Evaluation Portfolio Performance Using Jensen's Alpha

What is Jensen's Alpha?

Jensen’s Alpha is a portfolio performance evaluation metric that assesses the extent to which an investment manager outperforms the theoretical expected return. This measure was developed by Michael Jensen in 1968 and introduced in his research paper published in the Journal of Finance. It was designed to evaluate the performance of mutual fund managers and their ability to generate abnormal returns above the market benchmark.

Jensen’s Alpha builds upon the traditional alpha concept, which compares portfolio performance to a market index (commonly the S&P 500, representing broad U.S. equity market performance). However, Jensen’s Alpha diverges by using the theoretical return estimated via the Capital Asset Pricing Model (CAPM) as its benchmark instead of a market index.

How To Calculate Jensen's Alpha?

Jensen’s Alpha does not rely on superficial realized return measures for calculation. Instead, Jensen's alpha formula compares the actual return to the theoretical expected return derived from the CAPM model, as per the following equation:

Where:

• ERp: Theoretical expected return of the portfolio.

• Rf: Risk-free rate of return.

• βp: Beta coefficient, measuring the portfolio’s sensitivity to systematic market risk.

• ERm: Expected market return.

• ERm-Rf: Market risk premium (excess return of the market over the risk-free rate).

Jensen aimed to test the hypothesis that fund managers cannot consistently outperform the market or add long-term value to investors. His research was grounded in the Efficient Market Hypothesis (EMH), which posits that all known information is already reflected in asset prices, leaving no room for managers to exploit "hidden" data.

To test this, Jensen introduced an additional term (α) into the CAPM equation:

Here, α represents the manager’s ability to generate abnormal returns. Rearranging the equation gives Jensen’s Alpha formula:

In practice, ERp and ERm are often replaced with the portfolio’s actual realized return and the market portfolio’s realized return, respectively.

Jensen’s study concluded that only 3 out of 115 mutual funds exhibited statistically significant positive alpha, reinforcing the EMH by suggesting that fund managers lack persistent informational advantages.

Risk-free Rate in Jensen’s Alpha Calculation

The risk-free rate represents the return on an investment with theoretically zero default risk. It is typically proxied by the yield or coupon on 10-year U.S. Treasury bonds, as the U.S. government is deemed unlikely to default.

In Jensen’s Alpha, a higher risk-free return increases the expected return threshold for portfolio managers. Since investing in risky assets inherently demands returns above risk-free return, managers must outperform this baseline to justify their risk exposure.

How to Interpret Jensen’s Alpha Values?

In simple terms, Jensen's alpha reading is interpreted as follows:

• Positive alpha: This means that the investment manager achieved an actual return higher than expected (beat the market).

• Zero alpha: This indicates that the fund manager achieved a return equal to the expected return of the broader market. This is common in index-tracking funds (known as index funds), which primarily aim to achieve a return as close as possible to an index such as the S&P 500.

• Negative alpha: This indicates that the investment manager achieved a return lower than the market average or what is expected from it. In other words, if the investor had invested in a fund that tracks the S&P 500, they would have achieved a higher return than the investment manager could have achieved through managing a fund, which may involve risky strategies.

Taking this into account, assuming a fund achieves a Jensen's alpha of +1%, it exceeded the expected return of the market by one percentage point. The opposite is true if the reading is -1%.

What are the Advantages and Limitations of Jensen’s Alpha?

Advantages

• Risk-Adjusted Metric: Incorporates systematic risk (beta coefficient) to evaluate performance beyond raw returns.

• Wide Adoption: Rooted in the CAPM framework, making it academically and practically recognized.

• Based on Quantitative Methods: The nature of Jensen's alpha makes it easy to integrate into widely used quantitative models, helping investors make informed and systematic decisions based on measurable criteria.

Limitations

• Reliance on CAPM assumptions: The model relies on beta coefficient as the sole determinant of expected returns. However, a set of other factors influence market performance, making it a limited measure of risk-adjusted returns.

