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Chapter 19 Performance Evaluation 1 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 2 Outline Introduction Importance of measuring portfolio risk Traditional performance measures Performance evaluation with cash deposits and withdrawals Performance evaluation when options are used 3 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 4 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 5 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 E (U ) f E ( R), 2 6 A Lesson from History The 1968 Bank Administration Institute’s Measuring the Investment Performance of Pension Funds concluded: 1) Performance of a fund should be measured by computing the actual rates of return on a fund’s assets 2) These rates of return should be based on the market value of the fund’s assets 7 A Lesson from History (cont’d) 3) Complete evaluation of the manager’s performance must include examining a measure of the degree of risk taken in the fund 4) 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 8 A Lesson from A Few Mutual Funds Thetwo key points with performance evaluation: • The arithmetic mean is not a useful statistic in evaluating growth • Dollars are more important than percentages Considerthe historical returns of two mutual funds on the following slide 9 A Lesson from A Few Mutual Funds (cont’d) 44 Wall Mutual 44 Wall Mutual Year Street Shares Year Street 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% 10 A Lesson from A Few Mutual Funds (cont’d) Mutual Fund Performance $200,000.00 $180,000.00 $160,000.00 Ending Value ($) $140,000.00 44 Wall $120,000.00 Street $100,000.00 Mutual $80,000.00 Shares $60,000.00 $40,000.00 $20,000.00 $- 77 80 83 86 19 19 19 19 Year 11 A Lesson from A Few Mutual Funds (cont’d) Wall Street and Mutual Shares both had 44 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 12 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 13 Why the Arithmetic Mean Is Often Misleading (cont’d) The proper measure of average investment return over time is the geometric mean: 1/ n n GM Ri 1 i 1 where Ri the return relative in period i 14 Why the Arithmetic Mean Is Often Misleading (cont’d) Thegeometric 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 15 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? 16 Why the Arithmetic Mean Is Often Misleading (cont’d) Example Solution: The geometric mean is computed as follows: 1/ n n GM Ri 1 i 1 (0.60)(1.50)(1.00) 1 0.10 10% 17 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 18 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: $40, 000, 000 $250, 000 $40, 250, 000 12% $40, 250, 000 44% 12.10% 19 Traditional Performance Measures Sharpe and Treynor measures Jensen measure Performance measurement in practice 20 Sharpe and Treynor Measures The Sharpe and Treynor measures: R Rf Sharpe measure R Rf Treynor measure where R average return R f risk-free rate standard deviation of returns beta 21 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 22 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? 23 Sharpe and Treynor Measures (cont’d) Example (cont’d) Solution: First compute the average excess return for Stock XYZ: 1.86% 5.09% 1.99% 1.72% R 4 0.88% 24 Sharpe and Treynor Measures (cont’d) Example (cont’d) Solution (cont’d): Next, compute the Sharpe and Treynor measures: R Rf 0.88% Sharpe measure 0.29 3.07% R Rf 0.88% Treynor measure 0.73 1.20 25 Jensen Measure Jensen measure stems directly from the The CAPM: Rit R ft i Rmt R ft 26 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 27 Jensen Measure (cont’d) TheJensen measure is generally out of favor because of statistical and theoretical problems 28 Performance Measurement in Practice Academic issues Industry issues 29 Academic Issues use of traditional performance The measures relies on the CAPM Evidence continues to accumulate that may ultimately displace the CAPM • APT, multi-factor CAPMs, inflation-adjusted CAPM 30 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” Practicalperformance measures typically involve a comparison of the fund’s performance with that of a benchmark 31 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 32 33 Performance Evaluation With Cash Deposits & Withdrawals Introduction Dailyvaluation method Modified Bank Administration Institute (BAI) Method An example An approximate method 34 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 35 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 36 Daily Valuation Method (cont’d) The daily valuation method solves for R: n Rdaily Si 1 i 1 MVEi where S MVBi 37 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 38 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 39 Modified BAI Method (cont’d) It solves for R: n MVE Fi (1 R ) wi i 1 where F the sum of the cash flows during the period MVE market value at the end of the period, including accrued income F0 market value at the start of the period CD Di wi CD CD total number of days in the period Di number of days since the beginning of the period in which the cash flow occurred 40 An Example Aninvestor has an account with a mutual fund and “dollar cost averages” by putting $100 per month into the fund Thefollowing slide shows the activity and results over a seven-month period 41 42 An Example (cont’d) Thedaily valuation method returns a time- weighted return of 40.6% over the seven- months period • See next slide 43 44 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) 45 46 An Approximate Method Proposed by the American Association of Individual Investors: P 0.5(Net cash flow) R 1 1 P0 0.5(Net cash flow) where net cash flow is the sum of inflows and outflows 47 An Approximate Method (cont’d) Using the approximate method in Table 19- 6: P 0.5(Net cash flow) R 1 1 P0 0.5(Net cash flow) 5,500.97 0.5( 4, 200) 1 7,550.08 0.5(-4, 200) 0.395 39.5% 48 Performance Evaluation When Options Are Used Introduction Incremental risk-adjusted return from options Residual option spread Final comments on performance evaluation with options 49 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 50 Introduction (cont’d) Consider two alternative methods when options are included in a portfolio: • Incremental risk-adjusted return (IRAR) • Residual option spread (ROS) 51 Incremental Risk-Adjusted Return from Options Definition An IRAR example IRAR caveats 52 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 53 Definition (cont’d) Usethe 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 54 Definition (cont’d) 55 Definition (cont’d) The IRAR calculation: IRAR ( SH o SH u ) o where SH o Sharpe measure of the optioned portfolio SH u Sharpe measure of the unoptioned portfolio o standard deviation of the optioned portfolio 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 57 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 about 12% per year to the overall performance of the fund 59 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 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 stock A positive ROS does not necessarily mean that the incremental return is appropriate given the risk 62 Residual Option Spread (cont’d) Theresidual option spread (ROS) calculation: n n ROS Got Gut t 1 t 1 where Gt Vt / Vt 1 Vt value of portfolio in Period t 63 Residual Option Spread (cont’d) Theworksheet to calculate the ROS for the previous example is shown on the next slide TheROS translates into a dollar differential of $1,452 64 65 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 66 The M2 Performance Measure (cont’d) Calculate the risk-adjusted portfolio return as follows: benchmark Rrisk-adjusted portfolio Ractual portfolio portfolio benchmark 1 Rf portfolio 67 Final Comments IRAR and ROS both focus on whether an optioned portfolio outperforms an unoptioned portfolio • Can overlook subjective considerations such as portfolio insurance 68

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