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) ≡ 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
i 1
Some Definitions
 2  i 1
n
xi
 x 

x
   x 
n
i
n
n
2
 = √ 2
(Xi1- µi )
n
  x, y  
 x   ( x)  y   ( y ) 
i 1
n
1,2 
September 9, 2014
1,2
12
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
Mean
Asset 2 (μ2, σ2)
Asset 1 (μ1, σ1)
Standard Deviation = √(Variance)
September 15, 17, 2015
Mean
Portfolio (μP, σP)
Asset 2 (μ2, σ2)
Asset 1 (μ1, σ1)
σ
Where should the portfolio be in the diagram?
September 15, 17, 2015
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
Mean
This is the main contribution of Markowitz and
Is usually referred to as “mean-variance” theory
σ
September 15, 17, 2015
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
Mean
More risk
Less risk
σ
September 15, 17, 2015
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
Mean
Tangency picks out a specific
portfolio
All portfolios below the line are now feasible
σ
September 15, 17, 2015
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 µ1 + (1 – λ) µ2 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) = λ2Var(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) = λ2Var(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) = λ2Var(1)
– Taking square roots of both sides:
– Standard Deviation (New Portfolio) = λ*Stddev(1)
September 15, 17, 2015
September 15, 17, 2015