Topic 10 (Ch. 24) Portfolio Performance Evaluation

1. Topic 10 (Ch. 24) Portfolio Performance Evaluation • Measuring investment returns • The conventional theory of performance evaluation • Market timing • Performance attribution procedures

2.  Measuring Investment Returns • One period:  Find the rate of return (r) that equates the present value of all cash flows from the investment with the initial outlay.

3. Example: Consider a stock paying a dividend of $2 annually that currently sells for $50. You purchase the stock today and collect the $2 dividend, and then you sell the stock for $53 at year-end.

4. Multiperiod: • Arithmetic versus geometric averages: • Arithmetic averages:

5. Geometric averages: The compound average growth rate, rG, is calculated as the solution to the following equation: In general: where rtis the return in each time period.

6. Geometric averages never exceed arithmetic ones: Consider a stock that doubles in price in period 1 (r1 = 100%) and halves in price in period 2 (r2= -50%). The arithmetic average is: rA = [100 + (-50)]/2 = 25% The geometric average is: rG = [(1 + 1)(1 - 0.5)]1/2 – 1 = 0

7. The effect of the -50% return in period 2 fully offsets the 100% return in period 1 in the calculation of the geometric average, resulting in an average return of zero. This is not true of the arithmetic average. In general, the bad returns have a greater influence on the averaging process in the geometric technique. Therefore, geometric averages are lower.

8. Generally, the geometric average is preferable for calculation of historical returns (i.e. measure of past performance), whereas the arithmetic average is more appropriate for forecasting future returns: Example 1: Consider a stock that will either double in value (r = 100%) with probability of 0.5, or halve in value (r = -50%) with probability 0.5.

9. Suppose that the stock’s performance over a 2-year period is characteristic of the probability distribution, doubling in one year and halving in the other. The stock’s price ends up exactly where it started, and the geometric average annual return is zero: which confirms that a zero year-by-year return would have replicated the total return earned on the stock.

10. However, the expected annual future rate of return on the stock is not zero. It is the arithmetic average of 100% and -50%: (100 - 50)/2 = 25%. There are two equally likely outcomes per dollar invested: either a gain of $1 (when r = 100%) or a loss of $0.50 (when r = -50%). The expected profit is ($1 - $0.50)/2 = $0.25, for a 25% expected rate of return. The profit in the good year more than offsets the loss in the bad year, despite the fact that the geometric return is zero. The arithmetic average return thus provides the best guide to expected future returns.

11. Example 2: Consider all the possible outcomes over a two-year period:

12. The expected final value of each dollar invested is: (4 + 1 + 1 + 0.25)/4 = $1.5625 for two years, again indicating an average rate of return of 25% per year, equal to the arithmetic average. Note that an investment yielding 25% per year with certainty will yield the same final compounded value as the expected final value of this investment: (1 + 0.25)2 = 1.5625.

13. The arithmetic average return on the stock is: [300 + 0 + 0 + (-75)]/4 = 56.25% per two years, for an effective annual return of 25% since: (1 + 25%)(1 + 25%) – 1 = 56.25%. In contrast, the geometric mean return is zero since: [(1 + 3)(1 + 0)(1 + 0)(1 – 0.75)]1/4 = 1.0 Again, the arithmetic average is the better guide to future performance.

14. Dollar-weighted returns versus time-weighted returns: Example:

15. Dollar-weighted returns: Using the discounted cash flow (DCF) approach, we can solve for the average return over the two years by equating the present values of the cash inflows and outflows:

16. This value is called the internal rate of return, or the dollar-weighted rate of return on the investment. It is “dollar weighted” because the stock’s performance in the second year, when two shares of stock are held, has a greater influence on the average overall return than the first-year return, when only one share is held.

17. Time-weighted returns: Ignore the number of shares of stock held in each period. The stock return in the 1st year: The stock return in the 2nd year:

18.  The time-weighted (geometric average) return is: This average return considers only the period-by-period returns without regard to the amounts invested in the stock in each period. Note that the dollar-weighted average is less than the time-weighted average in this example because the return in the second year, when more money is invested, is lower.

19. Note: For an investor that has control over contributions to the investment portfolio, the dollar-weighted return is more comprehensive measure. Time-weighted returns are more likely appropriate to judge the performance of an investor that does not control the timing or the amount of contributions.

20.  The Conventional Theory of Performance Evaluation • Several risk-adjusted performance measures: • Sharpe’s measure: Sharpe’s measure divides average portfolio excess return over the sample period by the standard deviation of returns over that period. It measures the reward to (total) volatility trade-off. Note: The risk-free rate may not be constant over the measurement period, so we are taking a sample average, just as we do for rP.

21. Treynor’s measure: Like Sharpe’s, Treynor’s measure gives excess return per unit of risk, but it uses systematic risk instead of total risk. • Jensen’s measure: Jensen’s measure is the average return on the portfolio over and above that predicted by the CAPM, given the portfolio’s beta and the average market return. Jensen’s measure is the portfolio’s alpha value.

22. Information ratio: The information ratio divides the alpha of the portfolio by the nonsystematic risk of the portfolio. It measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio. Note: Each measure has some appeal. But each does not necessarily provide consistent assessments of performance, since the risk measures used to adjust returns differ substantially.

23. Example: Consider the following data for a particular sample period: The T-bill rate during the period was 6%.

24. Sharpe’s measure: Treynor’s measure:

25. Jensen’s measure: Information ratio:

26. The M2measure of performance • While the Sharpe ratio can be used to rank portfolio performance, its numerical value is not easy to interpret. We have found that SP = 0.69 and SM = 0.73. This suggests that portfolio P under-performed the market index. But is a difference of 0.04 in the Sharpe ratio economically meaningful? We often compare rates of return, but these ratios are difficult to interpret.

