Economics 434 Theory of Financial Markets

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Economics 434Theory of Financial Markets
Professor Edwin T Burton Economics
Department The University of Virginia
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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
  Harrys story
• Bill Sharpe (and Lintner and Mossin, etal)
• Put Tobins Result in Equilbrium
• The Rise of Beta
• The Insignificance of own variance

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Modern Portfolio Theory

• Randomness
• Construction of efficient (best) portfolios

Harry Markowitz Winner of Nobel Prize, 1990
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Consider, again, the ½, ½ Portfolio
Mean of X1
Mean of P
Mean of X2
Where P ½ X1 1/2 X2
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But, What About Standard Deviation?

• Is It Linear?
• Consider the formula for variance
• (? (Xi MeanX)2)/n
• (?(Xi-X)2

n
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New Symbols
?X1 Mean X1
?2 X1 VarianceX1 ?21
Standard Deviation X1 ?1
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Variance of a Portfoliowith two assets
? P ? (P - ?P)2
n
??1(X1- ?1) ?2(X2 - ?2)2
n
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Variance with two assets
??1(X1- ?1) ?2(X2 - ?2)2
?2P
n
? (?1)2X1 - ?12 (?2)2X2 - ?22
2?1?2(X1 - ?1)(X2 - ?2)
n
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Variance with 2 Assets - Continued
? (?1)2X1 - ?12 (?2)2X2 - ?22
2?1?2(X1 - ?1)(X2 - ?2)
n
n
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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
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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
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Variance with 2 Assets - Continued
(?1)2?12 (?2)2?22 2?1?2?1,2?1?2
?1,2
?1,2 ?
where
?1?2
What Happens if ? 1?
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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)
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If ? 1
?P (??1?1 ?2?2)
If ?? ? 1
?P ? (??1?1 ?2?2)
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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
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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
Where P ½ X1 ½ X2
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If ? 1
Then all the portfolios are here
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If ? ? 1
Then all the portfolios are here
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This Means the boundaryof the possible
portfolioslooks like this
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This is very convenient
What is This?
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This is very convenient
Mean
Maximizes Utility
Standard Deviation
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Tobins Result

• If there is a riskless asset
• It changes the feasible set
• All optimum portfolios contain
• The risk free asset and/or
• The portfolio E
• .in some combination.
• The Mutual Fund Theorem

James Tobin, Prof of Economics Yale
University Winner of Nobel Prize in Economics 1981
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The risk free asset
Mean
The one with the highest mean
Standard Deviation
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Combine with Risky Assets
Mean
?
Risky Assets
Risk Free Asset
Standard Deviation
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Recall the definition of the variance of a
Portfoliowith two assets
? P2 ? (P - ?P)2
n
??1(X1- ?1) ?2(X2 - ?2)2
n
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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
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If ?1 is zero
? P2 (?1)2?12 (?2)2?22 2?1?2?1,2?1?2
If one of the standard deviations is equal to
zero, e.g. ?1 then
? P2
(?2)2?22
(?2)?2
? P
Which means that
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Combine with Risky Assets
Mean
Risk Free Asset
Standard Deviation
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Combine with Risky Assets
Mean
The New Feasible Set
E
Always combines the risk free asset With a
specific asset (portfolio) E
Risk Free Asset
Standard Deviation
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Tobins Result
Mean
Use of Leverage
E
Risk Free Asset
Standard Deviation
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The End