1. Chapter 19 Performance Evaluation

2. - From Candide by Voltaire
And with that they clapped him into irons and hauled him off to the barracks. There he was taught “right turn,” “left turn,” and “quick march,” “slope arms,” and “order arms,” how to aim and how to fire, and was given thirty strokes of the “cat.” Next day his performance on parade was a little better, and he was given only twenty strokes. The following day he received a mere ten and was thought a prodigy by his comrades.
- From Candide by Voltaire

3. Outline Introduction Importance of measuring portfolio risk
Traditional performance measures
Performance evaluation with cash deposits and withdrawals
Performance evaluation when options are used

4. Introduction
Performance evaluation is a critical aspect of portfolio management
Proper performance evaluation should involve a recognition of both the return and the riskiness of the investment

5. Importance of Measuring Portfolio Risk
Introduction
A lesson from history: the 1968 Bank Administration Institute report
A lesson from a few mutual funds
Why the arithmetic mean is often misleading: a review
Why dollars are more important than percentages

6. Introduction
When two investments’ returns are compared, their relative risk must also be considered
People maximize expected utility:
A positive function of expected return
A negative function of the return variance

7. A Lesson from History
The 1968 Bank Administration Institute’s Measuring the Investment Performance of Pension Funds concluded:
Performance of a fund should be measured by computing the actual rates of return on a fund’s assets
These rates of return should be based on the market value of the fund’s assets

8. A Lesson from History (cont’d)
Complete evaluation of the manager’s performance must include examining a measure of the degree of risk taken in the fund
Circumstances under which fund managers must operate vary so great that indiscriminate comparisons among funds might reflect differences in these circumstances rather than in the ability of managers

9. A Lesson from A Few Mutual Funds
The two key points with performance evaluation:
The arithmetic mean is not a useful statistic in evaluating growth
Dollars are more important than percentages
Consider the historical returns of two mutual funds on the following slide

10. A Lesson from A Few Mutual Funds (cont’d)
Year
44 Wall Street
Mutual Shares
1975
184.1%
24.6%
1982
6.9
12.0
1976
46.5
63.1
1983
9.2
37.8
1977
16.5
13.2
1984
-58.7
14.3
1978
32.9
16.1
1985
-20.1
26.3
1979
71.4
39.3
1986
-16.3
16.9
1980
36.1
19.0
1987
-34.6
6.5
1981
-23.6
8.7
1988
19.3
30.7
Mean
19.3%
23.5%

11. A Lesson from A Few Mutual Funds (cont’d)

12. A Lesson from A Few Mutual Funds (cont’d)
44 Wall Street and Mutual Shares both had good returns over the 1975 to 1988 period
Mutual Shares clearly outperforms 44 Wall Street in terms of dollar returns at the end of 1988

13. Why the Arithmetic Mean Is Often Misleading
The arithmetic mean may give misleading information
E.g., a 50% decline in one period followed by a 50% increase in the next period does not return 0%, on average

14. Why the Arithmetic Mean Is Often Misleading (cont’d)
The proper measure of average investment return over time is the geometric mean:

15. Why the Arithmetic Mean Is Often Misleading (cont’d)
The geometric means in the preceding example are:
44 Wall Street: 7.9%
Mutual Shares: 22.7%
The geometric mean correctly identifies Mutual Shares as the better investment over the 1975 to 1988 period

16. Why the Arithmetic Mean Is Often Misleading (cont’d)
Example
A stock returns –40% in the first period, +50% in the second period, and 0% in the third period.
What is the geometric mean over the three periods?

17. Why the Arithmetic Mean Is Often Misleading (cont’d)
Example
Solution: The geometric mean is computed as follows:

18. Why Dollars Are More Important than Percentages
Assume two funds:
Fund A has $40 million in investments and earned 12% last period
Fund B has $250,000 in investments and earned 44% last period

19. Why Dollars Are More Important than Percentages
The correct way to determine the return of both funds combined is to weigh the funds’ returns by the dollar amounts:

20. Traditional Performance Measures
Sharpe and Treynor measures
Jensen measure
Performance measurement in practice

21. Sharpe and Treynor Measures
The Sharpe and Treynor measures:

22. Sharpe and Treynor Measures (cont’d)
The Treynor measure evaluates the return relative to beta, a measure of systematic risk
It ignores any unsystematic risk
The Sharpe measure evaluates return relative to total risk
Appropriate for a well-diversified portfolio, but not for individual securities

23. Sharpe and Treynor Measures (cont’d)
Example
Over the last four months, XYZ Stock had excess returns of 1.86%, -5.09%, -1.99%, and 1.72%. The standard deviation of XYZ stock returns is 3.07%. XYZ Stock has a beta of 1.20.
What are the Sharpe and Treynor measures for XYZ Stock?

