1. Portfolio Performance Evaluation
Portfolio Construction, Management, & Protection, 5e, Robert A. Strong
Copyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.
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2. 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
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3. Importance of Measuring Portfolio Risk
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
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4. A Lesson from History: The 1968 Bank Administration Institute Report
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
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5. A Lesson from History: The 1968 Bank Administration Institute Report (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 greatly that indiscriminate comparisons among funds might reflect differences in these circumstances rather than in the ability of managers
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6. 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
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7. A Lesson from a Few Mutual Funds (cont’d)
23.5%
30.7
6.5
16.9
26.3
14.3
37.8
12.0
Mutual Shares
19.3%
19.3
–34.6
–16.3
–20.1
–58.7
9.2
6.9
44 Wall Street
Mean
1988
1987
1986
1985
1984
1983
1982
Year
8.7
-23.6
1981
19.0
36.1
1980
39.3
71.4
1979
16.1
32.9
1978
13.2
16.5
1977
63.1
46.5
1976
24.6%
184.1%
1975
Change in net asset value, January 1 through December 31.
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8. A Lesson from a Few Mutual Funds (cont’d)
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9. 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
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10. Why the Arithmetic Mean Is Often Misleading: A Review
The arithmetic mean may give misleading information
e.g., a 50 percent decline in one period followed by a 50 percent increase in the next period does not produce an average return of zero
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11. Why the Arithmetic Mean Is Often Misleading: A Review (cont’d)
The proper measure of average investment return over time is the geometric mean:
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12. Why the Arithmetic Mean Is Often Misleading: A Review (cont’d)
The geometric means in the preceding example are:
44 Wall Street: 7.9 percent
Mutual Shares: 22.7 percent
The geometric mean correctly identifies Mutual Shares as the better investment over the 1975 to 1988 period
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13. Why the Arithmetic Mean Is Often Misleading: A Review (cont’d)
Example
A stock returns –40% in the first period, +50% in the second period, and 0% in the third period. [The average rate of return is 3.3%]
What is the geometric mean over the three periods?
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14. Why the Arithmetic Mean Is Often Misleading: A Review (cont’d)
Example
Solution: The geometric mean is computed as follows:
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15. Why Dollars Are More Important Than Percentages
Assume two funds:
Fund A has $40 million in investments and earned 12 percent last period
Fund B has $250,000 in investments and earned 44 percent last period
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16. Why Dollars Are More Important Than Percentages (cont’d)
The correct way to determine the return of both funds combined is to weigh the funds’ returns by the dollar amounts:
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17. Traditional Performance Measures
Sharpe Measure
Treynor Measures
Jensen Measure
Performance Measurement in Practice
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18. Sharpe and Treynor Measures
The Sharpe and Treynor measures:
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19. Sharpe and Treynor Measures (cont’d)
The Sharpe measure evaluates return relative to total risk
Appropriate for a well-diversified portfolio, but not for individual securities
The Treynor measure evaluates the return relative to beta, a measure of systematic risk
It ignores any unsystematic risk
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20. Sharpe and Treynor Measures (cont’d)
Example
Over the last four months, XYZ Stock had excess returns of 1.86 percent, –5.09 percent, –1.99 percent, and 1.72 percent. The standard deviation of XYZ stock returns is 3.07 percent. XYZ Stock has a beta of 1.20.
What are the Sharpe and Treynor measures for XYZ Stock?
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21. Sharpe and Treynor Measures (cont’d)
Example (cont’d)
Solution: First, compute the average excess return for Stock XYZ:
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22. Sharpe and Treynor Measures (cont’d)
Example (cont’d)
Solution (cont’d): Next, compute the Sharpe and Treynor measures:
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23. Jensen Measure The Jensen measure stems directly from the CAPM:
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24. 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
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25. Academic Issues Regarding Performance Measures
The use of Treynor and Jensen performance measures relies on measuring the market return and CAPM
Difficult to identify and measure the return of the market portfolio
Evidence continues to accumulate that may ultimately displace the CAPM
Arbitrage pricing model, multi-factor CAPMs, inflation-adjusted CAPM
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26. 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
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27. Industry Issues (cont’d)
“Fama’s return 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
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29. Industry Issues (cont’d)
Diversification is the difference between the return corresponding to the beta implied by the total risk of the portfolio and the return corresponding to its actual beta
Diversifiable risk decreases as portfolio size increases, so if the portfolio is well diversified the “diversification return” should be near zero
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30. Industry Issues (cont’d)
Net selectivity measures the portion of the return from selectivity in excess of that provided by the “diversification” component
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31. Volume Weighted Average Price
Portfolio managers want to minimize the impact of their trading on share price
A buy order increases demand and price
Portfolio managers frequently take the opposite position in the pre-market trading
Volume weighted average price compares the average market price to the average price paid by (or received) by a manager
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32. Trading Efficiency Impacts on Performance (cont’d)
Implementation shortfall is the difference between the value of a “on-paper” theoretical portfolio and portfolio purchased
“On-paper” portfolio is based upon price of last reported trade
Actual purchases are likely to lead to higher costs due to
Commissions
“Bid-Ask” Spread – the prior trade may have been at a bid, while your trade may be at the ask price (reverse if selling)
Market trend – are prices moving higher?
