The portfolio weight is the initial fraction xi of an investor’s money invested in each asset. Portfolio weights add up to 1.
The expected return of a portfolio is the weighted average of the expected returns of the investments within it, using the portfolio weights.
To find the risk of a portfolio, we need to know the degree to which stock returns move together. Covariance and correlation measure the co-movement of returns.
The covariance between returns Ri and Rj is defined by
and is estimated from historical data using
The correlation is defined as the covariance of the returns divided by the standard deviation of each return. The correlation is always between −1 and +1. It represents the fraction of the volatility due to risk that is common to the securities.
The variance of a portfolio depends on the covariance of the stocks within it.
For a portfolio with two stocks, the portfolio variance is
If the portfolio weights are positive, as we lower the covariance or correlation between the two stocks in a portfolio, we lower the portfolio variance.
The variance of an equally weighted portfolio is
Diversification eliminates independent risks. The volatility of a large portfolio results from the common risk between the stocks in the portfolio.
Each security contributes to the volatility of the portfolio according to its total risk scaled by its correlation with the portfolio, which adjusts for the fraction of the total risk that is common to the portfolio.
Efficient portfolios offer investors the highest possible expected return for a given level of risk. The set of efficient portfolios is called the efficient frontier. As investors add stocks to a portfolio, the efficient portfolio improves.
An investor seeking high expected returns and low volatility should invest only in efficient portfolios.
Investors will choose from the set of efficient portfolios based on their risk tolerance.
Investors may use short sales in their portfolios. A portfolio is short those stocks with negative portfolio weights. Short selling extends the set of possible portfolios.
Portfolios can be formed by combining the risk-free asset with a portfolio of risky assets.
The expected return and volatility for this type of portfolio is
The risk–return combinations of the risk-free investment and a risky portfolio lie on a straight line connecting the two investments.
The goal of an investor who is seeking to earn the highest possible expected return for any level of volatility is to find the portfolio that generates the steepest possible line when combined with the risk-free investment. The slope of this line is called the Sharpe ratio of the portfolio.
The risky portfolio with the highest Sharpe ratio is called the efficient portfolio. The efficient portfolio is the optimal combination of risky investments independent of the investor’s appetite for risk. An investor can select a desired degree of risk by choosing the amount to invest in the efficient portfolio relative to the risk-free investment.
The beta of an investment with a portfolio is
Beta indicates the sensitivity of the investment’s return to fluctuations in the portfolio’s return.
A portfolio is efficient when E[Ri ] = ri for all securities. The following relationship therefore holds between beta and expected returns for traded securities:
Three main assumptions underlie the Capital Asset Pricing Model (CAPM):
Investors trade securities at competitive market prices (without incurring taxes or transaction costs) and can borrow and lend at the risk-free rate.
Investors choose efficient portfolios.
Investors have homogeneous expectations regarding the volatilities, correlations, and expected returns of securities.
Because the supply of securities must equal the demand for securities, the CAPM implies that the market portfolio of all risky securities is the efficient portfolio.
Under the CAPM assumptions, the capital market line (CML), which is the set of portfolios obtained by combining the risk-free security and the market portfolio, is the set of portfolios with the highest possible expected return for any level of volatility.
The CAPM equation states that the risk premium of any security is equal to the market risk premium multiplied by the beta of the security. This relationship is called the security market line(SML), and it determines the required return for an investment:
The beta of a security measures the amount of the security’s risk that is common to the market portfolio or market risk. Beta is defined as follows:
The beta of a portfolio is the weighted-average beta of the securities in the portfolio.
