P2.T8. Risk Management & Investment Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com
Bodie, Chapter 24: Portfolio Performance Evaluation DIFFERENTIATE BETWEEN THE TIME-WEIGHTED AND DOLLAR-WEIGHTED RETURNS OF A PORTFOLIO AND THEIR APPROPRIATE USES.... 3 DESCRIBE AND DISTINGUISH BETWEEN RISK-ADJUSTED PERFORMANCE MEASURES, SUCH AS SHARPE S MEASURE, TREYNOR S MEASURE, JENSEN S MEASURE (JENSEN S ALPHA), AND INFORMATION RATIO.... 5 DESCRIBE THE USES FOR THE MODIGLIANI-SQUARED AND TREYNOR S MEASURE IN COMPARING TWO PORTFOLIOS, AND THE GRAPHICAL REPRESENTATION OF THESE MEASURES.... 6 DETERMINE THE STATISTICAL SIGNIFICANCE OF A PERFORMANCE MEASURE USING STANDARD ERROR AND THE T-STATISTIC.... 7 2
Bodie, Chapter 24: Portfolio Performance Evaluation Differentiate between the time-weighted and dollar-weighted returns of a portfolio and their appropriate uses. Describe and distinguish between risk-adjusted performance measures, such as Sharpe s measure, Treynor s measure, Jensen s measure (Jensen s alpha), and information ratio. Describe the uses for the Modigliani-squared and Treynor s measure in comparing two portfolios, and the graphical representation of these measures. Determine the statistical significance of a performance measure using standard error and the t-statistic. Explain the difficulties in measuring the performances of hedge funds. Explain how changes in portfolio risk levels can affect the use of the Sharpe ratio to measure performance. Describe techniques to measure the market timing ability of fund managers with a regression and with a call option model and compute return due to market timing. Describe style analysis. Describe and apply performance attribution procedures, including the asset allocation decision, sector and security selection decision and the aggregate contribution. Differentiate between the time-weighted and dollar-weighted returns of a portfolio and their appropriate uses. Time-weighted Return The time-weighted return (TWR), also known as the geometric average, is the preferred industry standard for measuring the portfolio performance. The return value is not affected by cash inflows or outflows over different time periods and depends only on the length of investment time horizon. Analysts compare TWR with a benchmark, such as S&P500 Index to analyze whether the portfolio has out-performed or under-performed the benchmark. Time-weighted return is calculated using the following formula: = [(1 + )(1 + ) (1 + )]. 1 = time weight (geometric) return, = single period returns 3
Dollar-weighted Return Dollar weighted return (DWR), also known as Internal rate of Return (IRR), calculates the rate of return at which the present value of cash inflows equals the present value of cash outflows. This return is both time-weighted and dollar-weighted, i.e. cash deposits and withdrawals at different periods affect the return value. DWR is most suitable to show clients the performance of their own funds. NOTE: For investments over a single period without any interim cash outflows or inflows, both TWR and DWR are exactly the same. Example: A client invests $100,000 in a fund at. By the end of the year (at ), the value of his investment rises to $105,000. At, the client invests $95,000 more in the fund, taking the total value of amount invested to $200,000. By the end of second year (at ), the portfolio value rises to $220,000. What is the time-weighted and dollar-weighted return on the client s portfolio over the 2-year horizon? At the end of year one, both TWR and DWR for will be 5%. TWR DWR = (.. ) =. % $, + $, ( + ) = $, ( + ) =. % The annualized DWR is higher than TWR because the fund manager had more money to invest in the second year in which the portfolio generated a higher return. 4
