Chapter 19 Performance Evaluation

- 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

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

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

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

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

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

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

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

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%

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

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

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

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

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

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?

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

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

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:

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

Sharpe and Treynor Measures The Sharpe and Treynor measures:

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

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?

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

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

Jensen Measure The Jensen measure stems directly from the CAPM:

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

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

Performance Measurement in Practice Academic issues Industry issues

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

Definition (cont’d)

Definition (cont’d) The IRAR calculation:

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

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

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

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

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

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

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

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

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