Learning Module 6: Analysis of Active Portfolio Management

Learning Module 6 Analysis of Active Portfolio Management EXAMPLE 1 Value Added and Country Equity Markets Consider the MSCI EAFE Index as the benchmark for an actively managed portfolio that includes allocations to individual countries, as given in the following exhibit. The portfolio (both benchmark and managed) weights are for the beginning of 2018. The portfolio manager actively changes country allocations but does not engage in security selection. Benchmark Weight Portfolio Weight 2018 Return Country 17% 16% -7.6% United Kingdom 25% 14% -9.0% Japan 11% 8% -3.5% France 9% 24% -15.8% Germany 38% 38% -0.1% Other Countries Source: Data from MSCI 1. Which countries have the largest overweight and largest underweight in the managed portfolio compared with the benchmark portfolio? What are the active weights for these two countries? Solution to 1: Germany has the largest overweight at 249+15%, and Japan has the largest underweight at 14-25-11%. 2. Using active weights and total returns, what was the value added of the managed portfolio over the benchmark portfolio in the calendar year 2018? Solution to 2: The value added is-0.01(-7.6) 0.11 (-9.0)-0.03(-3.5)+0.15(-15.8) -1.2%. Note that the "Other Countries" active weight is zero, so this asset does not contribute anything to the portfolio's active return. The value added can also be calculated using relative returns in Equation 3 with the same net result. Although this scaling convention is common in practice, multiplying monthly returns by a factor of 12 for averages and the square root of 12 for standard deviations ignores the multiplicative (i.e., compound) nature of returns over time. Simple multiplication factors (e.g., 250 and the square root of 250 for annualizing trading day returns) are only technically correct if the underlying returns are independent and continuously compounded or logarithmic. Similarly, annualized compound returns for the two values in the numerator of the Sharpe ratio (i.e., the portfolio return and the riskless rate) may be used instead of the annualized difference of arithmetic returns. The various methodologies produce slightly different results but should not be a serious problem as long as comparisons between different portfolios use the same approach. EXAMPLE 2 Adjusting Risk and Return Using the Sharpe Ratio Consider an investor choosing between two risky portfolios: a large-cap stock portfolio and a small-cap stock portfolio. Although forecasts about the future are subjective, suppose for simplicity that the investor expects that the future statistics will be those in the following table, with a risk-free rate of 2.3%. The forecasted 0.42 Sharpe ratio of the small-cap portfolio is higher than the 0.40 ratio of the large-cap portfolio, but suppose the investor does not want the high 19.2% volatility associated with the small-cap stocks. Large Cap Small Cap Expected return 8.2% 10.3% Expected volatility 14.6% 19.2% Sharpe ratio 0.40 0.42 1. What percentage of the portfolio would an investor need to hold in cash to reduce the risk of a portfolio invested in the small-cap portfolio and cash to the same risk level as that of the large-cap portfolio? Solution: We want to reduce the 19.2% volatility to 14.6% by adding cash. The weight of small-cap stocks in the combined portfolio must therefore be 14.6/19.2 = 76%, leaving a 24% weight in risk-free cash. With that amount of cash, the volatility of the combined portfolio will be 0.76(19.2%) 14.6%, the same as the large-cap portfolio. 2. Based on your answer to 1, calculate the Sharpe ratio of the small-cap plus cash portfolio. Solution: The Sharpe ratio of the combined portfolio is unaffected by the amount in cash, so it remains 0.42. 3. Compare the expected return of the small-cap plus cash portfolio with the expected return of the large-cap portfolio. Solution: The expected return of the combined portfolio is 0.76(10.3%) +0.24(2.3%) = 8.4%, 20 basis points (bps) higher than the 8.2% expected return on the large-cap portfolio, but with the same risk as the large-cap portfolio. To reconfirm, the Sharpe ratio of the combined portfolio is (8.4%-2.3%)/14.6% = 0.42, the same as the original 0.42 value, EXAMPLE 7 Ex Post Performance Consider an active management strategy that includes BR 100 investment decisions (e.g.. 100 individual stocks, whose active returns are uncorrelated and annual rebalancing), an expected information coefficient of IC 0.05, a transfer coefficient of TC = 0.80, and an annualized active risk of 64.0%. Thus, the expected value added according to the fundamental law is E(R) (TC) (IC) BR 0.80 0.05×100×4.0% 1.6% 1. Suppose that the realized information coefficient in a given period is -0.10, instead of the expected value of IC = 0.05. In the absence of constraint-induced noise, what would be the value-added that period? Solution: The value-added, without including constraint-induced noise (which has an expected value of zero) is E(RIC) (TC) (IC) BR 0.80×(-0.100×100×40% =-3.2%. In other words, conditional on the actual information coefficient, the investor should expect an active return that is negative because the realized information coefficient is negative. 2. Suppose that the actual return on the active portfolio was 2.6%. Given the -0.10 realized information coefficient, how much of the forecasted active return was offset by the noise component? Solution: The noise portion of the active return is the difference between the actual active return and the forecasted active return:-26-(-3.2) 0.6%. In other words, the noise component helped offset the negative value added from poor return forecasting. Of course, the constraint-induced noise component could just as easily have gone the other way, exacerbating the negative value added. Note that the negative realized active return of -2.6% is well within the range associated with the tracking error (active risk) of 4.0% per period. 3. What percentage of the performance variance (ie., tracking risk squared) in this strategy over time is attributed to variation in the realized information coefficient (Le, forecasting success), and what percentage of performance variance is attributed to constraint-induced noise? Solution: Given the transfer coefficient of TC 0.80, TC² 64% of the variation in performance over time is attributed to the success of the forecasting pro- cess, leaving 36% due to constraint-induced noise. EXAMPLE 10 Limitations of the Fundamental Law 1. Consider an active portfolio management strategy of selecting individual stocks in the S&P 500 on a monthly basis. The investor does a quick calculation of the fundamental law based on an information coefficient of IC = 0.05 and BR 12 x 500 6,000, giving an astounding information ratio of IR = 3.87. In other words, at an active portfolio risk of 3.0%, the expected active return would be 3.87(3.0) 11.6%. Provide at least two different explanations of why the information ratio in this example could be too high based on practical limitations of the fundamental law. Solution: Potential answers include the following: 1. Cross-sectional dependence: The active returns on the 500 stocks in the S&P 500 are probably correlated, so the number of independent monthly decisions is lower than 500. For example, the investor could be forecasting the outperformance of all the stocks in a given industrial sector and the underperformance of all the stocks in another sector. 2. Time-series dependence: The decisions on any particular stock may be correlated from month to month. For example, the forecasting process might be based on the earnings yield (reported EPS over price), which changes slowly over time. A stock that is forecasted to outperform in one month is likely to retain the outperformance forecast for several months in a row. 3. Uncertainty: Although an information coefficient of 0.05 appears to be modest, the basic form of the fundamental law does not account for uncertainty in the information coefficient or the likelihood that the information coefficient changes over tirne and could be different for different sets of stocks. 4. Constraints: An answer that involves accounting for such constraints as long only or turnover limits using a transfer coefficient is a weaker answer because the impact of constraints and the transfer coefficient is a well-known refinement of the fundamental law, even though it does not appear to be used in this example.