Approximation assumption returns are roughly normally distributed
Normal distribution characterized by two parameters mean and variance (i.e. return and risk)
Holding different combinations (portfolios) of assets affects the possible combinations of return and risk an investor can obtain
2 asset model
proportion of stock 1 held in portfolio
1-proportion of stock 2 held in portfolio
Joint distribution of the returns on the two stocks
Return to a portfolio is denoted by z with
Average return to the portfolio is
Variance of the portfolio is
We can derive the relationship between the mean of the portfolio and its variance by noting that
Substituting for in the expression for the variance of the portfolio we find
To portfolio spreadsheet
Choose portfolio which minimizes the variance of the portfolio subject to generating a specified average return
Have to perform the optimization since you can no longer solve for the weights from the specification of the relationship between the averages
As with the two asset case yields a quadratic relationship between average return to the portfolio and its variance which is called the mean-variance frontier
Frontier indicates possible combinations of risk and return available to investors when they hold efficient portfolios (i.e. those that minimize the risk associated with getting a specific return
Optimal portfolio choice can be determined by confronting investor preferences for risk versus return with possibilities
20 CAPM 21 CAPM
Two fund theorem
Introduce possibility of borrowing or lending without risk
Let rf denote the risk-free rate of return
Historically around 1.5
The two fund theorem then states that there exists a portfolio of risky assets (which we will denote by S) such that all efficient combinations or risk and return (i.e. those which minimize risk for a given rate of return) can be obtained by putting some fraction of wealth in S while borrowing or lending at the risk-free rate. The portfolio S is called the market portfolio.
22 CAPM 23 CAPM
Implications of the two fund theorem for asset prices
In equilibrium asset prices will adjust until all portfolios lie on the security market line
Implications for asset market equilibrium
Risk-averse investors require higher returns to compensate for bearing increased risk
Idiosyncratic risk versus market risk
Equilibrium risk vs. return relationships
Market risk of asset i is defined as the ratio of the covariance between asset i and the market portfolio to the variance of the market portfolio
Since iSiS i S (where iS is the correlation coefficient between asset i and the market portfolio S) we can write
Finally since the returns on all assets must be perfectly correlated with those on the market portfolio (in equilibrium) we know that iS1 so that
Since the equation for the market line is
it follows that the predicted equilibrium return on a given asset i will be
The term rS-rf is called the market risk premium since it measures the additional return over the risk-free rate required to get investors to hold the riskier market portfolio.
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