Portfolio Performance Evaluation

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Topic 11 Portfolio Performance Evaluation

• Measuring investment returns
• The conventional theory of performance evaluation
  
• Market timing
• Performance attribution procedures

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 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.

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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.
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• Multiperiod
• Example

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• 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

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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.
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• Time-weighted returns
• Ignore the number of shares of stock held in
  each period.
• The stock return in the 1st year
• The stock return in the 2nd year

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• ? The time-weighted return is
• This average return considers only the
  period-by-period returns without regard to the
  amounts invested in the stock in each period.

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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.
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• Arithmetic versus geometric averages
• Arithmetic averages
• The arithmetic average of the two annual
  returns, 10 and 5.66

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• Geometric averages
• The compound average growth rate, rG, is
  calculated as the solution to the following
  equation
• In general
• where rt is the return in each time period.

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• Geometric averages never exceed arithmetic ones
• Consider a stock that doubles in price in
  period 1 (r1 100) and halves in price in
  period 2 (r2 -50).
• The arithmetic average is
• rA 100 (-50)/2 25
• The geometric average is
• rG (1 1)(1 - 0.5)2 1 0

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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.
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• 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.

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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.
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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.
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Example 2 Consider all the possible
outcomes over a two-year period
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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.
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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.
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 The Conventional Theory of Performance
Evaluation

• 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 rP.

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• 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.

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• Appraisal ratio
• The appraisal 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.

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Example Consider the following data for a
particular sample period
The T-bill rate during the period was
6.
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• Sharpes measure
• Treynors measure

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• Jensens measure
• Appraisal ratio

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The M2 measure of performance

• While the Sharpe ratio can be used to rank
  portfolio performance, its numerical value is not
  easy to interpret.
• We have found that SP 0.69 and SM 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.

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To compute the M2 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 SP500). Because the
market index and portfolio P have the same
standard deviation, we may compare their
performance simply by comparing returns.
This is the M2 measure
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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 M2 measure 26.7
28 -1.3.
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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 M2 measure is then the
vertical distance (i.e., the difference in
expected returns) between portfolios P and M.
P will have a negative M2 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).
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Appropriate performance measures in 3 scenarios
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.
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• Jane's portfolio 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.

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• Determine the benchmark (alternative) portfolio
  that Jane would have held if she had chosen a
  passive strategy, such as the SP 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.

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• 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 ?(eP) is the diversifiable risk.

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The ratio ?P/?(eP) 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 eP
is just diversifiable risk that can be avoided.

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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, ?(eP). Therefore,
the ratio of ?P /?(eP) is the natural
benefit-to-cost ratio for portfolio P.
This performance measurement is the appraisal
ratio.
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• Janes choice portfolio is one of many portfolios
  combined into a large investment fund
• ? The Treynor measure is the appropriate
  criterion.
• E.g.

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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.
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Suppose portfolio Q can be mixed with
T-bills. Specifically, if we invest wQ
in Q and wF 1 - wQ in T-bills, the resulting
portfolio, Q, will have alpha and beta values
proportional to Qs alpha and beta scaled down by
wQ 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.
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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.
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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 wQ Portfolio Q has an
alpha of which is less than that of
P.
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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 TP, 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.
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• An example
• Excess returns for portfolios P Q and the
  benchmark M over 12 months

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Performance statistics
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• Portfolio Q is more aggressive than P, in the
  sense that its beta is significantly higher (1.40
  vs. 0.69).
• On the other hand, from its residual
  standard deviation P appears better diversified
  (1.95 vs. 8.98).
• Both portfolios outperformed the benchmark
  market index, as is evident from their larger
  Sharpe measures (and thus positive M2) and their
  positive alphas.

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• 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 (0.51 vs. 0.45) and better M2
  (2.69 vs. 2.19).
• As an active portfolio to be mixed with the
  market index, P is preferable to Q, as is evident
  from its appraisal ratio (0.84 vs. 0.59).

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• 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 (5.40
  versus 4.00).
• 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.

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Relationships among the various performance
measures

• The relation between Treynors measure and
  Jensens ?

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• The relation between Sharpes measure and
  Jensens ?

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Performance measurement with changing portfolio
composition
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.
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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.
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Portfolio returns in last four quarters are
more variable than in the first four
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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.
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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.
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 Market Timing
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.
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• 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.

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No market timing, beta is constant
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• 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 rM is higher, resulting in
  the curved line.

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Market timing, beta increases with expected
market excess return
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Such a line can be estimated by adding a
squared term to the usual linear index model
where rP 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 (rM - rf)
is larger.
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• 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.

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Such a line appears in regression form as
where D is a dummy variable that equals 1
for rM gt rf 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.
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 Performance Attribution Procedures

• 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.

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• 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.

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• 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 SP 500 may be chosen as
  benchmark for equities.

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The bogey portfolio is set to have fixed
weights in each asset class, and its rate of
return is given by where wBi
weight of the bogey in asset class i.
rBi return on the benchmark portfolio of
that class over the
evaluation period.
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The portfolio managers choose weights in
each class (wPi) based on their capital market
expectations, and they choose a portfolio of the
securities within each class based on their
security analysis, which earns rPi over the
evaluation period. Thus, the return of the
managed portfolio will be
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• ? 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

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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.
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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.
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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.
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• Sector selection within the equity market

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• Portfolio attribution summary