24. Sharpe and Treynor Measures (cont’d)
Example (cont’d)
Solution: First compute the average excess return for Stock XYZ:

25. Sharpe and Treynor Measures (cont’d)
Example (cont’d)
Solution (cont’d): Next, compute the Sharpe and Treynor measures:

26. Jensen Measure
The Jensen measure stems directly from the CAPM:

27. Jensen Measure (cont’d)
The constant term should be zero
Securities with a beta of zero should have an excess return of zero according to finance theory
According to the Jensen measure, if a portfolio manager is better-than-average, the alpha of the portfolio will be positive

28. Jensen Measure (cont’d)
The Jensen measure is generally out of favor because of statistical and theoretical problems

29. Performance Measurement in Practice
Academic issues
Industry issues

30. Academic Issues
The use of traditional performance measures relies on the CAPM
Evidence continues to accumulate that may ultimately displace the CAPM
APT, multi-factor CAPMs, inflation-adjusted CAPM

31. Industry Issues
“Portfolio managers are hired and fired largely on the basis of realized investment returns with little regard to risk taken in achieving the returns”
Practical performance measures typically involve a comparison of the fund’s performance with that of a benchmark

32. Industry Issues (cont’d)
Fama’s decomposition can be used to assess why an investment performed better or worse than expected:
The return the investor chose to take
The added return the manager chose to seek
The return from the manager’s good selection of securities

34. Performance Evaluation With Cash Deposits & Withdrawals
Introduction
Daily valuation method
Modified Bank Administration Institute (BAI) Method
An example
An approximate method

35. Introduction
The owner of a fund often taken periodic distributions from the portfolio and may occasionally add to it
The established way to calculate portfolio performance in this situation is via a time-weighted rate of return:
Daily valuation method
Modified BAI method

36. Daily Valuation Method
The daily valuation method:
Calculates the exact time-weighted rate of return
Is cumbersome because it requires determining a value for the portfolio each time any cash flow occurs
Might be interest, dividends, or additions and withdrawals

37. Daily Valuation Method (cont’d)
The daily valuation method solves for R:

38. Daily Valuation Method (cont’d)
MVEi = market value of the portfolio at the end of period i before any cash flows in period i but including accrued income for the period
MVBi = market value of the portfolio at the beginning of period i including any cash flows at the end of the previous subperiod and including accrued income

39. Modified BAI Method The modified BAI method:
Approximates the internal rate of return for the investment over the period in question
Can be complicated with a large portfolio that might conceivably have a cash flow every day

40. Modified BAI Method (cont’d)
It solves for R:

41. An Example
An investor has an account with a mutual fund and “dollar cost averages” by putting $100 per month into the fund
The following slide shows the activity and results over a seven-month period

43. An Example (cont’d)
The daily valuation method returns a time-weighted return of 40.6% over the seven-months period
See next slide

45. An Example (cont’d) The BAI method requires use of a computer
The BAI method returns a time-weighted return of 42.1% over the seven-months period (see next slide)

47. An Approximate Method
Proposed by the American Association of Individual Investors:

48. An Approximate Method (cont’d)
Using the approximate method in Table 19-6:

49. Performance Evaluation When Options Are Used
Introduction
Incremental risk-adjusted return from options
Residual option spread
Final comments on performance evaluation with options

50. Introduction
Inclusion of options in a portfolio usually results in a non-normal return distribution
Beta and standard deviation lose their theoretical value of the return distribution is nonsymmetrical

51. Introduction (cont’d)
Consider two alternative methods when options are included in a portfolio:
Incremental risk-adjusted return (IRAR)
Residual option spread (ROS)

52. Incremental Risk-Adjusted Return from Options
Definition
An IRAR example
IRAR caveats

53. Definition
The incremental risk-adjusted return (IRAR) is a single performance measure indicating the contribution of an options program to overall portfolio performance
A positive IRAR indicates above-average performance
A negative IRAR indicates the portfolio would have performed better without options

54. Definition (cont’d) Use the unoptioned portfolio as a benchmark:
Draw a line from the risk-free rate to its realized risk/return combination
Points above this benchmark line result from superior performance
The higher than expected return is the IRAR

55. Definition (cont’d)

56. Definition (cont’d)
The IRAR calculation:

57. An IRAR Example
A portfolio manager routinely writes index call options to take advantage of anticipated market movements
Assume:
The portfolio has an initial value of $200,000
The stock portfolio has a beta of 1.0
The premiums received from option writing are invested into more shares of stock

59. An IRAR Example (cont’d)
The IRAR calculation (next slide) shows that:
The optioned portfolio appreciated more than the unoptioned portfolio
The options program was successful at adding about 12% per year to the overall performance of the fund

61. IRAR Caveats
IRAR can be used inappropriately if there is a floor on the return of the optioned portfolio
E.g., a portfolio manager might use puts to protect against a large fall in stock price
The standard deviation of the optioned portfolio is probably a poor measure of risk in these cases

62. Residual Option Spread
The residual option spread (ROS) is an alternative performance measure for portfolios containing options
A positive ROS indicates the use of options resulted in more terminal wealth than only holding stock
A positive ROS does not necessarily mean that the incremental return is appropriate given the risk

63. Residual Option Spread (cont’d)
The residual option spread (ROS) calculation:

64. Residual Option Spread (cont’d)
The worksheet to calculate the ROS for the previous example is shown on the next slide
The ROS translates into a dollar differential of $1,452

66. The M2 Performance Measure
Developed by Franco and Leah Modigliani in 1997
Seeks to express relative performance in risk-adjusted basis points
Ensures that the portfolio being evaluated and the benchmark have the same standard deviation

67. The M2 Performance Measure (cont’d)
Calculate the risk-adjusted portfolio return as follows:

68. Final Comments
IRAR and ROS both focus on whether an optioned portfolio outperforms an unoptioned portfolio
Can overlook subjective considerations such as portfolio insurance