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高级宏观经济代写-A13072W1

时间：2021-04-16

A13072W1

DEGREE OF MASTER IN FINANCIAL ECONOMICS

PRS Qualifying Examination in Management Studies

Asset Pricing

TRINITY TERM 2020

Monday 20th April 2020,

Time Allowed: FOUR Hours

You MUST upload your submission within 4 hours of accessing the paper

Please start the answer to each question on a separate page.

Answer FOUR questions, TWO from Part A, ONE from Part B and ONE from

Part C.

Candidates may use their own calculators.

The following models are permitted: Sharp EL-531, Casio FX-83, Casio FX-85, HP12C,

HP12 Platinum, Texas Instrument BA II, Texas Instrument BA II Plus.

Each type of calculator may have several specific models, which are denoted by an additional suffix, e.g. CASIO

FX-83GTX. All such models are permitted so long as they clearly belong to one of the permitted types.

Do not turn over until told that you may do so.

1

Part A

1. (a) Imagine you work as an Analyst in the derivative division of a large bank. One

of your clients faces some uncertainty around the COVID-19 situation and wants

to hedge their portfolio using put and call options. Your client asks why your

bank quotes the same price for otherwise identical European and American call

options but deems American put options to be more costly than their European

counterparts. Answer - without equations - both of his questions.

(b) Imagine you are consulting the government of a European country on capital market

regulation following the market downturn of March 2020. A minister suggests:

“during the crisis, some asset managers used credit default swaps (CDS) to make

billions while the markets crashed. These instruments have no legitimate use and

should be banned.” Do you agree? Discuss.

(c) Imagine you are called by an incoming MFE in week 4 of Michaelmas term. They

were reading through the lecture slides and did not understand the Arrow-Debreu

& Complete Markets Equivalence Theorem. Because of some notational changes,

the student cannot use past support classes and has to ask you. They sent you the

following equations (1− 5).

Bh(q) =

{

x ∈ RS∗ : xh0 − eh0 ≤ −qθh and xhs − ehs ≤ Asθh for all states s ∈ S

}

(1)

⇒ Bh(qˆ) =

{

x ∈ RS∗ : xh0 − eh0 ≤ −qˆθˆh and xhs − ehs ≤ Isθˆh for all states s ∈ S

}

(2)

⇒ Bh(qˆ) =

{

x ∈ RS∗ : xh0 − eh0 +

∑

s

qˆs

(

xhs − ehs

) ≤ 0} (3)

⇒ Bh(qˆ) =

{

x ∈ RS∗ : pi · (xh − eh) ≤ 0} ,with pi = (1, qˆ) (4)

⇒ Bh(qˆ) = Bh(pi) (5)

Please explain - line by line - what the intuition behind the proof is and what the

conditions for the transformations are, respectively. Also, please define the meaning

of the variables to help the student.

A13072W1 2

2. (a) Consider a CAPM economy with just three risky assets, A, B, and C, with expected

rates of return of 5%, 20%, and 10%, respectively. Two portfolios, 1 and 2, are known

to lie on the minimum variance frontier. They are defined by the portfolio weights

w1 =

(

wA1 , w

B

1 , w

C

1

)

= (0.4, 0.4, 0.2) and w2 =

(

wA2 , w

B

2 , w

C

2

)

= (−0.4, 0.2, 1.2),

respectively. What are the minimum and maximum possible values for the expected

rate of return on the market portfolio?

(b) Consider a different CAPM economy with three risky assets A, B, and C that are

uncorrelated with each other. We have:

i E(ri) σi

A 5% 8%

B 7% 10%

C 11% 10%

The risk free rate is 3%.

(i) What is the Sharpe Ratio of an efficient portfolio in this economy?

(ii) What is the equation of the Capital Market Line?

(iii) What is the composition of the optimal portfolio for an investor who requires an

expected return of 7%?

(c) Consider the stock with price p(γt, t) depending on the state of nature γt at time t.

The stock’s price in the current period t = 0 is p(γ0, 0) = 100 and can rise or fall

by 5% over the next three periods, so that the model is defined for T = {0, 1, 2, 3}

with u = 1.05 and d = 0.95. The risk-free interest rate is 1%, so that Rf = 1.01.

