15 CAPM and portfolio management


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1 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 1 15 CAPM and portfolio management 15.1 Theoretical foundation for meanvariance analysis We assume that investors try to maximize the expected utility, E[U(W)], where W is their wealth. Consider a Taylor expansion of the utility function around E[W] (the expected wealth): U(W) = U(E[W]) + U (E[W]) (W E[W]) U (E[W]) (W E[W]) 2 + n=3 1 n! U(n) (E[W]) (W E[W]) n
2 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 2 Next, take the expectation: E[U(W)] = U(E[W]) + U (E[W])E [(W E[W])] U (E[W])E [ (W E[W]) 2] + i nfty 1 n! U(n) (E[W])E [(W E[W]) n ] n=3 With a meanvariance analysis we stop at the second order. There are two cases where this can be justified: If W is normally distributed, then the first two moments characterize all the moments. If the utility is quadratic, then U(W) (n) = 0 for n 3.
3 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management Portfolio with minimum variance Simple case: 2 assets Consider the following two assets x 1 and x 2 with V ar[x 1 ] = σ1, 2 V ar[x 2 ] = σ2 2 and Cov[x 1, x 2 ] = σ 12 w 1 : the weight of the first asset in the portfolio 1 w 1 : the weight of the second asset in the portfolio Denote by σp 2 the variance of the portfolio: σp 2 = V ar[w 1 x 1 + (1 w 1 )x 2 ] = w1σ (1 w 1 ) 2 σ w 1 (1 w 1 )σ 12
4 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 4 We want to find the minimum variance portfolio: min w1 σp 2 The First Order Condition (FOC) is: 2w1σ 1 2 2(1 w1)σ (1 w1)σ w1( 1)σ 12 = 0 Solving for w1, we get w 1 = σ 2 2 σ 12 σ σ 2 2 2σ 12
5 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 5 Diversification principle: Look at the FOC when we take w 1 = 0: σ 2 p w 1 (w 1 = 0) = 2(σ 12 σ 2 2) = 2(ρ 12 σ 1 σ 2 σ2) 2 ( = 2σ 1 σ 2 ρ 12 σ ) 2 σ 1 If ρ 12 < 0 or if ρ 12 > 0 but σ 2 /σ 1 > ρ 12, then σ2 p w 1 (w 1 = 0) < 0 so I should increase w 1 (i.e. buying x 1 ). If ρ 12 > 0 but σ 2 /σ 1 < ρ 12, then σ2 p w 1 (w 1 = 0) > 0 so I should decrease w 1 (i.e. shortsell x 1 ).
6 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management Case with N assets Consider the following elements: w = (w 1, w 2,...,w N ) : portfolio weights x = (x 1, x 2,...,x N ) : asset returns x = ( x 1, x 2,..., x N ) : expected asset returns the variance matrix σ1 2 σ 12 σ 1N σ 12 σ2 2 σ 2N Σ = x p = w x: portfolio s return σ 2 p = w Σw: portfolio s variance σ 1N σ 2N σ 2 N
7 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 7 Define the following elements: i: N 1 vector of 1. A = i Σ 1 x B = x Σ 1 x C = i Σ 1 i D = BC A 2 (we can show that D > 0) Show that w Σw is the variance of the portfolio.
