# Calculating Beta and Jensen's Alpha

Jensen's Alpha is complemented by other risk-adjusted performance metrics such as the Sharpe Ratio and the Treynor Ratio, offering a comprehensive view of portfolio performance.

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2/3/20246 min read

## What is Beta?

Beta is a measure of a security's sensitivity to market movements. It quantifies the relationship between the returns of an individual stock or portfolio and the returns of the overall market. By calculating Beta, investors can assess the level of systematic risk associated with an investment.

The formula to calculate Beta is as follows:

Beta = Covariance(Stock Returns, Market Returns) / Variance(Market Returns)

Where:

- Covariance(Stock Returns, Market Returns) measures the extent to which the returns of the stock move in relation to the market returns.

- Variance(Market Returns) represents the variability of the market returns. A Beta value of 1 indicates that the stock or portfolio moves in line with the market. A Beta greater than 1 suggests that the investment is more volatile than the market, while a Beta less than 1 indicates lower volatility compared to the market.

## Interpreting Beta

Understanding the implications of different Beta values is crucial for investors. Here's a breakdown of how to interpret Beta:

- Beta = 1: The stock or portfolio has the same level of volatility as the market.

- Beta > 1: The stock or portfolio is more volatile than the market. It tends to magnify market movements, both on the upside and downside.

- Beta < 1: The stock or portfolio is less volatile than the market. It exhibits relatively smaller price fluctuations compared to the overall market. It's important to note that Beta only captures systematic risk and does not account for unsystematic or idiosyncratic risk specific to a particular security or portfolio.

## What is Jensen's Alpha?

Jensen's Alpha, also known as the Jensen Performance Index, is a risk-adjusted measure that evaluates the excess returns generated by an investment compared to its expected returns. It helps investors determine whether a portfolio manager or investment strategy has outperformed or underperformed the market, considering the level of risk taken. The formula to calculate Jensen's Alpha is as follows:

Jensen's Alpha = Actual Portfolio Return - [Risk-Free Rate + Beta * (Market Return - Risk-Free Rate)]

Where:

- Actual Portfolio Return represents the realized return of the portfolio.

- Risk-Free Rate refers to the return on a risk-free investment, such as government bonds.

- Beta is the Beta value of the portfolio. - Market Return represents the average return of the market.

A positive Jensen's Alpha indicates that the portfolio has outperformed the market, while a negative value suggests underperformance. It is important to note that Jensen's Alpha considers both systematic and unsystematic risk.

## Interpreting Jensen's Alpha

When interpreting Jensen's Alpha, it is crucial to consider the following points:

- Positive Alpha: A positive Alpha indicates that the portfolio has delivered excess returns above what would be expected given its level of risk. This suggests that the portfolio manager or investment strategy has added value.

- Negative Alpha: A negative Alpha suggests that the portfolio has underperformed the market, considering its level of risk. This could indicate that the portfolio manager or investment strategy has not been able to generate excess returns. However, it is important to remember that Jensen's Alpha is not the sole determinant of investment performance. It is just one measure among many that investors should consider when evaluating investment strategies.

## Theoretical Framework of Jensen's Alpha: Principles and Economic Analysis

### Principles of the CAPM Model

The Capital Asset Pricing Model (CAPM) is fundamental in the modern understanding of the relationship between risk and return. Established in the 1960s, CAPM formulates that the expected return of any investment should be directly proportional to the level of market risk it assumes, compared to the risk-free rate of return. The CAPM equation is:

Ri = Rf + βi(Rm - Rf)

Where Ri is the expected return of asset i, Rf represents the risk-free rate of return, Rm is the expected market return, and βi measures the sensitivity of asset i's return to market movements. This model introduces the notion that systematic risk, or market risk, is the only type of risk that should generate expectations of additional returns.

## Jensen's Alpha and CAPM

Jensen's Alpha is derived from analyzing the performance of an investment in the context of CAPM, calculating the difference between the observed actual return and the expected return based on the assumed risk (beta). The formula for Jensen's Alpha is:

αi = Ri - (Rf + βi(Rm - Rf))

This calculation allows investors and analysts to evaluate a fund manager's ability to generate added value, regardless of the overall market movement. A positive Alpha indicates that the manager has achieved a return higher than predicted by CAPM, suggesting exceptional skill in investment selection or market timing.