Calculate the risk-neutral probabilities and find the value at t = 0 of a European

call option paying max[(p(γT , T )−K), 0] with a strike price K = 101.

A13072W1 3 turn over

3. Consider the three-period model illustrated below. The first (second) number in

each tuple is the value of security A (B) at the respective node. In addition to

these two securities, there is a risk free bond with price 100 at t = 0. The risk

free rate of return, per period, is 10%.

(76, 56)

(89, 34)

(80, 80)

(142, 8)

(79, 50)

(101, 94)

(91, 106)

(61, 34)

(a) Calculate the state prices for all states of nature as well as the unconditional

risk-neutral probabilities at t = 0. [6 marks]

(b) Calculate the price (at t = 0) of a 2-period European call option on asset B with

a strike price of 40. [4 marks]

(c) Confirm your result by calculating the replicating portfolio strategy and the price

of a 2-period European call option on asset B with a strike price of 40. [6 marks]

(d) Suppose that the American counterpart to the option in (b) and (c) is also traded

in the market. How big is the early exercise premium for this option? [6 marks]

(e) Which are the conditions to implement risk-neutral valuation in this question?

[6 marks]

A13072W1 4

Part B

4. (a) Assume there is an economy with k states of nature and where the following asset

pricing formula holds:

Pa =

k∑

s=1

pismsXsa = E [mXa]

Let an individual in this economy have the utility function ln(C0) + E[δln(C1)] ,

and let C∗0 be her equilibrium consumption at date 0 and C

∗

s be her equilibrium

consumption at date 1 in state s, s = 1, ...k. Denote the date 0 price of Arrow secu-

rity s as ps, and derive an expression for it in terms of the individual’s equilibrium

consumption.

(b) Consider an economy with k=2 states of nature, a “good” state and a “bad” state.

There are two assets, a risk-free asset with Rf = 1.05 and a second risky asset that

pays cashflows

X2 =

[

10

5

]

The current price of the risky asset is 6.

(i) Solve for the prices of the Arrow securities p1 and p2 and the risk-neutral prob-

abilities of the two states.

(ii) Suppose that the physical probabilities of the two states are pi1 = pi2 = 0.5.

What is the stochastic discount factor for the two states?

(iii) Consider a one-period economy with two end-of-period states. An option con-

tracts pays 3 in state 1 and 0 in state 2 and has a current price of 1. A forward

contract pays 3 in state 1 and -2 in state 2. What are the one-period risk-free

return and the risk- neutral probabilities of the two states?

A13072W1 5 turn over

5. Consider a three period world as indicated in the following diagram:

At every node the two possibilities have risk-neutral probabilities 1

2

. We describe the

one period spot interest rate at node sj, j = 0, 1, 2 as rj if the price of an one period

0 -coupon bond delivering $1 for sure in the two nodes immediately succeeding sj is

pj =

1

1+rj

. Finally, suppose that r0 = 10%, r1 = 11%, r2 = 9%

(a) What is the price of a two-year 0-coupon bond at t = 0 that pays 1 in s3, s4, s5 and

0 in s1, s2?

(b) Find the yield y of the two-year 0-coupon bond. Is the yield curve upward-sloping

or downward-sloping?

(c) Now suppose that r0 = 10%, r1 = 11%, r2 = 10%. Re-do parts (a) and (b).

(d) Based on your findings in (a) - (c) what should you expect about the one-period

spot interest rate so that the yield curve to be upward sloping?

(e) What is a state price and how is it calculated? How are state prices related to

risk-neutral probabilities? Be precise!

