Chapter 13
Performance Evaluation
and Risk Management
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Active and Passive Portfolio
Management
Performance evaluation
Comparing performance
measures
Investment risk management
More on computing value-at-risk
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It is Not the Return On My Investment...
“It is not the return on my investment that I
am concerned about. It is the return of my
investment!”
– Will Rogers
However,
“We’ll GUARANTEE you a 25% return of
your investment!”
– Tom and Ray Magliozzi
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Passive Portfolio Management
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Investors can either follow an active strategy or a passive strategy
after they form their portfolio.
Investors who choose passive management do not change the
composition of their portfolio. A Buy and Hold strategy involves
buying financial instruments, forming the portfolio, and then
holding the chosen securities in the original percentage of the
portfolio and not changing the composition of the portfolio.
Another technique of passive portfolio management is buying
shares of a market index fund that mimics a market index (e.g.,
S&P/TSX Composite Index) and holding it.
Investors do not pay transaction fees for frequent buying and
selling. They also do not spend a lot of time gathering information
to find underpriced securities.
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Active Portfolio Management
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In contrast to passive management, active management involves
changing the composition of portfolios.
Some active managers follow a market timing strategy. There are
mutual fund managers who specialize in market timing. This strategy
involves using various techniques and predicting future movements
in financial instruments, especially stocks, and buying and selling
them at the right time. Also, market timers may advocate moving into
a stock market or an industry at a particular time.
Other managers attempt to find undervalued and overvalued
securities, buy the undervalued ones and short sell the overvalued
ones to try and obtain abnormal profits.
Active management is costly, time consuming, and risky. Investors
need to evaluate managers of actively managed funds to find out
whether or not their strategies are successful. The Sharpe ratio,
Treynor ratio, and all the other risk measures are important tools for4
these investors.
Performance Evaluation Measures
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Performance evaluation is a term for assessing how
well a money manager achieves a balance between high
returns and acceptable risks.
The raw return on a portfolio, RP, is simply the total
percentage return on a portfolio.
The raw return is a naive performance evaluation
measure because:
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The raw return has no adjustment for risk.
The raw return is not compared to any benchmark, or standard.
Therefore, the usefulness of the raw return on a
portfolio is limited.
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Performance Evaluation Measures
The Sharpe Ratio
The Sharpe ratio is a reward-to-risk ratio that
focuses on total risk.
 It is computed as a portfolio’s risk premium
divided by the standard deviation for the
portfolio’s return.
R R
Sharperatio 
p
f
σp
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Performance Evaluation Measures
The Treynor Ratio
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The Treynor ratio is a reward-to-risk ratio that
looks at systematic risk only.
It is computed as a portfolio’s risk premium
divided by the portfolio’s beta coefficient.
Treynorratio 
Rp  R f
βp
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Performance Evaluation Measures
Jensen’s Alpha
 Jensen’s alpha is the excess return above or below
the security market line. It can be interpreted as a
measure of how much the portfolio “beat the
market.”
 It is computed as the raw portfolio return less the
expected portfolio return as predicted by the CAPM.
αp  Rp  R f  βp  ER M  R f  
“Extra” Actual
Return return
CAPM Risk-Adjusted ‘Predicted’ Return
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Jensen’s Alpha
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Another Method to Calculate Alpha
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Recall that the characteristic line graphs the relationship between the
return of an investment (on the y-axis) and the return of the market
or benchmark (on the x- axis).
The slope of this line represents the investment’s beta.
With only a slight modification, we can extend this approach to
calculate an investment’s alpha.
Consider the following equation,
ER P   R f  ER M  R f  β P
or, in termsof risk premiums:
ER P,RP   ER M,RP  β P
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Another Method to Calculate Alpha
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Suppose an actively managed fund took on the same amount of risk as the market
(i.e., a beta of 1), but the fund earned exactly 2% more than the market every
period?
