MATH 373 Test 4 Fall 2017jbeckley/WD/MA 373/F17/Math...Cv v §· ¨¸ ©¹ §· ¨¸ ©¹ ¦ ¦ 1 2...

May 3, 2018 Copyright Jeffrey Beckley 2017, 2018

MATH 373

Test 4

Fall 2017 December 12, 2017

1. A three year bond has an annual coupon of 40 and a maturity value of 1100.

Calculate the Modified Convexity for this bond at an annual effective interest rate of 6.5%.

Solution:

2

1 2 32

3

3

( )( 1)

40(1)(2)(1.065) 40(2)(3)(1.065) 1140(3)(4)(1.065)(1.065)

40 1100(1.065)

10.07065

t

t

t

t

v C t t vModCon

C v

a


2. Stanley Insurance Company has agreed to pay Jacqueline 400,000 at the end of each year for

three years. Stanley wants to use the following three bonds to exactly match the payments to

Jacqueline.

a. Bond A is a one year bond with a maturity value of 1000, annual coupons of 70, and a

price of 1005.

b. Bond B is a two year bond with annual coupons of 200 and a maturity value of 1000. It

sells to yield an annual effective interest rate of 8%.

c. Bond C is a three year bond with annual coupons of 270. The maturity value and the

price of this bond is 3000.

Calculate the cost to buy the bonds to exactly match the payments. Assume that we can

purchase partial bonds.

Solutions:

Time Year 1 Year 2 Year 3

Payments 400,000 400,000 400,000

Bond A 1070 0 0

Bond B 200 1200 0

Bond C 270 270 3270

Let A be the number of Bond A purchased. Let B be the number of Bond B purchased. Let C be

the number of Bond C purchased.

400,0003270( ) 400,000 122.3242

3270

400,000 (270)(122.3242)1200( ) 270( ) 400,000 305.8104

1200

1070( ) 200( ) 270( ) 400,000

400,000 200(305.8104) 270(122.3242)285.8041

1070

Cos ( )(1005

C C

B C B

A B C

A

Total t A

22 2

2

) ( )(Price of B) ( )(3000)

1 (1.08)Price of B = 200 1000 200 1000(1.08) 1213.99

0.08

(285.8041)(1005) (305.8104)(1213.99) (122.3242)(3000) 1,025,456.49

B C

a v


3. A 30 year bond matures for 10,000 and has semi-annual coupons of 300.

Calculate the Modified duration at an annual effective interest rate of 10.25%.

Solution:

(2)0.5

6060

(1.1025) 1 0.052

1 (1.05)Price 300 10,000(1.05) 6214.14

0.05

i

Modified Duration =

0.5 1.5 8 30

0.5 1 1.5 30 30

300(0.5) 300(1) 300(1.5) ... 300(30) (10,000)(30) 1

6214.14 1.1025

150(1.1025) 300(1.1025) 450(1.1025) ... 9000(1.1025) (10,000)(30) 1

6214.14 1.1025

1

t

t

t

t

C tv v v v v vv

C v

v

1 2 3 60 30

60 30

60 0.05 60 0.05

60 60

50(1.05) 300(1.05) 450(1.05) ... 9000(1.05) 10,000(30) 1

6214.14 1.1025

150150 60 1.05 (300,000) 1.1025

0.05

(6214.14) 1.1025

1 (1.05) 150 1 (1.05)150

0.05 0.05

v

a a

306060(1.05) 300,000 1.10250.05

9.641(6214.14) 1.1025


4. A three year bond with annual coupons of 300 matures for 4000.

Calculate the Macaulay Convexity of this bond at an annual effective rate of 5%.

Solution:

2

2 1 2 2 2 3

3

3

( )

300(1 )(1.05) 300(2 )(1.05) 4300(3 )(1.05)

300 4000(1.05)

34,804.664728.14654

4272.324803

t

t

t

t

C t vMacCon

C v

a


5. The Moses Life Insurance Company owns the following two bonds:

Price Macaulay Duration

Macaulay Convexity

Bond A 60,000 A 90

Bond B 40,000 12 125

Using an interest rate of 8%, the Modified Convexity of this bond portfolio is 98.654.

Determine A.

Solution:

2 2

2 2

2

Bond A

90

(1.08) (1.08)

Bond B

12 125117.4554

(1.08) (1.08)

90(60,000) (40,000)(117.4554)

(1.08)98.654

60,000 40,000

90

A A B B

A B

MacDur MacCon AModCon

MacDur MacConModCon

A

P C P CModConPort

P P

A

2

2

98.654(100,000) (40,000)(117.4554)86.1197

(1.08) 60,000

86.1197(1.08) 90 10.45A


6. Billie has promised to make a payment of 1,000,000 to Bokun at the end of 7.5 years. Billie

wants to immunize this payment using Reddington Immunization and the following two zero

coupon bonds:

a. Bond 1 is a zero coupon bond with a maturity value of 10,000 at the end of 4 years.

b. Bond 2 is a zero coupon bond with a maturity value of 15,000 at the end of 8 years.