A13072W1 6

Part C

6. (a) The dynamic dividend discount model can be approximated with log linear approx-

imation and transformed to this equation:

var (pt − dt) ≈ cov

(

pt − dt,

∞∑

j=0

ρj∆dt+j+1

)

− cov

(

pt − dt,

∞∑

j=0

ρj∆rt+j+1

)

(6)

Explain how this equation relates excess price (or price-dividend ratio) volatility to

return predictability. Hint: the answer should include a statement about empirical

evidence. [2 marks]

(b) Assume that a researcher has T = 600 observations and runs a predictive regression

of the form rt+1 = α + βxt + t+1 and gets these estimates:

rt+1 = 0.01 + 0.9xt + t+1 (7)

The variable xt is the (demeaned) dividend-price ratio. At the same time, he esti-

mates this AR(1) for the predictive variable xt+1 = φxt + ut+1 with this result:

xt+1 = 0.998xt + ut+1 (8)

The estimated coefficient β suffers from the so called Stambaugh bias, which can be

approximated by E[βˆ − β] = γE[φˆ− φ] with E[φˆ− φ] = −(1+3φ)

T

. Use the value for

γ = −90 and the estimated value φˆ for φ and calculate the value of βStambaugh that

corrects for the bias according to the suggestions of Stambaugh (1999). [2 marks]

(c) Impose an economic restriction on the behaviour of the dividend-price ratio that

excludes explosive bubbles and adjust for the possible bias in βˆ after taking this

economic assumption into account in the sense of Lewellen (2004). [4 marks]

(d) Expectation Hypothesis makes a statement about the relationship between expected

excess returns of zero-coupon bonds and yields:

Et [rn,t+1 − y1,t] = ν (9)

Use the definition of the excess holding period return rn,t+1 − y1,t = sn,t − (n −

1)(yn−1,t+1−yn,t) and show what Expectation Hypothesis implies on the relationship

between term spread snt and expected long-term yields. Hint: state the relation

formally and describe in words (one sentence is enough). [2 marks]

(e) The Vasicek model of interest rates assumes that the stochastic discount factor

follows this equation of motion:

−mt+1 = λ

2

2

+ zt + λt+1 (10)

A13072W1 7 turn over

zt+1 = (1− φ)θ + φzt + σt+1, (11)

where mt+1 is the stochastic discount factor, zt is a state variable, λ is a risk premium

and t+1 is a mean zero normally distributed random noise with variance one. Use

the Vasicek model to derive the expression of a forward rate, f1,t+1, which at time

t fixes the interest rate for one period from t + 1 to t + 2 in terms of the model

parameters and the state variable zt. [5 marks]

A13072W1 8 last page

学霸联盟

DEGREE OF MASTER IN FINANCIAL ECONOMICS

PRS Qualifying Examination in Management Studies

Asset Pricing

TRINITY TERM 2020

Monday 20th April 2020,

Time Allowed: FOUR Hours

You MUST upload your submission within 4 hours of accessing the paper

Please start the answer to each question on a separate page.

Answer FOUR questions, TWO from Part A, ONE from Part B and ONE from

Part C.

Candidates may use their own calculators.

The following models are permitted: Sharp EL-531, Casio FX-83, Casio FX-85, HP12C,

HP12 Platinum, Texas Instrument BA II, Texas Instrument BA II Plus.

Each type of calculator may have several specific models, which are denoted by an additional suffix, e.g. CASIO

FX-83GTX. All such models are permitted so long as they clearly belong to one of the permitted types.

Do not turn over until told that you may do so.

1

Part A

1. (a) Imagine you work as an Analyst in the derivative division of a large bank. One

of your clients faces some uncertainty around the COVID-19 situation and wants

to hedge their portfolio using put and call options. Your client asks why your

bank quotes the same price for otherwise identical European and American call

options but deems American put options to be more costly than their European

counterparts. Answer - without equations - both of his questions.

(b) Imagine you are consulting the government of a European country on capital market

regulation following the market downturn of March 2020. A minister suggests:

“during the crisis, some asset managers used credit default swaps (CDS) to make

billions while the markets crashed. These instruments have no legitimate use and

should be banned.” Do you agree? Discuss.

(c) Imagine you are called by an incoming MFE in week 4 of Michaelmas term. They

were reading through the lecture slides and did not understand the Arrow-Debreu

& Complete Markets Equivalence Theorem. Because of some notational changes,

the student cannot use past support classes and has to ask you. They sent you the

following equations (1− 5).

