Introduction
Portfolio theory is a fundamental aspect of modern investment theory that helps investors construct portfolios to maximize expected return based on a given level of market risk. Here we explore key concepts like expected return, portfolio variance, the Capital Asset Pricing Model (CAPM), and the Beta coefficient.
Expected Return and Portfolio Variance
1. Expected Return
The expected return of a portfolio is the weighted average of the expected returns of the individual assets in the portfolio. The formula is:
E(Rp) = ∑ (wi * E(Ri))
Where:
- wi = Weight of each asset in the portfolio,
- E(Ri) = Expected return of each asset.
2. Portfolio Variance
Portfolio variance is a measure of the dispersion of the returns of a portfolio. It is calculated as:
σ²(p) = ∑∑ (wi * wj * σi * σj * ρij)
Where:
- wi, wj = Weights of assets i and j,
- σi, σj = Standard deviations of assets i and j,
- ρij = Correlation coefficient between the returns of assets i and j.
Capital Asset Pricing Model (CAPM)
The CAPM is a model that describes the relationship between systematic risk and expected return for assets, particularly stocks. It is widely used throughout finance for the pricing of risky securities and generating expected returns for assets given the risk of those assets and the cost of capital. The formula is:
E(Ri) = Rf + βi * (E(Rm) – Rf)
Where:
- E(Ri) = Expected return on the capital asset,
- Rf = Risk-free rate,
- βi = Beta of the investment,
- E(Rm) = Expected return of the market.
Beta Coefficient
The Beta coefficient measures the volatility of an investment in comparison to the market as a whole. It is used in the CAPM to determine a stock’s expected rate of return. The Beta is calculated using regression analysis on the historical returns of the stock and the market. The formula is:
β = Covariance(Return of Stock, Return of Market) / Variance(Return of Market)
Conclusion
Understanding these fundamental concepts of portfolio theory can greatly enhance an investor’s ability to make informed decisions about their investment strategies. By applying these formulas, investors can assess risk, return, and the optimal asset allocation to achieve their investment objectives.
