Posted: December 8th, 2013

longer numeric/expression

12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 1/10
Final Exam Part 2 (longer numeric/expression answers)
The due date for this exam is Sun 8 Dec 2013 5:00 PM PST.
This part of the exam consists of several “questions” each of which has several “parts”.
Coursera won’t let me label them question 1, part a b c, but keep in mind that that is the
structure of the exam.
You will have two chances. Once you start, you must complete the exam within 24 hours, and
this includes both submissions. Detailed explanations and answers will show up after the exam
deadline has passed, and I hope you will come back and look at them.
In accordance with the Coursera Honor Code, I (Dimosthenis Christopoulos) certify
that the answers here are my own work.
Question 1
Question 1, Part a.
An investor has preferences with a minimum subsistence level of consumption , so she
maximizes
Suppose consumption follows
Find the stochastic discount factor for this case, i.e. (some expression involving ) = where
prices assets by . Enter your expression as an algebraic
expression, using “c” for , “g” for , “rho” for , “e^(x)” for exponentiation and standard
algebraic symbols. Throughout this problem, assume and don’t worry about what
happens otherwise.
Hint: the answer is for power utility (h=0)
h
E ∫ dt
∞
t=0
e
−ρt
(ct − h)
1−γ
1 − γ
= μdt + σd .
dct
ct
zt
c Λt
Λt pt = Et ∫ ds
∞
s=0
Λt+s
Λt
xt+s
ct γ ρ e
x
c > h
e
−ρtc
−γ
t
Time remaining
23:59:4812/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 2/10
Preview
Preview
Help
 Click here for more on this paper>>>>
Question 2
Question 1, Part b.
Find the real interest rate . You should have three terms. Enter your formula as an algebraic
expression. Use “g” for , “c” for , and otherwise spell Greek letters as above.
Hint: the classic power utility formula was
Help
Question 3
Question 1, Part c.
Enter a numerical value for the risk free rate, in percent, if (5%), , ,
, (1%), (5%). (This is just a second chance to get it right if you had
trouble entering the formula into Coursera.)
Be sure to enter your answer in percent; for example, if you calculate that , enter 2
and not .02.
Question 4
Question 1, Part d.
r
f
t
γ ct
r = − ( ) = ρ + γμ − γ(γ + 1)
f 1
dt
Et
dΛ
Λ
1
2
σ
2
ρ = 0.05 γ = 2 ct = 10
h = 5 μ = 0.01 σ = 0.05
r = .02
f12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 3/10
Preview
Find the maximum instantaneous Sharpe ratio among all assets priced by this discount factor.
Express your answer in annual units, i.e. . Again use “g” for ,
“c” for , “sigma” for , “mu” for , (the parameters of the consumption process) and ignore
any subscripts.
 Click here for more on this paper>>>>
Help
Question 5
Question 1, Part e.
Enter the numerical value of the maximum Sharpe ratio, if (5%), , ,
, (1%), (5%). (This is just a second chance to get it right if you had
trouble entering the formula into Coursera.)
Question 6
Question 2, Part a.
The binomial model. Suppose that there are two states tomorrow, up and down, and each can
happen with probability 1/2. Consumption is today, in the up
state and in the down state. Assume (i.e., ),
.
Find the price of a bond — an asset that pays 1 in each state. Enter a number, accurate to two
decimal places.
SR = (E(dR) − r )/σ(dR)
f γ
ct σ μ
ρ = 0.05 γ = 2 ct = 10
h = 5 μ = 0.01 σ = 0.05
ct = 1 ct+1 (u) = 3/2 = 1.5
ct+1 (d) = 3/4 = 0.75 γ = 1 u(c) = log(c)
β = 112/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 4/10
Question 7
Question 2, part b.
Find the price of asset A that pays in the up state and in the down state. Find
the price of asset B that pays in the up state and in the down state. Enter the
price of asset A an asset B, as decimals, accurate to two decimal points, separated by a space.
