1. Portfolio Performance Evaluation
Chapter 24

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 management leads to measurement problems

3. Dollar- and Time-Weighted Returns
Dollar-weighted returns
Internal rate of return considering the cash flow from or to investment
Returns are weighted by the amount invested in each stock
Time-weighted returns
Not weighted by investment amount
Equal weighting

4. Text Example of Multiperiod Returns
Period Action
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 $108 per share

5. Dollar-Weighted Return
Period Cash Flow
share purchase
1 +2 dividend share purchase
2 +4 dividend shares sold
Internal Rate of Return:

6. Time-Weighted Return
Simple Average Return:
(10% %) / 2 = 7.83%

7. Averaging Returns Arithmetic Mean: Text Example Average:
( ) / 2 = 7.81%
Geometric Mean:
Text Example Average:
[ (1.1) (1.0566) ]1/2 - 1
= 7.83%

8. Geometric & Arithmetic Means Compared
Past Performance - generally the geometric mean is preferable to arithmetic
Predicting Future Returns from historical returns
Use a weighted average of arithmetic and geometric averages of historical returns if the forecast period is less than the estimation period
Use geometric is the forecast and estimation period are equal

9. 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) / p

10. Factors Leading to Abnormal Performance
Market timing
Superior selection
Sectors or industries
Individual companies

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

12. 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

13. M2 Measure: Example
Managed Portfolio: return = 35% standard deviation = 42%
Market Portfolio: return = 28% standard deviation = 30% T-bill return = 6%
Hypothetical Portfolio:
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

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

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

16. Appraisal Ratio = ap / s(ep)
Appraisal Ratio divides the alpha of the portfolio by the nonsystematic risk
Nonsystematic risk could, in theory, be eliminated by diversification

17. 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 risk

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. Market Timing
Adjusting portfolio for up and down movements in the market
Low Market Return - low ßeta
High Market Return - high ßeta

20. Example of Market Timing
rp - rf * * * * * * * * * * * * * * * * * * * * *
rm - rf * *
Steadily Increasing the Beta

21. 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

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

23. 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

24. Formula for Attribution
Where B is the bogey portfolio and p is the managed portfolio

25. Contributions for Performance
Contribution for asset allocation (wpi - wBi) rBi
+ Contribution for security selection wpi (rpi - rBi)
= Total Contribution from asset class wpirpi -wBirBi

26. Complications to Measuring Performance
Two major problems
Need many observations even when portfolio mean and variance are constant
Active management leads to shifts in parameters making measurement more difficult
To measure well
You need a lot of short intervals
For each period you need to specify the makeup of the portfolio

27. Style Analysis Based on regression analysis
Examines asset allocation for broad groups of stocks
More precise than comparing to the broad market
Sharpe’s analysis: 97.3% of returns attributed to style