Albert Lee Chun Portfolio Management Portfolio Management 3-228-07 Albert Lee Chun Lecture 11 Evaluation of Portfolio Performance 2 Dec 2008 Albert Lee Chun Portfolio Management Introduction As portfolio managers, how can we evaluate the performance of our portfolio? We know that there are 2 major requirements of a portfolio managers performance: 1. The ability to derive above-average returns conditioned on risk taken, either through superior market timing or superior security selection. 2. The ability to diversify the portfolio and eliminate non-systematic risk, relative to a benchmark portfolio. wealth Albert Lee Chun Portfolio Management Today Performance Measurement Risk Adjusted Performance Measures Measures of Sharpe, Treynor and Jensen Measures of Skill and Timing Attribution de performance Concept de mesures ajustes pour le risque Mesures de Sharpe, Treynor et Jensen Mesure des habilits de timing wealth Albert Lee Chun Portfolio Management Averaging Returns Arithmetic Mean: == nttnrr1Geometric Mean: 1 ) 1 (/ 11 ((

+ = [=nnt tr rExample: (.10 + .0566) / 2 = 7.83% [ (1.1) (1.0566) ]1/2 - 1 = 7.808% Example: 17-3 wealth Albert Lee Chun Portfolio Management Geometric Average The arithmetic average provides unbiased estimates of the expected return of the stock. Use this to forecast returns in the next period. The fixed rate of return over the sample period that would yield the terminal value is know as the geometric average. The geometric average is less than the arithmetic average and this difference increases with the volatility of returns. The geometric average is also called the time-weighted average (as opposed to the dollar weighted average), because it puts equal weights on each return. 17-4 wealth Albert Lee Chun Portfolio Management Dollar- and Time-Weighted Returns Dollar-weighted returns Internal rate of return. Returns are weighted by the amount invested in each stock. Time-weighted returns Not weighted by investment amount. Equal weighting Geometric average 17-5 wealth Albert Lee Chun Portfolio Management Example: Multiperiod Returns Period Action 0 Purchase 1 share of Eggberts Egg Co. at $50 1 Purchase 1 share of Eggberts Egg Co. at $53 Eggbert pays a dividend of $2 per share 2 Eggbert pays a dividend of $2 per share Sell both shares for $108 17-6 wealth Albert Lee Chun Portfolio Management Period Cash Flow 0 -50 share purchase 1 +2 dividend -53 share purchase 2 +4 dividend + 108 shares sold % 117 . 7) 1 (112) 1 (51502 1= +++= r r rInternal Rate of Return: Dollar-Weighted Return Dollar Weighted: The stocks performance in the second year, when we own 2 shares, has a greater influence on the overall return. wealth Albert Lee Chun Portfolio Management Time-Weighted Return % 66 . 5532 53 54% 10502 50 5321=+ ==+ =rr[ (1.1) (1.0566) ]1/2 - 1 = 7.808% 17-8 Time Weighted: Each return has equal weight in the geometric average. Geometric Mean: wealth Albert Lee Chun Portfolio Management Performance Measurement wealth Albert Lee Chun Portfolio Management Early Performance Measure Techniques Portfolio evaluation before 1960 Once upon a time, investors evaluated a portfolios performance based purely on the basis of the rate of return. Research in the 1960s showed investors how to quantify and measure risk. Grouped portfolios into similar risk classes and compared rates of return within risk classes. wealth Albert Lee Chun Portfolio Management Peer Group Comparisons This is the most common manner of evaluating portfolio managers. Collects returns of a representative universe of investors over a period of time and displays them in a box plot format. Example: US Equity with Cash relative to peer universe of US domestic equity managers. Issue: There is no explicit adjustment for risk. Risk is only considered implicitly. wealth Albert Lee Chun Portfolio Management wealth Albert Lee Chun Portfolio Management Treynor Portfolio Performance Measure wealth Albert Lee Chun Portfolio Management Treynor (1965) Treynor (1965) developed the first composite measure of portfolio performance that included risk. He introduced the portfolio characteristic line, which defines