(Hint: Notice that with the mean and variance of the two asset payoffs is the same.)
Question 8
Question 2, part c.
Find the prices of contingent claims to the up state and down state respectively. Enter them as
two numbers separated by a space, each accurate to two decimals.
Question 9
Question 2, part d
Find a set of risk-neutral probabilities that you could use to price assets in this
circumstance. Report as two numbers, accurate to two decimal places,
separated by a space
Question 10
Question 3, part a)
x = 1 x = −1
x = −1 x = 1
π = 1/2
π (u), (d)
∗ π
∗
π (u), (d)
∗ π
∗12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 5/10
Suppose there is a single excess return , and an information variable which can take
two values, with equal probability. Use values
and
.
The notation means the same thing.
What is the unconditional Sharpe ratio of a constant weight portfolio, i.e. just holding Enter
one number, accurate to one decimal.
Hint: You can find unconditional variances with
or (which follows)
Question 11
Question 3, part b)
 Click here for more on this paper>>>>
Well, let’s see if we can do better by putting more money in to the high conditional Sharpe ratio
market. Find which allows you to characterize the conditional mean variance frontier as
. What does “find mean? Well, and there being only
one excess return, this is as general as portfolios get. So, find and enter in state 1 and state
2, separated by a space, accurate to two decimal points. Hint: Use a defining property of .
R
e
t+1 zt
zt = 1, 2
E(R ∣ = 1) = E( ∣ = 2) = 8%
e
t+1 zt R
e
t+1 zt
σ(R ∣ = 1) = 16%, σ( ∣ = 2) = 24%
e
t+1 zt R
e
t+1 zt
Et(Rt+1 ) = E(Rt+1 ∣ zt)
R ?
e
σ (x) = E( ) − E(x = E( ( )) − E( (x)
2 x
2
)
2 Et x
2 Et )
2
σ (x) = [ (x)] + E[ (x)]
2 σ
2 Et σ
2
t
R ,
e∗
t+1
R = δ
emv
t+1 R
e∗
t+1 R
e∗ R =
e∗
t+1 wtR
e
t+1
wt
R
e∗12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 6/10
Question 12
Question 3, part c)
Now find the Sharpe ratio of the unconditional mean-variance frontier. Report its numerical value
to two decimal places.
Question 13
Question 3, part d)
So, finally, what do you actually do? Find the weights in the underlying security of an
unconditional mean-variance efficient investment with unconditional mean equal to 8%. How
much do you put in — in if state 1 happens and how much for state 2? Enter two
numbers, separated by a space, accurate to two decimals.
Question 14
Question 4, part a)
Suppose the log prices of one-, two- and three-year zero-coupon bonds are ,
, . Find today’s log one-year yield and the two- and threeyear log forward rates . Enter three numbers, separated by a space, accurate to
two decimals, in net (not percent) units. For all of question 4, if the answer cannot be determined
from the given information, enter “99”.
R
e
wt wtR
e
t+1
p = −0.10
(1)
t
p = −0.30
(2)
t p = −0.40
(3)
t y
(1)
t
f ,
(2)
t f
 Click here for more on this paper>>>>
(3)
t12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 7/10
Question 15
Question 4, part b)
According to the expectations hypothesis, what is the expected value of interest rates in years
, , and , , ? Enter three numbers, separated
by a space, accurate to two decimals. Enter the rates in net, not percent units.
Question 16
Question 4, part c)
According to the expectations hypothesis, what is the expected value of forward rates at time
? What are and ? Enter two numbers separated by a space,
accurate to two decimals and in net units. (Two numbers because you already answered
.)
Question 17
Question 4, part d)
Specializing the Fama-Bliss regression slope coefficients to 0 (future spot regression) and 1
(excess return regression), and ignoring the constant (i.e. set it to zero), what do Fama and Bliss
say the expected one year rate and the expected return on two year bonds
are? Enter two numbers, separated by a space, in net units.
t + 1 t + 2 t + 3 Et(y )
(1)
t+1 Et(y )
(1)
t+2 Et(y )
(1)
t+3
t + 1 Et(f )
(2)
t+1 Et(f )
(3)
t+1
Et(y )
(1)
t+1
y
(1)
t+1 Et(rx )
(2)
t+112/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 8/10
Question 18
Question 5 part a)
You have a set of 3 excess returns . You form the covariance matrix
. You eigenvalue decompose , and the result is
You want to form a one-factor model that captures the most variance, i.e. maximizes in factor
model regressions.
What weights do you use to form factors from the return data — what is in ? Enter
three numbers, separated by a space, accurate to two decimal points.
Question 19
Question 5, part b)
What loadings will your factor model have on the first factor? If you run a regression of each
return on the first factor, what values of will you recover? Enter
three numbers, separated by a space, accurate to two decimal points.
Question 20
R (3 × 1)
e
t+1
Σ = cov(R )
eR
e′ Σ = QΛQ
′
Q = =
⎡
⎣
⎢
0.58
0.58
0.58
−0.71
0.00
0.71
−0.41
0.81
−0.41
 Click here for more on this paper>>>>
⎤
⎦
⎥
⎡
⎣
⎢⎢
1/√3
1/√3
1/√3
−1/√2
0.00
1/√2
−1/√6
2/√6
−1/√6
⎤
⎦
⎥⎥
Λ =
⎡
⎣
⎢
3.00
0
0
0
1.00
0
0
0
0.06
⎤
⎦
⎥
R
2
w ft = w
′R
e
t
R = a + + ,
(ei)
t bift ε
(i)
t bi12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 9/10
Question 5, part c)
What value will you achieve in the regression of on the first factor? Enter your answer
as a decimal.
Question 21
Question 6 part a)
Let’s solve a simple term structure model. Suppose the state variable follows an MA(1),
and the discount factor is
Remember, is known at time , so , not 0, and not . Also, this
will be a two state variable model, as both and or equivalently and describe
where we are at any given date.
Find the price of a one-period bond, and hence the yield . To check your answer, enter the
numerical value of if , , and and
hence . Enter as a percent (10, not 0.10), accurate to two decimal places.
Question 22
Question 6 part b)
Find the price of a two-year bond, and hence the two-year forward rate, , under the
same assumptions as the previous question. To check your answer, calculate the forward rate
in the given parameter configuration, and enter it as a percent, accurate to two decimal
R
2 R
(e1)
t
Xt
Xt = εt + θεt−1
Mt+1 = e
−Xt− −λ
1
2
λ
2X 2
t σ
2
ε Xtεt+1
εt t Et(εt) = εt σt(εt) = 0 σ
2
ε
Xt εt−1 εt εt−1
y
(1)
t
y
(1)
t θ = 1 λ = 50, σε = 0.10 εt = 0.01, εt−1 = 0.01
Xt = 0.02 y
(1)
t
p
(2)
t f
(2)
t
f
(2)
t12/7/13 Exam| AssetPricing
https://class.coursera.org/assetpricing-001/quiz/attempt?quiz_id=441 10/10
places.
Question 23
Question 6 part c)
Now find the expected value of the one-year yield one year ahead, , and the expected
return of the two year bond . To check all your answers, find the relationship between
the forward rate and these two quantities, and make sure it holds. Stop and savor how the
market price of risk controls the split of the forward rate between expected interest rate
changes and risk premium. To check your answer, enter and , as percent
values, separated by a space.
In accordance with the Coursera Honor Code, I (Dimosthenis Christopoulos) certify
that the answers here are my own work.
Save Answers
You cannot submit your work until you agree to the Honor Code. Thanks!
Ety
(1)
t+1
Etr
(2)
t+1
f
(2)
t
λ
Ety
(1)
t+1 Etr
(2)
t+1
Submit Answers
 Click here for more on this paper>>>>