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Chapter 19 Performance Evaluation Prof. Rushen Chahal Prof. Rushen Chahal 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 Prof. Rushen Chahal 2 Outline • Introduction • Importance of measuring portfolio risk • Traditional performance measures • Performance evaluation with cash deposits and withdrawals • Performance evaluation when options are used Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 8 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 Prof. Rushen Chahal 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% Prof. Rushen Chahal 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 Prof. Rushen Chahal 11 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 14 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 Prof. Rushen Chahal 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? Prof. Rushen Chahal 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% Prof. Rushen Chahal 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 Prof. Rushen Chahal 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% Prof. Rushen Chahal 19 Traditional Performance Measures • Sharpe and Treynor measures • Jensen measure • Performance measurement in practice Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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? Prof. Rushen Chahal 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% Prof. Rushen Chahal 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 Prof. Rushen Chahal 25 Jensen Measure • The Jensen measure stems directly from the CAPM: Rit R ft i Rmt R ft Prof. Rushen Chahal 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 Prof. Rushen Chahal 27 Jensen Measure (cont’d) • The Jensen measure is generally out of favor because of statistical and theoretical problems Prof. Rushen Chahal 28 Performance Measurement in Practice • Academic issues • Industry issues Prof. Rushen Chahal 29 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 Prof. Rushen Chahal 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” • Practical performance measures typically involve a comparison of the fund’s performance with that of a benchmark Prof. Rushen Chahal 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 Prof. Rushen Chahal 32 Prof. Rushen Chahal 33 Performance Evaluation With Cash Deposits & Withdrawals • Introduction • Daily valuation method • Modified Bank Administration Institute (BAI) Method • An example • An approximate method Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 36 Daily Valuation Method (cont’d) • The daily valuation method solves for R: n Rdaily Si 1 i 1 MVEi where S MVBi Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 40 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 Prof. Rushen Chahal 41 Prof. Rushen Chahal 42 An Example (cont’d) • The daily valuation method returns a time- weighted return of 40.6% over the seven- months period – See next slide Prof. Rushen Chahal 43 Prof. Rushen Chahal 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) Prof. Rushen Chahal 45 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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% Prof. Rushen Chahal 48 Performance Evaluation When Options Are Used • Introduction • Incremental risk-adjusted return from options • Residual option spread • Final comments on performance evaluation with options Prof. Rushen Chahal 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 Prof. Rushen Chahal 50 Introduction (cont’d) • Consider two alternative methods when options are included in a portfolio: – Incremental risk-adjusted return (IRAR) – Residual option spread (ROS) Prof. Rushen Chahal 51 Incremental Risk-Adjusted Return from Options • Definition • An IRAR example • IRAR caveats Prof. Rushen Chahal 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 Prof. Rushen Chahal 53 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 Prof. Rushen Chahal 54 Definition (cont’d) Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 57 Prof. Rushen Chahal 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 Prof. Rushen Chahal 59 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 62 Residual Option Spread (cont’d) • The residual 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 64 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 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 Prof. Rushen Chahal 68