Describe and distinguish between risk-adjusted performance measures, such as Sharpe s measure, Treynor s measure, Jensen s measure (Jensen s alpha), and information ratio. Evaluating standalone performance based on average returns is not very useful when it comes to performance measurement and evaluation, and returns must be adjusted for risk before they can be compared meaningfully. Four commonly used risk-adjusted portfolio performance evaluation measures are: 1. Sharpe ratio: This ratio is calculated by dividing the average excess returns (returns above the risk-free rate) by the standard deviation of returns. It s a measure of how much extra reward for every unit of additional risk (reward to volatility trade-off). = average return on portfolio = average risk free rate h = ( ) = standard deviation of portfolio returns 2. Treynor ratio: This is similar to Sharpe ratio and is calculated by dividing the average excess returns (returns above the risk-free rate) by the portfolio s systematic risk (beta). Like sharpe s measure, the higher the treynor s measure, the better the portfolio performance. = systematic risk of portfolio = ( ) 3. Jensen s alpha: This measure calculates the returns of a portfolio that are in excess of the theoretical expected returns as estimated by CAPM, after accounting for the risk-free rate, average market return and the portfolio beta. A positive Alpha indicates that the portfolio has performed better than its projected risk-adjusted return. = Jensen s alpha = average market return = [ + ] 4. Information ratio: is calculated by dividing the portfolio s alpha ( ) by the portfolio s tracking error (nonsystematic risk). The information ratio measures abnormal returns per unit of risk that can, in principle, be eliminated through diversification by holding a portfolio replicating the market index. ( ) = tracking error = ( ) 5
Example: Using the following data for Portfolio P and the Market Index M, and given a t-bill rate of 5%, calculate Sharpe ratio, Treynor ratio, Jensen s alpha and information ratio for the portfolio Portfolio P Market M Average return (r) 25% 20% Beta 1.2 1.0 Standard deviation 18.0% 15.0% Tracking error 6.0% 0 h = (0.25 0.05) 0.18 = 1.11 = (0.25 0.05) 1.2 = 0.17 or 17% h = 0.25 [0.05 + 1.2 (0.20 0.05)] = 0.02 or 2% = 0.02 0.06 = 33.33 Describe the uses for the Modigliani-squared and Treynor s measure in comparing two portfolios, and the graphical representation of these measures. Modigliani-squared (M 2 ) can be simply defined as the excess returns earned by the adjusted portfolio. = = return on adjusted portfolio The Modigliani-Squared (M2) measure is closely related to the Sharpe measure. It requires adjusting the portfolio such that its standard deviation is identical to that of the market portfolio. When the portfolio has a standard deviation that is higher than that of the Market index, amount invested in the portfolio is decreased to match the standard deviation of the Market index. Excess amount is then invested into T-bills. On the other hand, if the portfolio standard deviation is lower than that of the Market index, funds are borrowed at the risk free rate to make additional investment in the portfolio up to the amount where the standard deviation of the portfolio matches the standard deviation of the Market index. Uses of Modigliani-squared: Compared to the Sharpe ratio M2 is easier to interpret when used to rank portfolio performance Adjusted returns in M2 can be compared to benchmark returns to gauge the portfolio s performance 6
Uses of Treynor s measure: The Treynor ratio is a more suitable measure to compare performance if the portfolios being compared are sub-portfolios of a larger, diversified investment fund. The ratio helps weigh mean excess return of a sub-portfolio against its systematic risk, rather than against total risk, to evaluate contribution to performance Determine the statistical significance of a performance measure using standard error and the t-statistic. The standard error of the alpha estimate in a security characteristic line (SCL) is simply the sample estimate of nonsystematic risk divided by the root of number of observations. ( ) = ( ) The t-statistic for the alpha estimate is: ( ) = ( ) Example: Shawn Douglas is evaluating the performance of a portfolio that has a measured alpha of 0.4% with a standard error of 3% using 240 monthly observations. To assess whether this alpha is a result of the portfolio manager s ability at a 5% significance level, Shawn conducts a t-test: ( ) = 0.4 240 3 = 2.07 Since the calculated t-statistic of 2.07 is higher than the t-value of 1.96 (that coincides with a 5% significance level), Shawn can conclude that the alpha generated by the portfolio manager is a result of her superior return generation ability. 7