IAPM June 202 Second Semester Solutions The calculations are given below. A good answer requires both the correct calculations and an explanation of the calculations. Marks are lost if explanation is absent. 5. a. r p = X A r A + X C r C = 0.5 0.0 + 0.5 0.2 = 0. σ 2 p = X 2 Aσ 2 A + X 2 Cσ 2 C + 2X A X C ρ AC σ A σ C = 0.5 2 0.06 2 + 0.5 2 0.07 2 + 2 0.5 0.5 0.06 0.07 = 0.000025 b. r p = X A r A + X B r B = 0.0 + 2 0.4 = 0.2667 σ 2 p = XAσ 2 2 A + XBσ 2 2 B + 2X A X B ρ AB σ A σ B 2 2 = 0.06 2 2 + 0.08 2 2 + 2 0 0.06 0.08 = 0.0024 c. r p = X A r A + X B r B + X C r C = 0.0 + 0.4 + 0.2 = 0.2 σ 2 p = XAσ 2 2 A + XBσ 2 2 B + XCσ 2 2 C + 2X A X B ρ AB σ A σ B + 2X A X C ρ AC σ A σ C + 2X B X C ρ BC σ B σ C 2 2 2 = 0.06 2 + 0.08 2 + 0.07 2 + 2 0 0.06 0.07 +2 0.06 0.07 + 2 0.5 0.08 0.07 = 0.004 d. r p = 6 0 0.0 + 8 0 0.4 4 0.2 = 0.24 0
6 σ 2 p = 0 6 +2 0 = 0.0064 2 0.06 2 8 + 0 2 0.08 2 + 4 0 4 0.06 0.07 + 2 0 2 0.07 2 6 + 2 0 8 0 8 0 0.06 0.07 0 4 0.5 0.08 0.07 0 e. The logic is: invest e.g. 00 in risky assets. 0% borrowing means that 0 is borrowed money which is formally an investment of - 0 in the risk-free asset. The protfolio proportions then satisfy X P + X f =, Solving this pair of equations gives Using these X f X P X P = 0 2, X f = 9 2 r P = 9 2 0.05 + = 0 00. 0 0.2 = 0.5 2 σ 2 p = 2 0 0.004 2 = 0.0027 ii. a. σ p = 0.5 X A 2 0.06 2 + X A 2 0.07 2 + 2 X A X A 0.060.07 r P = X A 0.0 + X A 0.2 The assets have perfect negative correlation. The minimum variance portfolio is σ C 0.07 X A = = σ A + σ C 0.06 + 0.07 = 7, X C = 7 with expected return b. σ p = r P = 7 7 0.0 + 0.2 = 0.09 0.5 X B 2 0.08 2 + X B 2 0.07 2 + 2 X B X B 0.50.080.07 r P = X B 0.4 + X B 0.2 2
X A=0,X C= r = 0.09 X A=,X C=0 0.45 0.4 Risk free asset must have same return as the mvp with assets A and C otherwise arbitrage will occur. In this case the MV P has a return of r = 0.09 which is much higher than the rist-free rate of 0.05. iii. Only A and B. Use the frontier to talk about a move. More risk averse means indifference curve becomes steeper. Tangency point moves round the frontier to the left. This implies proportion of A rises, proportion of B falls. 6. i Based on pages 5-54 of module lecture notes. ii Based on pages 0-07 Markovitz model and pages 52-54 CAPM model of module lecture notes. iii Based on pages 54-6 of module lecture notes. iv This requires individual interpretation of the meaning of CAPM and the expression of personal opinions. An answer should at least make reference to the Separation Theorem and the role of the market portfolio. 7. i. The single index model assumes that the return on any asset i is determined by the process r i = α i + β i r I + ε i, where r i is the return on asset i and r I is the return on an index. The terms α i and β i are constants, and ε i is a random error term. What this model says is that the return on the asset is linearly related to a single common influence and that this influence is summarized by the return, r I, on an index. This return is the aggregate variable. Furthermore, the return on the asset is not completely determined by the return on the index so there is some residual variation unexplained by the index - the random error, ε i. As will be shown
