**Jensen’s alpha **is a popular performance evaluation metric, which was developed by the American financial economist Michael Jensen in the late 60s. It is a *risk-adjusted performance measure* (other examples include the Sharpe ratio and Treynor ratio) and is theoretically linked to the capital asset pricing model (or the CAPM).

#### Contents

## Jensen’s alpha formula

We can state the **Jensen’s alpha formula **as follows:

where α* _{i}* is the Jensen’s alpha for asset

*i*,

*E[R*is the expected return on asset

_{i}]*i*,

*β*is the beta of asset

_{i}*i*,

*R*is the risk-free rate of return, and

_{f}*E[R*is the expected return on the market portfolio. Note that this measure is forward-looking as it is based on expected returns. You can also compute the realized Jensen’s alpha, which is backward-looking, using the same equation but with historical returns instead of expected returns.

_{m}]### Example

Let’s suppose an actively-managed fund generated a return of 20% last year. The market return and the risk-free rate were 10% and 2% over the same period, respectively. Furthermore, the fund’s beta is 1.4. Then, we have:

α_{i} = 20% − [ 2% + 1.4 (10% *− 2%*) ]

*= 20% − [ 2% + 11.2% ] = 6.8%*

**Test your knowledge**

You’ve identified a mutual fund that is expected return 36% per year according to your estimates. The fund’s beta is 2.8. If you believe that the stock market will generate %16 per year, and if the risk-free rate is 4%, what is the alpha of this fund?

Note: You can use the calculator below to verify your answer. The solution is provided at the bottom of this page.

## Jensen’s alpha calculator

In order to use our **Jensen’s alpha calculator**, you need to enter the asset’s return, its beta, and the market return. Note that the calculator works with both expected returns and historical returns over a single period.

We’ve got calculators for other popular risk-adjusted performance measures as well.

## Relationship with the CAPM

We can think of an asset’s Jensen’s alpha as the difference between its return and what its return should be according to the CAPM. In other words, if an asset’s return is exactly as what the CAPM predicts, then its Jensen’s alpha is zero: α* _{i}*=0. If it exceeds the level predicted by the CAPM, α

*is positive. And, if it falls short of CAPM’s prediction, α*

_{i}*is negative. This is illustrated in Figure 1 as follows.*

_{i}Figure 1 plots the security market line (SML), which is simply a depiction of the CAPM equation. The figure also shows asset A. The Jensen’s alpha for asset A (α* _{A}*) is simply the

*vertical distance*between this asset’s return and the return of another asset that has the same beta as A but that lies on the SML.

Clearly, asset A has a positive α* _{A}*. That is, its return is higher than an asset that has the same beta and that lies on the SML. In general, for any risky asset such as a stock, mutual fund, etc., α

*is*

_{i}*positive*if the asset lies

*above*the SML and

*negative*if it lies

*below*the SML. Of course, the asset’s α

*would be zero if it lied on the SML.*

_{i}In practice, the realized alpha for a risky asset over a specific period can be calculated by regressing the excess returns on the risky asset on the excess market returns. In this case, α* _{i}* is the intercept estimate of this regression model. It would be important to test its statistical significance to check whether or not it is statistically different than zero at conventional levels.

##### Summary

In this lesson, we have discussed the Jensen’s alpha formula α* _{i}* and its link to the CAPM. Note that α

*can also be calculated based on multi-factor models such as the Fama-French three-factor model.*

_{i}Regardless of the specific asset pricing model used, a positive α* _{i}* signifies outperformance and a negative α

*implies underperformance on a risk-adjusted basis. Actively managed funds exist to provide their investors with a positive α*

_{i}*. And, the more positive the α*

_{i}*is, the better the fund’s performance is.*

_{i}As a final word, it might be worth highlighting that the fact that a stock or fund yielded a positive α* _{i}* in the past (say the last five years) would by no means guarantee that the same asset would yield a positive α

*in the future as well.*

_{i}Further reading:

Jensen (1968) ‘The Performance of Mutual Funds in the Period 1945-1964,’ The Journal of Finance, Vol. 23 (2), pp. 389-416.

##### What is next?

This lesson is part of our free course on investments.

**Next lesson**: We will introduce another popular risk-adjusted performance measure: the Sharpe ratio.**Previous lesson**: We focused on the arbitrage pricing theory (APT), highlighting its links to factor models.

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**Solution for the “test your knowledge” exercise**

Using the calculator above we find that the fund’s alpha is *−*1.6%:

We can verify this figure using the formula as well:

α* _{i} = 36% − [ 4% + 2.8 (16% − 4%) ]* =

*−*1.6%

Overall, even though the fund’s expected return is much higher than that of the market (36% vs 16%), its Jensen’s alpha is negative because of its high beta (2.8). According to CAPM, this fund should yield a return of:

*4% + 2.8 (16% − 4%)* = 37.6%

in order to justify its high degree of systematic risk.