Portfolio Performance Evaluation
Theoretically correct measures are difficult to construct
Different statistics or measures are appropriate for different types of
investment decisions or portfolios
Many industry and academic measures are different
The nature of active management leads to measurement problems
Dollar- and Time-Weighted Returns
Internal rate of return considering the cash flow from or to investment
Returns are weighted by the amount invested in each stock
Not weighted by investment amount
Expressing Average Returns – what is the average return per year?
Easy example: Suppose you buy a stock for $50, sell for $53 one period
later, and get a $1 dividend. What is your average return?
Now, consider multiple periods, where cash is added / withdrawn:
0 Purchase 1 share at $50
1 Purchase 1 share at $53
Stock pays a dividend of $2 per share
2 Stock pays a dividend of $2 per share
Stock is sold at $54 per share (so you receive a total of $108)
There are several ways of representing your “average” return for each time
(1) Dollar-Weighted Return:
Process: Add up the total cash flows (positive or negative) in each period, enter them
into your calculator as cash flows, and then computer the IRR. This is similar to
calculating the YTM for a bond.
Period Cash Flow
0 -50 share purchase
1 +2 dividend -53 share purchase
2 +4 dividend + 108 shares sold
Internal Rate of Return = Dollar-weighted Return:
50 IRR r 7.117%
(1 r )1 (1 r ) 2
(2) Time-Weighted Return:
This assumes that your average return is purely based on the price movements for one
share, from one period to the next. No consideration is given to the number of shares
held, or whether you are buying or selling.
53 50 2
54 53 2
After you have calculated the time-weighted average return in each period, you need to
average them together. There are two ways of doing this:
(a) Arithmetic Mean: Example Average:
rt (.10 + .0566) / 2 = 7.83%
t 1 n
(b) Geometric Mean: Example Average:
r (1 rt )
[(1.1) (1.0566)] 1/2 – 1 = 7.81%
Comparison of Geometric and Arithmetic Means
Past Performance - generally the geometric mean is preferable to
arithmetic – it tells you the constant rate of return needed each year to get
the actual overall holding period return. Due to compounding, it will also
always be less.
Predicting Future Returns- generally the arithmetic average is preferable
to geometric. The arithmetic mean is an unbiased estimate of the portfolio‟s
expected future return – it doesn‟t take compounding into account, so is better as an
estimator of return in some future period.
- Geometric return has a downward bias
Rule: the geometric mean will be less than the arithmetic mean (unless there is zero
volatility), and this difference will be greater for more volatility
Big Question: What is “abnormal”?
Abnormal performance is measured:
Market model / index model adjusted
Generally, we can gauge abnormal performance with broad measures that measure
reward to risk in some way (e.g. the Sharpe Measure, Treynor Measure, Jensen‟s Alpha,
and the Appraisal Ratio). Which measure is most appropriate depends on the “scenario”
that applies. Also, these measures presume that the distribution of returns are constant.
There are also several performance measures that break down abnormal performance into
factors such as market timing and security selection (e.g. performance attribution).
Factors That Lead to Abnormal Performance
- Sectors or industries
- Individual companies
Reward to Risk Measures
1) Sharpe Index
where rP = Average (geometric or arithmetic) return on the portfolio
rF = Average risk free rate
P = Standard deviation of portfolio return
Scenario: If your portfolio is the entire risky investment that you are considering.
Essentially, the Sharpe Index involves calculating the CAL slope of your
portfolio, and then comparing it to other funds / portfolios. Selecting portfolios
based on the Sharpe measure is the same thing as choosing the highest sloped
CAL in earlier chapters.
M2 Measure - This is related to and expands on the Sharpe Ratio
Developed by Modigliani and Modigliani
Equates the volatility of the managed portfolio with the market by creating a
hypothetical portfolio made up of T-bills and the managed portfolio
If the risk is lower than the market, leverage is used and the hypothetical portfolio is
compared to the market
Your portfolio Portfolio: average return = 35%
standard deviation = 42%
Market Portfolio: average return = 28%
standard deviation = 30% T-bill return = 6%
30/42 = .714 in P (1-.714) or .286 in T-bills
(.714) (.35) + (.286) (.06) = 26.7%
Since this return is less than the market, the managed portfolio underperformed.