Liquidity impact – your demand may exceed share available a lower price
Or, shares sold may exceed the number purchased at higher price
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33. Dollar-Weighted and Time-Weighted Rates of Return
The dollar-weighted rate of return is analogous to the internal rate of return in corporate finance
It is the rate of return that makes the present value of a series of cash flows equal to the cost of the investment:
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34. Dollar-Weighted and Time-Weighted Rates of Return (cont’d)
The time-weighted rate of return measures the compound growth rate of an investment
It eliminates the effect of cash inflows and outflows by computing a return for each period and linking them (like the geometric mean return):
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35. Dollar-Weighted and Time-Weighted Rates of Return (cont’d)
The time-weighted rate of return and the dollar-weighted rate of return will be equal if there are no inflows or outflows from the portfolio
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36. Performance Evaluation with Cash Deposits and Withdrawals
The owner of a fund often takes 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
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37. 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 to or withdrawals
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38. Daily Valuation Method (cont’d)
The daily valuation method solves for R:
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39. 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
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40. Modified Bank Administration Institute (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
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41. Modified Bank Administration Institute (BAI) Method (cont’d)
It solves for R:
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42. 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
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43. Table19-4 Mutual Fund Account Value
Date
Description
$ Amount
Price
Shares
Total Shares
Value
January 1
balance forward
$7.00 1,
$7,560.08
January 3
purchase
100
14.286 1,
$7,660.08
February 1
$7.91
12.642 1,
$8,755.89
March 1
$7.84
12.755 1,
$8,778.40
March 23
liquidation
5,000
$8.13
$4,103.11
April 3
$8.34
11.900
$4,309.09
May 1
$9.00
11.111
$4,750.10
June 1
$9.74
10.267
$5,240.67
July 3
$9.24
10.823
$5,071.64
August 1
$9.84
10.163
$5,500.97
August 1 account value: $ x $9.84 = $5,500.97
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44. An Example (cont’d)
The daily valuation method returns a time-weighted return of 40.6 percent over the seven-month period
See next slide
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45. Table19-5 Daily Valuation Worksheet
Date
Sub Period
MVB
Cash Flow
Ending Value
MVE
MVE/MVB
January 1
$7,560.08
January 3 1
100
$7,660.08
1.00
February 1 2
$8,755.89
$8,655.89
1.13
March 1 3
$8,778.40
$8,678.40
0.991
March 23 4
5,000
$4,103.11
$9,103.11
1.037
April 3 5
$4,309.09
$4,209.09
1.026
May 1 6
$4,750.10
$4,650.10
1.079
June 1 7
$5,240.67
$5,140.67
1.082
July 3 8
$5,071.64
$4,971.64
0.949
August 1 9
$5,500.97
$5,400.97
1.065
Product of MVE/MVB values = 1.406;  R = 40.6%
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46. An Example (cont’d) The BAI method requires use of a computer
The BAI method returns a time-weighted return of 42.1 percent over the seven-month period (see next slide)
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47. 7/4/2018
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48. An Approximate Method
Proposed by the American Association of Individual Investors:
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49. Intuition
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50. An Approximate Method (cont’d)
Using the approximate method in Table 17-6:
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51. Performance Evaluation When Options Are Used
Inclusion of options in a portfolio usually results in a non-normal return distribution
Beta and standard deviation lose their theoretical value if the return distribution is nonsymmetrical
Consider two alternative methods when options are included in a portfolio:
Incremental risk-adjusted return (IRAR)
Residual option spread (ROS)
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52. Incremental Risk-Adjusted Return from Options
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
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53. Incremental Risk-Adjusted Return from Options (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
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54. Incremental Risk-Adjusted Return from Options (cont’d)
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55. Incremental Risk-Adjusted Return from Options (cont’d)
The IRAR calculation:
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56. 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
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57. Table19-9 Modified BAI Method Worksheet (Using Interest Rate of 42.1%)
Week
Unoptioned Portfolio
Long Stock Plus Option Premiums
Less
Short Optionsa
Equals
Optioned Portfolio
$200,000
$214,112
$14,112 1
190,000
203,406
8,868
194,538 2
195,700
209,508
10,746
198,762 3
203,528
217,888
14,064
203,824 4
199,528
213,530
11,220
202,310 5
193,474
207,124
7,758
199,366 6
201,213
215,409
10,614
204,795 7
207,249
221,871
13,164
208,707
aOption values are theoretical Black-Scholes values based on an initial OEX value of 300, a striking price of 300, a risk-free interest rate of 8 percent, an initial life of 90 days, and a volatility of 35 percent. Six OEX calls are written.
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58. 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 percent per year to the overall performance of the fund
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59. Risk-free rate = 8%/year = 0.15%/week
Table IRAR Worksheet
Unoptioned Portfolio
Relative Return
Optioned Portfolio
$200,000
0.9500
$190,000
1.0300
$194,538
0.9727
$195,700
1.0400
$198,762
1.0217
$203,528
0.9800
$203,824
1.0255
$199,457
$202,310
0.9926
$193,474
0.9700
$199,366
0.9854
$201,213
$204,795
1.0272
$207,249
$208,707
1.0191
Mean = 1.007
Mean =
Standard Deviation =
Standard Deviation =
Risk-free rate = 8%/year = 0.15%/week
SHu = ( ) ÷ =
SHo = ( ) ÷ =
IRAR = ( – ) x = per week, or about 12% per year
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60. 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
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61. 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 the stock
A positive ROS does not necessarily mean that the incremental return is appropriate given the risk
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62. Residual Option Spread (cont’d)
The residual option spread (ROS) calculation:
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63. 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
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64. Residual Options Spread Worksheet
Unoptioned Portfolio
(0.95)(1.03)(1.04)(0.98)(0.97)(1.04)(1.03) =
Optioned Portfolio
(0.9727)(1.0217)(1.0255)(0.9926)(0.9854)(1.0272)(1.0191) =
ROS = – =
Given an initial investment of $200,000, the ROS translates into a dollar differential of $200,000 x = $1,452.
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65. Final Comments
IRAR and ROS both focus on whether an optioned portfolio outperforms an unoptioned portfolio
Can overlook subjective considerations such as portfolio insurance
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