This Session
Investments
Session 15 – Performance Evaluation
Prof. Nuno Fernandes
Nfernandes@fcee.ucp.pt
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Why do we need performance analysis?
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The risks associated
measurement.
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The importance of benchmarking.
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Risk-adjusted returns.
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Stock-picking and market timing.
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Different
tools
performance
Prof. Nuno Fernandes
Averaging Returns
r =
∑
t =1
performance
analyze
portfolio
Investments 2005/2006
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Motivation for Performance Analysis
Arithmetic Mean:
n
to
with
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For example, let us say that you decide
to invest in a diversified equity portfolio
with average risk. You see that your
return was 20%.

Is this good or bad?
rt
n
Geometric Mean:
1/ n
⎡n
⎤
r = ⎢∏(1+ rt )⎥ −1
⎣ t =1
⎦
Prof. Nuno Fernandes
Investments 2005/2006
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Benchmark Returns


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The investor has to compare the returns
of his/her manager with the returns that
would have been obtained had he/she
invested in an alternative portfolio with
identical risk.
Performance must be evaluated on a
relative basis; not on absolute basis!
Investments 2005/2006
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Analyzing Performance Statistics
In performance analysis you need to
make relevant comparisons.
Prof. Nuno Fernandes
Prof. Nuno Fernandes
5
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Mutual funds with the highest average
rate of return might not have the
highest rank because…
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A highly aggressive fund may earn
higher returns than a less aggressive
fund but the higher returns may not be
sufficient to compensate for the extra
risk taken
Prof. Nuno Fernandes
Investments 2005/2006
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1
The Basic Problems
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A Fund’s Performance Against Its Peers
60
There are two basic problems with
performance analysis:
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40
9 Even if the mean and variance of returns
are assumed to be constant, there is the
need to obtain many observations to
achieve significant results (skills vs good
luck)
9 In the case of active management (as
opposed to passively managed funds)
shifting parameters and styles could make
evaluation quite difficult.
Prof. Nuno Fernandes
Investments 2005/2006
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20
10
0
-10
-20
Year1
Year 2 Year 3 Year 4 Year 5 Year 6 Year 7
5th Percentile
7
Prof. Nuno Fernandes
Median
95th Percentile
Fund
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Sharpe Ratio
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One simple way to investigate the fund’s
performance is to consider risk-adjusted
returns…remember that the CAPM tells us that
the more market risk you take on, the higher
should be the return.

A widespread measure is the Sharpe ratio:
⎛R − R ⎞
⎜ P
f ⎟⎠
Sharpe = ⎝
σ
P
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Reward to total variability trade-off.
Risk Adjusted Measures
Prof. Nuno Fernandes
Investments 2005/2006
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Sharpe Ratio
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The MM-M (M2) Performance Measure
⎛R − R ⎞
⎜ P
f ⎟⎠
Sharpe = ⎝
σ
P
Prof. Nuno Fernandes
Prof. Nuno Fernandes
Investments 2005/2006
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The Modigliani-Modigliani measure, M2 , is a
variant of the Sharpe ratio.
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It focuses on total volatility but the riskadjusted return is, essentially, a differential
return relative to the benchmark returns.
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The measure takes into account the fund’s
average return and determines what it would
have been if the fund (under investigation) had
the same degree of total risk as the market
portfolio (for example, Wilshire 5000 or S&P
500 indices).
Prof. Nuno Fernandes
Investments 2005/2006
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2
M2 Performance Measure
M2 = R
P*
−R
E(R)
M2 Performance Measure
CML
M
M
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The first step is to generate the so-called
“adjusted portfolio”, P*.
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This portfolio is made up of two parts:
CAL(P)
M2
P
P*
9 The managed portfolio, P; and
9 A position in T-Bills in a way that the total volatility
of the adjusted portfolio matches the market index’s
volatility.
Prof. Nuno Fernandes
Investments 2005/2006
σ
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Treynor’s Measure
Prof. Nuno Fernandes
The Treynor’s measure gives you the
excess return per unit of risk.
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However, unlike the Sharpe Ratio, it
uses the systematic risk instead of total
risk.
Prof. Nuno Fernandes
Investments 2005/2006
P
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Prof. Nuno Fernandes
Like Jack Treynor and William Sharpe, Michael
Jensen recognized the CAPM’s implications for
performance measurement.
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Alpha is average fund’s return above the
predicted return from the CAPM, given the
portfolio’s Beta and average market return.
Prof. Nuno Fernandes
Investments 2005/2006
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Appraisal Ratio
Appraisal =
⎡
⎤
= R − ⎢ R + β ⎛⎜ R − R ⎞⎟⎥
P ⎣ f
P⎝ M
f ⎠⎦