Assuming an interest rate of 8%, determine the number of Bond 1 which Billie should purchase.

Assume that we can purchase partial bonds.

Solution:

2

2 1

7.5

We can use the shortcut to do this problem.

Amount of Money to be used to Purchase Bond 1 = ( )

8 7.5(1,000,000)(1.08) 70,182.98683

8 4

Price of

Bond Liability

Bond Bond

D DPVofLiability

D D

4Bond 1 = (10,000)(1.08) 7350.298527

Amount of Money 70,182.98683Number of Bonds = 9.5483

Price of Bond 1 7350.298527


7. A perpetuity due pays 100 at the start of each year.

Calculate the Macaulay Duration at an interest rate of 6%.

Solution:

0 1 2

0 1 2

1 2 2

0 1 2

( ) (100)(0) (100)(1) (100)(2) ...

(100) (100) (100) ...

100 100 11

(100)(1) (100)(2) ...

100(100) (100) (100) ... (1 )(1 )

11

0.06 16.671.06

t

t

t

t

C t v v v vMacDur

C v v v v

v v i i i

v v v ii

i


8. Graham owns a bond with a price of 130,000 at an annual effective yield rate of 7%. The bond

has a Modified Duration of 14.0187 and a Modified Convexity of 100 at an annual effective

interest rate of 7%.

Graham estimates that price of this bond at an interest rate of i using the first order Macaulay

approximation to be 117,886.61.

Determine .i

Solution:

00

1

15

First Order Macaulay Approximation

1( ) ( )

1

( ) 14.0187 (1.07) ( ) (14.0187)(1.07) 15

1.07 1.07 117,886.61( ) 117,886.61 (130,000)

1 1 130,0

MacDuri

P i P ii

ModDur V MacDur MacDur MacDur

P ii i

1

15

0.99350046600

1.071 1.077 0.077

0.993500466i i


9. You are given the following spot interest rates:

Time (t) tr

0.5 2.1%

1.0 2.3%

1.5 2.6%

2.0 3.0%

2.5 3.3%

3.0 3.7%

3.5 4.2%

4.0 4.8%

4.5 5.5%

5.0 6.0%

Megan purchases a two year par value bond with semi-annual coupons at a rate of 8%

convertible semi-annually. The par value of the bond is 10,000.

Calculate the price of the bond.

Solution:

0.5 1 1.5 2

(10,000)(0.08 / 2) 400

400 400 400 10,40010,974.76

(1.021) (1.023) (1.026) (1.03)

Coupon

P


10. The following three bonds are priced using the same spot interest rates:

a. Bond A is a one year bond with a maturity value of 1000, annual coupons of 70, and a

price of 1005.

b. Bond B is a two year bond with annual coupons of 200 and a maturity value of 1000. It

sells to yield an annual effective interest rate of 8%.

c. Bond C is a three year bond with annual coupons of 270. The maturity value and the

price of the bond are 3000.

Kristin has a three year annuity immediate with annual payments of 1000. Using the same spot

interest rates, determine the accumulated value of Kristin’s annuity.

Solution:

1

First we need to find the spot rates using bootstrapping. Then, using the spot rates, we

will find the present value of the annuity. Finally, we will find the accumulated value

of the annuity.

1070

1 r1

22 2

2 21 2

0.5

22

2

10701005 1 0.06467662

1005

200 1200 1 (1.08)Price of B = 200 1000 200 1000(1.08) 1213.99

1 (1 ) 0.08

1200 200 12001213.99 1026.14 1 0.081402

(1 ) 1.06467662 1026.14

r

a vr r

rr

2 3 3 2

1 2 3 3

1/3

3

2 3 2

1 2 3

270 270 3270 3270 270 2703000 3000 2515.5202

1 (1 ) (1 ) (1 ) 1.06467662 (1.081402)

32701 0.091373

2515.5202

1000 1000 1000 1000 1000 1000

1 (1 ) (1 ) 1.06467662 (1.081402)

r r r r

r

PVr r r

3

3 3

3

2563.64(1.091373)

(1 ) (2563.64)(1.091373) 3332.55AV PV r


11. You are given:

a. [0,1] 0.05f

b. [1,2] 0.06f

c. [2,3] 0.07f

d. [3,4] 0.08f

Calculate the price of a zero coupon bond that matures for 100,000 at the end of four years.