8 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 8 We want to characterize the meanvariance frontier (finding the portfolio with the lowest variance for a given expected return) The problem is This is a constrained optimization We write the Lagrangian min 2 w Σw subject to i w = 1 w 1 x w = µ
9 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 9 L = 1 2 w Σw + γ(1 i w) + λ(µ x w) The FOCs are L w = L/ w 1 L/ w 2. = Σw γi λ x = 0 (1) L/ w N L = 1 i w = 0 (2) γ L λ = µ x w = 0 (3)
10 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 10 From equation (1), we get w = γσ 1 i + λσ 1 x We need to solve for γ and λ From equation (2), we know that 1 i w = 0 1 i [γσ 1 i + λσ 1 x] = 0 1 γi Σ 1 i λi Σ 1 x = 0 1 γc λa = 0 (4) From equation (3), we know that µ x w = 0 µ x [γσ 1 i + λσ 1 x] = 0 µ γ x Σ 1 i λ x Σ 1 x = 0 µ γa λb = 0 (5)
11 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 11 We can solve (4) and (5) for γ and λ. We get λ = Cµ A D γ = B Aµ D It follows that the optimal portfolio is [ ] [ ] B Aµ Cµ A w = Σ 1 i + Σ 1 x D D [ BΣ 1 i = D AΣ 1 x ] [ ] CΣ 1 x + D D AΣ 1 i µ D = g + hµ where g = 1 D [ BΣ 1 i AΣ 1 x ] h = 1 D [ CΣ 1 x AΣ 1 i ]
12 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 12 Variance of the portfolio when w = w σ 2 p = w Σw = w Σ(γΣ 1 i + λσ 1 x) = w [ γσσ 1 i + λσσ 1 x ] = w [γi + λ x] = γ w i }{{} =1 = γ + λµ +λ w x }{{} =µ = B Aµ D + = B 2Aµ + Cµ2 D ( ) Cµ A µ D parabola
13 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 13 Expected return and variance combination for ω=ω * µ 0 0 σ 2
14 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 14 We can find the global minimal variance σ 2 p µ = 2Cµ 2A D = 0 µ g = A C What is this variance? (σp) 2 g = B 2AA + C ( A C C D = BC 2A2 + A 2 = = 1 C CD BC A2 CD ) 2
15 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 15 Minimum variance portfolio Efficient portfolios A/C µ Inefficient portfolios 0 0 1/C σ 2
16 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 16 What is γ and λ for this µ g? λ g γ g = C ( A C) A D = B A( ) A C D = 0 expected return constraint not binding = 1 C What is the portfolio with minimum global variance? w g = γ g Σ 1 i + λ g Σ 1 x = 1 C Σ 1 i + 0Σ 1 x = 1 C Σ 1 i = Σ 1 i i Σ 1 i
17 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 17 If we go back to w (optimal portfolio for a given expected return): w = γσ 1 i + λσ 1 x ( ) ( ) Σ 1 i Σ 1 x = γc +λa C A } {{ } } {{ } =w g =w d We see that w is a combination of: The portfolio with the lowest global variance but lowest expected return (w g ). A second portfolio (w d ) that will increase expected return but will increase the variance.
18 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 18 But what is γc + λa? γc + λa = ( ) ( ) B Aµ Cµ A C + A D D BC ACµ = D BC A2 = D = 1 + ACµ A2 D Conclusion: (γc) and (λa) are the two shares of a portfolio
19 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management Covariance properties of minimal variance portfolios w g has a covariance constant with every asset or portfolio (= 1/C): Cov(x g, x p ) = E [ w g(x x)(x x) w p ] = w ge[(x x)(x x) ]w p = w gσw p ( ) i Σ 1 = Σw p C = i w p C = 1 C
20 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 20 Covariance of portfolio w d with any other portfolio: Cov(x d, x p ) = E [w d(x x)(x x) w p ] = w dσw p = xσ 1 A Σw p = xw p A = x p A We see that the expected return of any portfolio will be proportional to its covariance with w d since x p = ACov(x d, x p ).
21 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 21 Consider a portfolio a on the minimum variance frontier (w a = (1 a)w g + aw d ). What is the covariance between a and another portfolio p? Cov(x a, x p ) = (1 a)cov(x g, x p ) + acov(x d, x p ) If x p = x a, then = (1 a) 1 C + a x p A Cov(x a, x p ) = V ar(x a ) = 1 a C + a A x a
22 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 22 As long as a is not the minimum variance portfolio, it s possible to find a portfolio z that has a zero covariance with a: Cov(x a, x z ) = 1 a C x z = a 1 a + a x z A = 0 A C This is the expected return of a portfolio with zero covariance with any portfolio on the minimum variance frontier. We saw previously that V ar(x a ) = 1 a C + a A x a = a x z A + a A x a = a A ( x a x z ) using the result from previous slide
23 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 23 Next, define Using previous results β pa Cov(x a, x p ) V ar(x a ) β pa = 1 a + A x a C p a ( x A a x z ) [ 1 a ] β pa ( x a x z ) = A a C + A x a p [ ] (1 a)a + ac xp = A a = 1 a a A C + x p CA } {{ } = x z x p = x z + β pa ( x a x z )
24 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 24 The last equality is the CAPM equation without a riskfree asset. It is telling us that the expected return on any portfolio p (i.e. x p ) is equal to the expected return on a portfolio uncorrelated with portfolio a (i.e. x z ) plus β pa times the excess return of a over z. Portfolio a is a portfolio on the minimum variance frontier. Portfolio z is a portfolio uncorrelated with portfolio a.
