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程序代写案例-FIN 448

时间：2021-03-31

FIN 448: Midterm 2019

90 min

April 23, 2019

1. Do not open the exam until everyone gets a copy and starts at the same time.

2. Write your answers in the space provided.

3. Show your work. Significant partial credit will be given for correct approach.

4. If for some question you need a number that you couldn’t calculate from the previous

question simply plug in the variable and write down the formulas you would solve.

This will give you significant partial credit.

Points Max

Part 1 15

Part 2 16

Part 3 8

Part 4 8

Total 47

The following Simon Code of Academic Integrity will be enforced:

“Every Simon student is expected to be completely honest in all academic matters. Simon

students will not in any way misrepresent their academic work or attempt to advance their

academic position through fraudulent or unauthorized means. No Simon student will be

involved knowingly with another student’s violation of this standard of honest behavior.”

Clearly Print Your Name:

1

Part 1

True or False (3pts). Evaluate whether the following statements are true of false.

1. 1pt (True / False) Suppose that there is no inflation (and there will be no inflation in

the future). Then for a long-term investor long-term bonds are safer than short-term

bonds.

Solution. This statement is true. Without inflation long-term bonds move wealth

across time preserving its purchasing power. Rolling over short-term bonds exposes

you to the fluctuations of the interest rates, thus, introduces uncertainty about the

long-term wealth through the interest rate risk.

2. 1pt (True / False) Continuously compounded yield curve is higher than semi-annually

compounded yield curve (when all rates are positive).

Solution. This statement is false. The same price corresponds to a smaller interest

rate rate the higher is the frequency of compounding.

3. 1pt (True / False) Time series of repo rates is not enough to estimate the parameters

of the Vasicek Model.

Solution. This statement is true. You need bond prices to estimate the two risk-neutral

parameters of the model r¯∗ and γ∗.

2

Multiple Choice (4pts). Answer the following multiple choice questions. Explain your

choice in a few sentences. You will receive 50% of the point for the correct answer and 50%

points for the correct explanation.

1. 2pts Consider a newly issued 3-year FRN with quarterly payments and a newly is-

sued 3-year coupon paying bond with quarterly coupons being determined by today’s

forward rates. Compare their prices:

(a) FRN is more expensive than the coupon bond

(b) FRN has the same price as the coupon bond

(c) FRN is cheaper than the coupon bond

Solution. The correct answer is (b). They are both priced at par.

2. 2pts Consider again a newly issued 3-year FRN with quarterly payments and a newly

issued 3-year coupon paying bond with quarterly coupons being determined by today’s

forward rates. Compare their durations:

(a) Duration of the FRN is larger than duration the coupon bond

(b) Duration of the FRN is the same as duration the coupon bond

(c) Duration of the FRN is smaller than duration the coupon bond

Solution. The correct answer is (c). Duration of the FRN is 0.25 (0ime till the next

payment) but the duration of the coupon bond will be close to 3 because of the large

principal at the end.

3

Short Answer (8pts). Answer the following questions.

1. 2pts Write down at least two equivalent statements of the expectation hypothesis.

Solution. Long spot rates are just the average of future expected spot rates. Forward

rates are equal to expected future spot rates. Expected holding period returns have to

be equal across maturities.

2. 2pts Name two reasons why banks were manipulating LIBOR.

Solution. To change the payoffs of derivatives linked to LIBOR and to appear strong

to their creditors during ’08-’09 crisis.

3. 2pts Suppose you test in the Treasury data whether 1-year forward rate f(0, 1, 2)

predicts 1-year spot rate r(1, 2) 1-year ahead. What coefficient b will you find in the

following regression? Why?

r(1, 2)− r(0, 1) = a+ b[f(0, 1, 2)− r(0, 1)] + ε

Solution. The coefficient will be close to 0. That’s because the forward rates do not

predict future spot rates, instead they predict excess returns.

4. 2pts Suppose that the 5-year LIBOR swap rate is 4%. What would be the price of a

5-year fixed coupon bond issued by the LIBOR quality bank with a 4% coupon rate?

Explain.

Solution. The price should be par since the swap rate is simply the par rate of the

LIBOR quality bonds.

