Portfolio Performance Measures
• Objective: Evaluation of active portfolio man- agement. 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 under- valued stocks. They invest in the securi- ties 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.
• 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 activi- ties.
• In the real world, however, it is possible that the market is not in equilibrium and prof- itable 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 nec- essarily the best trader, since the portfolio might carry a higher level of risk than that borne by other traders.
Sharpe Ratio
• Sharpe, W. F. (1966). Mutual Fund Perfor- mance. Journal of Business 3, 119-138.
• 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 re- spective mean return and standard deviation as in the following table.
Portfolio µˆp ˆσp
A 25 30
B 10 7
• Portfolio A has both the higher mean re- turn and the greater standard deviation (i.e., risk). Which has a superior risk/return rela- tionship?
• The answer depends on the risk–free rate.
• Recall that, for each portfolio of risky assets, Q, and risk–free rate, rf, the transformation line, which describes the risk/return trade–
off for portfolio combinations (p) of Q and the risk–free asset, is
µp = rf +
[µQ − rf
σQ
]
| {z }
=θQ
σp.
• Any µp − σp combination on the transforma- tion 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 relation- ship if θA > θB and vice versa.
• For rf = 4, θB > θA, 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
xµB + (1 − x)rf
= µ! A
⇒ x = µA − rf
µB − rf
= 25 − 4
10 − 4 = 3.5.
The resulting portfolio has standard devia- tion
σp = 3.5σB = 24.5 < 30 = σA.
• Similarly, using a portfolio weight of x = 30/7 = 4.2857, we obtain a portfolio vari- ance of 30 (= σA), but the resulting portfo- lio has mean µp = 4.2857×10+(1−4.2857)×
4 = 29.7142 > 25 = µA.
0 5 10 15 20 25 35 40 0
5 15 20 30
Standard deviation
Mean return
Risk−free rate r
f = 4
µA
µB
σA σB
rf + θA σp rf + θB σp
rf
Portfolio A
Portfolio B
0 5 10 15 20 25 30 35 40 0
5 10 15 20 25 30
Risk−free rate r
f = 7
Mean return
Standard deviation
rf
A
B
transformation line for B transformation
line for A
• On the other hand, if rf = 7, A outperforms B.
• Thus, the Sharpe–Ratio for a portfolio Q of risky assets is defined as
S
Q=
µQ−rfσQ
.
• If portfolio A has a higher Sharpe Ratio than portfolio B, we can, by appropriately borrow- ing or lending at the risk–free rate, always construct portfolio combinations of portfo- lio 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 ex- cess return per unit of risk.
• 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.
Treynor Ratio
• Treynor, J. L. (1966). How to Rate Man- agement of Investment Funds. Harvard Busi- ness Review 63, 63-75.
• This measure is directly related to the Cap- ital 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 effi- ciently diversified.
• Thus, only systematic risk is of importance.
• The Standard CAPM implies that the ex- pected excess return, µQ − rf, of a portfolio Q of risky assets is given by
µQ − rf = βQ(µM − rf),
where µM is the mean return of the market portfolio M , and βQ = COV(rQ, rM)/σM2 .
• Here, βQ measures the non-diversifiable, or systematic, risk of portfolio Q, i.e., its co- variance with the market.
• 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.,
µi − rf
βi = µM − rf, for all i.
In (βi, µi)–space, all assets plot along a straight line with intercept rf and slope µM − rf (se- curity market line).
The Security Market Line
βM = 1 µM
rf
beta expected return
Note that the linearity implies for a portfolio Q with weights xi, i = 1, . . . , N , that
µQ =
∑N i=1
xiµi =
∑N i=1
xirf +
∑N i=1
xiβi
| {z }
=βQ
(µM − rf)
= rf + βQ(µM − rf),
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=
µQ−rfβQ
.
• If T > µM − rf, the fund manager outper- forms the benchmark. Values of T can also be used to rank individual investment man- ager’s portfolios.
• In practice, it is important to be aware of the fact that the measure may crucially de- pend on the benchmark, which may affect the ranking of different portfolios.
• Also, if the benchmark is chosen by the man- ager, he/she may have incentives to select a benchmark which displays low correlation with the managed portfolio (low beta, hence high value of T ).
Jensen’s α
• Jensen, M. C. (1968). The performance of Mutual Funds in the Period 1945-1964.
Journal of Finance 23, 389-416.
• This measure also comes directly from the CAPM.
• If the average value of asset i’s excess re- turn (i.e., rit−rf,t) is completely explained by the CAPM risk premium (i.e., βi[rM,t−rf,t]), then the intercept term in the time series regression
(rit−rf,t) = αi+βi(rM,t−rf,t)+ϵi,t, t = 1, . . . , T, is zero for each asset i.
• If αi > 0, portfolio i earns an abnormal high return, relative to the risk adjusted return predicted by the CAPM. On the other hand, a negative αi 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 ϵit in the above regression equation and estimat- ing the parameters by OLS, a test for abnor- mal return can easily be constructed using standard distributional results.
• Note that
Ti = αi
βi + (µM − rf),
so that, if βi > 0, an abnormal positive re- turn according to the Treynor Ratio also im- plies an abnormal positive return according to Jensen’s α.
• These performance measures may often be used only on the basis of an ad–hoc argu- ment.
• If the CAPM holds (i.e., the market portfo- lio 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
µi = rf + βi(µm − rf)
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.