below, this process for the generation of returns greatly simplifies the calculation of variance of return on a portfolio. The single-index model is completed by adding to the specification in three assumptions on the structure of the errors, ε i :. The expected error is zero: E [ε i ] = 0, i =,..., N; 2. The error and the return on the index are uncorrelated: E [ε i r I r I ] = 0, i =,..., N;. The errors are uncorrelated between assets: E [ε i ε j ] = 0, i =,..., N, j =,..., N, i j. ii. a. The expected returns on the three assets are b. r i = α i + β i r I r A = 0.2 +.2 0 = 2.2 r B = 0. + 0.8 0 = 8. r C = 0.2 +. 0 = 0.8 σ 2 i = β 2 i σ 2 I + σ 2 εi σ 2 A =.2 2 25 2 + 2 2 = 904 σ 2 B = 0.8 2 25 2 + 2 = 409 σ 2 C =. 2 25 2 + 2 = 757.25 iii. The portfolio frontier for A and B is generated from r p = X A 2.2 + X A 8. The portfolio beta is β p = X A.2 + X A 0.8 The portfolio variance σ 2 p = β 2 pσ 2 I + XAσ 2 2 εa + X A 2 σ 2 ε B = X A.2 + X A 0.8 2 25 2 + X A 2 2 2 + X A 2 2 Would you short sell A? This takes you off the left-hand end of the curve. But since this does not bend back there is no problem with this strategy. 4
4 2 0 iv. Consistent with CAPM? The CAPM returns from the security market line are r i = r f + β i [ r I r f ] r A = 5 +.2 0 5 = r B = 5 + 0.8 0 5 = 9 r C = 5 +. 0 5 = 0.5 They are not consistent. Logic: if CAPM is true, then single index will hold if index is the market as it is here. Single index can be true even if CAPM is not. Here, CAPM is not verified. But this does not establish that single index is true. 8. i. Based on pages 2-242 of mudule lecture notes. ii. This is the basic binomial valuation argument. It is not put-call parity. Hold call short D units of the stock. Payoff from portfolio in up state is: S=8, E=0, us= 2, ds=9. Pu = Vu - DuS = max{2-0,0} - D2 = 2 - D2 Payoff in down state Pd = Vd - DdS = max{9-0, 0} - D9 = 0 - D9 This is risk-free if Pu = Pd 2 D2 = 0 D9 5
or D = 2. c. The portfolio costs Vc - DS = Vc - 6 Hence V c 6 = + r So V c = 6 + r 8. It delivers Pu = Pd = - 8 8 for sure. Noting that d = 9/8 so =r > 9/8, hence Vc>0. iii. All binomial questions should be accompanied by a correctly labelled binomial tree. S = 20, us = 24, u =.2, ds = 8, d = 0.9, E = 22, R =., q = R d. 0.9.2 0.9 = 2. The payoffs are V u = max{e us,0}= max{22 24,0}=0 V d = max{e ds,0}= max{22 8,0}=4 u d = The option value is V = 2. 0 + 4 =.22 iii. a. S = 20, u 2 S = 24, d 2 S = 8, u 2 20 = 24 0.5 24 u = =.0954 20 d 2 20 = 8 0.5 8 d = = 0.9486 20 R =. 0.5 =.0488 q = R d u d.0488 0.9486 =.0954 0.9486 = 0.68256 uds=.0954 0.9486 20 = 20.782 6
The payoffs are V uu = max{e uus, 0} = max{22 24, 0} = 0 V ud = max{e uds, 0} = max{22 20.782, 0} =.28 V dd = max{e dds,0}= max{22 8,0}=4 This gives the option value V = R 2 q 2 V u + 2q qv ud + q 2 V dd =.0488 2 0 + 2 0.6825 0.6825.28 + 0.6825 2 4 = 0.84646 b. All binomial questions should be accompanied by a correctly labelled binomial tree. S = 20, u S = 24, d S = 8, The payoffs are u 20 = 24 / 24 u = =.0627 20 d 20 = 8 / 8 d = = 0.9654 20 R =. / =.02 q = R d u d.02 0.9654 =.0627 0.9654 = 0.6875 u 2 ds =.0627.0627 0.9654 20 = 2.805 ud 2 S =.0627 0.9654 0.9654 20 = 9.809 V uuu = max{e uus, 0} = max{22 24, 0} = 0 V uud = max{e uus, 0} = max{22 2.805, 0} = 0.95 V udd = max{e uds, 0} = max{22 9.809, 0} = 2.9 V ddd = max{e dds,0}= max{22 8,0}=4 7
This gives the option value V = R 0 + q 2 qv uud + q q 2 V ddu + q V ddd =.02 0 + 0.6875 2 0.6875 0.95 + 0.6875 0.6875 2 2.9 + 0.6875 4 = 0.59068 8