4

Part 2 (16 pts)

Suppose that the semi-annually compounded Treasury yield curve today looks like

6mo 1y 18mo 2y

2% 2.25% 2.5% 3%

1. 4pts What is 12-18 months semi-annually compounded forward rate today, i.e., what

is f2(0, 1, 1.5)?

Solution. Using the forward discount factor we can write(

1 +

f2(0, 1, 1.5)

2

)−1

= F (0, 1, 1.5) =

B(0, 1.5)

B(0, 1)

=

(1 + 2.25%/2)2

(1 + 2.5%/2)3(

1 +

f2(0, 1, 1.5)

2

)−1

= 0.985217173684379

hence f2(0, 1, 1.5) ≈ 3.00%

2. 4pts Suppose that 6 months ago you bought a 18-24 months FRA with a forward rate

of 2% and notional $100. What is the value of this FRA now?

Solution. The value is

1

(1 + 2.5%/2)2

· f2(0, 1, 1.5)− 2%

2

· 100 ≈ 0.4817

5

3. 4pts What is the current 18 months swap rate?

Solution. First, let’s compute the prices of ZCB’s:

B(0, 0.5) = (1 + 2%/2)−1 ≈ 0.9901, B(0, 1) = (1 + 2.25%/2)−2 ≈ 0.97787,

B(0, 1.5) = (1 + 2.5%/2)−3 ≈ 0.96341.

And the 18-month swap c rate solves

c

2

[B(0, 0.5) +B(0, 1) +B(0, 1.5)] +B(0, 1.5) = 1

so c ≈ 2.495%

4. 4pts What is the current price of the last payment of the 18 months swap with notional

$100?

Solution. The last payment of the swap is 100 · r2(1,1.5)−2.495%

2

. Its price is the same as

the price of 100 · f2(0,1,1.5)−2.495%

2

since the difference between those is an FRA whose

price is 0. And the latter price is

1

(1 + 2.5%/2)3

· 100 · 3%− 2.495%

2

≈ 0.24326

6

Part 3 (8 pts)

Suppose that the price of a 6-month T-Bill is 99.005, the price of the 1-year 2% T-Note

is 99.496 and the price of a 1.5 year 4% T-Note is 101.442.

1. 4pts Bootstrap the 1-year continuously compounded spot rate.

Solution. The one-year ZCB solves

1 · 0.99005 + (1 + 100)B(0, 1) = 99.496 ⇒ B(0, 1) ≈ 0.97530

and the continuous spot rate is

r(0, 1) = −ln(B(0, 1)) ≈ 2.5%

2. 4pts Bootstrap the 1.5-year continuously compounded spot rate.

Solution. The 1.5-year ZCB solves

2 · 0.99005 + 2 · 0.97530 + (2 + 100) ·B(0, 1.5) = 101.442

so the price of the ZCB is

B(0, 1.5) ≈ 0.95599

and the continuous spot rate is

r(0, 1.5) = −ln(B(0, 1.5))/1.5 ≈ 3%

7

Part 4 (8 pts)

Suppose you have a portfolio which pays

6mo 1y 18mo 2y

200× r2(0,0.5)

2

200× r2(0.5,1)

2

100× r2(1,1.5)

2

100× r2(1.5,2)

2

where r2(0, T ) is the semi-annually compounded spot rate.

1. 8pts Replicate this portfolio using swaps and zero-coupon bonds. Suppose that swaps

have semi-annual payments and that the 1-year swap rate is 2%, 1.5-year swap rate is

2.5%, and 2-year swap rate is 3%.

Solution. Break the initial portfolio into

6mo 1y 18mo 2y

100× r2(0,0.5)−3%

2

100× r2(0.5,1)−3%

2

100× r2(1,1.5)−3%

2

100× r2(1.5,2)−3%

2

100× r2(0,0.5)−2%

2

100× r2(0,0.5)−2%

2

0 −0

100× 5%

2

100× 5%

2

100× 3%

2

100× 3%

2

The first line is a payoff of an 2-year swap with notional 100.

The second line is a payoff of a 1-year swap with notional 100.