If we graph this hypothetical situation, the slope (i.e., beta) is still 1, but the x
intercept is now 2. This intercept value is the fund’s alpha.
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Estimating Alpha Using Regression
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Suppose we want to estimate a fund’s alpha.
Unlike the previous situation, we do know whether its returns are
consistently higher (or lower) than the market by a fixed amount.
In this case, we apply a simple linear regression.
 X-variable: the excess return of the market
 Y-Variable: the excess return of the investment
The intercept of this estimated equation is the fund’s alpha.
Consider the following Spreadsheet Analysis slide.
 In the spreadsheet example, the intercept estimate is -.0167, or 1.67 percent per year.
 The beta of this security is .96, which is the coefficient estimate
on the x variable.
How would you characterize the performance of this fund?12
Estimating Alpha Using Regression
13
The Information Ratio
Suppose a mutual fund reports a positive alpha.
 How do we know whether this alpha is statistically significantly different from zero
or simply represents a result of random chance?
 One way: Evaluate the significance level of the alpha estimate using a regression.
 Another way: Calculate the fund’s information ratio.
 The information ratio is a fund’s alpha divided by its tracking error.
 Tracking error measures the volatility of the fund’s returns relative to its benchmark.
 Consider the fund we evaluated in the Spreadsheet Analysis in the previous slide.
 Over the five years, the excess return differences are 2%, 4%,-20%,14%, and -10%.
 The tracking error is the standard deviation of these return differences, which is
13.2%.
 The information ratio for this fund is -1.67% divided by 13.2%, which is -0.13.
 The information ratio allows us to compare investments that have the same alpha: A
higher information ratio means a lower tracking error risk.
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R-Squared
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Recall that correlation measures how returns for a particular security
move relative to returns for another security. Correlation also plays
a key role in performance measurement.
Suppose a particular fund has had a large alpha over the past three
years.
 All else equal, we might say that this fund is a good choice.
 Suppose this fund is a sector-based fund that invests only in gold.
 It is possible that the large alpha is simply due to a run-up in gold
prices over the period and is not reflective of good management
or future potential.
To evaluate this type of risk we can calculate R-squared, which is
simply the squared correlation of the fund to the market.
For example, if this fund’s correlation with the market was .60, then
the R-squared value is .36.
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R-Squared
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R-squared represents the percentage of the fund’s movement that can
be explained by movements in the market.
Because correlation ranges only from -1 to +1, R-squared values will
range from 0 to 100 percent.
 An R-squared of 100 indicates that all movements in the security
are driven by the market, indicating a correlation of -1 or +1.
 A high R-squared value (say greater than .80) might suggest that
the performance measures (such as alpha) are more representative
of potential longer term performance.
Excel reports correlation as “Multiple R” and R-squared as “R
Square.”
Verify that the squared value for “Multiple R” equals “R Square” in
the previous Spreadsheet Analysis slide.
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How do you interpret an R-squared of 0.549?
Performance Evaluation Measures
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M2 Measure
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It is the excess return of a hypothetical portfolio over
the market portfolio.
For a portfolio p, we can create a hypothetical portfolio
by combining a percentage of portfolio p with risk free
asset (T-bill) such that the hypothetical portfolio will
have the same standard deviation as the market portfolio.
We can then compare two same risk portfolios using
their returns.
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M2 = Rhp- Rm
where hp represents the hypothetical portfolio and m represents
the market index.
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13- 17
Portfolio Performance Measurement
Table 13.1 and Table 13.2
© 2009 McGraw-Hill Ryerson
Limited
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Investment Performance Measurement on the
Web Stock Fund, a small-cap value fund.
Consider the information below for the Fidelity Low-Priced
•
• We obtained the numbers from www.morningstar.com by entering the fund’s ticker symbol
(FLPSX) and following the “Ratings and Risk” link.
• How do you judge the manager’s of this fund? For this fund:
― Beta is 1.10, so the degree of market risk is above average.
― Alpha is 3.51 percent, which could indicate superior past performance.
