Solutions to Financial Assignment 6

Suggestion Solution of Assignment 6 for SEEM 3590 Investment Science Problem 1 (6pts): (a). The Capital market line has the following formula: r = rf + Thus, we have r = 0.07 + (b). (i) 0.39 = 0.07 +

σ 2

rM − rf σ σM

0.23 − 0.07 σ σ = 0.07 + 0.32 2

⇒ σ = 64%

(ii) We construct a portfolio by allocating weight w to market portfolio, (1 − w) to risk-free asset. Solve 0.07w + 0.23(1 − w) = 0.39, yielding w = −1, Hence, borrow $1000 at the risk-free rate; invest $2000 in the market. (c). 300(1 + 7%) + 700(1 + 23%) = 1182 Problem 2 (6pts):

1 (a). Since M = 2 (A + B), we have rM = 1 (rA + rB ) 2 2 σM

σAM σBM ⇒ βA

1 2 2 = var(rM ) = (σA + 2σAB + σB ), 4 1 = cov(rA , rM ) = cov(rA , (rA + rB )) = 2 1 = cov(rB , rM ) = cov(rB , (rA + rB )) = 2 2 2σA + 2σAB σAM = = 2 2 2 σM σA + 2σAB + σB 2σ 2 + 2σAB σBM = 2 B 2 2 σM σA + 2σAB + σB

1 2 (σ + σAB ), 2 A 1 2 (σ + σAB ), 2 B

⇒ βB =

(b). According to the CAPM, we have ri = rf + βi (rM − rf ) 5 ⇒ rA = 0.1 + (0.18 − 0.1) = 20%, 4 3 ⇒ rB = 0.1 + (0.18 − 0.1) = 16%, 4

1

Problem 3 (5pts): Since the market portfolio in normalized form is x = (x1 , x2 , ..., xn ), we have rM = since these n assets are mutually uncorrelated. Then, σiM

2 σM 2 = cov(ri , rM ) = xi σi , n n i=1 xi ri ,

= var(rM ) = σiM 2 = σM

2 x2 σj , j

⇒ βi =

j=1 2 xi σi n 2 2 j=1 xj σj

Problem 4 (8pts): (a). The market consists of $150 in capital value of A and $300 capital value of B. Hence, the market return is rM = (b).

2 σM

300 1 2 150 rA + rB = rA + rB = 13% 450 450 3 3

= =

1 2 σ + 9 A 1 2 σ + 9 A

4 2 2 σAB + σB 9 9 4 2 2 ρAB σA σB + σB = 0.0081 9 9

⇒ σM = 0.09 (c). 2 1 2 σAM = σA + ρAB σA σB = 0.0105 3 3 ⇒ βA =

σAM 2 σM

= 1.2693

(d). Since Simpleland satisfies the CAPM exactly, stocks A and B plot on the security market line. Specifically, rA − rf = βA (rM − rf ) Hence, rf = 0.0625

2

Problem 5 (5pts): E(s2 ) = E[ 1 n−1 1 n−1

n

r) (ri − ˆ 2 ] = E[

i=1 n

1 n−1

n

(ri −

i=1...