a relation between the rate of return on a specific portfolio and the rate of return on the market portfolio. The beta is the slope that measures the volatility of the portfolios returns relative to the market. Alpha represents unique returns for the portfolio. As the portfolio becomes diversified, unique risk diminishes. ( ) t p p p , t M, t p,R R c | o + + =Albert Lee Chun Portfolio Management A risk-adjusted measure of return that divides a portfolio's excess return by its beta. The Treynor Measure is given by Treynor Measure pf pp r R = T|The Treynor Measure is defined using the average rate of return for portfolio p and the risk-free asset. wealth Albert Lee Chun Portfolio Management Treynor Measure pf pp r R = T|A larger Tp is better for all investors, regardless of their risk preferences. Because it adjusts returns based on systematic risk, it is the relevant performance measure when evaluating diversified portfolios held in separately or in combination with other portfolios. wealth Albert Lee Chun Portfolio Management Treynor Measure Beta measures systematic risk, yet if the portfolio is not fully diversified then this measure is not a complete characterization of the portfolio risk. Hence, it implicitly assumes a completely diversified portfolio. Portfolios with identical systematic risk, but different total risk, will have the same Treynor ratio! Higher idiosyncratic risk should not matter in a diversified portfolio and hence is not reflected in the Treynor measure. A portfolio negative Beta will have a negative Treynor measure. Also known as the Treynor Ratio. wealth Albert Lee Chun Portfolio Management T-Lines 17-18 Q has higher alpha, but P has steeper T-line. P is the better portfolio. wealth Albert Lee Chun Portfolio Management Sharpe Portfolio Performance Measure wealth Albert Lee Chun Portfolio Management Similar to the Treynor measure, but uses the total risk of the portfolio, not just the systematic risk. The Sharpe Ratio is given by The larger the measure the better, as the portfolio earned a higher excess return per unit of total risk. Sharpe Measure pf ppr R = Sowealth Albert Lee Chun Portfolio Management Sharpe Measure It adjusts returns for total portfolio risk, as opposed to only systematic risk as in the Treynor Measure. Thus, an implicit assumption of the Sharpe ratio is that the portfolio is not fully diversified, nor will it be combined with other diversified portfolios. It is relevant for performance evaluation when comparing mutually exclusive portfolios. Sharpe originally called it the "reward-to-variability" ratio, before others started calling it the Sharpe Ratio. wealth Albert Lee Chun Portfolio Management SML vs. CML Treynors measure uses Beta and hence examines portfolio return performance in relation to the SML. Sharpes measure uses total risk and hence examines portfolio return performance in relation to the CML. For a totally diversified portfolio, both measures give equal rankings. If it is not a diversified portfolio, the Sharpe measure could give lower rankings than the Treynor measure. Thus, the Sharpe measure evaluates the portfolio manager in terms of both return performance and diversification. wealth Albert Lee Chun Portfolio Management Price of Risk Both the Treynor and Sharp measures, indicate the risk premium per unit of risk, either systematic risk (Treynor) or total risk (Sharpe). They measure the price of risk in units of excess returns per each unit of risk (measured either by beta or the standard deviation of the portfolio). T = r Rp p f p | p p f pS r R o = Albert Lee Chun Portfolio Management Jensen Portfolio Performance Measure wealth Albert Lee Chun Portfolio Management Alpha is a risk-adjusted measure of superior performance This measure adjusts for the systematic risk of the portfolio. Positive alpha signals superior risk-adjusted returns, and that the manager is good at selecting stocks or predicting market turning points. Unlike the Sharpe Ratio, Jensens method does not consider the ability of the manager to diversify, as it is only accounts for systematic risk. Jensens Alpha ( ) t p t f p p r, , t