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markowitz - Chapter 11 of Corporate Finance Modern...

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Chapter 11 of Corporate Finance Modern Portfolio Theory & Capital Asset Pricing Theory (CAPM) Lecture Notes for Actsc 372 - Fall 2011 Ken Seng Tan Department of Statistics and Actuarial Science University of Waterloo K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 1 Topics Covered Risk and return (reward) tradeoffs, investment opportunity/feasible set and efficient portfolio two risky assets N risky assets one risk-free and one risky asset one risk-free and N risky assets Modern Portfolio Theory Markowitz portfolio theory (1952) mean-variance efficient portfolio Nobel Prize for Economics 1990 Capital Asset Pricing Theory (CAPM) William Sharpe (1964) Nobel Prize for Economics 1990 John Lintner (1965), Jan Mossin (1966) K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 2 Types of Securities Treasury Bills (T-bills) Bonds Government bonds (Treasury bonds) Corporate bonds Stocks large, medium, small caps, index (S&P 500, TSX 300) riskfree (or riskless) vs risky assets riskfree rates: r f rate of return for investing in T-bills returns from other assets are commonly expressed as a “spread" in excess of the riskfree return; i.e. r asset = r f + risk premium K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 3
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Return Statistics R A : rate of return r.v. (over one holding period) for asset A p i = Pr( R A = x i ) , Expected return: E( R A ) = μ A = i p i x i , Variance: Var( R A ) = E[( R A μ A ) 2 ] = i p i ( x i μ A ) 2 = E[( R A ) 2 ] (E[ R A ]) 2 = σ 2 A Standard Deviation: s.d. ( R A ) = Var( R A ) = σ A , Covariance: Cov( R A , R B ) = E[( R A μ A )( R B μ B )] E( R A R B ) E( R A )E( R b ) Correlation: ρ AB = ρ = Cov( R A , R B ) σ A σ B K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 4 Estimating Return Statistics State Probability Stock A ( R A ) Stock B ( R B ) Recession 0.3 -2.0% 5.0% Normal 0.6 9.2% 6.2% Boom 0.1 15.4% 7.4% E( R A ) = 0 . 3( 0 . 02) + 0 . 6(0 . 092) + 0 . 1(0 . 154) = 6 . 46% = μ A Var( R A ) = 0 . 3( 0 . 02 μ A ) 2 + 0 . 6(0 . 092 μ A ) 2 + 0 . 1(0 . 154 μ A ) 2 = 0 . 003397 = E[( R A ) 2 ] [E( R A )] 2 = 0 . 00757 0 . 0646 2 σ A = Var( R A ) = 0 . 003397 = 5 . 83% E( R B ) = 5 . 96% s.d. ( R B ) = 0 . 72% Cov( R A , R B ) = 0 . 000412 ρ = Cov( R A , R B ) σ A σ B = 0 . 98 K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 5 Historical Estimates Recall that if { ˆ r i ; i = 1 , . . . , N } is the empirical rate of return in period i Then sample mean and sample variance are estimated as: E( R ) = 1 N N i =1 ˆ r i = ˆ μ Var( R ) = 1 N 1 N i =1 r i ˆ μ ) 2 = 1 N 1 N i =1 ˆ r 2 i N ˆ μ 2 = ˆ σ 2 Similarly for estimating sample covariance and sample correlation K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 6
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Portfolio of Two Assets with Return r.v. ( R 1 , R 2 ) Dollar amount w 1 in asset 1 and w 2 in asset 2 Portfolio P Asset 1 Asset 2 dollar amt. invested w 1 w 2 w = w 1 + w 2 % amt. invested x 1 = w 1 w x 2 = w 2 w x 1 + x 2 = 1 Let R P be the rate of return r.v. for the above portfolio, then R P = dollar return over the period initial investment amount = 1 w ( w 1 R 1 + w 2 R 2 ) = x 1 R 1 + x 2 R 2 E( R P ) = μ P = E( x 1 R 1 + x 2 R 2 ) = x 1 μ 1 + x 2 μ 2 Var( R P ) = σ 2 P = Var( x 1 R 1 + x 2 R 2 ) = x 2 1 σ 2 1 + x 2 2 σ 2 2 + 2 x 1 x 2 · ρσ 1 σ 2 K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 7 Risk and Reward: Two Assets Case μ 1 8% σ 1 12% μ 2 14% σ 2 20%
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