Topic 6 EfficientDiversification

Topic 6 EfficientDiversification - Topic6:...

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1 Topic 6: Efficient Diversification FIN 4504 Aaron Gubin
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2 Overview • In previous lecture we learned how to combine one   risky asset (or portfolio) and a risk-free asset to get  the CAL. • We also learned why and how different investors  choose different combinations (different weights) • This lecture we will learn how to combine more than  one  risky asset together.
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3 Overview • Question: Why is combining risky assets any  different than combining a risky asset and a risk-free  one? E(r) . A σ . B
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4 Diversification • Magic of diversification: A B (A + B) Volatile Volatile Less Volatile
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5 Combining Two Risky Assets ( ) ( ) ( ) where 1. P A A B B A B E r w E r w E r w w = + + = 2 2 2 , ( ) ( ) 2 P A A B B A B A B w w w w σ = + + Covariance: It measures how 2 stocks move together. , , , 1 1 ( )( ) 1 N A B A t A B t B t r r r r N = = - - - , ( )[ ( ) ( )][ ( ) ( )] A B A A B B s p s r s E r r s E r = - - , , Also A B A B A B ρ σ σ =
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6 Combining Two Risky Assets 2 2 2 , ( ) ( ) 2( )( ) P A A B B A A B B A B w w w w σ ρ = + + Correlation: Another way to measure how 2 stocks move together. , , and -1 1. A B A B A B σ σ =
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7 Combining Two Risky Assets 2 2 , ( ) ( ) 2( )( ) P A A B B A A B B A B w w w w σ ρ = + + Example : 2 risky assets: E(r A )= 9% σ A = 12% and E(r B )= 18% σ B = 25% and ρ A,B = -0.5 Draw the potential portfolio combinations using various weights. Use: ( ) ( ) ( ) where 1. P A A B B A B E r w E r w E r w w = + + = Ex: For w A =0.25 and w B =0.75: ( ) 0.25 0.09 0.75*0.18 0.1575 P E r = × + = 2 2 (0.25 0.12) (0.75 0.25) 2 0.25 0.12 0.75 0.25 ( 0.5) = 0.1744 P = × + × + × × × × × -
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8 Combination of two risky assets 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 STD E(r) A B MVP
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9 Combining Two Risky Assets 2 2 , ( ) ( ) 2( )( ) P A A B B A A B B A B w w w w σ ρ = + + Example : 2 risky assets: E(r A )= 9% σ A = 12% and E(r B )= 18% σ B = 25% Draw the potential portfolio combinations using various weights and correlations . Use: ( ) ( ) ( ) where 1. P A A B B A B E r w E r w E r w w = + + =
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10 Combinations of two risky assets 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 STD E(r) corr=1 corr=0.5 corr=0 corr=-0.5 corr=-1 A B
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11 Combining Two Risky Assets 2 2 , ( ) ( ) 2( )( ) P A A B B A A B B A B w w w w σ ρ = + + How is this possible: 2 2 A,B If 1 ( ) ( ) 2( )( ) P A A B B A A B B w w w w = ⇒ = + + Note the similarity with a 2 + b 2 + 2ab = (a + b) 2 2 A,B If 1 ( ) P A A B B A A B B w w w w = ⇒ = + = + A,B For 1 P A A B B w w < ⇒ < +
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12 Optimal Risky Portfolio With a  Risk-Free Asset E(r A )= 9% σ A = 12% E(r B )= 18% σ B = 25% ρ A,B = 0.3 and r f =4% 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.1 0.2 0.3 0.4 0.5 STD E(r) A B ORP
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13 Applications with two risky assets 1) 2 risky assets with a certain correlation. What is the expected return of the efficient portfolio given the standard deviation? What are the corresponding weights? Combination of two risky assets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.1 0.2 0.3 0.4 STD E( r )
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14 Applications with two risky assets 2) 2 risky assets with a certain correlation.
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This note was uploaded on 12/03/2009 for the course FIN 4504 taught by Professor Banko during the Fall '08 term at University of Florida.

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Topic 6 EfficientDiversification - Topic6:...

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