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Department of Economics, SUFFOLK UNIVERSITY Jonathan Haughton, Fall 2014 ECONOMICS 826: Financial Economics ASSIGNMENT 5 Answers to this assignment are due back by Wednesday, October 8, 2014. Question 1: Measuring betas 1. A market has only the following three risky assets: E(r) (% per month) Risk (i.e. σi) , % Covariance with market (σim) Asset 1 2.03 2 1.12 Asset 2 1.79 1 0.90 Asset 3 1.49 1 0.62 Market 0.92 portfolio The market portfolio has 4% invested in Asset 1, 76% in Asset 2, and 20% in Asset 3. a. What is the expected return of this portfolio? b. What are the betas of the three risky assets? c. If the riskless rate of interest is 0.8% (per month), are these three securities priced correctly? Question 2: Allais Paradox Assume that we are in the world of cumulative prospect theory (CPT), and that the value function is as follows: , ≥ 0 () ≔ { −(−) , < 0 Assume too that the probability weighting function is given by: () ≔ ( +(1−) )1/ . If λ=2.25, α=β=0.8, and γ=0.7, show that the Allais Paradox is resolved. You will remember that the four lotteries in the Allais Paradox are as follows: Lottery A Lottery B Lottery C Lottery D State Outcome State Outcome State Outcome State Outcome 1-33 2500 1-100 2400 1-33 2500 1-33 2400 34-99 2400 100 0 34-100 0 34-90 0 100 2400 Allais found that most people prefer B to A, and C to D, even though this behavior is not rational (in the sense of being consistent with expected utility theory). [Time needed: 5 hours] 1|Page Question 3: CPT Here is exercise 2.10 from the Hens and Rieger book, along with their answer to the question. After working through this, answer the subsequent questions, which are variants on this theme: “Let us assume that a value function v is given by v(x):=x and a weighting function w is w(F)=√F. A lottery is described by the probability measure p:=a(x)dx where the probability density a is given by , 0 ≤ < 1 (): = {2 − , 1 ≤ < 2 0 , ℎ. Compute the Cumulative Prospect Theory (CPT) value of this lottery. Use this to compute the certainty equivalent. Explain the difference between the certainty equivalent and the expected value.” [Time needed: 5 hours] 2|Page Now try the following. a. Let us assume that a value function v is given by v(x):=x and a weighting function w is w(F)=√F. A lottery is described by the probability measure p:=a(x)dx where the probability density a is given by , 0 ≤ < 0.5 , 0.5 ≤ < 2 0.5 (): = { 2.5 − , 2 ≤ < 2.5 0 , ℎ Graph this probability density and verify that it integrates to 1. b. Compute the CPT value of this lottery (or at least set up an expression, like the last equation in the sample answer). c. Find the expected value of this lottery. [Should not be too hard!] [Time needed: 5 hours] 3|Page