• Sensitivity to Estimation Error: Because Jensen's Alpha is based on quantitative models, which are not free of estimation or forecasting errors.

• Inability to Capture Complex Market Dynamics: Jensen's alpha and the CAPM model assume a linear relationship between returns and risks. The relationship may change from time to time and may not be a simple linear one.

Using Jensen’s Alpha for Investment Decisions

There are many aspects for using Jensen's alpha for investment decisions systematically and effectively, taking into account the risks surrounding the market. The key aspects of using Jensen's alpha can be summarized as follows:

• Portfolio Performance Evaluation: Jensen's alpha helps investors determine whether an asset manager has added value beyond what would be expected based on the level of market risk and the risk-free return. It also enables us, based on the objective for which it was used, to measure the exceptional skill of an investment manager and their ability to outperform their peers.

• Comparing Different Funds: Investors can use Jensen's alpha to compare options among actively managed funds. For example, if two funds have the same beta coefficient but different alpha values, the fund with the higher alpha is often more efficient in terms of generating returns at the same level of risk.

• Evaluating Fund Prices: Because Jensen's alpha measures an asset's performance relative to its expected return, it can be used to identify unfairly priced assets. If the alpha is consistently high, it may indicate that the fund is undervalued and has upside potential, while a consistently negative alpha may indicate that the fund is overvalued.

• Risk Management: Jensen's alpha can also be used as a risk management tool by ensuring that superior returns to benchmarks are not the result of excessive risk tolerance. If an investment is generating high returns but Jensen's alpha is relatively low or negative, this may indicate that the returns are the result of excessive risk exposure rather than skilful fund management.

Jensen's Alpha vs. Sharpe ratio

Jensen's alpha aims to evaluate performance based on risk-weighted measures and is based on the CAPM model, which typically uses expected returns in the broader market.

In contrast, although the Sharpe ratio is also a risk-weighted measure, its calculation methodology differs fundamentally. It is calculated by subtracting the actual or expected return from the risk-free rate of return by the standard deviation of the return. Furthermore, risk in the Sharpe ratio is measured by the standard deviation of the portfolio's returns, not the correlation with the broader market or a specific index.

Thus, the key differences between Jensen's alpha and the Sharpe ratio can be summarized as follows:

• Risk Measurement: Jensen's alpha measures risk based on systematic risk represented by the beta coefficient, while the Sharpe ratio takes into account the total risk to the portfolio itself, measured by the standard deviation of the return, including both systematic and unsystematic risk.

• Benchmark Reliance: Jensen's alpha requires a benchmark to estimate the expected return, making it useful in evaluating portfolio managers' performance against various benchmarks as needed. The Sharpe ratio, on the other hand, does not require a benchmark, making it more suitable for comparing different portfolios.

• Interpretation: A positive Jensen's alpha indicates that the portfolio has outperformed the investment manager's expectations, while a higher Sharpe ratio indicates better performance relative to the risk taken, regardless of market performance.

• Applications: Jensen's alpha is most effective when evaluating the performance of active portfolio management, while the Sharpe ratio is more suitable for comparing different investment strategies or independent portfolios.

Conclusion

Jensen's alpha is a risk-weighted return measure that helps investors make investment decisions based on systematic, quantitative methods, while not using superficial and sometimes misleading figures. Investment managers can beat the market by claiming to possess special skills or knowledge, hence the need to create Jensen's alpha, which enables them to express these skills in numbers.

What distinguishes Jensen's alpha is its reliance on one of the most well-known quantitative models, the Capital Asset Pricing Model, in addition to considering the risk-free return. This enables investors to construct a quantitative formula based on systematic foundations. Despite its advantages, Jensen's alpha has its own limitations, including its assumption of a simple, linear, single-factor relationship between returns and risk, and its sensitivity to the resulting estimation error.