M, t f, t p,R r R c | o + + = wealth Albert Lee Chun Portfolio Management Multifactor Jensens Measure Measure can be extended to a multi-factor setting, for example: ( ) t pp pt fpp HML SML r,3 2, t M,1t f, t p,R r R c | | | o + + + + = Albert Lee Chun Portfolio Management Information Ratio wealth Albert Lee Chun Portfolio Management Information Ratio 1 Using a historical regression, the IR takes on the form where the numerator is Jensens alpha and the denominator is the standard error of the regression. Recalling that co o p pIR =( ) t p t f p p r, , t M, t f, t p,R r R c | o + + = Note that the risk here is nonsystematic risk, that could, in theory, be eliminated by diversification. Albert Lee Chun Portfolio Management Information Ratio 2 ERbppR R = IRoMeasures excess returns relative to a benchmark portfolio. Sharpe Ratio is the special case where the benchmark equals the risk-free asset. Risk is measured as the standard deviation of the excess return (Recall that this is the Tracking Error) For an actively managed portfolio, we may want to maximize the excess return per unit of nonsystematic risk we are bearing. wealth Albert Lee Chun Portfolio Management Portfolio Tracking Error Excess Return relative to benchmark portfolio b Average Excess Return Variance in Excess Difference Tracking Error t b t p t R R ER, , === Tt tERTER11( )21211= = Tt t ER ER ERTo2ER ER o o =wealth Albert Lee Chun Portfolio Management Information Ratio Excess return represents managers ability to use information and talent to generate excess returns. Fluctuations in excess returns represent random noise that is interpreted as unsystematic risk. Information to noise ratio. Annualized IR p pIR T = IRERbppR R = IRoAlbert Lee Chun Portfolio Management Information Ratios wealth Albert Lee Chun Portfolio Management M2 Measure wealth Albert Lee Chun Portfolio Management M2 Measure Developed by Leah and her grandfather Franco Modigliani. M2 = rp*- rm rp* is return of the adjusted portfolio that matches the volatility of the market index rm. It is mixed with a position in T-bills. If the risk of the portfolio is lower than that of the market, one has to increase the volatility by using leverage. Because the market index and the adjusted portfolio have the same standard deviation, we may compare their performances by comparing returns. wealth Albert Lee Chun Portfolio Management M2 Measure: Example Managed Portfolio: return = 35% st dev = 42% Market Portfolio: return = 28% st dev = 30% T-bill return = 6% Hypothetical Portfolio: 30/42 = .714 in P (1-.714) or .286 in T-bills Return = (.714) (.35) + (.286) (.06) = 26.7% Since the return of the portfolio is less than the market, M2 is negative, and the managed portfolio underperformed the market. 17-35 wealth Albert Lee Chun Portfolio Management M2 of Portfolio P 17-36 17-36 wealth Albert Lee Chun Portfolio Management Excess Returns for Portfolios P and Q and the Benchmark M wealth Albert Lee Chun Portfolio Management Performance Statistics 17-38 wealth Albert Lee Chun Portfolio Management Which Portfolio is Best? It depends. If P or Q represent the entire portfolio, Q would be preferable based on having higher sharp ratio and a better M2. If P or Q represents a sub-portfolio, the Q would be preferable because it has a higher Treynor ratio. For an actively managed portfolio, P may be preferred because its information ratio is larger (that is it maximizes return relative to nonsystematic risk, or the tracking error). wealth Albert Lee Chun Portfolio Management Style Analysis wealth Albert Lee Chun Portfolio Management Style Analysis Introduced by William Sharpe 1992 study of mutual fund performance 91.5% of variation in return could be explained by the funds allocations to bills, bonds and stocks Later studies show that 97% of the variation in return could be explained by the funds allocation to set of different asset classes. 17-41 wealth Albert Lee Chun Portfolio Management Sharpes Style Portfolios for the Magellan Fund 17-42 Monthly returns on Magellan Fund over five year period. Regression coefficient only positive for 3. They explain 97.5% of Magellans returns. 