(2) Appraisal Ratio = e
where P = your portfolio‟s alpha coefficient, and
(e P ) = standard deviation of the portfolio‟s residuals (i.e. amount of non-
Scenario: You mix your portfolio with a passive index portfolio. The appraisal
ratio tells you how much you “enhance” the CAL of the passive portfolio with
your actively managed “sub”- portfolio.
To calculate the appraisal ratio, you first need to calculate P . This is known as
“Jensen‟s alpha” and, as in the CAPM and market model chapters, measures your
portfolio‟s return in excess of what would have been expected (as defined by your
Jensen‟s alpha = P = rp - [ rf + ßp ( rm - rf)] or, the intercept of the
market model regression
where P = Alpha for the portfolio
rP = Average return on the portfolio
ßp = Weighted average Beta
rF = Average risk free rate
rM = Avg. return on market index port.
(3) Treynor Measure
where rp = Average return on the portfolio
rf = Average risk free rate
ßp for portfolio, P= Weighted average of individual asset ’s.
Scenario: Your portfolio is combined with other actively managed small
portfolios that make up a larger investment fund. Since the large investment fund
is completely diversified, systematic risk is all that is relevant for your portfolio –
i.e., how much risk it adds to the larger fund.
Note that a higher portfolio P will make the Treynor measure higher.
Which Measure is Appropriate?
It depends on investment assumptions
1) If the portfolio represents the entire investment for an individual, Sharpe
Index compared to the Sharpe Index for the market.
2) If many alternatives are possible, use the Jensen or the Treynor
measure. The Treynor measure is more complete because it adjusts for
Assumptions underlying measures limit their usefulness
When the portfolio is being actively managed, basic stability requirements
are not met
Secondary question: Are portfolio returns really based on ability? Returns will depend
on other things besides selection ability!
Practitioners often use benchmark portfolio comparisons to measure
Main idea: Adjusting portfolio for up and down movements in the market. Changing
portfolio composition in response to changes in market conditions will make the
measures discussed above unreliable (since you are changing the distribution of your
Low Market Return - low ßeta – e.g. you anticipate a poor stock market,
and move money into bonds. This results in a lower portfolio beta. When
the stock market does poorly, your portfolio does not drop by as much.
High Market Return - high ßeta – e.g., you anticipate a better market and
move money back into stocks. This results in a higher beta. When the
market does improve, your portfolio rises up by more.
Example of Market Timing
rp - rF
* * **
* * *
** * * r m - rf
* * *
Steadily Increasing the Beta
Good investment performance depends on being in the right asset (e.g. stocks vs. bonds
etc) and in the right securities at the right time. Performance attribution separates over /
under performance into these two components, so you can see how much asset allocation
added / subtracted from your portfolio‟s performance, and likewise how much security
picking added / subtracted.
Decomposing overall performance into components
Components are related to specific elements of performance
- Broad Allocation
- Security Choice
- Up and Down Markets
Process of Attributing Performance to Components
Set up a „Benchmark‟ or „Bogey‟ portfolio
Use indexes for each component
Use target weight structure
Process of Attributing Performance to Components
Calculate the return on the „Bogey‟ and on the managed portfolio
Explain the difference in return based on component weights or selection
Summarize the performance differences into appropriate categories
Formula for Performance Attribution:
“Bogey” return: rB wBi rBi
Your portfolio‟s return: rp wpi rpi
n n n
rp rB wpi rpi wBi rBi ( wpi rpi wBi rBi )
i 1 i 1 i 1
Contributions for Performance
Contribution for asset allocation (wpi – wBi) rBi
+ Contribution for security selection wpi (rpi – rBi)
= Total Contribution from asset class wpirpi –wBirBi
“i” is each asset class. If you have 2 asset classes (say stocks and bonds), then the
contribution for asset selection involves calculating (wpi – wBi) rBi for stocks and
bonds and then summing them up. The contribution for security selection is a
Complications to Measuring Performance
Two major problems
- Need many observations even when portfolio mean and variance are
- Active management leads to shifts in parameters making measurement
To measure well:
- You need a lot of short intervals
- For each period you need to specify the makeup of the portfolio