longer numeric/expression answers

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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 Course ra Honor Code , I (Dimosthe nis Christopoulos) ce rtify

that the answe rs he re 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 ( − hct ) 1−γ

1 − γ

= μdt + σd . dct

ct zt

c Λt

Λt = dspt Et ∫ ∞

s=0 Λt+s Λt

xt+s

ct γ ρ e x

c > h

e−ρt c −γ t

Time remaining 23:59:48

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Preview

Help

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

= − ( ) = ρ + γμ − γ(γ + 1)rf 1 dt

Et dΛ Λ

1 2

σ 2

ρ = 0.05 γ = 2 = 10ct h = 5 μ = 0.01 σ = 0.05

= .02rf

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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.

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) − )/σ(dR)rf γ

ct σ μ

ρ = 0.05 γ = 2 = 10ct h = 5 μ = 0.01 σ = 0.05

= 1ct (u) = 3/2 = 1.5ct+1 (d) = 3/4 = 0.75ct+1 γ = 1 u(c) = log(c)

β = 1

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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

(u), (d)π ∗ π ∗

(u), (d)π ∗ π ∗

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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 numbe r, accurate to one decimal.

Hint: You can find unconditional variances with

or (which follows)

Question 11

Question 3, part b)

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

= 1, 2zt

E( ∣ = 1) = E( ∣ = 2) = 8%Ret+1 zt R e t+1 zt

σ( ∣ = 1) = 16%, σ( ∣ = 2) = 24%Ret+1 zt R e t+1 zt

( ) = E( ∣ )Et Rt+1 Rt+1 zt

?Re

(x) = E( ) − E(x = E( ( )) − E( (x)σ2 x2 )2 Et x2 Et )2

(x) = [ (x)] + E[ (x)]σ2 σ2 Et σ2t

,Re∗t+1 = δRemvt+1 R

e∗ t+1 R

e∗ =Re∗t+1 wt R e t+1

wt

R e∗

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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

numbe rs, 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 three-

year log forward rates . Enter thre e numbe rs, 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 wt R e t+1

= −0.10p (1) t

= −0.30p (2) t = −0.40p

(3) t y

(1) t

,f (2) t f

(3) t

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Question 15

Question 4, part b)

According to the expectations hypothesis, what is the expected value of interest rates in years

, , and , , ? Enter thre e numbe rs, 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 numbe rs 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 numbe rs, 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 (r )Et x

(2) t+1

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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

thre e numbe rs, 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

thre e numbe rs, separated by a space, accurate to two decimal points.

Question 20

(3 × 1)Ret+1 Σ = cov( )ReRe ′ Σ = QΛQ′

Q = = ⎡ ⎣⎢

0.58 0.58 0.58

−0.71 0.00 0.71

−0.41 0.81

−0.41

⎤ ⎦⎥

⎡ ⎣ ⎢⎢

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 =f t w ′ R

e t

= a + + ,R (ei) t bi f t ε

(i) t bi

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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

= + θXt εt εt−1

=Mt+1 e − − −λX t

1 2

λ2 X 2t σ 2 ε X tεt+1

εt t ( ) =Et εt εt ( ) = 0σt εt σ2ε Xt εt−1 εt εt−1

y (1) t

y (1) t θ = 1 λ = 50, = 0.10σε = 0.01, = 0.01εt εt−1

= 0.02Xt y (1) t

p (2) t f

(2) t

f (2) t

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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 Course ra Honor Code , I (Dimosthe nis Christopoulos) ce rtify

that the answe rs he re are my own work.

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You cannot submit your work until you agree to the Honor Code. Thanks!

Et y (1) t+1

Et r (2) t+1

f (2) t

λ

Et y (1) t+1 Et r

(2) t+1

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