Portfolio Performance Measures

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1 Portfolio Performance Measures Objective: Evaluation of active portfolio management. A performance measure is useful, for example, in ranking the performance of mutual funds. Active portfolio managers attempt to beat the market by identifying over- and undervalued stocks. They invest in the securities they deem to be undervalued and in some cases short-sell the ones they believe are overvalued. By contrast, passive fund managers adopt a buy-and-hold strategy in which their goal is to mimic the performance of a market index.

2 If the CAPM holds, the best strategy is to buy-and-hold the market portfolio (passive portfolio management), and active portfolio management is useless or even harmful, due to the costs of trading and research activities. In the real world, however, it is possible that the market is not in equilibrium and profitable opportunities arise. Any index of portfolio performance would have to measure the actual returns of funds relative to some risk/return relationship. A manager who consistently turns in a higher return than all other managers is not necessarily the best trader, since the portfolio might carry a higher level of risk than that borne by other traders.

3 Sharpe Ratio Sharpe, W. F. (1966). Mutual Fund Performance. Journal of Business 3, As in the Standard (Sharpe Lintner) CAPM, it is assumed that all investors are able to invest funds at a common risk free rate and to borrow funds at the same rate (at least to the desired extent). Consider two portfolios, A and B, with respective mean return and standard deviation as in the following table. Portfolio ˆµ p ˆσ p A B 10 7 Portfolio A has both the higher mean return and the greater standard deviation (i.e., risk). Which has a superior risk/return relationship? The answer depends on the risk free rate.

4 Recall that, for each portfolio of risky assets, Q, and risk free rate, r f, the transformation line, which describes the risk/return trade off for portfolio combinations (p) of Q and the risk free asset, is [ ] µq r f µ p = r f + σ p. } σ {{ Q } =θ Q Any µ p σ p combination on the transformation line can be achieved by investing in the portfolio of risky assets Q plus borrowing or lending at the risk free rate. Portfolio A has superior risk/return relationship if θ A > θ B and vice versa.

5 For r f = 4, θ B > θ A, so that B exhibits superior performance. The logic is as follows. We can borrow at the risk free rate to obtain a portfolio with the same expected return but lower risk than A. The amount to borrow can be obtained from the condition! xµ B + (1 x)r f = µ A x = µ A r f = 25 4 µ B r f 10 4 = 3.5. The resulting portfolio has standard deviation σ p = 3.5σ B = 24.5 < 30 = σ A. Similarly, using a portfolio weight of x = 30/7 = , we obtain a portfolio variance of 30 (= σ A ), but the resulting portfolio has mean µ p = ( ) 4 = > 25 = µ A.

6 30 Risk free rate r f = 4 µ A Portfolio A r f + θ B σ p 20 Mean return µ B 15 Portfolio B r f + θ A σ p 5 r f σ B Standard deviation σ A

7 30 Risk free rate r f = 7 25 A 20 transformation line for A Mean return r f 5 B transformation line for B Standard deviation

8 On the other hand, if r f = 7, A outperforms B. Thus, the Sharpe Ratio for a portfolio Q of risky assets is defined as S Q = µ Q r f σ Q. If portfolio A has a higher Sharpe Ratio than portfolio B, we can, by appropriately borrowing or lending at the risk free rate, always construct portfolio combinations of portfolio A and the risk free rate such that the resulting portfolio has the same mean as B but lower risk, or the same risk but higher mean. The measure can also be interpreted as excess return per unit of risk.

9 In practical applications, the mean return and the standard deviation are estimated from historical data over the period of interest (for which the comparison is to be made), and the risk free rate is chosen accordingly.

10 Treynor Ratio Treynor, J. L. (1966). How to Rate Management of Investment Funds. Harvard Business Review 63, This measure is directly related to the Capital Asset Pricing Model (CAPM). It is assumed that the funds to be evaluated form only a (small) part of the investor s portfolio, e.g., a specialized fund (industry, sector, type of security). The overall portfolio is assumed to be efficiently diversified. Thus, only systematic risk is of importance.

11 The Standard CAPM implies that the expected excess return, µ Q r f, of a portfolio Q of risky assets is given by µ Q r f = β Q (µ M r f ), where µ M is the mean return of the market portfolio M, and β Q = COV(r Q, r M )/σ 2 M. Here, β Q measures the non-diversifiable, or systematic, risk of portfolio Q, i.e., its covariance with the market.

12 Thus, according to the CAPM, the excess return per unit of systematic risk is the same and equal to the excess return on the market portfolio for each asset or portfolio i, i.e., µ i r f β i = µ M r f, for all i. In (β i, µ i ) space, all assets plot along a straight line with intercept r f and slope µ M r f (security market line). The Security Market Line expected return µ M r f β M = 1 beta

13 Note that the linearity implies for a portfolio Q with weights x i, i = 1,..., N, that µ Q = N i=1 x i µ i = N i=1 x i r f + = r f + β Q (µ M r f ), N x i β i (µ M r f ) i=1 } {{ } =β Q i.e., we can obtain the beta of a portfolio of assets by calculating the weighted average of the betas of the portfolios components. The Treynor Ratio of portfolio Q is defined by excess return per unit of systematic risk, T Q = µ Q r f β Q. If T > µ M r f, the fund manager outperforms the benchmark. Values of T can also be used to rank individual investment manager s portfolios.

14 In practice, it is important to be aware of the fact that the measure may crucially depend on the benchmark, which may affect the ranking of different portfolios. Also, if the benchmark is chosen by the manager, he/she may have incentives to select a benchmark which displays low correlation with the managed portfolio (low beta, hence high value of T ).

15 Jensen s α Jensen, M. C. (1968). The performance of Mutual Funds in the Period Journal of Finance 23, This measure also comes directly from the CAPM. If the average value of asset i s excess return (i.e., r it r f,t ) is completely explained by the CAPM risk premium (i.e., β i [r M,t r f,t ]), then the intercept term in the time series regression (r it r f,t ) = α i +β i (r M,t r f,t )+ϵ i,t, t = 1,..., T, is zero for each asset i.

16 If α i > 0, portfolio i earns an abnormal high return, relative to the risk adjusted return predicted by the CAPM. On the other hand, a negative α i implies that the portfolio has underperformed, relative to this standard. The α coefficient in the regression equation is called Jensen s α. Assuming normally distributed errors ϵ it in the above regression equation and estimating the parameters by OLS, a test for abnormal return can easily be constructed using standard distributional results. Note that T i = α i β i + (µ M r f ), so that, if β i > 0, an abnormal positive return according to the Treynor Ratio also implies an abnormal positive return according to Jensen s α.

17 These performance measures may often be used only on the basis of an ad hoc argument. If the CAPM holds (i.e., the market portfolio is efficient), and we have identified the relevant market portfolio correctly (or if the CAPM does not hold but we are using the tangency portfolio as benchmark), then we have seen that the CAPM relation µ i = r f + β i (µ m r f ) is a simple algebraic fact and holds for any asset or portfolio, so that the Treynor Ratio would always be equal to the excess return on the market portfolio and Jensen s α would always be equal to zero. Thus, if these measures indicate abnormal returns, we can conclude that either we have incorrectly measured the market portfolio or the CAPM does not hold. Nevertheless, in real world applications, where markets are not in equilibrium, the indices may still provide useful information.

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