Understanding Alpha & Beta in Portfolio Theory

Introduction

In portfolio theory, alpha and beta are two of the most fundamental metrics for evaluating investment performance.

What is Beta?

Beta measures the systematic risk of an asset relative to the market. A beta of 1.0 means the asset moves perfectly with the market.

Beta Formula

Beta is calculated as:

$\beta = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$

Where:

• $R_i$ is the return of the asset
• $R_m$ is the return of the market

Interpreting Beta

What is Alpha?

Alpha measures the excess return of an investment relative to its expected return given its beta.

Alpha Formula

$\alpha = R_i - [R_f + \beta(R_m - R_f)]$

Where $R_f$ is the risk-free rate.

Practical Implications

Understanding alpha and beta helps portfolio managers:

• Hedge systematic risk using derivatives
• Identify skill vs luck in fund management
• Construct portfolios with desired risk profiles

Conclusion

Alpha and beta remain cornerstone metrics in quantitative finance. Mastering them is essential for anyone pursuing systematic investing.