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Transcript

Chapter 19 Performance Evaluation Portfolio Construction, Management, & Protection, 5e, Robert A. Strong Copyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved. 1 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 2 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 E (U ) f E ( R), 2 3 A Lesson from History: The 1968 Bank Administration Institute Report 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 4 A Lesson from History: The 1968 Bank Administration Institute Report (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 greatly that indiscriminate comparisons among funds might reflect differences in these circumstances rather than in the ability of managers 5 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 6 A Lesson from a Few Mutual Funds (cont’d) Year 44 Wall Street Mutual Shares Year 44 Wall Street Mutual Shares 2010 184.1% 24.6% 2017 6.9 12.0 2011 46.5 63.1 2018 9.2 37.8 2012 16.5 13.2 2019 –58.7 14.3 2013 32.9 16.1 2020 –20.1 26.3 2014 71.4 39.3 2021 –16.3 16.9 2015 36.1 19.0 2022 –34.6 6.5 2016 -23.6 8.7 2023 19.3 30.7 Mean 19.3% 23.5% Change in net asset value, January 1 through December 31. 7 A Lesson from a Few Mutual Funds (cont’d) Ending Value ($) Mutual Fund Performance $200,000.00 $180,000.00 $160,000.00 $140,000.00 $120,000.00 $100,000.00 $80,000.00 $60,000.00 $40,000.00 $20,000.00 $- 44 Wall Street Mutual Shares Year 8 A Lesson from a Few Mutual Funds (cont’d) 44 Wall Street and Mutual Shares both had good returns over the 2010 to 2023 period. Mutual Shares clearly outperforms 44 Wall Street in terms of dollar returns at the end of 2023. 9 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 10 Why the Arithmetic Mean Is Often Misleading: A Review (cont’d) The proper measure of average investment return over time is the geometric mean: 1/ n GM Ri 1 i 1 where Ri the return relative in period i n 11 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 2010 to 2023 period 12 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? 13 Why the Arithmetic Mean Is Often Misleading: A Review (cont’d) Example Solution: The geometric mean is computed as follows: GM Ri i 1 n 1/ n 1 (0.60 )(1.50 )(1.00 )1/ 3 1 0.0345 3.45 % 14 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 15 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: $40, 000, 000 $250, 000 $40, 250, 000 12% $40, 250, 000 44% 12.10% 16 Traditional Performance Measures Sharpe Measure Treynor Measures Jensen Measure Performance Measurement in Practice 17 Sharpe and Treynor Measures The Sharpe and Treynor measures: Sharpe measure Treynor measure R Rf R Rf where R average return R f risk-free rate standard deviation of returns beta 18 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 19 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? 20 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% 21 Sharpe and Treynor Measures (cont’d) Example (cont’d) Solution (cont’d): Next, compute the Sharpe and Treynor measures: Sharpe measure Treynor measure R Rf R Rf 0.88% 0.29 3.07% 0.88% 0.73 1.20 22 Jensen Measure The Jensen measure stems directly from the CAPM: Rit R ft i Rmt R ft 23 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 24 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 25 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 26 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 27 28 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 29 Industry Issues (cont’d) Net selectivity measures the portion of the return from selectivity in excess of that provided by the “diversification” component 30 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 31 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 32 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: C3 C1 C2 cost 2 (1 R) (1 R) (1 R)3 33 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): time - weighted return (1 R1 )(1 R2 )(1 R3 )(1 R4 ) 1 34 Dollar-Weighted and Time-Weighted Rates of Return (cont’d) The time-weighted rate of return and the dollarweighted rate of return will be equal if there are no inflows or outflows from the portfolio 35 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 timeweighted 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 to or withdrawals 37 Daily Valuation Method (cont’d) The daily valuation method solves for R: n Rdaily Si 1 i 1 MVEi where S MVBi 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 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 40 Modified Bank Administration Institute (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 CD CD total number of days in the period Di number of days since the beginning of the period wi in which the cash