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# Treynor Index and Jensen Index

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The Treynor index is a risk-adjusted measure of performance that standardizes the risk premium of a portfolio with the portfolio’s systematic risk or beta coefficient. The Treynor index is similar to the Sharpe index except that Treynor uses the beta coefficient rather than the standard deviation of the portfolio to measure risk. The Treynor index uses only nondiversifiable risk, whereas the Sharpe index includes the total risk of the portfolio. Treynor’s index is determined as follows:

Treynor index = (portfolio return - risk-free rate)/ portfolio beta coefficient

The Treynor index is useful when it is compared with the market, or with other portfolios to determine superior performance. For example, if a portfolio has a total return of 9 percent and the risk-free rate is 3.5 percent with a portfolio beta of 1.1, the Treynor index is 0.05:

Treynor index p = (rp - rf)/βp = (0.09 - 0.035)/1.1 = 0.05

Comparing this portfolio with the market during the same period, in which the market return is 8.7 percent and has a beta coefficient of 1, the Treynor index for the market is

Treynor index m = (rp - rf )/βp = (0.087 - 0.035)/1 = 0.052

Thus the portfolio return is inferior to that of the market (0.05 versus 0.052). The portfolio return per unit of diversifiable risk is less than that of the market.

## Jensen Index

The Jensen index is a risk-adjusted measure of performance that compares realized returns with returns that should have been earned per unit of nondiversifiable risk. Michael Jensen’s performance index is based on the capital asset pricing model and differs from the Sharpe and Treynor measures. The Jensen index compares excess return with returns that should have been earned in the market based on the nondiversifiable risk of the portfolio. The result can be positive, negative, or zero. A positive result indicates that performance of the portfolio was superior to that of the market. A negative result indicates that the portfolio underperformed the market, and a zero indicates identical performance to that of the market. The Jensen index is determined as follows:

Jensen index = (total portfolio return - risk-free rate) - [portfolio β * (market return - risk-free rate)]

For example, a portfolio with a return of 12 percent and a 1.3 beta coefficient when the market return is 9 percent and the risk-free rate is 4 percent results in a Jensen index of

Jensen index = (Rp - rf ) - [βp * (rm - rf )] = (0.12 - 0.04) - [1.3(0.09 - 0.04)] = 0.015

This portfolio outperformed the market on a risk-adjusted basis. Unlike the Sharpe and Treynor indexes, the Jensen index adjusts for the market return, which allows you to compare the portfolio return with that of the market in one computation. Because the Sharpe index uses the standard deviation as a measure of total risk, it reveals a poorly diversified portfolio, which will have a large standard deviation. Apoorly diversified portfolio may not be uncovered by the beta coefficient, which is used in the Treynor and Jensen indexes.

A problem that occurs in comparing the performance of a portfolio to that of the market is determining which market index is the most appropriate to use. If a portfolio has small-cap, largecap, and foreign stocks, then the use of the S&P 500 Index is problematic because it does not have the same composition as the portfolio for comparison purposes. This problem is compounded by the fact that beta coefficients based on different market indexes differ and may produce biased beta coefficients that distort the performance evaluations. A solution to the benchmark index is to determine a weighted-average return of the portfolio and compare it with the weighted-average returns of the corresponding benchmark indexes, as illustrated in Table 12–4.

Using the weighted-average returns of the corresponding benchmark indexes for comparison with the portfolio returns makes the result more meaningful. The portfolio underperformed the corresponding weighted average of the benchmark indexes by 0.22 percent (3.92% - 3.7%) on a non-risk-adjusted basis. The efficient market hypothesis assumes that even if your portfolio obtains returns that are superior on a risk-adjusted basis to those of the market, you cannot expect to consistently repeat the superior returns.

Table 12-4
Weighted-Average Returns

As can be expected, security analysts have not really embraced these academic theories, especially the efficient market hypothesis. Their view is that academicians are so immersed in their own research that they would not be able to recognize an undervalued stock even if it was brought to their attention. The ongoing battle between the analysts and the academicians is of little importance. What is important for investors is an awareness of these theories from several practical points of view.

The degree of efficiency of the market determines your investment strategy with regard to the selection of stocks and the length of time to hold those stocks. If you believe that the market is efficient and that all information is reflected in the price of the stock, your strategy might be to select quality stocks with good future earnings and hold them for long periods (the buy-and-hold strategy). On the other hand, if you believe that the markets are inefficient, you can use technical analysis to determine which stocks to buy and sell over shorter periods and fundamental analysis to select undervalued stocks to buy and hold for longer periods. The degree of efficiency is debatable.

The CAPM suggests that investors diversify their investments to eliminate unsystematic risk. The returns earned by most investments will be consistent with the returns of the market and the related amount of risk. Bearing this in mind, the investment strategy you choose should be consistent with your objectives.

No known method consistently beats the markets over long periods. Anomalies in the efficient market theory exist, but because of the competitive nature of the markets, they have not consistently earned abnormal returns for long periods. Yet, at the other extreme, overwhelming support of the efficient market hypothesis paralyzes investors into thinking that no research is valuable.

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