### Economic Interpretation of Alpha

From an economic perspective, the existence of a consistent positive Alpha challenges the efficient market hypothesis, which holds that all assets are correctly priced and that it is impossible to consistently outperform the market through stock selection or market timing. Jensen's Alpha suggests that, under certain circumstances, managers may possess information, intuition, or analytical skills that allow them to identify undervalued investment opportunities or predict market changes before most investors.

## Applications and Case Studies of Jensen's Alpha

### Fund Selection for Analysis

The practical application of Jensen's Alpha covers a wide range of investment contexts, from evaluating mutual funds and ETFs to analyzing quantitative investment strategies and individually managed portfolios. Advanced methodology and qualitative analysis complement the calculation of Alpha, providing deep insights into management ability and the effectiveness of various investment strategies.

• Diversity of Strategies: Including funds with different investment approaches, from active management to passive strategies, to evaluate how Alpha varies in different contexts.

• Variety of Asset Classes: Examining funds that invest in stocks, bonds, emerging markets, and more, provides a richer understanding of how Jensen's Alpha reflects management ability across different markets.

• Geographical Exposure: Including funds that focus on specific regions or have global exposure helps analyze the impact of geographical factors on Alpha generation.

### Empirical Study of Real Funds

Conducting empirical studies on Jensen's Alpha involves rigorous statistical analysis of performance data. These studies typically examine:

• Historical Performance: Analyzing the historical performance of funds relative to their risk-adjusted benchmarks to determine the existence of positive Alpha.

• Alpha Consistency: Evaluating whether a fund's positive Alpha is a consistent result over time or is attributed to random factors or market anomaly periods.

• Contributing Factors: Identifying strategies, investment decisions, or market conditions that may have contributed to the generation of positive Alpha, thus providing a deeper understanding of management ability.

### Results Analysis and Discussion

Once Jensen's Alpha is calculated for selected funds, interpreting the results requires detailed analysis considering:

• Market Context: Placing the fund's performance and its Alpha within the context of market conditions during the study period.

• Investment Strategy: Examining how the fund's investment strategy and asset allocation decisions influenced its ability to generate positive Alpha.

• Comparison with Related Metrics: Contrasting Jensen's Alpha with other risk-adjusted performance metrics to obtain a holistic assessment of fund management effectiveness.

### Limitations and Considerations

It is crucial to recognize the inherent limitations in calculating Jensen's Alpha, including dependence on the underlying CAPM model and the need for accurate and complete data. Additionally, interpreting Alpha should be done cautiously, considering that superior past performance does not guarantee similar future results.

## Detailed Methodology for Calculating and Applying Jensen's Alpha

### Detailed Formula and Components

The extended formula of Jensen's Alpha remains true to its original purpose: measuring the excess return of a portfolio beyond what the CAPM predicts, given its beta. However, in advanced applications, meticulous selection of components such as:

• Risk-Free Rate (Rf): The choice of the instrument representing the risk-free rate may vary depending on the economic context and the reference currency. Typically, short-term treasury bonds of the country in question are used.

• Market Return (Rm): Properly defining the market index used as Rm is essential, choosing the one that best represents the investment universe of the analyzed portfolio.

• Portfolio Beta (β): Beta reflects the portfolio's volatility relative to the market. Precise beta calculation is crucial, considering not only historical volatility but also portfolio sectorial and geographical exposures.

### Risk Adjustments

Advanced methodology also incorporates adjustments for non-systematic risks, acknowledging that CAPM beta does not capture all the risks to which a portfolio may be exposed. This includes:

• Specific Sector or Company Risk: Adjustments to consider concentration in specific sectors or exposure to risks inherent in certain companies.

• Geographical Risk: For portfolios with international exposure, the Alpha is adjusted to reflect political, currency exchange, and economic risks specific to the invested regions.

## Comparison with Other Performance Metrics

Jensen's Alpha is complemented by other risk-adjusted performance metrics such as the Sharpe Ratio and the Treynor Ratio, offering a comprehensive view of portfolio performance. While Jensen's Alpha focuses on relative performance to the market given a level of risk, the Sharpe Ratio evaluates excess return per unit of total volatility, and the Treynor Ratio measures excess return per unit of systematic risk.