― Standard deviation is 22.90 percent.
― Mean of -1.08 percent is the geometric return of the fund over the past three years.
― Sharpe ratio is -.14.
― R-squared is 91, which means that 91% of its returns are driven by the market’s return.
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Comparing Performance Measures
Because the performance rankings can be substantially different,
which performance measure should we use?
Sharpe ratio:
Appropriate for the evaluation of an entire portfolio.
 Penalizes a portfolio for being undiversified, because in general, total
risk  systematic risk only for relatively well-diversified portfolios.
Treynor ratio and Jensen’s alpha:
 Appropriate for the evaluation of securities or portfolios for possible
inclusion into an existing portfolio.
 Both are similar, the only difference is that the Treynor ratio
standardizes returns, including excess returns, relative to beta.
 Both require a beta estimate (and betas from different sources can
differ a lot).
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Global Investment Performance Standards
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Comparing investments can be difficult: various performance measures can
provide different rankings.
Even if the various performance measures provide a similar ranking, we
could still make an incorrect choice in selecting an investment manager. How
is this possible?
Performance measures often rely on self-reported returns (many funds do
not have public portfolios).
The CFA Institute developed the Global Investment Performance Standards
(GIPS).
Goal: Provide a consistent method to report portfolio performance to
prospective (and current) clients
By standardizing the way portfolio performance is reported, GIPS provide
investors with the ability to make comparisons across managers.
 Firms are not required by law to comply with GIPS, compliance is
voluntary.
 Firms that do comply, however, are recognized by the CFA Institute. 21
Sharpe-Optimal Portfolios
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Allocating funds to achieve the highest possible Sharpe
ratio is said to be Sharpe-optimal.
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To find the Sharpe-optimal portfolio, first look at the plot
of the possible risk-return possibilities, i.e., the
investment opportunity set.
Expected
Return
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
Standard deviation
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Sharpe-Optimal Portfolios
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The slope of a straight line drawn from the risk-free rate to
where the portfolio plots gives the Sharpe ratio for that
portfolio.
Expected
Return
Rf
A
×
ER A   R f
slope 
σA
Standard deviation
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The portfolio with the steepest slope is the Sharpe-optimal
portfolio.
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Sharpe-Optimal Portfolios
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Solving for a Sharpe-Optimal Portfolio
Two asset portfolio return is given by:
weight in asset B
PortfolioRe turn: E(R p )  ω A E(R A )  (1 ω A )E(R B )
PortfolioV ariance: σ P2  ω 2A σ 2A  (1 ω A ) 2 σ B2  2ω A (1 ω A )σ A σ B CORR(R A , R B )
SharpeRatio 
E(R p ) - rf
σP

ω A E(R A )  (1 ω A )E(R B ) - rf
ω 2A σ 2A  (1 ω A ) 2 σ B2  2ω A (1 ω A )σ A σ B CORR(R A , R B )
So, now our job is to choose the weight in asset A that
maximizes the Sharpe Ratio.
We could use calculus to do this, or we could use Excel.
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Using Excel to Solve the Sharpe-Optimal Portfolio
Suppose we enter the following into a spreadsheet, guessing that
W_A = 0.25 is a “good” portfolio.
Using formulas for portfolio return and standard deviation, we can
compute a Sharpe Ratio:
Data
Inputs:
ER(A):
ST D(A):
ER(B):
ST D(B):
CORR(A,B):
R_f:
0.12
0.15
0.06
0.10
0.10
0.04
W_A: 0.250
ER(P): 0.075
ST D(P): 0.087
Sharpe
Ratio:
0.402
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Using Excel to Solve the Sharpe-Optimal Portfolio
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Now, we let Excel solve for the weight in portfolio A
that maximizes the Sharpe Ratio.
We use the Solver, found under Tools.
