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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 mean-variance 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 mean-variance 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. short-sell 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 mean-variance 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 risk-free 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 risk-free 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 risk-free 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 risk-free 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 risk-free 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 risk-free 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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