flow occurred 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 42 Table19-4 Mutual Fund Account Value Date Description $ Amount Price Shares $7.00 Total Shares Value 1,080.011 $7,560.08 January 1 balance forward January 3 purchase 100 $7.00 14.286 1,094.297 $7,660.08 February 1 purchase 100 $7.91 12.642 1,106.939 $8,755.89 March 1 purchase 100 $7.84 12.755 1,119.694 $8,778.40 March 23 liquidation 5,000 $8.13 -615.006 504.688 $4,103.11 April 3 purchase 100 $8.34 11.900 516.678 $4,309.09 May 1 purchase 100 $9.00 11.111 527.789 $4,750.10 June 1 purchase 100 $9.74 10.267 538.056 $5,240.67 July 3 purchase 100 $9.24 10.823 548.879 $5,071.64 August 1 purchase 100 $9.84 10.163 559.042 $5,500.97 August 1 account value: $559.042 x $9.84 = $5,500.97 43 An Example (cont’d) The daily valuation method returns a timeweighted return of 40.6 percent over the seven-month period • See next slide 44 Table19-5 Date Sub Period Daily Valuation Worksheet MVB Cash Flow January 1 Ending Value MVE MVE/MVB $7,560.08 January 3 1 $7,560.08 100 $7,660.08 $7,560.08 1.00 February 1 2 $7,660.08 100 $8,755.89 $8,655.89 1.13 March 1 3 $8,755.89 100 $8,778.40 $8,678.40 0.991 March 23 4 $8,778.40 5,000 $4,103.11 $9,103.11 1.037 April 3 5 $4,103.11 100 $4,309.09 $4,209.09 1.026 May 1 6 $4,309.09 100 $4,750.10 $4,650.10 1.079 June 1 7 $4,750.10 100 $5,240.67 $5,140.67 1.082 July 3 8 $5,240.67 100 $5,071.64 $4,971.64 0.949 August 1 9 $5,071.64 100 $5,500.97 $5,400.97 1.065 Product of MVE/MVB values = 1.406; R = 40.6% 45 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) 46 47 An Approximate Method Proposed by the American Association of Individual Investors: P1 0.5(Net cash flow) R 1 P0 0.5(Net cash flow) where net cash flow is the sum of inflows and outflows 48 Intuition The approximation formula can be rearranged and written as: Net Cash Flow Net Cash Flow P1 (1 R) P0 (1 R) 2 2 The initial value is compounded forward at the interest rate R because P0 is a "time-zero" value. The rest of the cash flows is split into two: half is modeled as being paid at the beginning (and thus needs to be compounded forward at the rate R), while the other half is modeled as being paid at the end and thus does not need any compounding since its "spot on the timeline" is the same as the final portfolio value P1. 49 An Approximate Method (cont’d) Using the approximate method in Table 17- 6: P1 0.5(Net cash flow) R 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% 50 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) 51 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 52 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 53 Incremental Risk-Adjusted Return from Options (cont’d) 54 Incremental Risk-Adjusted Return from Options (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 55 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 56 Table19-9 Modified BAI Method Worksheet (Using Interest Rate of 42.1%) Unoptioned Portfolio Long Stock Plus Option Premiums 0 $200,000 $214,112 $14,112 $200,000 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 Week Less Short Optionsa Equals Optioned Portfolio 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. 57 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 11.3 percent per year to the overall performance of the fund 58 Table19-10 IRAR Worksheet Unoptioned Portfolio Relative Return Optioned Portfolio Relative Return $200,000 0.9500 $200,000 $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 0.9800 $202,310 0.9926 $193,474 0.9700 $199,366 0.9854 $201,213 1.0400 $204,795 1.0272 $207,249 1.0300 $208,707 1.0191 Mean = 1.007 Mean = 1.0063 Standard Deviation = 0.0350 Standard Deviation = 0.0206 Risk-free rate = 8%/year = 0.15%/week SHu = (0.0057 - 0.0015) ÷ 0.0350 = 0.1200 SHo = (0.0063 - 0.0015) ÷ 0.0206 = 0.2330 IRAR = (0.2330 – 0.1200) x 0.0206 = 0.0023 per week, or about 12% per year 59 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 60 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 61 Residual Option Spread (cont’d) The residual option spread (ROS) calculation: n n t 1 t 1 ROS Got Gut where Gt Vt / Vt 1 Vt value of portfolio in Period t 62 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 63 Residual Options Spread Worksheet Unoptioned Portfolio • (0.95)(1.03)(1.04)(0.98)(0.97)(1.04)(1.03) = 1.03625 Optioned Portfolio • (0.9727)(1.0217)(1.0255)(0.9926)(0.9854)(1.0272)(1.0191) = 1.04351 ROS = 1.04351 – 1.03625 = 0.00726 Given an initial investment of $200,000, the ROS translates into a dollar differential of $200,000 x 0.00726 = $1,452. 64 Final Comments IRAR and ROS both focus on whether an optioned portfolio outperforms an unoptioned portfolio • Can overlook subjective considerations such as portfolio insurance 65