Solving for the Optimal Sharpe Ratio
Given the data inputs below, we can use the
SOLVER function to find the Maximum Sharpe Ratio:
Data
Inputs:
ER(A):
STD(A):
ER(B):
STD(B):
CORR(A,B):
R_f:
0.12
0.15
0.06
0.10
0.10
0.04
W_A:
ER(P):
STD(P):
Sharpe
Ratio:
Changer
Cell:
0.700
0.102
0.112
0.553
Target Cell
Well, the guess
of 0.25 was a
tad low….
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Investment Risk Management
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Investment risk management concerns a money manager’s
control over investment risks, usually with respect to potential
short-run losses.
We will focus on what is known as the Value-at-Risk (VaR)
approach.
Value-at-Risk (VaR) is a technique of assessing risk by stating the
probability of a loss that a portfolio may experience within a fixed
time horizon.
If the returns on an investment follow a normal distribution, we
can state the probability that a portfolio’s return will be within a
certain range, if we have the mean and standard deviation of the
portfolio’s return.
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Example: VaR Calculation
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Suppose you own an S&P 500 index fund.
What is the probability of a return of -7% or worse in a
particular year?
That is, one year from now, what is the probability that
your portfolio value is down by 7 percent (or more)?
First, the historic average return on the S&P index is
about 13%, with a standard deviation of about 20%.
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A return of -7 percent is exactly one standard deviations
below the average, or mean (i.e., 13 – 20 = -7).
We know the odds of being within one standard deviation of
the mean are about 2/3, or 0.67.
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Example:VaR Calculation
In this example, being within one standard deviation of the
mean is another way of saying that:
 Prob(13 – 20  RS&P500  13 + 20)  0.67 or
 Prob (–7  RS&P500  33)  0.67
That is, the probability of having an S&P 500 return between -7%
and 33% is 0.67.
•
So, the return will be outside this range one-third of the time.
•
When the return is outside this range, half the time it will be above
the range, and half the time below the range.
•
Therefore, we can say: Prob (RS&P500  –7)  1/6 or 0.17
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Example: A Multiple Year VaR
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Once again, you own an S&P 500 index fund.
Now, you want to know the probability of a
loss of 30% or more over the next two years.
As you know, when calculating VaR, you use
the mean and the standard deviation.
To make life easy on ourselves, let’s use the
one year mean (13%) and standard deviation
(20%) from the previous example.
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Example: A Multiple Year VaR
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Calculating the two-year average return is easy, because
means are additive. That is, the two-year average return
is:
13 + 13 = 26%
Standard deviations, however, are not additive.
Fortunately, variances are additive, and we know that
the variance is the squared standard deviation.
The one-year variance is 20 x 20 = 400. The two-year
variance is:
400 + 400 = 800.
Therefore, the 2-year standard deviation is the square
root of 800, or about 28.28%.
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Example: A Multiple Year VaR
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The probability of being within two standard deviations
is about 0.95.
Armed with our two-year mean and two-year standard
deviation, we can make the probability statement:
Prob(26 – 228  RS&P500  26 + 228)  .95
Prob (–30  RS&P500  82)  .95
•
•
The return will be outside this range 5 percent of the
time. When the return is outside this range, half the time
it will be above the range, and half the time below the
range.
So, Prob (RS&P500  –30)  2.5%.
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Computing Other VaRs.
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In general, for a portfolio, if T is the number of
σp,T  σp  T
years, ERp,T   ERp  T
Using the procedure from before, we make make
probability statements. Three very useful one are:

ProbR
ProbR

Prob R p,T  ER p  T  2.326  σ p T  1%
p,T
p,T

 ER p  T  1.96  σ p T  2.5%

 ER p  T  1.645  σ p T  5%
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Useful Websites
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www.stanford.edu/~wfsharpe (visit Professor Sharpe’s
homepage)
www.morningstar.com (Comprehensive source of
investment information)
www.gloriamundi.org (learn all about Value-at-Risk)
www.garp.org (The Global Association of Risk
Professionals)
www.riskmetrics.com (Check out risk grades)
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