Problem Set 6 - Solutions

Transcription

1 ECO573 Financial Economics Problem Set 6 - Solutions 1. Debt Restructuring CAPM. a Before refinancing the stoc the asset have the same beta: β a = β e = 1.2. After restructuring the company has the same beta as before β a = 1.2. b The beta of debt is 0. Therefore the beta of the stoc can be derived as follows by linearity of betas: β a = 1 2 β e implying that β e = 2.4. c Before restructuring: Er e r f = 1.2r M r f = 11% implying that r M = 10%. Then using CAPM again r e = r f + 2.4r M r f = 17%. 2. Dividends Prices. By CAPM the expected rate of return on the stoc is Er e = 3% % 3% = 10.2%. Then we can also write that 1 + Er e = P t+1 + d t+1 P t hence P t+1 = 1 + Er e P t d t+1 = = CARA-Normal. Clearly in a CARA-normal framewor the investor has mean variance preferences over consumption. And the same decomposition of the investor program as in the lectures shows that we can separate the choice of the optimal return the choice of total investment so that the optimal return is the result of optimizing with mean-variance preferences over returns. That gives us the two fund theorem. We do the calculations of the CCAPM with a slightly different method from the class using directly the decomposition of the investor program. So let c i R be a solution of the overall program of the investor we can write that R = θ i R + θ f i Rf where θ i = 1 θ f i that θ i : max θ i E exp { ρ 1 i e i t+1 + e i t c i θ i R + 1 θ i R f }. R is a normally distributed vector with mean ER variance Γ. So when e i t+1 is certain we can rewrite the objective function as: ρ 1 i e i t+1 + e i t c i θ i ER + 1 R f 1 2 ρ 2 i e i t c i 2 θ iγθ i And the first order condition with respect to θ i gives: Aggregating that gives us: ρ i ER R f = e i t c i cov R θ i R. ρer R f = e t c cov R R M. 1

2 We can write the same pricing formula for the maret return: ρer M R f = e t c var R M taing the ratio we obtain the CAPM pricing formula: ER R f = cov R R M ρer M R f. var R M Now rewriting the FOC in a vector form we have: Γθ i = ρ i e i t c i Hence the individual portfolio is given by: ER R f 1 K θ i = ρ i Γ 1 ER R e i t c f 1 K i θ f i = 1 θ i. We can rewrite the expression above in terms of total portfolio rather than weights since θ i = e i t c i θ i we have: θ i = ρ i ρ θ M where θ M is the maret portfolio. We see how each agent is holding a fraction of the maret portfolio that depends on her level of ris aversion. Now in the case where e i t+1 is normally distributed we write the objective function as: ρ 1 i E e i t+1 + e i t c i θ i ER + 1 R f 1 2 ρ 2 i var e i t+1 + e i t c i θ i R the FOC with respect to θ i now gives: ρ i ER R f = cov R e i t+1 + e i t c i θ i R. Aggregating: ρer R f = cov R e i t+1 + e t c R M = cov R c t+1 which is the CCAPM pricing formula in the CARA normal framewor: ER R f = cov R c t+1 ρ yielding: ER R f = cov R c t+1 ER R f. cov R M c t+1 4. Efficient Frontier a Two Fund Theorem. a We have: R θ s = θ d s q θ = q θ R s q θ = θ R s. Hence ER θ = Eθ R varr θ = θ Γθ. 2

3 b The set of attainable returns is the set R = { r R S r = θ R θ = 1 }. It is easy to see that it is convex every return in R has a nonnegative variance. To find the efficient frontier as defined in the class we can minimize on R the variance of returns for any given mean m in R. This is exactly what the program is doing. This is the set of points that could be optimal for a consumer with mean-variance preferences. c Letting 2λ 2µ be the Lagrange multipliers the program becomes: min θ Γθ + 2λ m Eθ R + 2µ 1 θ θ d The first order conditions for the Lagrangian in b are given by the equation: Together with the constraints Γθ λer µ1 K = 0. θ ER = m θ = 1 they provide a system of necessary sufficient conditions in θ λ µ because the objective function is strictly convex. Let θ 0 be a solution of this program for m 0 θ 1 a solution for m 1 m 0. Then for any m 2 there exists a unique real number α such that αm αm 1 = m 2. We will show that the portfolio θ 2 = αθ αθ 1 is a solution of our program for m 2. TThe fact that θ 0 θ 1 are solutions implies that there exist scalars λ 0 λ 1 µ 0 µ 1 such that θ 0 µ 0 λ 0 solves the system of equations above for m = m 0 θ 1 µ 1 λ 1 solves the system of equations for m = m 1. We will show that this implies that θ 2 µ 2 λ 2 solves the same system for m = m 2 where λ 2 = αλ αλ 1 µ 2 = αµ αµ 1. First it is easy to see that θ2 = α θ0 + 1 α θ1 = 1. Second: Finally: θ 2 ER = αθ 0 ER + 1 αθ 1 ER = αm αm 1 = m 2. Γθ 2 λ 2 ER µ 2 1 K = α Γθ 0 λ 0 ER µ 0 1 K + 1 α Γθ 1 λ 1 ER µ 1 1 K = 0. 3

4 e To do that we solve the program. The FOC for the Lagrangian is Γθ = λer + µ1 K where 1 K is the vector of dimension K with ones everywhere. Together with the two constraints we have a system with K + 2 equations K + 2 unnowns θ λ µ. Then we have θ = Γ 1 λer + µ1 K. Replacing in the constraints yields: λ ER Γ 1 ER +µ ER Γ 1 1 K = m a b λ 1 KΓ 1 ER +µ 1 K Γ 1 1 K = 1. =b c The determinant of this system is = ac b 2 > 0 inverting we have: λ = cm b µ = a bm. Then multiply the first-order condition to find that at the optimal weight vector we obtain for a mean m a minimal variance of: σ 2 = θ Γθ = λeθ R + µθ 1 K = λm + µ. Hence we obtained the following equation for σ as a function of m: or σ 2 = cm2 + a 2bm c σ2 m b 2 = c c. 2 This characterizes the efficient frontier it is the equation of a hyperbola in the σ m space. 5. CAPM in Incomplete Marets. a We can write the program of the agent as follows: u i tc + βeu i t+1 e t c R max c R s.t. R R where R = { ρ R S ρ = θ f R f + θ R θ f + θ = 1 }. And letting c i Ri be a maximizing pair we have R = arg max 2a i E R e i t c i E R 2 + var R. R R Also note that when R is pinned down we can recover c by the first FOC of the first program: u i t c i = βe R u i t+1 e i t c i R. 4

5 b We wrote R as the optimal portfolio for an investor with mean-variance preferences so the two-fund theorem of the course applies. c We can rewrite the second program as { max E u i t+1 e i t c i θ i R + 1 } R f. θ i θ f i And taing the FOC with respect to multiplying by a i : { }} E R R f a i e i t c i {θ i R + θ fi Rf = 0 Aggregating but remember that the maret portfolio is not determined by a fixed supply here it is endogenous: E { R R f a e t c R M} = 0 Hence: ER R f a e t c ER M = e t c cov R R M The same expression must hold for R M so: ER M R f a e t c ER M = e t c var R M And that gives us the CAPM pricing equation: ER R f = cov R R M var R M ER M R f. 5