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Outline-5 - Ec 173AFINANCIAL MARKETS L ECTURE OUTLINE...

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Ec 173A—FINANCIAL MARKETS LECTURE OUTLINE Foster, UCSD July 16, 2010 TOPIC 5 – MODERN PORTFOLIO THEORY A. Introduction to Modern Portfolio Theory (MPT) 1. Background: a) Harry Markowitz, 1952 [Nobel, 1990]. b) The assumptions of MPT. 2. Portfolio Rates of Return: a) Consider a portfolio of one stock and one bond. b) Portfolio ex ante rate of return. b b s s p r r r ϖ ϖ + = . Proof: c) Portfolio returns and expected returns. . 3. Portfolio Risk: a) Covariance and correlation of ex ante rates of return. NOTATION -- for j { s, b } Symb ol Definition Notes P j Beginning price of stock or bond n j Number of units of asset in portfolio V j Total value of asset in portfolio V j = n j P j CF j Cash flows per unit of asset r j Ex ante rate of return on asset r j = (CF j + P j )/ P j V p Beginning portfolio price (value) V p = V s + V b ω j Proportion of portfolio value in asset ω j = V j /V p ; Σω j = 1 b b s s p b b b s s s p b b b b b s s s s s p p p p r r V r P n r P n V P P P CF n P P P CF n V V CF r ϖ ϖ + = + = + + + = + = ) ( ) ( ) ( ) ( ) ( ) ( b b s s p p r r r or r E μ ϖ μ ϖ μ + =
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Ec 173A MPT OUTLINE p. 2 of 11 b) Portfolio risk. 4. Example: [Table 1] 5. Generalization to N assets: a) Consider a portfolio with asset classes X i , i = 1...N. b) Ex ante portfolio return r p is a linear combination of asset returns r i . c) Information problems. 6. Principles of Diversification: a) Reduce risk thru portfolio diversification. b) Example. [Table 2] c) Notes. [Fig. 1] Table 1 State of World Pr( j) r s r b j=1: Drought 0.2 0 0.01 0.02 j=2: Normal 0.6 0 0.08 0.08 0.2 0 0.12 0.03 (r) = μ 0.070 0 0.058 0 (r) = σ 0.042 9 0.027 1 (r σ s , r b ) = 0.00046 (r ρ s , r b ) = 0.3952 Table 2 Portfolio Return Portfolio Risk = (r σ p ) ω s ω b (r μ p ) = 0 ρ = +1 ρ = 1 ρ 100 % 70% 50% 44 % 30% 0% 0% 30% 50% 56 % 70% 100 % 10.0 % 8.80 % 8.00 % 7.75 % 7.20 % 6.00 % 0.090 0 0.066 4 0.057 0 0.055 7 0.055 9 0.070 0 0.090 0 0.084 0 0.080 0 0.078 8 0.076 0 0.070 0 0.090 0 0.042 0 0.010 0 0.000 0 0.022 0 0.070 0 ) ( ) ( ) , ( ) , ( ) ( ) ( ) Pr( ) , ( 1 b s b s b s b s K j bj sj b s r r r r r r n Correlatio r r j r r r r riance Cova σ σ σ ρ μ μ σ = - = = = = = + = = = i j j i j i N i i i p N i i i p N i i i p r r r r Variance Risk Portfolio r r return Expected r r return Portfolio ) , ( 2 ) ( ) ( : ) ( ) ( : ; : 1 2 2 2 1 1 σ ϖ ϖ σ ϖ σ μ ϖ μ ϖ Var(r p ) = σ 2 (r p ) = ω s 2 σ 2 (r s ) + ω b 2 σ 2 (r b ) + 2ω s ω b σ(r s ,r b )
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Ec 173A MPT OUTLINE p. 3 of 11 7. Hedging: • B • S (r μ p ) 10 9 8 7 6 0 7 9 (r σ ) ρ = −1 Fig. 1 -- Portfolio = 0 ρ = ρ
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Ec 173A MPT OUTLINE p. 4 of 11 B. Efficient Portfolios 1. Two Risky Assets: [Fig. 2] 2.
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