The third line is a payoff of four different ZCBs.

8

学霸联盟

90 min

April 23, 2019

1. Do not open the exam until everyone gets a copy and starts at the same time.

2. Write your answers in the space provided.

3. Show your work. Significant partial credit will be given for correct approach.

4. If for some question you need a number that you couldn’t calculate from the previous

question simply plug in the variable and write down the formulas you would solve.

This will give you significant partial credit.

Points Max

Part 1 15

Part 2 16

Part 3 8

Part 4 8

Total 47

The following Simon Code of Academic Integrity will be enforced:

“Every Simon student is expected to be completely honest in all academic matters. Simon

students will not in any way misrepresent their academic work or attempt to advance their

academic position through fraudulent or unauthorized means. No Simon student will be

involved knowingly with another student’s violation of this standard of honest behavior.”

Clearly Print Your Name:

1

Part 1

True or False (3pts). Evaluate whether the following statements are true of false.

1. 1pt (True / False) Suppose that there is no inflation (and there will be no inflation in

the future). Then for a long-term investor long-term bonds are safer than short-term

bonds.

Solution. This statement is true. Without inflation long-term bonds move wealth

across time preserving its purchasing power. Rolling over short-term bonds exposes

you to the fluctuations of the interest rates, thus, introduces uncertainty about the

long-term wealth through the interest rate risk.

2. 1pt (True / False) Continuously compounded yield curve is higher than semi-annually

compounded yield curve (when all rates are positive).

Solution. This statement is false. The same price corresponds to a smaller interest

rate rate the higher is the frequency of compounding.

3. 1pt (True / False) Time series of repo rates is not enough to estimate the parameters

of the Vasicek Model.

Solution. This statement is true. You need bond prices to estimate the two risk-neutral

parameters of the model r¯∗ and γ∗.

2

Multiple Choice (4pts). Answer the following multiple choice questions. Explain your

choice in a few sentences. You will receive 50% of the point for the correct answer and 50%

points for the correct explanation.

1. 2pts Consider a newly issued 3-year FRN with quarterly payments and a newly is-

sued 3-year coupon paying bond with quarterly coupons being determined by today’s

forward rates. Compare their prices:

(a) FRN is more expensive than the coupon bond

(b) FRN has the same price as the coupon bond

(c) FRN is cheaper than the coupon bond

Solution. The correct answer is (b). They are both priced at par.

2. 2pts Consider again a newly issued 3-year FRN with quarterly payments and a newly

issued 3-year coupon paying bond with quarterly coupons being determined by today’s

forward rates. Compare their durations:

(a) Duration of the FRN is larger than duration the coupon bond

(b) Duration of the FRN is the same as duration the coupon bond

(c) Duration of the FRN is smaller than duration the coupon bond

Solution. The correct answer is (c). Duration of the FRN is 0.25 (0ime till the next

payment) but the duration of the coupon bond will be close to 3 because of the large

principal at the end.

3

Short Answer (8pts). Answer the following questions.

1. 2pts Write down at least two equivalent statements of the expectation hypothesis.

Solution. Long spot rates are just the average of future expected spot rates. Forward

rates are equal to expected future spot rates. Expected holding period returns have to

be equal across maturities.

2. 2pts Name two reasons why banks were manipulating LIBOR.

Solution. To change the payoffs of derivatives linked to LIBOR and to appear strong

to their creditors during ’08-’09 crisis.

3. 2pts Suppose you test in the Treasury data whether 1-year forward rate f(0, 1, 2)

predicts 1-year spot rate r(1, 2) 1-year ahead. What coefficient b will you find in the

following regression? Why?

r(1, 2)− r(0, 1) = a+ b[f(0, 1, 2)− r(0, 1)] + ε

Solution. The coefficient will be close to 0. That’s because the forward rates do not

predict future spot rates, instead they predict excess returns.

4. 2pts Suppose that the 5-year LIBOR swap rate is 4%. What would be the price of a

5-year fixed coupon bond issued by the LIBOR quality bank with a 4% coupon rate?

Explain.

Solution. The price should be par since the swap rate is simply the par rate of the

LIBOR quality bonds.

