Jensen’s alpha formula and calculator

jensen's alpha formula and calculator

When evaluating the performance of mutual funds, ETFs, or your own portfolio, it is vital to do that on a risk-adjusted basis. That is, it would be misleading to compare investment opportunities on the basis of returns only as higher returns normally require bearing more risk. In this post, we discuss one of the most popular risk-adjusted performance measures: Jensen’s alpha. In particular, we cover Jensen’s alpha formula, offer a calculator, and explain the link between this performance measure and the CAPM.

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Jensen’s alpha formula

Jensen’s alpha takes its name from the American financial economist Michael Jensen. You can find a reference to Jensen’s original paper where he introduces this portfolio performance measure at the bottom of this page (see further reading). As a side note, Michael Jensen is also well known for his joint work with William Meckling on the agency costs that arise due to the separation of ownership and control.

We can state Jensen’s alpha formula as follows:

αi = Ri − [ Rf + βi (Rm Rf) ]

where αi is the Jensen’s alpha for asset i, Ri is the return on asset i, βi is the beta of asset i, Rf is the risk-free rate of return, and Rm is the return on the market portfolio.

You may have noticed that the term in the squared brackets is the asset’s return according to the CAPM. Therefore, we can think of Jensen’s alpha as the difference between the asset’s actual return and what its return should be according to the CAPM. In other words, if an asset’s return turned out to be exactly as what the CAPM predicted, its Jensen’s alpha would be zero. If it exceeded the level predicted by the CAPM, αi would be positive. And, if it fell short of CAPM’s prediction, αi would be negative.

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 can calculate this fund’s Jensen’s alpha as:

αi = 20% − [ 2% + 1.4 (10% − 2%) ]

= 20% − [ 2% + 11.2% ] = 6.8%

Jensen’s alpha calculator

In this section, you can find our Jensen’s alpha calculator. This can be used to calculate the realized Jensen’s alpha whereby the inputs are historical returns and historical beta. It can also be used to calculate the expected Jensen’s alpha whereby the inputs are expected returns and beta.

Relationship with the CAPM

We have already pointed out that Jensen’s alpha can be interpreted as the difference between an asset’s actual return and what that return should be according to the CAPM. Now, we delve deeper into the relationship between Jensen’s alpha and CAPM.

Figure 1 plots the security market line (SML), which is simply a depiction of the CAPM equation. The figure also shows asset A, which is lying above the SML. Jensen’s alpha for asset A (αA) is simply the vertical distance between asset A and another asset that has the same beta as A but that lies on the SML.

Clearly, asset A has a positive Jensen’s alpha. 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., Jensen’s alpha is positive if the asset lies above the SML and negative if it lies below the SML. Of course, the asset’s Jensen’s alpha would be zero if it lied on the SML.

jensen's alpha formula and calculator
Figure 1: Jensen’s alpha and the security market line

In practice, the realized Jensen’s 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, Jensen’s alpha 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 post, we discuss the popular Jensen’s alpha measure in detail by presenting Jensen’s alpha formula and highlighting its link to the CAPM. In particular, Jensen’s alpha is a risk-adjusted performance measure (other such measures include Sharpe ratio, Treynor ratio, and so on). It captures the excess return an asset generates beyond what is predicted by the CAPM. Of course, Jensen’s alpha can also be calculated based on multi-factor models such as the three-factor model or the five-factor model developed by Fama and French.

Regardless of the specific asset pricing model used to calculate Jensen’s alpha, a positive Jensen’s alpha signifies outperformance and a negative Jensen’s alpha implies underperformance on a risk-adjusted basis. Actively managed funds exist to provide their investors with a positive alpha. And, the more positive the alpha is, the better the fund’s performance is.

As a final word, it might be worth highlighting that the fact that a stock or fund yielded a positive Jensen’s alpha in the past (say the last five years) would by no means guarantee that the same asset would yield a positive Jensen’s alpha in the future as well.

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 post is part of our free course on investments. The previous post in the course discussed the arbitrage pricing theory (APT). In the next post, we will introduce another popular risk-adjusted performance measure: the Sharpe ratio.

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Jensen’s alpha formula and calculator

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