Performance Evaluation Measures, I.

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2. Performance Evaluation Measures, I. The raw returnon a portfolio, RP, is simply the total percentage return on a portfolio. The raw returnis a naive performance evaluation measure because: The raw return has no adjustment for risk. The raw return is not compared to any benchmark, or standard. Therefore, the usefulness of the raw returnon a portfolio is limited.

3. Performance Evaluation Measures, II. The Sharpe Ratio The Sharpe ratio is a reward-to-risk ratio that focuses on total risk. It is computed as a portfolio’s risk premium divided by the standard deviation of the portfolio’s return.

4. Performance Evaluation Measures, III. The Treynor Ratio The Treynor ratio is a reward-to-risk ratio that looks at systematic risk only. It is computed as a portfolio’s risk premium divided by the portfolio’s beta coefficient.

5. Performance Evaluation Measures, IV. Jensen’s Alpha Jensen’s alpha is the excess return above or below the security market line. It can be interpreted as a measure of by how much the portfolio “beat the market.” It is computed as the raw portfolio return less the expected portfolio return as predicted by the CAPM. “Extra” Return Actual return CAPM Risk-Adjusted ‘Predicted’ Return

6. Jensen’s Alpha

7. Another Method to Calculate Alpha, I. Recall that the characteristic line graphs the relationship between the return of an investment (on the y-axis) and the return of the market or benchmark (on the x- axis). The slope of this line represents the investment’s beta. With only a slight modification, we can extend this approach to calculate an investment’s alpha. Consider the following equation,

8. Another Method to Calculate Alpha, II. Suppose an actively managed fund took on the same amount of risk as the market (i.e., a beta of 1), but the fund earned exactly 2% more than the market every period? If we graph this hypothetical situation, the slope (i.e., beta) is still 1, but the x intercept is now 2. This intercept value is the fund’s alpha.

9. Estimating Alpha Using Regression, I. Suppose we want to estimate a fund’s alpha. Unlike the previous situation, we do know whether its returns are consistently higher (or lower) than the market by a fixed amount. In this case, we apply a simple linear regression. X-variable: the excess return of the market Y-Variable: the excess return of the investment The intercept of this estimated equation is the fund’s alpha. Consider the following Spreadsheet Analysis slide. In the spreadsheet example, the intercept estimate is -.0167, or -1.67 percent per year. The beta of this security is .96, which is the coefficient estimate on the x variable. How would you characterize the performance of this fund?

10. Estimating Alpha Using Regression, II.

11. The Information Ratio Suppose a mutual fund reports a positive alpha. How do we know whether this alpha is statistically significantly different from zero or simply represents a result of random chance? One way: Evaluate the significance level of the alpha estimate using a regression. Another way: Calculate the fund’s information ratio. The information ratio is a fund’s alpha divided by its tracking error. Tracking error measures the volatility of the fund’s returns relative to its benchmark. Consider the fund we evaluated in the Spreadsheet Analysis in the previous slide. Over the five years, the excess return differences are 2%, 4%, -20%, 14%, and -10%. The tracking error is the standard deviation of these return differences, which is 13.2%. The information ratio for this fund is -1.67% divided by 13.2%, which is -0.13. The information ratio allows us to compare investments that have the same alpha: A higher information ratio means a lower tracking error risk.

12. R-Squared, I. Recall that correlation measures how returns for a particular security move relative to returns for another security. Correlation also plays a key role in performance measurement. Suppose a particular fund has had a large alpha over the past three years. All else equal, we might say that this fund is a good choice. Suppose this fund is a sector-based fund that invests only in gold. It is possible that the large alpha is simply due to a run-up in gold prices over the period and is not reflective of good management or future potential. To evaluate this type of risk we can calculate R-squared, which is simply the squared correlation of the fund to the market. For example, if this fund’s correlation with the market was .60, then the R-squared value is .36.

13. R-Squared, II. R-squared represents the percentage of the fund’s movement that can be explained by movements in the market. Because correlation ranges only from -1 to +1, R-squared values will range from 0 to 100 percent. An R-squared of 100 indicates that all movements in the security are driven by the market, indicating a correlation of -1 or +1. A high R-squared value (say greater than .80) might suggest that the performance measures (such as alpha) are more representative of potential longer term performance. Excel reports correlation as “Multiple R” and R-squared as “R Square.” Verify that the squared value for “Multiple R” equals “R Square” in the previous Spreadsheet Analysis slide. How do you interpret an R-squared of 0.549?

14. Investment Performance Measurement on the Web • Consider the information below for the Fidelity Low-Priced Stock Fund, a small-cap value fund. • We obtained the numbers from www.morningstar.com by entering the fund’s ticker symbol (FLPSX) and following the “Ratings and Risk” link. • How do you judge the manager’s of this fund? For this fund: • Beta is 1.10, so the degree of market risk is above average. • Alpha is 3.51 percent, which could indicate superior past performance. • Standard deviation is 22.90 percent. • Mean of -1.08 percent is the geometric return of the fund over the past three years. • Sharpe ratio is -.14. • R-squared is 91, which means that 91% of its returns are driven by the market’s return.

15. Comparing Three Well-Known Performance Measures: Be Careful

16. Comparing Performance Measures, I. Because the performance rankings can be substantially different, which performance measure should we use? Sharpe ratio: Appropriate for the evaluation of an entire portfolio. Penalizes a portfolio for being undiversified, because, in general, total risk  systematic risk only for relatively well-diversified portfolios.

