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:
𝑥𝛼
,𝑥 ≥ 0
𝑣(𝑥) ≔ {
𝛽
−𝜆(−𝑥)
,𝑥 < 0
Assume too that the probability weighting function is given by:
𝑤(𝑝) ≔
𝑝𝛾
(𝑝𝛾 +(1−𝑝)𝛾 )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
34-99
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
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!]
[Time needed: 5 hours]
3|Page