Treynor Index and Jensen Index

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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