# Jensen Ratio – What Is It And How Is It Calculated? (Jensen’s Performance Index)

Last Updated on May 18, 2022 by Quantified Trading

As an investor, you would want to know whether your investmentâ€™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 Ratio** (**Jensenâ€™s Performance Index)** is one of them. But what is the Jensen Ratio?

**Named after the renowned economist Michael Jensen who formulated it in 1968, Jensenâ€™s Performance Index, or simply called Jensenâ€™s Alpha or Jensen’s Ratio is a formula for calculating a portfolioâ€™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 portfolioâ€™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, letâ€™s look into the ratios and trading strategy and system performance metricsÂ that can be used to evaluate the risk-adjusted performance of an investment. In this post, our focus is on the Jensenâ€™s measure, and we will be discussing it under the following headings:

- What does Jensenâ€™s Ratio mean?
- How is the Jensenâ€™s alpha calculated?
- The Significance of Jensenâ€™s alpha
- How Jensenâ€™s alpha is used in financial analysis
- The criticisms of the Jensenâ€™s measure

Table of contents:

**What does Jensenâ€™s Ratio mean****? **

By way of definition, the Jensenâ€™s alpha, or Jensenâ€™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 portfolioâ€™s or investmentâ€™s beta and the average market return.

The Jensenâ€™s measure is a statistical measurement of the portion of a securityâ€™s or portfolioâ€™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 Jensenâ€™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 managerâ€™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 portfolioâ€™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 managerâ€™s skills.

**How is the Jensenâ€™s alpha calculated?**

As you can deduce from our discussion so far, Jensenâ€™s alpha, also known as the Jensenâ€™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:

Î± = R_{p} â€“ [R_{f} + Î²(R_{m} â€“ R_{f})]

Where:

Î± = Jensenâ€™s alpha

R_{p} = Portfolioâ€™s Realized Return

R_{f} = Risk-Free Rate

Î² = Beta of the Portfolio

R_{m} = Expected Market Return

R_{f} = Risk-Free Rate

Note that the portfolioâ€™s minimum expected return can be written as:

E(R) = R_{f} + Î²(R_{m} â€“ R_{f})

Hence,

Î± = R_{p} â€“ E(R)

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

Jensenâ€™s Alpha = Portfolioâ€™s Realized Return â€“ [Risk-Free Rate + Beta of the Portfolio X (Expected Market Return â€“ Risk-Free Rate)]

Or

Jensenâ€™s Alpha = Portfolioâ€™s Realized Return â€“ Expected Return

**The Significance of Jensenâ€™s alpha **

Jensenâ€™s alpha can have a positive or negative value. A positive value suggests that the portfolioâ€™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 Jensenâ€™s alpha works, you need to realize that the higher a portfolioâ€™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 â€” as reflected by a positive alpha value â€” the portfolio can be said to have performed very well, earning more than the level predicted by the market. The portfolioâ€™s manager can be said to have beat the market.

**How Jensen Ratio is used in financial analysis**

In finance, Jensenâ€™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. Jensenâ€™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 portfolioâ€™s rate of return is attributable to the managerâ€™s ability to deliver above-average returns, adjusted for market risk. The higher the ratio, the better the risk-adjusted returns.

Jensenâ€™s measure measures a fund managerâ€™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 Jensenâ€™s Performance Index is not a ratio â€” it is a whole value with a unit.

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

**Example 1:**

Letâ€™s say a mutual fund realized a return of 16% last year, while the relevant market index returned 10%. If the fundâ€™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% â€“ [2% + 1.4x (10% â€“ 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%, whatâ€™s the performance of the portfolio if its beta is 2.5 and the risk-free rate is 2%?

Alpha = 18% â€“ [2% + 2.1x (10% â€“ 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 Ratio**

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 Ratio** 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.