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In the asset price model, verify that E[S(t)] = Soeht E[S(t)²] = S²e(²µ+o²) t Var[S(t)] = S²e²ut (eo²t - 1)

Essentials Of Investments

11th Edition

ISBN:9781260013924

Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Chapter1: Investments: Background And Issues

Section: Chapter Questions

Problem 1PS

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In the asset price model, verify that
E[S(t)] = Soeht
E[S(t)²] = S²e(²µ+o²)t
Var[S(t)] = Se²ut (eo²t - 1)

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2. Suppose that you have a riskfree asset and N risky assets for investment. The rate of return on the riskfree asset is r,, while the (Nx1) vector of the rate of return on the N risky assets is r, which is multivariate normal, i.e., r N(u, E). Your utility function for a portfolio that consists of the riskfree asset and the N risky asset is u(r,)=r,-=o, 2 Suppose that the sum of investment proportions on the riskfree and risky assets is one. Answer the following question. A. What is your optimal investment proportion in the risky assets? How is your investment on the riskfree asset affected by different values of 2? B. Suppose that there is only one risky asset i. Show the effects of the Sharpe ratio (4,/0, ) on the investment proportion in the risky asset.
3. A market consists of two risky assets and no risk-free asset. Let R₁ and R₂ denote the return on each of the risky assets. Using market data the following have been estimated: E[R₁] = 0.10, E[R₂] = 0.15, o² = Var(R₁) = 0.1², o² = Var(R₂) = 0.2² and p1,2 = -1/2 where P1,2 denotes the correlation coefficient for R₁ and R₂. (i) Given that an investor is targeting a total expected return of portfolio, what is the minimum variance that can be achieved? = 0.125 on a (ii) Determine the global minimum variance portfolio and the expected return and variance of return on this portfolio. (iii) Using your answers to parts (i) and (ii) make a rough sketch of the minimum- variance set in - o2 space. You should indicate the efficient frontier and the global minimum variance portfolio. (iv) Without further calculation, explain how you would expect the variance of return calculated in (ii) to change if the two risky assets were independent.
2. We know the following two premiums: E(r) – E(˜Â) = 0.4 and E(rÂ) — rƒ = 0.1. (a) Calculate ₁. (b) If security B has BB : = 1 and E(rB) - rf = 0.5. What is aß?
Assume that the variables I, N, and PV represent the interest rate, investment or deposit period, and present value of the amount deposited or invested, respectively. Which equation best represents the calculation of a future value (FV) using: Compound interest? O FV = PV / (1+1)N O FV = PV + (PV XIX N) O FV = PV x (1 + I)N Simple interest? OFV = PV + (PV XIX N) O FV = PV XIX N O FV = PV/(PV XIX N)
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The forward contract value today is denoted as a.Vo(0,T) b.Fo(0,T) c.So(1 + r)^T d.Fo(0,T)/(1 + r)^(T – t)
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Consider the following contingent claim whose value at maturity date T is given by f(T,Sp) = min(Sr S;) where T, is some intermediate betweent (current time) and T (maturity date). Sr, and Sr denote the prices of the underlying asset at time To and time T respectively. We assume that the asset is non-dividend paying and its asset price satisfies dS; = uS,dt + oS,dW, with u = 0.05, a = 0.2. The riskfree interest rate is 6% per annum. Take t = 0, T, = 0.5 and T = 1.5. The current price of the asset is So = $25. (a) Calculate the price of the contingent claim at time Ta using replication technique. Express your answer in terms of ST.. (©Hint: The value of S, is known at time To. To find the price, treat T, as the "current time"). (b) Using the result of (a), find the current price of this contingent claim using risk neutral valuation principle

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