## Defination and Application of Jensen’s Alpha

#### Introduction

We know that, the higher the risk associated with an asset, the greater is the expected return from that asset. Thus, high returns are assumed to be the rewards for those aggressive investors who are willing to invest in the riskier assets. It makes the calculation of an asset’s or a portfolio’s risk-adjusted performance critical in making investment decisions. A Jensen’s Alpha also known as “Alpha /Jensen’s Measure / Jensen’s Performance Index” is a measure of portfolio performance. The portfolio should be well-diversified to leverage unsystematic risk.

#### Definition : Jensen’s Alpha

- Jensen’s Measure or Jensen’s Alpha was developed by American economist Michael Jensen in 1968 . It is used to calculate the return on a portfolio in excess of its theoretical expected return.
- Alpha is the difference between actual returns of a fund and the expected return- that could have been earned on a benchmark portfolio with the same amount of market risk (e.g. the same Beta).
- Jensen’s measure measures a fund manager’s performance against the returns that could have been expected from a market-related investment. In short, Jensen’s alpha tries to explain whether an investment has performed better or worse than its beta value would suggest. The ratio determines how much active management can increase returns above those that are purely a reward for bearing market risk.
- Therefore, It becomes an important ratio in the mutual fund analysis to interpret which funds to recommend to investors. Apart from Alpha, other important ratios like Sharpe ratio, Treynor ratio which we use in the analysis are covered in our earlier articles.

#### Formula

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

2. Aj = Rp – [ Rf + Bp * ( Rm – Rf )]

Where,

- Aj = Jensen’s Alpha
- Rp = Expected Portfolio Return
- Rf = Risk-Free Rate
- Bp = Beta of the Portfolio
- Rm = Expected Market Return

#### how does this formula work?

- The above formula is used to calculate the difference between the excess return of an asset amd its expected return which was calculated theoreotically. The formula holds good for any type of asset including securities, bonds, stocks and derivatives.
- We compute the above mentioned theoretical expected return using the capital asset pricing model (CAPM). It is a financial model which calculates the expected return of a security based on average market return, risk-free interest rate and the beta of the security as a multiplier.
- Beta is the volatility of an asset in comparison to the overall market factors. While the Alpha represents the excess return an asset generated over the return calculated using the CAPM. An asset generates a return that is either more or less than what we calculate using the CAPM.
- If Asset generates return > Expected theoretical return of CAPM, Positive Alpha = Better performance of the asset compared to expectations
- If Asset generates return < Expected theoretical return of CAPM, Negative Alpha = Poor performance of the asset compared to expectations

#### an illustration

- A portfolio has realized a return of 16%. The approximate market index returned 12%. The beta of the fund versus the same index is 1.3 and the risk-free rate is 4%.
- Jensen’s Alpha = 16 – [ 4 + 1.3 * (12 – 4 ) ] = 1.6%
- Given, the Beta of 1.3, the fund is expected to be risky than the market index and thus earn more. A positive alpha is an indication that the portfolio manager earned substantial return to be compensated for the additional risk taken over the course over the year.
- If the fund would have returned 14%, the computed alpha would be -0.4%. A negative alpha indicates that the investor was not earning enough returns for the quantum of risk which was borne.

#### Risk-Reward tradeoff

- A positive alpha shows that the fund manager’s stock selection skill has delivered superior risk-adjusted returns. While comparing two funds with similar beta ratios, investors would prefer the one with the higher alpha. Because it indicates the greater reward at the same level of risk.
- When we measure the performance of return, Jensen’s alpha takes an investment’s risk profile into account. Thus, it gives us an overall picture of a portfolio or stock’s performance on a risk-adjusted basis. This helps investors to gauge the value which a fund manager adds or detracts from a portfolio, and helps in the comparison of funds.
- For example, if two investments deliver the same levels of return, we would go with the more stable, less-risky option. Since that particular fund outperforms when we consider the risk as a performance measure.

#### why is Jensen’s index important?

- For every investor, it is important to understand the risks they would be taking when they invest in a particular asset. For this, they need a properly calculated measure of the total return of an investment against the risk involved in it. The aim of investors is to go for securities that offer maximum returns with minimal risks.
- It means that between two mutual fund schemes that are offering similar returns, the one with less risk would be more lucrative for investors than the one with higher risk.
- The Jensen’s Alpha can help investors determine if the return an asset is generating on average is acceptable compared to the risks it is offering. It is risk-adjusted return. A positive alpha indicating an excess returns, which investors are looking for when they are using this formula.
- So, if you are considering some investment options, make sure you have calculated the risk-adjusted returns these options offer to understand what you are really getting into. The higher the alpha value, the more lucrative an option is. If you are dealing with options that generate a negative alpha value, investing in them might not be a wise choice.

#### summary

- The 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).
- Alpha represents the performance of a portfolio relative to a benchmark. So it often represents the value that a portfolio manager adds to or subtracts from a fund’s return.
- In other words, alpha is the return on an investment that is not a result of general movement in the greater market.
- An alpha of zero would indicate that the portfolio or fund is tracking perfectly with the benchmark index (Passive fund management).

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