Lecture 6-4.ppt - Lecture 6 1 Introduction to Asset...

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Lecture 6 Lecture 6 Introduction to Introduction to Asset Allocation Asset Allocation and Modern and Modern Portfolio Theory Portfolio Theory 1
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Subjects Subjects Risk aversion Asset allocation Two risky assets Many risky assets A risky asset and a risk free asset Many risky assets and a risk free asset 2
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Introduction Introduction We will discuss modern portfolio theory. The main assumption: investors are mean-variance investors. Developed by Markowitz at the late 1950s (Nobel prize in 1990). 3
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Risk aversion Risk aversion 0 0 U U U W 4
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Risk aversion Risk aversion Million NOK 2 0 ½ ½ 5
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Indifference curve Indifference curve 6
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Indifference curves – which investor is more which investor is more risk averse risk averse ? ? 7
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Measuring risk aversion Measuring risk aversion How is risk aversion measured in practice? Of course we cannot ask people for the slopes of their indifference curves or for their coefficient of risk aversion.. One example: 9
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Mean-Variance criteria Mean-Variance criteria Security 1 dominates security 2 if • E(r 1 )>E(r 2 ) and σ 1 ≤σ 2
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Return distribution Return distribution Mean Variance Std. Deviation Covariance Correlation S s s r p r E ) ( r VAR S s s s r E r p r VAR 2 2 1 12 2 1 2 1 2 , 1 ) , ( r r COV 2 , 2 1 , 1 12 2 1 , r E r r E r p r r Cov s s s s
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Return distribution and Return distribution and past returns past returns The various moments of the return distribution characterize the distribution and serve us in making investment decisions. These moments are not observable. We can estimate them based on past returns .
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Estimating the moments Estimating the moments Arithmetic Mean Variance Std. Dev . Covariance Correlation T t t r r T r Var 1 2 1 1 ) ( r Var t t r T r 1 2 , 2 1 1 , 1 12 2 1 1 1 , r r r r T r r Cov t T t t 2 1 12 2 1 2 1 12 , r r Cov
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Sample Variance General Motors Stock Price 1962-2008
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Two risky assets Two risky assets A B E(r ( 3.5% 6.7% Var (r ( 0.00454 0.00802 STD (r ( 6.742% 8.956% COV(r A ,r B ( 0.001492 ρ (r A r B ( 0.247 15
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µ 6.7% 3.5% B 6.7% 8.95 % σ A Two risky assets Two risky assets Can we say which is better, A or B? 16
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Two risky assets Two risky assets Suppose we invest 70% in A and 30% in B % 43 . 4 % 7 . 6 3 . 0 % 5 . 3 7 . 0 ) ( P r E 2 P 17
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Two risky assets Two risky assets We have a portfolio with expected return of 4.43% (vs. 3.5% of A) and standard deviation of 5.97% (vs. 6.74% of A).
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