to keep track of portfolio composition and
changes in portfolio mean and risk.
Slide 57
57 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, but partially, between T-bills and the market. Market
Timing
Slide 58
58 Suppose that an investor holds only the market- index
portfolio and T-bills. If the weight of the market were constant,
say, 0.6, then portfolio beta would also be constant, and the
security characteristic line (SCL) would plot as a straight line
with slope 0.6.
Slide 59
59 No market timing, beta is constant:
Slide 60
60 If the investor could correctly time the market and shift
funds into it in periods when the market does well. If bull and
bear markets can be predicted, the investor will shift more into
the market when the market is about to go up. The portfolio beta
and the slope of the SCL will be higher when r M is higher,
resulting in the curved line.
Slide 61
61 Market timing, beta increases with expected market excess
return:
Slide 62
62 Such a line can be estimated by adding a squared term to the
usual linear index model: where r P is the portfolio return, and a,
b, and c are estimated by regression analysis. If c turns out to be
positive, we have evidence of timing ability, because this last
term will make the characteristic line steeper as (r M - r f ) is
larger.
Slide 63
63 A similar and simpler methodology suggests that the beta of
the portfolio take only two values: a large value if the market is
expected to do well and a small value otherwise.
Slide 64
64 Such a line appears in regression form as: where D is a
dummy variable that equals 1 for r M > r f and zero otherwise.
Hence, the beta of the portfolio is b in bear markets and b + c in
bull markets. Again, a positive value of c implies market timing
ability.
Slide 65
65 Example: Regressing the excess returns of portfolios P and Q
on the excess returns of M and the square of these returns: we
derive the following statistics:
Slide 66
66 The numbers in parentheses are the regression estimates from
the single variable regression (reported in Table 24.3).
Slide 67
67 Portfolio P shows no timing (c P = 0). The results for
portfolio Q reveal that timing has, in all likelihood, successfully
been attempted (c Q = 0.10). The evidence thus suggests successful
timing (positive c) offset by unsuccessful stock selection
(negative a). Note that the alpha estimate, a, is now -2.29% as
opposed to the 5.26% estimate derived from the regression equation
that did not allow for the possibility of timing activity.
Slide 68
68 Portfolio managers constantly make broad-brush asset
allocation decisions as well as more detailed sector and security
allocation decisions within asset class. Performance attribution
studies attempt to decompose overall performance into discrete
components that may be identified with a particular level of the
portfolio selection process. Performance Attribution
Procedures
Slide 69
69 The difference between a managed portfolios performance and
that of a benchmark portfolio then may be expressed as the sum of
the contributions to performance of a series of decisions made at
the various levels of the portfolio construction process. For
example, one common attribution system decomposes performance into
3 components: broad asset market allocation choices across equity,
fixed-income, and money markets. industry (sector) choice within
each market. security choice within each sector.
Slide 70
70 The attribution method explains the difference in returns
between a managed portfolio, P, and a selected benchmark portfolio,
B (called the bogey). Suppose that the universe of assets for P and
B includes n asset classes such as equities, bonds, and bills. For
each asset class, a benchmark index portfolio is determined. For
example, the S&P 500 may be chosen as benchmark for
equities.
Slide 71
71 The bogey portfolio is set to have fixed weights in each
asset class, and its rate of return is given by: where w Bi :
weight of the bogey in asset class i. r Bi : return on the
benchmark portfolio of that class over the evaluation period.
Slide 72
72 The portfolio managers choose weights in each class (w Pi )
based on their capital market expectations, and they choose a
portfolio of the securities within each class based on their
security analysis, which earns r Pi over the evaluation period.
Thus, the return of the managed portfolio will be:
Slide 73
73 The difference between the two rates of return is: We can
decompose each term of the summation into a sum of two terms as
follows: Contribution from asset allocation: + Contribution from
selection: = Total contribution from asset class i:
Slide 74
74
Slide 75
75 Example: Consider the attribution results for a portfolio
which invests in stocks, bonds, and money market securities. The
managed portfolio is invested in the equity, fixed-income, and
money markets with weights of 70%, 7%, and 23%, respectively. The
portfolio return over the month is 5.34%.
Slide 76
76 Bogey Performance and Excess Return ComponentBenchmark
Weight Return of Index during Month (%) Equity (S&P
500)0.605.81 Bonds (Barclays Aggregate Bond Index) 0.301.45 Cash
(money market )0.100.48 Bogey = (0.60 5.81 ) + (0.30 1.45) + (0.10
0.48) = 3.97% Return of managed portfolio5.34% - Return of bogey
portfolio3.97% Excess return of managed portfolio1.37%
Slide 77
77 Note: The bogey portfolio is comprised of investments in
each index with the following weights: 60%: equity 30%: fixed
income 10%: cash (money market securities). These weights are
designated as neutral or usual. They depend on the risk tolerance
of the investor and must be determined in consultation with the
client.
Slide 78
78 This would be considered a passive asset-market allocation.
Any deviation from these weights must be justified by a belief that
one or another market will either over- or underperform its usual
risk-return profile.
Slide 79
79 A. Contribution of Asset Allocation to Performance
(1)(2)(3)(4) (5) = (3) (4) ActualBenchmarkContribution to Weight in
ExcessMarket Performance Market WeightReturn (%)(%)
Equity0.700.600.105.810.5810 Fixed-income0.070.30 -0.231.45 -0.3335
Cash0.230.100.130.480.0624 Contribution of asset allocation
0.3099
Slide 80
80 B. Contribution of Selection to Total Performance
(1)(2)(3)(4) (5) = (3) (4) PortfolioIndexExcess Performance
PortfolioContribution Market(%) Weight(%)
Equity7.285.811.470.701.03 Fixed- income 1.891.450.440.070.03
Contribution of selection within markets 1.06