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⎛R − R ⎞
⎜ P
f ⎟⎠
Treynor = ⎝
β
P
Jensen’s alpha:
Jensen = α
σ
Treynor’s Measure
Jensen’s Measure

P
Investments 2005/2006
⎛R − R ⎞
⎜ P
f ⎟⎠
 Treynor’s measure: Treynor = ⎝
β
P

σ
M
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αP
σ (ε P )
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Appraisal Ratio divides the alpha of the
portfolio by the nonsystematic risk (from the
CAPM equation, where Jensen’s alpha is
estimated)
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Nonsystematic risk could, in theory, be
eliminated by diversification
Prof. Nuno Fernandes
Investments 2005/2006
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3
About Portfolio Performance Measures
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To adequately evaluate a portfolio, must
analyze both risk and return
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SHARPE measures risk-premium per unit of
total risk
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TREYNOR measures risk-premium per unit of
systematic risk
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Jensen’s alpha measures risk-adjusted returns
for both portfolios and individual assets
Prof. Nuno Fernandes
Investments 2005/2006
Which Measure Should We Use?
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To decide on compensation:
9 Use the Jensen measure, which should provide you
the amount you are willing to pay the manager
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To decide on optimal portfolio choices:
9 Use Sharpe ratio if portfolio represents entire
investment
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Use Treynor’s measure for a fund that is just
one sub-portfolio out of a large set of
passively-managed portfolios
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Additional tools are available for measuring a
manager’s market timing skills
Prof. Nuno Fernandes
Investments 2005/2006
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Performance Attribution
Decomposing overall performance into
components
Components are related to specific
elements of performance
Example components
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Performance Attribution
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9 Broad Allocation
9 Industry
9 Security Choice
9 Up and Down Markets - Timing
Prof. Nuno Fernandes
Investments 2005/2006
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Prof. Nuno Fernandes
Investments 2005/2006
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Market Timing
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A good market timer structures a
portfolio to have a relatively high Beta
when the market is expected to rise and
low Beta when the market is expected to
drop.
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In other words, the market timer wants
to do the following strategy:
Market Timing
9 Hold a high-Beta portfolio when RM > R f
9 Hold a low-Beta portfolio when RM < R f
Prof. Nuno Fernandes
Investments 2005/2006
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Prof. Nuno Fernandes
Investments 2005/2006
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4
Superior Stock Selection
Market Timing
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
R −R
P
f
If the fund manager really has good timing
abilities (good and accurate forecasts of
market movements), then the portfolio will do
better than a benchmark portfolio that has a
constant Beta (that is equal to the average
Beta of the timer’s portfolio).
Alpha
R −R
M
f
To really “time the market”, the fund manager
must change:
9 the average Beta of the risky securities held in the
portfolio; or
9 the relative amounts invested in the risk-free asset
and the risky assets.
Prof. Nuno Fernandes
Investments 2005/2006
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The previous exhibit indicates that the
relationship between the portfolio’s excess
returns and the market’s excess return was
linear.
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This result suggests that the portfolio’s Beta
was, roughly speaking, the same during the
entire period under consideration.
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Superior Market Timing
Stock Selection
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Prof. Nuno Fernandes
R −R
P
f
Alpha
R −R
M
f
In this case, it appears that the investment
manager successfully identified and invested in
some underpriced securities (alpha is positive).
Prof. Nuno Fernandes
Investments 2005/2006
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Prof. Nuno Fernandes
Investments 2005/2006
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Market Timing
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The previous exhibit indicates that the
relationship between the portfolio’s excess
returns and the market’s excess return was not
linear.
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The exhibit suggests that the portfolio
consisted of high-Beta securities during periods
when the market return was high and low-Beta
securities when the market dropped.
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In this case, it appears that the investment
manager successfully identified market timing
(alpha is positive).
Prof. Nuno Fernandes
Investments 2005/2006
Bogey
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Prof. Nuno Fernandes
Investments 2005/2006
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Process of Attributing Performance
to Components
Process of Attributing Performance
to Components
Set up a ‘Benchmark’ or ‘Bogey’ portfolio
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Use indexes for each component
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Use target weight structure
Prof. Nuno Fernandes
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Calculate the return on the ‘Bogey’ and on