Bh(q) =

{

x ∈ RS∗ : xh0 − eh0 ≤ −qθh and xhs − ehs ≤ Asθh for all states s ∈ S

}

(1)

⇒ Bh(qˆ) =

{

x ∈ RS∗ : xh0 − eh0 ≤ −qˆθˆh and xhs − ehs ≤ Isθˆh for all states s ∈ S

}

(2)

⇒ Bh(qˆ) =

{

x ∈ RS∗ : xh0 − eh0 +

∑

s

qˆs

(

xhs − ehs

) ≤ 0} (3)

⇒ Bh(qˆ) =

{

x ∈ RS∗ : pi · (xh − eh) ≤ 0} ,with pi = (1, qˆ) (4)

⇒ Bh(qˆ) = Bh(pi) (5)

Please explain - line by line - what the intuition behind the proof is and what the

conditions for the transformations are, respectively. Also, please define the meaning

of the variables to help the student.

A13072W1 2

2. (a) Consider a CAPM economy with just three risky assets, A, B, and C, with expected

rates of return of 5%, 20%, and 10%, respectively. Two portfolios, 1 and 2, are known

to lie on the minimum variance frontier. They are defined by the portfolio weights

w1 =

(

wA1 , w

B

1 , w

C

1

)

= (0.4, 0.4, 0.2) and w2 =

(

wA2 , w

B

2 , w

C

2

)

= (−0.4, 0.2, 1.2),

respectively. What are the minimum and maximum possible values for the expected

rate of return on the market portfolio?

(b) Consider a different CAPM economy with three risky assets A, B, and C that are

uncorrelated with each other. We have:

i E(ri) σi

A 5% 8%

B 7% 10%

C 11% 10%

The risk free rate is 3%.

(i) What is the Sharpe Ratio of an efficient portfolio in this economy?

(ii) What is the equation of the Capital Market Line?

(iii) What is the composition of the optimal portfolio for an investor who requires an

expected return of 7%?

(c) Consider the stock with price p(γt, t) depending on the state of nature γt at time t.

The stock’s price in the current period t = 0 is p(γ0, 0) = 100 and can rise or fall

by 5% over the next three periods, so that the model is defined for T = {0, 1, 2, 3}

with u = 1.05 and d = 0.95. The risk-free interest rate is 1%, so that Rf = 1.01.

Calculate the risk-neutral probabilities and find the value at t = 0 of a European

call option paying max[(p(γT , T )−K), 0] with a strike price K = 101.

A13072W1 3 turn over

3. Consider the three-period model illustrated below. The first (second) number in

each tuple is the value of security A (B) at the respective node. In addition to

these two securities, there is a risk free bond with price 100 at t = 0. The risk

free rate of return, per period, is 10%.

(76, 56)

(89, 34)

(80, 80)

(142, 8)

(79, 50)

(101, 94)

(91, 106)

(61, 34)

(a) Calculate the state prices for all states of nature as well as the unconditional

risk-neutral probabilities at t = 0. [6 marks]

(b) Calculate the price (at t = 0) of a 2-period European call option on asset B with

a strike price of 40. [4 marks]

(c) Confirm your result by calculating the replicating portfolio strategy and the price

of a 2-period European call option on asset B with a strike price of 40. [6 marks]

(d) Suppose that the American counterpart to the option in (b) and (c) is also traded

in the market. How big is the early exercise premium for this option? [6 marks]

(e) Which are the conditions to implement risk-neutral valuation in this question?

[6 marks]

A13072W1 4

Part B

4. (a) Assume there is an economy with k states of nature and where the following asset

pricing formula holds:

Pa =

k∑

s=1

pismsXsa = E [mXa]

Let an individual in this economy have the utility function ln(C0) + E[δln(C1)] ,

and let C∗0 be her equilibrium consumption at date 0 and C

∗

s be her equilibrium

consumption at date 1 in state s, s = 1, ...k. Denote the date 0 price of Arrow secu-

rity s as ps, and derive an expression for it in terms of the individual’s equilibrium

consumption.

(b) Consider an economy with k=2 states of nature, a “good” state and a “bad” state.

There are two assets, a risk-free asset with Rf = 1.05 and a second risky asset that

pays cashflows

X2 =

[

10

5

]

The current price of the risky asset is 6.

(i) Solve for the prices of the Arrow securities p1 and p2 and the risk-neutral prob-

abilities of the two states.

