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Transcript 
Economics 434
Theory of Financial Markets
Professor Edwin T Burton
Economics Department
The University of Virginia
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Modern Portfolio Theory
 Three Significant Steps to MPT
 Harry Markowitz
 Mean Variance Analysis
 The Concept of an “Efficient Portfolio”
 James Tobin
 What Happens When You Add a “Risk Free Asset” to
Harry’s story
 Bill Sharpe (Treynor Lintner, Mossin, etal)
 Put Tobin’s Result in Equilbrium
 The Rise of Beta
 The Insignificance of “own variance”
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Need Some Statistics
 Mean, Variance, Covariance
 Standard Deviation
 Mean --- roughly means ‘average’
 Variance – squared differences summed up
 Covariance – two variable, take differences of each from their
means, multiply the differences observation by observation
 Standard Deviation = Square Root of Variance
 We will typically use normal distribution – but that is not
necessary. We could use other distributions
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Because of uncertainty, we need a
different utility function
 Why not just expected value?
 Instead, we use expected utility
 What’s the difference?
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
i 1
Some Definitions
 2  i 1
n
xi
 x 

x
   x 
n
i
n
n
2
 = √ 2
n
  x, y  
 x   ( x)  y   ( y ) 
i 1
n
1,2 
Economics 434 – Financial Market Theory
1,2
12
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Modern Portfolio Theory
 Randomness
 Construction of efficient
(best) portfolios
Harry Markowitz
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
How Do You Create A Portfolio?
Try It with Two Assets
Mean
“X1”
“X2”
Where are the Portfolios That Can
Be Created from Just These Two
Assets ?
Standard Deviation
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Back to the ½, ½ Portfolio
Mean of
X1
Mean of P
Mean of
X2
Economics 434 – Financial Market Theory
Where P = ½ [X1] + 1/2 [X2]
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Variance of a Portfolio
with two assets
 P2 =  (P - P)2
n
= {1(X1- 1) + 2(X2 - 2)}2
n
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Variance – with two assets
2
P
2
{
(X

)
+

(X

)}
1
1
1
2
2
2
=
n
= { (1)2[X1 - 1]2 + (2)2[X2 - 2]2}
+ 212(X1 - 1)(X2 - 2)}
n
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Variance with 2 Assets Continued
=  {(1)2[X1 - 1]2 + (2)2[X2 - 2]2
n
+ 212(X1 - 1)(X2 - 2)}
n
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Variance with 2 Assets - Continued
= (1)212 + (2)222 + { 212(X1 - 1)(X2 - 2)}
n
= (1)212 + (2)222 + 212Cov (X1,X2)
= (1)212 + (2)222 + 2121,2
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Variance with 2 Assets - Continued
= (1)212 + (2)222 + 2121,2
Recall the definition of the correlation coefficient:
1,2 
1,2
12
= (1)212 + (2)222 + 2121,212
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Variance with 2 Assets - Continued
= (1)212 + (2)222 + 2121,212
where
1,2 
1,2
12
What Happens if  = 1?
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
If  = 1
(1)212 + (2)222 + 2121,212
becomes
= (1)212 + (2)222 + 21212
2P= (11 + 22)2
P= (11 + 22)
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
If  = 1
P= (11 + 22)
If   1
P  (11 + 22)
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Back to the ½, ½ Portfolio
If  = 1
Mean of
X1
Mean of P
Mean of
X2
Where P = ½ [X1] + 1/2 [X2]
1,2 = 1/21 + 1/22
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
Back to the ½, ½ Portfolio
If   1
1,2  1/21 + 1/22
Mean of
X1
Mean of P
Mean of
X2
Economics 434 – Financial Market Theory
Where P = ½ [X1] + ½ [X2]
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
If  = 1
Then all the portfolios are here
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
If   1
Then all the portfolios are here
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
This Means the “boundary”of the possible portfolios
looks like this:
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
This is very convenient
Mean
Maximizes
Utility
Standard Deviation
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
The End
Economics 434 – Financial Market Theory
Tuesday,
Tuesday,
September
August
24,
21,2010
2010
Tuesday,
Sept
25, 2012
Tuesday,
August
25,
2009
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