Economics 434 Financial Markets Professor Burton University of Virginia Fall 2015 September 15, 17, 2015

The Capital Asset Pricing Model Markowitz – mean, variance analysis Tobin – the role of the risk free rate Sharpe (and others) – beta and the market basket September 15, 17, 2015

Need Mathematical Concepts Mean Variance Covariance Correlation Coefficient September 15, 17, 2015

Symbols Mean [x] ≡ µ(x) ≡ µ x Variance [x] ≡ σ 2 (x) ≡ σ x 2 Covariance [x,y] ≡ σ x,y – If x and y are the same variable, then – σ y,y ≡ σ x,x ≡ σ x 2 ≡ σ y 2 Correlation coefficient ≡ ρ x,y September 15, 17, 2015

Some Definitions  1,2   1,2 1212  = √  2 (X i1 - µ i ) September 9, 2014

Harry Markowitz September 9, 2014

Mean-Variance (Harry Markowitz, 1955) Each asset defined as: – Probability distribution of returns – Mean and Variance of the distribution known – Assume no riskless asset (all variances > 0) Portfolio is – A collection of assets with a mean and a variance that can be calculated – Also an asset (no difference between portfolio and an asset) September 15, 17, 2015

Diagram with 2 Assets September 15, 17, 2015 Mean Standard Deviation = √(Variance) Asset 1 (μ 1, σ 1 ) Asset 2 (μ 2, σ 2 )

Now combine asset 1 and 2 into portolios consisting only of assets 1 and 2 September 15, 17, 2015 Mean Asset 1 (μ 1, σ 1 ) Asset 2 (μ 2, σ 2 ) Portfolio (μ P, σ P ) σ Where should the portfolio be in the diagram? Asset 1 (μ 1, σ 1 ) Asset 2 (μ 2, σ 2 ) Portfolio (μ P, σ P ) σ

An Efficient Portfolio Definition: – There is no other portfolio with: The same standard deviation, but higher mean The same mean, but lower standard deviation All efficient portfolios (there are infinitely many of them) lie on the “efficient frontier” September 15, 17, 2015

Efficient Frontier September 15, 17, 2015 Mean σ σ This is the main contribution of Markowitz and Is usually referred to as “mean-variance” theory

Investors will Choose some portfolio among those on the efficient frontier Those who wish less risk choose portfolios that are further to the left on the efficient frontier. These portfolios are those with lower mean and lower standard deviation Investors desiring more risk move to the right along the efficient frontier in search of higher mean, higher standard deviation portfolios September 15, 17, 2015

Portfolio Choice September 15, 17, 2015 Mean σ σ Less risk More risk

James Tobin (Yale) Suppose there is a riskless asset Such an asset with have a mean (the risk free rate) and zero variance of return There may be other riskless assets, but “the” riskless asset is the riskless asset with the highest mean return (which is the risk free rate) September 15, 17, 2015

Adding a Risk-Free Asset September 15, 17, 2015 Mean σ σ Tangency picks out a specific portfolio All portfolios below the line are now feasible

Why are portfolios below the line from the risk free rate tangent to the efficient frontier now feasible? The risk free rate has mean r and standard deviation zero: – Mean of any two assets is equal to: λ µ 1 + (1 – λ) µ 2 where 0 < λ < 1 Where λ is the proportion of the new portfolio that consists of asset 1 and (1 – λ) is the proportion of the new portfolio that consists of asset 2. – Variance (or standard deviation) is more complicated Var (New Portfolio) = λ 2 Var(1) +(1-λ) 2 Var(2) +2 λ(1-λ)Covar(1,2) September 15, 17, 2015

Proof that adding risk free asset creates a “straight line” boundary – Var (New Portfolio) = λ 2 Var(1) +(1-λ) 2 Var(2) +2 λ(1-λ)Covar(1,2) – But if asset 2 is the risk free asset then: Var(2) = 0 (by definition) Covar(1,2)= 0 since 2 never changes – Thus: Var (New Portfolio) = λ 2 Var(1) – Taking square roots of both sides: – Standard Deviation (New Portfolio) = λ*Stddev(1) September 15, 17, 2015