4

Part 2 (16 pts)

Suppose that the semi-annually compounded Treasury yield curve today looks like

6mo 1y 18mo 2y

2% 2.25% 2.5% 3%

1. 4pts What is 12-18 months semi-annually compounded forward rate today, i.e., what

is f2(0, 1, 1.5)?

Solution. Using the forward discount factor we can write(

1 +

f2(0, 1, 1.5)

2

)−1

= F (0, 1, 1.5) =

B(0, 1.5)

B(0, 1)

=

(1 + 2.25%/2)2

(1 + 2.5%/2)3(

1 +

f2(0, 1, 1.5)

2

)−1

= 0.985217173684379

hence f2(0, 1, 1.5) ≈ 3.00%

2. 4pts Suppose that 6 months ago you bought a 18-24 months FRA with a forward rate

of 2% and notional $100. What is the value of this FRA now?

Solution. The value is

1

(1 + 2.5%/2)2

· f2(0, 1, 1.5)− 2%

2

· 100 ≈ 0.4817

5

3. 4pts What is the current 18 months swap rate?

Solution. First, let’s compute the prices of ZCB’s:

B(0, 0.5) = (1 + 2%/2)−1 ≈ 0.9901, B(0, 1) = (1 + 2.25%/2)−2 ≈ 0.97787,

B(0, 1.5) = (1 + 2.5%/2)−3 ≈ 0.96341.

And the 18-month swap c rate solves

c

2

[B(0, 0.5) +B(0, 1) +B(0, 1.5)] +B(0, 1.5) = 1

so c ≈ 2.495%

4. 4pts What is the current price of the last payment of the 18 months swap with notional

$100?

Solution. The last payment of the swap is 100 · r2(1,1.5)−2.495%

2

. Its price is the same as

the price of 100 · f2(0,1,1.5)−2.495%

2

since the difference between those is an FRA whose

price is 0. And the latter price is

1

(1 + 2.5%/2)3

· 100 · 3%− 2.495%

2

≈ 0.24326

6

Part 3 (8 pts)

Suppose that the price of a 6-month T-Bill is 99.005, the price of the 1-year 2% T-Note

is 99.496 and the price of a 1.5 year 4% T-Note is 101.442.

1. 4pts Bootstrap the 1-year continuously compounded spot rate.

Solution. The one-year ZCB solves

1 · 0.99005 + (1 + 100)B(0, 1) = 99.496 ⇒ B(0, 1) ≈ 0.97530

and the continuous spot rate is

r(0, 1) = −ln(B(0, 1)) ≈ 2.5%

2. 4pts Bootstrap the 1.5-year continuously compounded spot rate.

Solution. The 1.5-year ZCB solves

2 · 0.99005 + 2 · 0.97530 + (2 + 100) ·B(0, 1.5) = 101.442

so the price of the ZCB is

B(0, 1.5) ≈ 0.95599

and the continuous spot rate is

r(0, 1.5) = −ln(B(0, 1.5))/1.5 ≈ 3%

7

Part 4 (8 pts)

Suppose you have a portfolio which pays

6mo 1y 18mo 2y

200× r2(0,0.5)

2

200× r2(0.5,1)

2

100× r2(1,1.5)

2

100× r2(1.5,2)

2

where r2(0, T ) is the semi-annually compounded spot rate.

1. 8pts Replicate this portfolio using swaps and zero-coupon bonds. Suppose that swaps

have semi-annual payments and that the 1-year swap rate is 2%, 1.5-year swap rate is

2.5%, and 2-year swap rate is 3%.

Solution. Break the initial portfolio into

6mo 1y 18mo 2y

100× r2(0,0.5)−3%

2

100× r2(0.5,1)−3%

2

100× r2(1,1.5)−3%

2

100× r2(1.5,2)−3%

2

100× r2(0,0.5)−2%

2

100× r2(0,0.5)−2%

2

0 −0

100× 5%

2

100× 5%

2

100× 3%

2

100× 3%

2

The first line is a payoff of an 2-year swap with notional 100.

The second line is a payoff of a 1-year swap with notional 100.

The third line is a payoff of four different ZCBs.

8

学霸联盟