Brief topic: Portfolio theories
Sharpe Ratio
Which risky portfolio on the efficient frontier should an investor ideally combine with a risk-free investment? A risk-averse investor prefers the portfolio that optimizes the risk/return ratio and thus maximizes the slope of the capital market line. This slope corresponds to the Sharpe ratio, which is the ratio of the risk premium to the standard deviation of a portfolio:
Sharpe ratio = risk premium / standard deviation = (expected return of the portfolio - risk-free interest rate) / standard deviation of the portfolio.
The optimal portfolio is therefore the one that achieves the highest Sharpe ratio and thus represents the tangential point on the efficient frontier. This tangential portfolio remains identical for all investors who have the same expectations of return, risk and covariance and forms the market portfolio. The individual risk appetite is only reflected in the relationship to the risk-free investment.
Example: A risk-averse investor with €100,000 could invest €10,000 in the market portfolio and €90,000 in a risk-free investment. A risk-averse investor, on the other hand, could invest € 120,000 in the market portfolio and take out a loan of € 20,000.
Beta
What are the characteristics of the market portfolio? The risk and return contribution of individual securities is central. The risk that a security contributes to the market portfolio is represented by the beta:
βᵢ = covariance of security i with the market portfolio / variance of the market portfolio.
A beta of 1 means that the security bears the average market risk. A security with beta > 1 has an above-average risk, with beta < 1 a below-average risk. As the market portfolio consists of the optimum combination of all risky securities, it itself has a beta of 1. For example: with a beta of 1.29 for Apple, a market increase of 1% means that the Apple return increases by an average of 1.29%. The beta of a portfolio is the weighted average of the betas of its individual securities.
Fama-French model
The three-factor model developed by Eugene Fama and Kenneth French extends the Capital Asset Pricing Model (CAPM) by adding two additional explanatory factors for share returns. The formula is
r = risk-free return + (beta of the market portfolio * market risk premium) + (beta of the size factor * size risk premium) + (beta of the book-to-market factor * book-to-market risk premium).
The risk premiums for these factors are based on the difference between market returns and risk-free returns, small-cap and large-cap returns and the difference between high BTM (value stocks) and low BTM (growth stocks). The factor sensitivities (betas) can be determined using econometric analyses.
Summary
- Only portfolios on the efficient frontier are optimal for investors.
- If risk-free investments and loans are possible, the choice of the optimal risky portfolio is independent of the investor's attitude to risk.
- All investors hold the same market portfolio as a risky investment.
- The personal risk appetite is reflected in the allocation between risk-free capital and the market portfolio.
- The expected return on a security depends on its beta and the general market risk premium.
- The Fama-French model supplements the market risk premium with two additional factors: size and book-to-market.
SourceBrealey, R., S. Myers, F. Allen, A. Edmans (2022): Principles of Corporate Finance, 14th edition, McGraw Hill, ISBN 1260013901 and lecture slides.