17. Comparing Performance Measures, II. Treynor ratio and Jensen’s alpha: Appropriate for the evaluation of securities or portfolios for possible inclusion into an existing portfolio. Both are similar, the only difference is that the Treynor ratio standardizes returns, including excess returns, relative to beta. Both require a beta estimate (and betas from different sources can differ a lot).

18. Global Investment Performance Standards Comparing investments can be difficult: various performance measures can provide different rankings. Even if the various performance measures provide a similar ranking, we could still make an incorrect choice in selecting an investment manager. How is this possible? Performance measures often rely on self-reported returns (many funds do not have public portfolios). The CFA Institute developed the Global Investment Performance Standards (GIPS). Goal: Provide a consistent method to report portfolio performance to prospective (and current) clients By standardizing the way portfolio performance is reported, GIPS provide investors with the ability to make comparisons across managers. Firms are not required by law to comply with GIPS, compliance is voluntary. Firms that do comply, however, are recognized by the CFA Institute.

19. Sharpe-Optimal Portfolios, I. Allocating funds to achieve the highest possible Sharpe ratio is said to be Sharpe-optimal. To find the Sharpe-optimal portfolio, first look at the plot of the possible risk-return possibilities, i.e., the investment opportunity set. Expected Return × × × × × × × × × × × × × × × × Standard deviation

20. Sharpe-Optimal Portfolios, II. The slope of a straight line drawn from the risk-free rate to where the portfolio plots gives the Sharpe ratio for that portfolio. A Expected Return × Rf Standard deviation • The portfolio with the steepestslope is the Sharpe-optimal portfolio.

21. Sharpe-Optimal Portfolios, III.

22. Example: Solving for a Sharpe-Optimal Portfolio From a previous chapter, we know that for a 2-asset portfolio: So, now our job is to choose the weight in asset S that maximizes the Sharpe Ratio. We could use calculus to do this, or we could use Excel.

23. Example: Using Excel to Solve for the Sharpe-Optimal Portfolio Suppose we enter the data (highlighted in yellow) into a spreadsheet. We “guess” that Xs = 0.25 is a “good” portfolio. Using formulas for portfolio return and standard deviation, we compute Expected Return, Standard Deviation, and a Sharpe Ratio:

24. Example: Using Excel to Solve for the Sharpe-Optimal Portfolio, Cont. Now, we let Excel solve for the weight in portfolio S that maximizes the Sharpe Ratio. We use the Solver, found under Tools. Well, the “guess” of 0.25 was a tad low….

25. Investment Risk Management Investment risk management concerns a money manager’s control over investment risks, usually with respect to potential short-run losses. We will focus on what is known as the Value-at-Risk (VaR) approach.

26. Value-at-Risk (VaR) Value-at-Risk (VaR) is a technique of assessing risk by stating the probability of a loss that a portfolio may experience within a fixed time horizon. If the returns on an investment follow a normal distribution, we can state the probability that a portfolio’s return will be within a certain range, if we have the mean and standard deviation of the portfolio’s return.

27. Example: VaR Calculation, I. Suppose you own an S&P 500 index fund. What is the probability of a return of -7% or worse in a particular year? That is, one year from now, what is the probability that your portfolio value is down by 7 percent (or more)?

28. Example: VaR Calculation, II. First, the historic average return on the S&P index is about 13%, with a standard deviation of about 20%. A return of -7 percent is exactly one standard deviation below the average, or mean (i.e., 13 – 20 = -7). We know the odds of being within one standard deviation of the mean are about 2/3, or 0.67. In this example, being within one standard deviation of the mean is another way of saying that: Prob(13 – 20  RS&P500  13 + 20)  0.67 or Prob (–7  RS&P500  33)  0.67

29. Example: VaR Calculation, III. That is, the probability of having an S&P 500 return between -7% and 33% is 0.67. So, the return will be outside this range one-third of the time. When the return is outside this range, half the time it will be above the range, and half the time below the range. Therefore, we can say: Prob (RS&P500  –7)  1/6 or 0.17

30. Example: A Multiple Year VaR, I. Once again, you own an S&P 500 index fund. Now, you want to know the probability of a loss of 30% or more over the next two years. As you know, when calculating VaR, you use the mean and the standard deviation. To make life easy on ourselves, let’s use the one year mean (13%) and standard deviation (20%) from the previous example.

31. Example: A Multiple Year VaR, II. Calculating the two-year average return is easy, because means are additive.That is, the two-year average return is: 13 + 13 = 26% Standard deviations, however, are not additive. Fortunately, variances are additive, and we know that the variance is the squared standard deviation. The one-year variance is 20 x 20 = 400. The two-year variance is: 400 + 400 = 800. Therefore, the 2-year standard deviation is the square root of 800, or about 28.28%.

32. Example: A Multiple Year VaR, III. The probability of being within two standard deviations is about 0.95. Armed with our two-year mean and two-year standard deviation, we can make the probability statement: Prob(26 – 228  RS&P500  26 + 228)  .95 orProb (–30 RS&P500 82)  .95 The return will be outside this range 5 percent of the time. When the return is outside this range, half the time it will be above the range, and half the time below the range. So, Prob (RS&P500  –30)  2.5%.

33. Computing Other VaRs. In general, for a portfolio, if T is the number of years, Using the procedure from before, we make make probability statements. Three very useful ones are: