Investments/Ivkovich Problem Set 5 Due: 02/23/2016

Instructions: Each group will turn in one solution to the problem set. Recall that late submission will not be allowed: problem sets must be submitted at the beginning of class on the due date. Please show your work: do not merely give a solution without adequate explanations and intermediate steps necessary to obtain the results. You will be graded not only on the basis of the accuracy of your results, but also on the clarity and completeness of your written solution. The total number of points for this problem set is 80.

1)(30 points) Consider the following description of the five assets in the economy:
asset 1 asset 2 asset 3 asset 4 asset 5
state 1 0 1 0 4 3
state 2 1 1 1 3 1
state 3 0 0 1 2 2
price 1/2 3/4 5/8 P4 1

a) (10 points): Use assets 1, 2, and 3 to construct the three Arrow-Debreu (A-D) securities for states 1, 2, and 3. For each of the three A-D securities, report (i) its composition in terms of assets 1, 2, and 3 (that is, the weights on 1, 2, and 3) and (ii) its price.

State 1:
q1*0+q2*1+q3*0 = 1
q1*1+ q2*1+q3*1= 0
q1*0+ q2*0+q3*1= 0
q1 = -1 , q2 = 1, q3 = 0
S1 = -1*1/2+1*3/4+0*5/8 = ¼
State 2:
q1*0+q2*1+q3*0 = 0
q1*1+ q2*1+q3*1= 1
q1*0+ q2*0+q3*1= 0
q1 = 1 , q2 = 0, q3 = 0
S2 = 1*1/2+0*3/4+0*5/8 = ½
State 3:
q1*0+q2*1+q3*0 = 0
q1*1+ q2*1+q3*1= 0
q1*0+ q2*0+q3*1= 1
q1 = -1 , q2 = 0, q3 = 1
S3 = -1*1/2+0*3/4+1*5/8 = 1/8

b) (10 points): Use A-D securities for states 1, 2, and 3 (constructed in (1a)) to price asset 4. Report the price P4 and show your work.

P4 = ¼*4+1/2*3+1/8*2 = 11/4
c) (10 points): Consider asset 5. Is there an arbitrage opportunity in this economy? If there is not, please explain why not. If there is, please describe the arbitrage strategy in terms of positions in the assets and the zero-cost riskless payoff to the strategy.

Yes.

2) (50 points) Consider two well-diversified portfolios A and B that conform to a two-factor model (the two shocks to factors, F1,t and F2,t,have zero means), that is, their alphas are equal to zero, and the risk-free rate (assume that you may both lend and borrow at the risk-free rate). All returns are expressed in percentage points:

rA,t= 10.4 + 0.5 F1,t + 0.7 F2,t ; rB,t= 20.3 + 1.5 F1,t + 1.4 F2,t ; rf = 3.

a) (10 points) Write the expressions for the expected rates of return on portfolios A and B according to the two-factor model (Hint: The general form is E(rX)= rf + ßX,1?1 +ßX,2?2;use all the specific quantities as given above).

b) (10 points) What is the risk premium per unit of factor-1 risk, ?1? What is the risk premium per unit of factor-2 risk, ?2 (Hint: Use the two expressions developed under a) to set up a system of equations and solve for ?1 and ?2)

c) (10 points) Construct a portfolio P-1 out of A and B that is not sensitive to F1. Report its weights wA and wB. How sensitive is P-1 to F2 (how many units of factor-2 risk)? Using these calculations, produce a factor portfolio for factor 2, FP2, by combining P-1 with the risk-free asset. Report the portfolio composition of FP2, that is, the weights that FP2 places on A, B, and the risk-free asset.