25 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management Introduction of a riskless asset Now assume there is one more asset. This asset is riskless and has a riskfree rate r f. The problem is now subject to 1 min w 2 w Σw w x + (1 w i)r f = µ
26 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 26 We form the Lagrangian L = 1 2 w Σw + λ(µ w x (1 w i)r f ) The FOCs are: L w = Σw + λ( x + ir f) = 0 (6) L λ = µ w x (1 w i)r f = 0 (7) In equation (6) we can solve for w: w = λσ 1 ( x ir f ) (8)
27 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 27 We can solve for λ using equation (7): µ = w x + (1 w i)r f µ = [λσ 1 ( x ir f )] x + (1 [λσ 1 ( x ir f )] i)r f µ = λ( x r f i )Σ 1 x + (1 λ( x r f i Σ 1 i))r f µ = λ( x Σ 1 x r f i Σ 1 x) + r f λ( x Σ 1 i r f i Σ 1 i)r f µ = λ(b r f A) + r f λ(a r f C)r f µ r f = λ(b 2Ar f + Crf) 2 λ = µ r f H (9) where H = B 2Ar f + Cr 2 f Equation (9) into equation (8): ( µ w = Σ 1 rf ( x ir f ) H )
28 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 28 The variance of this portfolio is σ 2 p = w Σw [ ( µ = w Σ Σ 1 rf ( x ir f ) H ( )] µ = w [( x rf ir f ) H = (w x }{{} =µ = (µ r f) 2 H }{{} w i r f ) (µ r f) H =1 )]
29 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 29 Expected return and variance combination with a riskless asset µ 0 0 σ 2
30 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 30 Recall the parabola when we have N risky assets: σ 2 p = B 2Aµ + Cµ2 D Minimum variance portfolio Efficient portfolios µ g = A/C µ Inefficient portfolios 0 0 1/C σ 2
31 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 31 In the µ σ plane we get Expected return and standard deviation combination A/C µ 0 0 1/C 1/2 σ
32 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 32 It can be argued that in equilibrium we should have r f < A/C Expected return and standard deviation combination A/C µ r f 0 0 1/C 1/2 σ
33 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 33 If we create a portfolio by combining the riskfree asset with a portfolio b on the frontier, we could get the following combination of µ and σ Expected return and standard deviation combination A/C b µ r f 0 0 1/C 1/2 σ
34 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 34 The portfolio x b would not be optimal. It is possible to get a higher µ for the same σ by switching from b to m (a portfolio that is tangent) Expected return and standard deviation combination m A/C µ r f 0 0 1/C 1/2 σ A combination of any other risky portfolio with the riskfree asset would give less µ for the same σ.
35 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 35 It follows that everyone should choose a portfolio which falls on the r f m line. If you want higher pair (µ, σ), you put more weight on m. If you want lower pair (µ, σ), you put more weight on riskfree asset. The relative proportion of the risky assets should be the same regardless of where you are on the r f m line. We refer to m as the market portfolio. Your risk aversion will determine where on the r f m line you are * Higher risk aversion close to r f * Low risk aversion close to m or beyond m.
36 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 36 What is the CAPM equation when we have a riskfree asset? consider the portfolio m (which is a portfolio on the frontier), then for a portfolio q Cov(x q, x m ) = w qσw m [ = w qσ Σ 1 ( x ir f ) µ ] m r f H = w q( x ir f ) µ m r f H = (µ q r f )(µ m r f ) H
37 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 37 But we also know from slide 28, when we take µ = µ m, that σm 2 = (µ m r f ) 2 H µ m r f σm 2 = H (µ m r f ) We can next combine the last equation with the Cov(x q, x m ) equation on the previous slide to get σ 2 m Cov(x q, x m ) = (µ q r f ) µ m r f µ q r f = Cov(x q, x m ) (µ σm } {{ 2 m r f ) } =β qm Conclusion: the expected return of an asset/portfolio: Does not depend on its variance. Depends only on its covariance with the market.
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