UTS16657 Chap.6 Risk Adjusted Performance Indices
Risk Adjusted Performance Indices
Once performance has to account for risk as well as return, several measures derived from investment theory become available, and they exist so that managers can be compared once risk has been taken into account. The subject uses three.
They share a numerator, the excess return over the risk-free rate, and they differ entirely in how they charge for risk.
The Jensen index determines the difference between the actual return of a portfolio and what its return would need to be for the portfolio to sit on the security market line.
It reports the answer in return units, so a positive value is the margin by which the portfolio beat what its market exposure alone entitled it to.
The subject is careful about its limits: it says nothing about how many securities were used, or how well spread they were, in producing that excess, and it leans heavily on the fund's own risk and on how the market itself performed rather than on skill alone.
The Treynor index divides the risk premium by the portfolio's beta, so it reports excess return per unit of market risk and treats only the non-diversifiable portion, ignoring diversifiable risk entirely.
That makes it the appropriate measure where the investment being evaluated represents a fraction of a larger portfolio, because the specific risk is already being diversified away elsewhere.
The Sharpe index divides by the standard deviation, which is the full amount of risk in the portfolio rather than only its risk against the market, so it is more meaningful when an entire portfolio is being measured and it is the most widely used of the three.
The interesting case is when they disagree, which happens when a portfolio carries a large amount of specific risk relative to its market exposure.
Treynor and Jensen do not charge for that risk and therefore flatter such a portfolio; Sharpe charges for all of it and does not. The disagreement is a finding rather than an error, and reading it correctly is what the chapter is for.
What this chapter covers
- 01
The inputs all three indices need, and matching the periods
- 02
Jensen as a distance from the security market line
- 03
Treynor as excess return per unit of beta
- 04
Sharpe as excess return per unit of total risk
- 05
A worked case where the three rank two portfolios differently
- 06
Choosing the index from whether the holding is whole or partial
- 07
Three ways these numbers are routinely misread
Two portfolios where the three indices do not agree
- +1Market risk premium: 8.40 − 3.85 = 4.55 percentage points. Both portfolios face the same premium and differ only in their exposure to it.
- +1Required returns: P needs 3.85 + 4.55(0.72) = 7.126%, and Q needs 3.85 + 4.55(1.32) = 9.856%. Q is required to earn more because it carries more market risk.
- +1Jensen: P is 9.18 − 7.126 = +2.054% and Q is 10.40 − 9.856 = +0.544%. Both added value and P added nearly four times as much.
- +1Treynor, in decimals: P is (0.0918 − 0.0385) ÷ 0.72 = 0.0740 and Q is (0.1040 − 0.0385) ÷ 1.32 = 0.0496, so P wins. Sharpe: P is 0.0533 ÷ 0.1245 = 0.4281 and Q is 0.0655 ÷ 0.1110 = 0.5901, so Q wins.
- +1Explain rather than re-check. P has a low beta and a high standard deviation, so much of its volatility is specific rather than market-driven, which means it is under-diversified. Treynor and Jensen do not charge for specific risk and therefore flatter P; Sharpe charges for all of it and does not.
Key terms
- Jensen index
- The difference between a portfolio's actual return and the return required for its beta, reported in percentage points. A positive value is the margin by which the portfolio beat the security market line.
- Treynor index
- The excess return of a portfolio over the risk-free rate divided by its beta, so it reports return per unit of market risk and ignores diversifiable risk entirely.
- Market risk premium
- The return on an appropriate market index less the risk-free rate over the same period. It is the compensation for carrying one unit of market risk and it is common to every asset.
- Required return
- The return an asset or portfolio should deliver given its beta, found as the risk-free rate plus the market risk premium multiplied by beta. It is the height of the security market line at that beta.
- Portfolio beta
- The weighted sum of the betas of the individual assets held. Unlike portfolio risk, it genuinely does average, which is why it can be computed without a covariance matrix.
- Risk-free rate
- The average return on risk-free bonds over the same period as the portfolio and market returns being measured. Drawing it from a different window silently produces a meaningless index.
Risk Adjusted Performance Indices FAQ
Can I compare a Treynor index with a Sharpe ratio?
No. The three indices are not on a common scale, because their denominators are different quantities measured in different units: one is a beta and the other a standard deviation. A Treynor of 0.0740 and a Sharpe of 0.5901 are not comparable numbers even for the same portfolio. You may rank portfolios within one index, and you may not rank the indices against each other.
What does a negative index actually mean?
It depends which one. A negative Jensen index says the portfolio returned less than its beta entitled it to; it does not say the portfolio lost money, and a single period is far too small a sample to call a manager incompetent.
A negative Treynor or Sharpe usually means the portfolio returned less than the risk-free rate, in which case the ratio is not meaningfully interpretable and the correct answer is to say so rather than rank it against positive ones.
Why does Jensen depend on the market and Sharpe does not?
Because Jensen is defined against the security market line, which is built from the risk-free rate, the market return and the portfolio's beta. Sharpe needs only the portfolio's own return, the risk-free rate and its own standard deviation, so no market index enters it at all.
That difference is worth noticing: it means Jensen can move because the market moved rather than because the manager did anything, which is one of the stated limitations of relying on it alone.
Which index does the subject say is the most widely used?
Sharpe. The stated reason is that the standard deviation measures the full amount of risk of a portfolio rather than only its risk against the market portfolio, and it also facilitates comparison of a portfolio's performance with the market. When an entire portfolio is being measured, the Sharpe index is described as the more meaningful measure.
Exam move
Compute all three on the same data set at least three times, because the disagreement case is the one that appears in exams and it only becomes readable once the arithmetic is automatic. Fix one convention and stick to it: this guide uses decimals throughout so the Treynor and the Sharpe of one portfolio sit on scales you can hold in mind together.
Then rehearse the choosing rule out loud, because a five-mark question here is almost always asking which index rather than what the numbers are.
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