This document discusses performance evaluation of investments and portfolios. It covers the following key points in 3 sentences: Abnormal performance is measured against benchmarks adjusted for market and risk factors. Various risk-adjusted return measures are used, including the Sharpe Ratio, Treynor Ratio, and Jensen's Alpha, to evaluate performance based on different investment assumptions. Performance attribution decomposes overall returns into components related to allocation decisions, security selection, and market conditions to explain sources of performance differences from a benchmark.

1. Performance
Evaluation

2. Introduction
• Complicated subject
• 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 managements leads to
measurement problems

3. Abnormal Performance
What is abnormal?
Abnormal performance is measured:
• Benchmark portfolio
• Market adjusted
• Market model / index model adjusted
• Reward to risk measures such as the Sharpe
Measure:
E (rp-rf) / sp

4. Factors That Lead to Abnormal
Performance
• Market timing
• Superior selection
– Sectors or industries
– Individual companies

5. Risk Adjusted Performance: Sharpe
1) Sharpe Index
rp - rf
rp = Average return on the portfolio
rf = Average risk free rate
p
= Standard deviation of portfolio
return
p
s
s

6. Risk Adjusted Performance: Treynor
2) Treynor Measure rp - rf
ßp
rp = Average return on the portfolio
rf = Average risk free rate
ßp = Weighted average b for portfolio

7. = rp - [ rf + ßp ( rm - rf) ]
3) Jensen’s Measure
p
p
rp = Average return on the portfolio
ßp = Weighted average Beta
rf = Average risk free rate
rm = Avg. return on market index port.
Risk Adjusted Performance: Jensen
= Alpha for the portfolio
a
a

8. M2 Measure
• 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

9. M2 Measure: Example
Managed Portfolio Market T-bill
Return 35% 28% 6%
Stan. Dev 42% 30% 0%
Hypothetical Portfolio: Same Risk as Market
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

10. T2 (Treynor Square) Measure
• Used to convert the Treynor Measure into
percentage return basis
• Makes it easier to interpret and compare
• Equates the beta of the managed portfolio with the
market’s beta of 1 by creating a hypothetical
portfolio made up of T-bills and the managed
portfolio
• If the beta is lower than one, leverage is used and
the hypothetical portfolio is compared to the market

11. T2 Example
Port. P. Market
Risk Prem. (r-rf) 13% 10%
Beta 0.80 1.0
Alpha 5% 0%
Treynor Measure 16.25 10
Weight to match Market w = bM/bP = 1.0 / 0.8
Adjusted Return RP
* = w (RP) = 16.25%
T2
P = RP
* - RM = 16.25% - 10% = 6.25%
T2
P = RP/Bp – RM = 13/.8 – 10 = 6.25%

12. 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 a or the Treynor measure
The Treynor measure is more complete
because it adjusts for risk

13. 13
• Other Measures of Risk
– A very popular measure of risk is called Value at
Risk or VAR. It is generally the amount you can
lose at a particular confidence interval. It is only
concerned with loss.
– At the 95% level, E(R) – 1.65(standard dev)
– At the 99% level, E(R) – 2.33(standard dev)

14. 14
– VAR Example
– Invest $100 at 10% with S.D. of 15%. What is the
95% Var over the year?
– 10% - 1.65(15%) = -14.75%
– Var = 100 * -.1475 = $14.75

15. Semi-variance or semi-standard
deviation
This statistic only measures variations below the
mean or some threshold.
Semi-variance = 1/n * (summation of the
average – rt)^2 where
Where:
n = the total number of observations below
the mean
rt = the observed value
average = the mean or target value of the data
set. Standard deviation is just the square root
of the above.

16. Semi-variance or semi-standard
deviation
Example:
You note the returns 3, 8, 3, 5, 11. Mean is 6. Thus
3 returns are less than the mean. The semi
variance is [(6-3)^2 + (6-3)^2 + (6-5)^2]/3
= 6.33 and the semi-standard deviation is 6.33^.5 = 2.51.
Compare this to the population standard deviation which
is [(6-3)^2 + (6-8)^2 + (6-3)^2 + (6-5)^2+(6-11)^2]/5 = 9.6.
Square root of 19.6 is 3.1%. Note that deviations above
the mean are not considered, thus the large deviation(11
compared to 6) is ignored and the semi-standard
deviation is actually smaller in this case.

17. Monte Carlo Methods
• Monte Carlo methods can be used to ask what if
questions based on thousands of simulations. The
range of values found during the simulations can
be used to estimate possible outcomes and thus
the risk associated with any position.
• For example, using the mean and standard
deviation of the stock market, one could simulate
possible return patterns over many years to see
how a portfolio may be affected.

18. Limitations
• Assumptions underlying measures limit their
usefulness
• When the portfolio is being actively managed,
basic stability requirements are not met
• Practitioners often use benchmark portfolio
comparisons to measure performance

19. Performance Attribution
• Decomposing overall performance into
components
• Components are related to specific elements of
performance
• Example components
– Broad Allocation
– Industry
– Security Choice
– Up and Down Markets

20. Process of Attributing Performance to
Components
Set up a ‘Benchmark’ or ‘Bogey’ portfolio
• Use indexes for each component
• Use target weight structure

21. Appropriate Benchmark
• Unambiguous – anyone can reproduce it
• Investable
• Measurable
• Specified in Advance

22. 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

23. Example, Performance Attributuion
Actual
Ret Act. Wt Bmk Wt Index Ret
Semi-C 2% 0.7 0.6 2.5
Energy 1% 0.2 0.3 1.2
Bio-Tech 0.50% 0.1 0.1 0.5
Actual: (0.70 ´ 2.0%) + (0.20 ´ 1.0%) + (0.10 ´
0.5%) = 1.65%
Benchmark: (0.60 ´ 2.5%) + (0.30 ´
1.2%) + (0.10 ´ 0.5%) = 1.91%
Underperformance = 1.91% -
1.65% = 0.26%

24. Security Selection
Portfolio Sector Excess Man. Port
Sector Performa
nce
Perform
ance
Perform
ance
Weight Contrib
ution
Semi-C 2.00% 2.50% -0.50% 0.7 -0.35%
Energy
1.00% 1.20% -0.20% 0.2 -0.04%
Bio-
Tech
0.50% 0.50% 0.00% 0.1 0.00%
Contribution of security
selection:
-0.39%

25. Sector Allocation
Actual Benchm
ark
Excess
Sector Weight Weight Weight Sector Ret. Contrib
ution
Semi-C 0.7 0.6 0.1 2.50% 0.25%
Energy 0.2 0.3 -0.1 1.20% -0.12%
Biotech 0.1 0.1 0 0.50% 0.00%
Contribution of sector
allocation:
0.13%

26. Lure of Active Management
Are markets totally efficient?
• Some managers outperform the market for extended
periods
• While the abnormal performance may not be too
large, it is too large to be attributed solely to noise
• Evidence of anomalies such as the turn of the year
exist
The evidence suggests that there is some role
for active management