Last Updated on 24 September, 2020 by Samuelsson

As an investor, you would want to know whether your investment’s return is worth the risk. There are different methods you can use to measure the risk-adjusted performance of your portfolio, and the Jensen’s Performance Index is one of them. But what is the Jensen formula?

Named after the renowned economist Michael Jensen who formulated it in 1968, Jensen’s Performance Index, or simply called Jensen’s alpha is a formula for calculating a portfolio’s returns after adjusting for risk. It measures the returns earned in excess of or below the expected return based on the Capital Asset Pricing Model (CAPM), given the portfolio’s beta, the overall market returns, and the risk-free rate of return.

After understanding the R-squared, Beta, and Standard Deviation in our earlier posts, let’s look into the ratios and metrics that can be used to evaluate the risk-adjusted performance of an investment. In this post, our focus is on the Jensen’s measure, and we will be discussing it under the following headings:

• What does Jensen’s alpha mean?
• How is the Jensen’s alpha calculated?
• The Significance of Jensen’s alpha
• How Jensen’s alpha is used in financial analysis
• The criticisms of the Jensen’s measure

## What does Jensen’s alpha mean?

By way of definition, the Jensen’s alpha, or Jensen’s measure, is a risk-adjusted performance measure that represents the average return on a portfolio or investment, above or below that predicted by the capital asset pricing model (CAPM), given the portfolio’s or investment’s beta and the average market return.

The Jensen’s measure is a statistical measurement of the portion of a security’s or portfolio’s return that is not explained by the market movement and how the portfolio is expected to move relative to the market movement. It shows how much the portfolio has performed above or below its expected performance given its risk level relative to the market.

This theoretical expected return/performance is predicted by a market model known as the capital asset pricing model (CAPM). The model uses statistical methods to estimate the appropriate risk-adjusted return of an asset. The Jensen’s alpha, simply referred to as alpha, measures how much the investment returned above or below this expected return.

For a managed portfolio, Jensen’s alpha shows the manager’s risk-adjusted performance relative to the market returns. It is the degree with which a manager outperforms or underperforms the market, putting into account the portfolio’s systemic risk relative to the broad market as measured by the beta. Beating the market is to deliver an alpha. Hence, alpha is a measure of a portfolio manager’s skills.

## How is the Jensen’s alpha calculated?

As you can deduce from our discussion so far, Jensen’s alpha, also known as the Jensen’s Performance Index, is a measure of the excess returns earned by the portfolio compared to returns suggested by the CAPM. So mathematically, alpha can be calculated from the CAPM formula.

The formula for Jensen’s alpha can be presented as follows:

α = Rp – [Rf + β(Rm – Rf)]

Where:

α = Jensen’s alpha

Rp = Portfolio’s Realized Return

Rf = Risk-Free Rate

β = Beta of the Portfolio

Rm = Expected Market Return

Rf = Risk-Free Rate

Note that the portfolio’s minimum expected return can be written as:

E(R) = Rf + β(Rm – Rf)

Hence,

α = Rp – E(R)

To put it in words, the formula goes like this:

Jensen’s Alpha = Portfolio’s Realized Return – [Risk-Free Rate + Beta of the Portfolio X (Expected Market Return – Risk-Free Rate)]

Or

Jensen’s Alpha = Portfolio’s Realized Return – Expected Return

## The Significance of Jensen’s alpha

Jensen’s alpha can have a positive or negative value. A positive value suggests that the portfolio’s return is more than the expected return, while a negative value shows that the portfolio earned less than the expected return. The more positive (bigger) the value of alpha, the better the return compared to the expected.

To understand how Jensen’s alpha works, you need to realize that the higher a portfolio’s risk (as measured by Beta), the greater the value of its expected return. What it means is that investors are supposed to be rewarded with a higher return for taking a bigger risk. So when a portfolio earns more than the expected risk-adjusted value — as reflected by a positive alpha value — the portfolio can be said to have performed very well, earning more than the level predicted by the market. The portfolio’s manager can be said to have beat the market.

## How Jensen’s alpha is used in financial analysis

In finance, Jensen’s alpha is used to determine the abnormal return of a security or portfolio of investment earned in excess of the expected return calculated from CAPM. Jensen’s alpha was first used as a measure in the evaluation of mutual fund managers by Michael Jensen in 1968. It measures how much of the portfolio’s rate of return is attributable to the manager’s ability to deliver above-average returns, adjusted for market risk. The higher the ratio, the better the risk-adjusted returns.

Jensen’s measure measures a fund manager’s performance against the returns that could have been expected from a market-related investment. As you know, there are other risk-adjustment metrics, which are ratios. For example, the Sharpe Ratio, or reward-to-variability ratio, is the slope of the capital allocation line (CAL), and the greater the slope (higher number) the better. But the Jensen’s Performance Index is not a ratio — it is a whole value with a unit.

Here are some real-world examples of how alpha is used:

Example 1:

Let’s say a mutual fund realized a return of 16% last year, while the relevant market index returned 10%. If the fund’s beta versus the market index is 1.4 and the risk-free rate is 2%, how did the fund perform relative to the market expectation?

Alpha = 16% – [2% + 1.4x (10% – 2%)] = 2.8%

So, the fund beat the market on a risk-adjusted basis.

Example 2:

Assuming a portfolio returned 18% and the relevant market index returned 10%, what’s the performance of the portfolio if its beta is 2.5 and the risk-free rate is 2%?

Alpha = 18% – [2% + 2.1x (10% – 2%)] = -0.8%

So, the fund underperformed.

Example 3:

There are 2 mutual funds, A and B, and each made a return of 25% in the past year. However, Mutual Fund A has an alpha value of 2.1%, while Mutual Fund B has an alpha value of 1.8%. Which one would you invest with?

Obviously, Mutual Fund A has a better risk-adjusted value.

## The criticisms of the Jensen’s measure

Followers of the efficient market hypothesis (EMH) believe that the market has already priced in all available information so there is no way portfolio managers can consistently outperform the market. They argue that any positive value of Jensen’s alpha is just from luck or random chance rather than the skill of the portfolio manager.

Expectedly, they use the fact that passive index funds perform better than most actively managed portfolios to justify their theory. 