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# Sharpe ratio, Treynor ratio, M2 and Jensen’s alpha

Performance evaluation refers to the measurement, attribution, and appraisal of investment results. It provides information that is important in taking timely corrective actions.

It has three stages:

• measurement is concerned with quantification of return and risk,
• attribution is concerned with finding the sources of performance, and
• appraisal is concerned with assessing investment skills.

Four ratios that are commonly used in performance appraisal include the Sharpe ratio, Treynor ratio, M2: risk-adjusted performance, and Jensen’s alpha. These are mainly based on the capital asset pricing model but multi-factor appraisal analysis is also common.

## Sharpe ratio

Since return is linked with risk, it is important to assess return with reference to risk. Sharpe ratio (also called rewards to risk ratio or rewards-to-variability ratio) is one such measure that works out the return in excess of the risk-free rate per unit of risk. It can also be defined as the portfolio’s risk premium divided by its risk.

It can be used to analyze both expected excess return (ex-ante basis) and realized excess return (ex-post)

$Sharpe\ Ratio=\frac{Expected\ Return\ -\ Risk\ Free\ Rate}{Standard\ Deviation}$

Sharpe ratio is the slope of the capital allocation line. A higher ratio is better. A negative ratio is meaningless.

However, the Sharpe ratio suffers from two limitations:

• it uses total risk (which is appropriate only if the investor has no other assets), and
• it does not provide any information other than the ranking of investments.

## Treynor ratio

Treynor ratio replaces the standard deviation in the numerator of the Sharpe ratio with beta coefficient. Hence, it addresses the first limitation of the Sharpe ratio.

Treynor ratio can be defined as the excess return per unit of systematic risk. It is calculated as follows:

$Treynor\ Ratio=\frac{Expected\ Return\ -\ Risk\ Free\ Rate}{Beta}$

### Limitations of Treynor ratio

The limitations of Treynor ratio include

• negative values of excess return (and/or beta) are meaningless,
• it just helps in ranking of investments.

M2 and Jensen’s alpha attempt to provide information about the extent of the overperformance or underperformance.

M2 (also called risk-adjusted performance measure) is based on the Sharpe ratio, ranks portfolios similarly, but provides information about the extent of performance (in percentage terms).

The basic concept behind M2 is the capital market theory. It determines a leverage ratio at which a portfolio’s total risk is equal to the market risk. This leverage is applied to the excess return to which the risk-free rate is added. It is then compared with actual return to determine performance differential.

$M^2=[E(R_i)-R_f]×\frac{σ_m}{σ_p}+R_f=Sharpe Ratio×σ_m+R_f$

M2 and Sharpe ratios rank portfolios identically because, in any given period, risk-free rate and the market volatility are constants.

Any excess of the risk-adjusted performance of the portfolio over the market return is referred to as M2 alpha.

## Jensen’s alpha

Jensen’s alpha represents any excess of the actual portfolio return over the risk-adjusted (expected) return which it ought to generate given its systematic risk. The expected return is calculated using CAPM. It represents the vertical distance between actual return and SML.

$\alpha_p=R_P-{R_f+\beta\times[{E(R}_m)-R_f)]}$

If multiple periods are involved, we would use the average risk-free rate. Positive (negative) alpha means that the portfolio has outperformed (underperformed) the market. Alpha allows us to both rank and quantify the performance of different investments.

## Example

Which of the following would most likely be positive even for a portfolio with return lower than the market?

A) Jensen’s alpha
C) Sharpe ratio

C is correct. Sharpe ratio is positive even if expected/actual return is lower than the market as long as it is greater than the risk-free rate.

If the Sharpe ratio is 0.40, the market standard deviation is 13%, the asset’s standard deviation is 15% and its beta coefficient is 1.15, its Treynor’s ratio would be:

A) 0.0452
B) 0.0522
C) 0.3261

B is correct. We first need to use the formula for the Sharpe ratio to find the excess return. Since the Sharpe ratio equals excess return divided by asset standard deviation, excess return equals the Sharpe ratio (0.40) multiplied by the asset standard deviation (15%). This gives us the excess return of 6%, which when divided by beta (1.15) yields a Treynor’s ratio of 0.0522.