Department of Economics, SUFFOLK UNIVERSITY
Jonathan Haughton
Fall 2012
ECONOMICS 826: Financial Economics
ASSIGNMENT 5
Answers to this assignment are due back by Thursday, October 11, 2012
Question 1: Allais Paradox
Assume that we are in the world of cumulative prospect theory (CPT), and that the value function is as
follows:
𝑥𝛼
,𝑥 ≥ 0
𝑣(𝑥) ≔ {
−𝜆(−𝑥)𝛽 , 𝑥 < 0
Assume too that the probability weighting function is given by:
𝑝𝛾
𝑤(𝑝) ≔ (𝑝𝛾 +(1−𝑝)1/𝛾 .
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
34-90
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).
Question 2: Quick concept checkers
Answer questions 1, 2, 4, 6, 11, 16 on pages 86-88 of the Hens and Rieger book. Note that for some of
the questions there is more than one correct answer.
[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
𝑥
, 𝑖𝑓 0 ≤ 𝑥 < 1
𝑎(𝑥): = {2 − 𝑥 , 𝑖𝑓 1 ≤ 𝑥 < 2
0
, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
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
𝑥
, 𝑖𝑓 0 ≤ 𝑥 < 0.5
, 𝑖𝑓 0.5 ≤ 𝑥 < 2
0.5
𝑎(𝑥): = {
2.5 − 𝑥 , 𝑖𝑓 2 ≤ 𝑥 < 2.5
0
,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
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!]
Question 4: Portfolios (again)
This is a modification of exercise 3.1 in the Hens and Rieger textbook. There are two risky assets, k =
1,2, and one risk-free asset that yields 2%. Short selling is not allowed. The expected returns on the
risk assets are μ1=6% and μ2=9% respectively. The covariance matrix is:
2%
−1.5%
𝐶𝑂𝑉 == (
).
−1.5%
5%
a. Calculate the minimum-variance portfolio. [Hint: Either use solver in Excel, or minimize the
variance of the portfolio subject to the shares summing to 1 (i.e. s.t. λ1+λ2=1). This is equivalent
to minimizing L’ COV L, where L is a column vector with λ1 and 1-λ1.]
b. Calculate the Tangent portfolio. [Hint: Apply equation 3.3. from the book; in effect we need L =
COV-1 (μ – rf1) where μ is a vector of returns and 1 a column of ones. You will then need to
normalize the L so that they add up to 1 – an application of two-fund separation.]
c. Assume the market portfolio has shares λM = (0.4, 0.6). Compute the betas for each asset.
[Hint: Find the covariance of asset returns with the market portfolio; and the variance of the
2
market portfolio. It may help to note that 𝜎𝑟1,𝑟𝑀 = 𝜆12 𝜎𝑟𝑖
+ 𝜆2 𝜎𝑟1,𝑟2 , where r1 are the returns
on asset 1, etc.]
d. Assume the excess return of the market portfolio is 3%. Determine the expected returns of the
two risky assets.
[Time needed: 5 hours]
3|Page