2.5 percent attributed to security selection within asset classes. wealth Albert Lee Chun Portfolio Management Fidelity Magellan Fund Returns vs Benchmarks 17-43 Fund vs Style and Fund vs SML Impact of positive alpha on abnormal returns. 19.19% wealth Albert Lee Chun Portfolio Management Average Tracking Error for 636 Mutual Funds 17-44 Bell shaped wealth Albert Lee Chun Portfolio Management Market Timing wealth Albert Lee Chun Portfolio Management Perfect Market Timing A manager with perfect market timing, that shifts assets efficiently across stocks, bonds and cash would have a return equal to | | 0 , , max t bt t st t pt RFR R RFR R RFR R + =Albert Lee Chun Portfolio Management Returns from 1990 - 1999 Year Lg Stocks T-Bills 1990 -3.20 7.86 1991 30.66 5.65 1992 7.71 3.54 1993 9.87 2.97 1994 1.29 3.91 1995 37.71 5.58 1996 23.07 5.58 1998 28.58 5.11 1999 21.04 4.80 17-47 wealth Albert Lee Chun Portfolio Management With Perfect Forecasting Ability Switch to T-Bills in 90 and 94 Mean = 18.94%, Standard Deviation = 12.04% Invested in large stocks for the entire period: Mean = 17.41% Standard Deviation = 14.11 17-48 wealth Albert Lee Chun Portfolio Management Performance of Bills, Equities and Timers Beginning with $1 dollar in 1926, and ending in 2005.... wealth Albert Lee Chun Portfolio Management Value of Imperfect Forecasting Suppose you are forecasting rain in Seattle. If you predict rain, you would be correct most of the time. Does this make you a good forecaster? Certainly not. We need to examine the proportion of correct forecasts for rain (P1) and the proportion of correct forecasts for sun (P2). The correct measure of timing ability is P = P1 + P2 1 An forecaster who always guesses correctly will show P1 = P2 = P =1, whereas on who always predicts rain will have P1 = 1, P2 = P = 0. wealth Albert Lee Chun Portfolio Management Identifying Market Timing If an investor holds only the market and the risk free security, and the weights remained constant, the portfolio characteristic line would be a straight line. Adjusting portfolio weights for up and down movements in market returns, we would have: Low Market Return - low weight on the market - low eta High Market Return high weight on the market - high eta 17-51 Henriksson (1984) showed little evidence of market timing. Evidence of market efficiency. wealth Albert Lee Chun Portfolio Management Characteristic Lines: Market Timing 17-52 No Market Timing Beta Increases with Return Two Values of Beta wealth Albert Lee Chun Portfolio Management Testing Market Timing The following regression equation, controls for the movements in bond and stock markets, and captures the superior market timing of managers Gamma was found to be equal to .3 and statistically significant, suggesting that TAA managers were able to time the markets. However, the study also found a negative alpha of -.5. ( ) ( ) ( )| | { } t t bt t stst s t bt b t ptRFR R RFR R RFR R RFR R RFR Rc | | o+ + + + = 0 , , maxAlbert Lee Chun Portfolio Management Performance Attribution Analysis wealth Albert Lee Chun Portfolio Management Selectivity The basic premise of the Fama method is that overall performance of a portfolio can be decomposed into a portfolio risk premium component and a selectivity component. Selectivity is the portion of excess returns that exceeds that which can be attained by an unmanaged benchmark portfolio. Overall performance = Portfolio Risk Premium + Selectivity wealth Albert Lee Chun Portfolio Management Overall performance = Portfolio Risk Premium+ Selectivity | | | | | | ) ( R - R R ) ( R = Rx p f x f p p pR | | + Overall Performance Portfolio Risk Premium Selectivity Selectivity measures the distance between the return on portfolio p and the return on a benchmark portfolio with beta equal to the beta of portfolio p. wealth Albert Lee Chun Portfolio Management Attribution Analysis Portfolio managers add value to their investors by 1) selecting superior securities 2) demonstrating superior market timing skills by allocating funds to different asset classes or market segments. Attribution analysis attempts to