the managed portfolio
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Explain the difference in return based on
component weights or selection
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Summarize the performance differences
into appropriate categories
Prof. Nuno Fernandes
Formula for Attribution
rB =
n
∑w
i =1
r p − rB =
n
∑ (w
i =1
pi
r & rp =
Bi Bi
∑w
i =1
pi
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Contributions for Performance
n
∑w
i =1
pi
r pi
Contribution for asset allocation
n
n
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r pi − ∑ w Bi rBi =
i =1
(wpi - wBi) rBi
+
Contribution for security selection
wpi (rpi - rBi)
=
Total Contribution from asset class wpirpi -wBirBi
r pi − w Bi rBi )
Where B is the bogey portfolio and p is the managed portfolio
Prof. Nuno Fernandes
Investments 2005/2006
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Prof. Nuno Fernandes
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Mutual Funds - Investment Policies
Style Analysis
Prof. Nuno Fernandes
Investments 2005/2006
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Money Market
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Fixed Income
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Equity
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Balance & Income
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Asset Allocation
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Indexed
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Specialized Sector
Prof. Nuno Fernandes
Investments 2005/2006
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Style Analysis
Analyzing a Portfolio Manager’s Style
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Based on regression analysis
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Examines asset allocation for broad
groups of stocks
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More precise than comparing to the
broad market
Prof. Nuno Fernandes
Investments 2005/2006
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Analyzing a Portfolio Manager’s Style
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Sharpe (1992) - model to analyze a
portfolio manager’s style (i.e., growth vs
value, etc.)
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Uses modest amount of public
information about funds
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Uses price indexes for 12 asset classes
as explanatory variables for a mutual
fund’s return
Prof. Nuno Fernandes
Investments 2005/2006
Analyzing a Portfolio Manager’s Style
Ri − RF = α i + β i1 ( RI1 − RF ) + β i 2 ( RI 2 − RF ) + ... + β i12 ( RI 12 − RF )
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Sample explanatory factors
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Uses APT framework
9 Soloman Brothers 90-day Treasury bill
index
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The factor loadings are estimates of the
weights that a fund invests in the twelve
asset categories
9 Lehman Brothers Intermediate-Term
Government Bond Index
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9 FTA Japan Index
9 Sharpe/BARRA Value Stock Index
Same type of analysis could be done
using a ‘rolling’ regression
Prof. Nuno Fernandes
Investments 2005/2006
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Rolling Style Analysis
Prof. Nuno Fernandes
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Prof. Nuno Fernandes
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Benefits From Using Quantitative
Management Style Analysis
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Investment holdings are usually not reported
publicly until months after they are made—too
late for investors to react in a timely manner
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Mutual funds can report misleading investment
goals. It can also provide better forecasts of
mutual fund’s risk/return than subjective
comments in newspapers, etc.
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Sharpe’s analysis: 97.3% of returns attributed
to style
Prof. Nuno Fernandes
Investments 2005/2006
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Empirical Evidence
Empirical Evidence
Prof. Nuno Fernandes
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Empirical Evidence
Prof. Nuno Fernandes
Investments 2005/2006
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Prof. Nuno Fernandes
Investments 2005/2006
9 Not enough to minimize diversifiable risk
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Basic problems with performance evaluation;
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The issue of skill against good luck;
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Mutual funds are usually able to reduce their
diversifiable risk
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Risk-adjusted returns (Sharpe ratio, Treynor’s
measure, Jensen measure, appraisal ratio)
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Investors can maintain their desired risk-class
by mutual fund investing
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Benchmark returns from benchmark portfolios
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Asset Allocation versus Stock Selection.
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Most investors should focus on a mutual fund’s
fees and favor funds charging smallest fees
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Learning Experience
Average American has only about 7 different
stocks
Prof. Nuno Fernandes
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Empirical Evidence
The Bottom Line
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Prof. Nuno Fernandes
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Prof. Nuno Fernandes
Investments 2005/2006
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8