(ii) Suppose that the physical probabilities of the two states are pi1 = pi2 = 0.5.

What is the stochastic discount factor for the two states?

(iii) Consider a one-period economy with two end-of-period states. An option con-

tracts pays 3 in state 1 and 0 in state 2 and has a current price of 1. A forward

contract pays 3 in state 1 and -2 in state 2. What are the one-period risk-free

return and the risk- neutral probabilities of the two states?

A13072W1 5 turn over

5. Consider a three period world as indicated in the following diagram:

At every node the two possibilities have risk-neutral probabilities 1

2

. We describe the

one period spot interest rate at node sj, j = 0, 1, 2 as rj if the price of an one period

0 -coupon bond delivering $1 for sure in the two nodes immediately succeeding sj is

pj =

1

1+rj

. Finally, suppose that r0 = 10%, r1 = 11%, r2 = 9%

(a) What is the price of a two-year 0-coupon bond at t = 0 that pays 1 in s3, s4, s5 and

0 in s1, s2?

(b) Find the yield y of the two-year 0-coupon bond. Is the yield curve upward-sloping

or downward-sloping?

(c) Now suppose that r0 = 10%, r1 = 11%, r2 = 10%. Re-do parts (a) and (b).

(d) Based on your findings in (a) - (c) what should you expect about the one-period

spot interest rate so that the yield curve to be upward sloping?

(e) What is a state price and how is it calculated? How are state prices related to

risk-neutral probabilities? Be precise!

A13072W1 6

Part C

6. (a) The dynamic dividend discount model can be approximated with log linear approx-

imation and transformed to this equation:

var (pt − dt) ≈ cov

(

pt − dt,

∞∑

j=0

ρj∆dt+j+1

)

− cov

(

pt − dt,

∞∑

j=0

ρj∆rt+j+1

)

(6)

Explain how this equation relates excess price (or price-dividend ratio) volatility to

return predictability. Hint: the answer should include a statement about empirical

evidence. [2 marks]

(b) Assume that a researcher has T = 600 observations and runs a predictive regression

of the form rt+1 = α + βxt + t+1 and gets these estimates:

rt+1 = 0.01 + 0.9xt + t+1 (7)

The variable xt is the (demeaned) dividend-price ratio. At the same time, he esti-

mates this AR(1) for the predictive variable xt+1 = φxt + ut+1 with this result:

xt+1 = 0.998xt + ut+1 (8)

The estimated coefficient β suffers from the so called Stambaugh bias, which can be

approximated by E[βˆ − β] = γE[φˆ− φ] with E[φˆ− φ] = −(1+3φ)

T

. Use the value for

γ = −90 and the estimated value φˆ for φ and calculate the value of βStambaugh that

corrects for the bias according to the suggestions of Stambaugh (1999). [2 marks]

(c) Impose an economic restriction on the behaviour of the dividend-price ratio that

excludes explosive bubbles and adjust for the possible bias in βˆ after taking this

economic assumption into account in the sense of Lewellen (2004). [4 marks]

(d) Expectation Hypothesis makes a statement about the relationship between expected

excess returns of zero-coupon bonds and yields:

Et [rn,t+1 − y1,t] = ν (9)

Use the definition of the excess holding period return rn,t+1 − y1,t = sn,t − (n −

1)(yn−1,t+1−yn,t) and show what Expectation Hypothesis implies on the relationship

between term spread snt and expected long-term yields. Hint: state the relation

formally and describe in words (one sentence is enough). [2 marks]

(e) The Vasicek model of interest rates assumes that the stochastic discount factor

follows this equation of motion:

−mt+1 = λ

2

2

+ zt + λt+1 (10)

A13072W1 7 turn over

zt+1 = (1− φ)θ + φzt + σt+1, (11)

where mt+1 is the stochastic discount factor, zt is a state variable, λ is a risk premium

and t+1 is a mean zero normally distributed random noise with variance one. Use

the Vasicek model to derive the expression of a forward rate, f1,t+1, which at time

t fixes the interest rate for one period from t + 1 to t + 2 in terms of the model

parameters and the state variable zt. [5 marks]

A13072W1 8 last page

学霸联盟