Economics 434: The Theory of Financial Markets Professor Burton Fall 2016 Oct 6, 2016
Prof Burton Talk in Winchester Wednesday Shenandoah Valley Museum: Business Advisory Luncheon Talk is about “Economic Growth: Where Has it Gone” Oct 6, 2016
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
Today: Markowitz on Mean-Variance Theory Oct 6, 2016
Need Mathematical Concepts Mean Variance Covariance Correlation Coefficient September 15, 17, 2015
Correlation coefficient ≡ ρx,y Symbols Mean [x] ≡ µ(x) ≡ µx Variance [x] ≡ σ2(x) ≡ σx2 Covariance [x,y] ≡ σx,y If x and y are the same variable, then σy,y ≡ σx,x ≡ σx2 ≡ σy2 Correlation coefficient ≡ ρx,y September 15, 17, 2015
= √ 2 1,2 1,2 12 Some Definitions (Xi1- µi ) 1,2 12 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
Consider the Case of Only Two Assets Use the mean/standard deviation diagram Identify an asset as a point in the diagram Given any two assets Where are all the portfolios that can be created from just two assets? Oct 6, 2016
Diagram with 2 Assets Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) Mean Standard Deviation = √(Variance) September 15, 17, 2015
Now combine asset 1 and 2 into portolios consisting only of assets 1 and 2 Mean Asset 2 (μ2, σ2) Asset 2 (μ2, σ2) Portfolio (μP, σP) Portfolio (μP, σP) Asset 1 (μ1, σ1) Asset 1 (μ1, σ1) σ σ Where should the portfolio be in the diagram? September 15, 17, 2015
First, where will the mean of a portfolio that has half of one asset and half of another? Portfolio (μP, σP) Portfolio (μP, σP) Where is the portfolio that Is half Asset 1, half Asset 2? Asset 1 (μ1, σ1) Asset 1 (μ1, σ1) σ σ Oct 6, 2016
Mean of a portfolio of two assets: 𝑛 𝑡=1 λ𝑋 1 + 1−λ( 𝑋 2 ) 𝑛 Where 0 ≤ λ ≤ 1 “n” is the number of time periods, beginning with t = 1 And X1 is the return of asset one and X2 the return of asset two (should be indexed by time) Thus Mean of Portfolio = λ times Mean of Asset 1 + (1 – λ) times Mean of Asset 2 Or µP = Portfolio Mean of Two Assets = λµ1 + (1-λ)µ2 Oct 6, 2016
Mean of ½ Asset 1 and ½ Asset 2 μ2 Asset 2 (μ2, σ2) Mean of half/half Asset 1 (μ1, σ1) μ1 Standard Deviation = √(Variance) September 15, 17, 2015
But where with the standard deviation of the half/half portfolio be? Mean Asset 2 (μ2, σ2) ? ? ? Asset 1 (μ1, σ1) Oct 6, 2016
Now, some mathematics: Oct 6, 2016
Variance of a Portfolio with two assets P2 = (P - P)2 n = {λ(X1- 1) + (1-λ)(X2 - 2)}2 n October 4, 2016
After some mild heavy lifting: σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ Cov(1,2) Note that Cov(1,2) ≡ σ1,2 So: σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ σ 1,2 Oct 6, 2016
Make use of correlation coefficient: σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ σ 1,2 ρ 1,2 ≡ σ 1,2 σ 1 σ 2 σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ ρ 1,2 σ 1 σ 2 Oct 6, 2016
σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ ρ 1,2 σ 1 σ 2 What happens when ρ1,2 = 0 ? σ 𝑃 2 = λ σ 1 + 1−λ σ 2 2 Now, take square roots of both sides: σ 𝑃 = λ σ 1 + 1−λ σ 2 If λ= ½, then σ 𝑃 = 1 2 σ 1 + 1 2 σ 2 Oct 6, 2016
So, the half/half case is: σ 𝑃 = 1 2 σ 1 + 1 2 σ 2 Mean μ2 Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) μ1 σ1 ½ σ1 + ½ σ2 σ2 Oct 6, 2016
If ρ < 1 σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ σ 1,2 σ 𝑃 2 = λ 2 σ 1 2 + 1−λ 2 σ 2 2 +2λ 1−λ σ 1,2 The right hand side will be smaller than before This implies a smaller variance of P And a smaller standard deviation Oct 6, 2016
So, if σ < 1 σ1 ½ σ1 + ½ σ2 σ2 Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) P will lie to the left of the Line joining the Assets Mean μ2 Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) μ1 σ1 ½ σ1 + ½ σ2 σ2 Oct 6, 2016
Portfolio Choice Mean More risk Less risk σ σ September 15, 17, 2015
Oct 6, 2016