27. To compute the M2measure, we imagine that a managed portfolio, P, is mixed with a position in T-bills so that the complete, or “adjusted,” portfolio (P*) matches the volatility of a market index (such as the S&P500). Because the market index and portfolio P* have the same standard deviation, we may compare their performance simply by comparing returns. This is the M2measure:

28. Example: P has a standard deviation of 42% versus a market standard deviation of 30%. The adjusted portfolio P* would be formed by mixing portfolio P and T-bills and : weight in P: 30/42 = 0.714 weight in T-bills: (1 - 0.714) = 0.286. The return on this portfolio P* would be: (0.286  6%) + (0.714  35%) =26.7% Thus, portfolio P has an M2 measure: 26.7 – 28 = -1.3%.

30. We move down the capital allocation line corresponding to portfolio P (by mixing P with T-bills) until we reduce the standard deviation of the adjusted portfolio to match that of the market index. The M2measure is then the vertical distance (i.e., the difference in expected returns) between portfolios P* and M. P will have a negative M2 measure when its capital allocation line is less steep than the capital market line (i.e., when its Sharpe ratio is less than that of the market index).

31. Appropriate performance measures in 3 scenarios Suppose that Jane constructs a portfolio (P) and holds it for a considerable period of time. She makes no changes in portfolio composition during the period. In addition, suppose that the daily rates of return on all securities have constant means, variances, and covariances. This assures that the portfolio rate of return also has a constant mean and variance. We want to evaluate the performance of Jane’s portfolio.

32. Jane's portfolio P represents her entire risky investment fund: We need to ascertain only whether Jane’s portfolio has the highest Sharpe measure. We can proceed in 3 steps: • Assume that past security performance is representative of expected performance, meaning that realized security returns over Jane’s holding period exhibit averages and covariances similar to those that Jane had anticipated.

33. Determine the benchmark (alternative) portfolio that Jane would have held if she had chosen a passive strategy, such as the S&P 500. • Compare Jane’s Sharpe measure to that of the best portfolio. In sum: When Jane’s portfolio represents her entire investment fund, the benchmark is the market index or another specific portfolio. The performance criterion is the Sharpe measure of the actual portfolio versus the benchmark.

34. Jane’s portfolio P is an active portfolio and is mixed with the market-index portfolio M: When the two portfolios are mixed optimally, the square of the Sharpe measure of the complete portfolio, C, is given by: where Pis the abnormal return of the active portfolio relative to the market-index, and (eP)is the diversifiable risk.

35. The ratio P/(eP)is thus the correct performance measure for P in this case, since it gives the improvement in the Sharpe measure of the overall portfolio. To see this result intuitively, recall the single-index model: If P is fairly priced, then P = 0, and ePis just diversifiable risk that can be avoided.

36. However, if P is mispriced, P no longer equals zero. Instead, it represents the expected abnormal return. Holding P in addition to the market portfolio thus brings a reward of P against the nonsystematic risk voluntarily incurred, (eP). Therefore, the ratio of P/(eP)is the natural benefit-to-cost ratio for portfolio P. This performance measurement is the information ratio.

37. Jane’s choice portfolio P is one of many portfolios combined into a large investment fund:  The Treynor measure is the appropriate criterion. E.g.:

39. Note: We plot P and Q in the expected return-beta (rather than the expected return-standard deviation) plane, because we assume that P and Q are two of many sub-portfolios in the fund, and thus that nonsystematic risk will be largely diversified away, leaving beta as the appropriate risk measure.

40. Suppose portfolio Q can be mixed with T-bills. Specifically, if we invest wQin Q and wF = 1 -wQin T-bills, the resulting portfolio, Q*, will have alpha and beta values proportional to Q’s alpha and beta scaled down by wQ: Thus, all portfolios Q* generated from mixing Q with T-bills plot on a straight line from the origin through Q. We call it the T-line for the Treynor measure, which is the slope of this line.

41. P has a steeper T-line. Despite its lower alpha, P is a better portfolio after all. For any given beta, a mixture of P with T-bills will give a better alpha than a mixture of Q with T-bills.

42. Suppose that we choose to mix Q with T-bills to create a portfolio Q* with a beta equal to that of P. We find the necessary proportion by solving for wQ: Portfolio Q* has an alpha of: which is less than that of P.

43. In other words, the slope of the T-line is the appropriate performance criterion for this case. The slope of the T-line for P, denoted by TP,is: Treynor’s performance measure is appealing because when an asset is part of a large investment portfolio, one should weigh its mean excess return against its systematic risk rather than against total risk to evaluate contribution to performance.

44. An example: Excess returns for portfolios P &Q and the benchmark M over 12 months:

45. Performance statistics:

46. Portfolio Q is more aggressive than P, in the sense that its beta is significantly higher (1.40 vs. 0.69). On the other hand, from its residual standard deviation P appears better diversified (1.95% vs. 8.98%). Both portfolios outperformed the benchmark market index, as is evident from their larger Sharpe measures (and thus positive M2)and their positive alphas.

47. Which portfolio is more attractive based on reported performance? • If P or Q represents the entire investment fund, Q would be preferable on the basis of its higher Sharpe measure (0.51 vs. 0.45) and better M2(2.69% vs. 2.19%). • As an active portfolio to be mixed with the market index, P is preferable to Q, as is evident from its information ratio (0.84 vs. 0.59).

48. When P and Q are competing for a role as one of a number of subportfolios, Q dominates again because its Treynor measure is higher (5.40 versus 4.00). Thus, the example illustrates that the right way to evaluate a portfolio depends in large part how the portfolio fits into the investor’s overall wealth.

49. Relationships among the various performance measures • The relation between Treynor’s measure and Jensen’s :

50. The relation between Sharpe’s measure and Jensen’s :