distinguish is the source of the portfolios overall performance. Total value added performance is the sum of selection and allocation effects. wealth Albert Lee Chun Portfolio Management ) (&11 11 1Bi Bini pi pini Bi Bini pi pi B pni pi pi pni Bi Bi Br w r wr w r w r rr w r r w r= = = = == == =Where B is the bogey portfolio and p is the managed portfolio. Formula for Attribution 17-58 Set up a Benchmark or Bogey portfolio wealth Albert Lee Chun Portfolio Management Allocation Effect Asset Allocation Effect Captures the managers decision to over or underweight a particular market segment i. Overweighting a segment i when the benchmark yield is high is rewarded. ( ) ( ) | |Bi Bi Pi i r w w E =wealth Albert Lee Chun Portfolio Management Selection Effect Security Selection Effect Captures the stock picking ability of the manager, and rewards the ability to form specific market segment portfolios. Rewards the manger for placing larger weights on those segments where his portfolio outperforms the benchmark portfolio in that particular segment. ( ) ( ) | |Bi Pi Pi i r r w E =Albert Lee Chun Portfolio Management Performance of the Managed Portfolio 17-61 wealth Albert Lee Chun Portfolio Management Performance Attribution 17-62 wealth Albert Lee Chun Portfolio Management Sector Selection within the Equity Market 17-63 wealth Albert Lee Chun Portfolio Management Portfolio Attribution: Summary 17-64 wealth Albert Lee Chun Portfolio Management Global Benchmark Problem (Optional) wealth Albert Lee Chun Portfolio Management Benchmark Error Market portfolio is difficult to approximate Benchmark error can effect slope of SML can effect calculation of Beta greater concern with global investing problem is one of measurement Note: Sharpe measure not as dependent on market portfolio as the Treynor measure and others relying on Beta. wealth Albert Lee Chun Portfolio Management Differences in Betas Two major differences in the various beta statistics: For any particular stock, the beta estimates change a great deal over time. There are substantial differences in betas estimated for the same stock over the same time period when two different definitions of the benchmark portfolio are employed. wealth Albert Lee Chun Portfolio Management wealth Albert Lee Chun Portfolio Management Global Benchmark Problem - SML wealth Albert Lee Chun Portfolio Management Global Benchmark Problem - SML wealth Albert Lee Chun Portfolio Management Global Benchmark Problem - SML wealth Albert Lee Chun Portfolio Management Bond Portfolio Performance Measures (Optional) wealth Albert Lee Chun Portfolio Management Bond Portfolio Measures Returns-Based Bond Performance Measurement Early attempts to analyze fixed-income performance involved peer group comparisons Peer group comparisons are potentially flawed because they do not account for investment risk directly. How did the performance levels of portfolio managers compare to the overall bond market? What factors lead to superior or inferior bond-portfolio performance? Albert Lee Chun Portfolio Management Fama-French Measure Fama and French extended their 3-factor equity pricing model with 2 additional factors to account for the return characteristics of bonds TERM captures the term premium in the slope of the yield curve. DEF captures the default premium in the credit spread between corporate bonds and treasuries. These two bond factors are the dominate drivers of bond portfolio returns. ( )jt t j j1 mt t j2 t j3 t j4 t j4 t jtR - RFR = + b R - RFR + b SMB + b HML + b TERM + b DEF + e ( ( Albert Lee Chun Portfolio Management Seven Bond Portfolios wealth Albert Lee Chun Portfolio Management Bond Performance Attribution A Bond Market Line Need a measure of risk such as beta coefficient for equities Difficult to achieve due to bond maturity and coupon effect on volatility of prices Composite risk measure is the bonds duration Duration replaces beta as risk measure in a bond market line wealth Albert Lee Chun Portfolio Management Bond Market Line wealth Albert Lee Chun Portfolio Management wealth Albert Lee Chun Portfolio Management Thats all for today! wealth