Solution:

100,00077,749.45

(1.05)(1.06)(1.07)(1.08)P


You are given the following spot interest rates and information for Questions 12-15:

Time t Spot Rate rt

1 0.053

2 0.057

3 0.063

4 0.071

5 0.080

The F&G Corporation borrows 700,000 from Abbott Bank. The terms of the loan are such that F&G will

pay a variable interest rate on the loan each year and will repay the 700,000 at the end of four years.

The variable interest rate will be the one year spot rate at the beginning of each settlement period.

F&G is uncomfortable with the risk of having a variable loan interest rate. Therefore, F&G purchases a

four year interest rate swap from M&J Investment Bank. At the end of each year, F&G will pay a fixed

rate to M&J Investment Bank. At the end of each year, M&J will pay the variable rate to F&G. The

amount of the payment will be based on the amount of the loan of 700,000.

12. Answer the questions below. These four parts are worth one question.

a. List the Payer under this agreement.

F&G Corporation

b. List the Receiver under this agreement.

M&J Investment Bank

c. State the Settlement Period under the agreement.

The settlement period is one year.

d. List the notional amount in the first year under this agreement.

The notional amount in the first year is 700,000.

13. Calculate the swap rate under this agreement.

Solution:

4

4

1 2 3 4

1 2 3 4

1 1 (1.071)0.069807645

(1.053) (1.057) (1.063) (1.071)

PR

P P P P


You are given the following spot interest rates and information for Questions 12-15:

Time t Spot Rate rt

1 0.053

2 0.057

3 0.063

4 0.071

5 0.080

The F&G Corporation borrows 700,000 from Abbott Bank. The terms of the loan are such that F&G will

pay a variable interest rate on the loan each year and will repay the 700,000 at the end of four years.

The variable interest rate will be the one year spot rate at the beginning of each settlement period.

F&G is uncomfortable with the risk of having a variable loan interest rate. Therefore, F&G purchases a

four year interest rate swap from M&J Investment Bank. At the end of each year, F&G will pay a fixed

rate to M&J Investment Bank. At the end of each year, M&J will pay the variable rate to F&G. The

amount of the payment will be based on the amount of the loan of 700,000.

14. Based on the spot interest rate curve at the time of the loan, what is the implied rate that F&G

would pay to Abbott Bank in the third year of the loan?

Solution:

[2,3]

3 33 2 3

3 2 [2,3] [2,3] 2 2

2

The implied rate is .

(1 ) (1.063)(1 ) (1 ) (1 ] 1 1 0.075102

(1 ) (1.057)

f

rr r f f

r

15. Calculate the net swap payment at the end of the first year. State who receives the payment

and who makes the payment.

Solution:

F&G Owes (notional amount)(fixed rate) (700,000)(0.069807645) 48,865.35

M&J Owes (notional amount)(variable rate) (700,000)(0.053) 37,100.00

F&G Pays 11,765.35 to M&J


16. You are given the following spot interest rates:

Time t Spot Rate rt

1 0.053

2 0.057

3 0.063

4 0.071

5 0.080

Thomas and Brett enter into a five year deferred interest rate swap. There will be no swap

during the first three years. The settlement periods will be one year. Under the swap, Thomas

agrees to pay a variable interest rate to Brett at the end of the fourth year and at the end of the

fifth year. The variable interest rate is based on the one year spot interest rate at the start of

each year.

In return, Brett agrees to pay a fixed rate to Thomas at the end of the fourth year and the fifth

year.

The notional swap amount is 100,000 for both settlement periods.

Calculate the swap rate for this swap.

Solution:

3 5

3 5

4 5

4 5

(1.063) (1.080)0.10547

(1.071) (1.080)

P PR

P P


17. The current spot interest rate curve is as follows:

t tr t

tr

0.25 2.50% 1.75 3.40%

0.50 2.65% 2.00 3.48%

0.75 2.79% 2.25 3.80%

1.00 2.92% 2.50 4.10%

1.25 3.10% 2.75 4.35%

1.50 3.25% 3.00 4.50%

Cai has a two year loan for 500,000 which has a variable interest rate that resets at the

beginning of each six month period. The interest rate will be the six month spot interest rate at

the beginning of each six month period.

Cai enters into an interest rate swap where she is the payer. The characteristics of the swap

mirror those of the loan.

Determine the six month swap rate that Cai will pay.

Solution:

2

2

0.5 1 1.5 2

0.5 1 1.5 2

1 1 (1.0348)0.0171956

(1.0265) (1.0292) (1.0325) (1.0348)

PR

P P P P


18. Sue and Gavin enter into a three year swap. Under the swap, Sue is the payer and Gavin is the

receiver. The swap has annual settlement periods. The variable rate to be paid is the one year

spot rate at the beginning of each year for the next three years.

The following is the spot interest rate curve:

Time t Spot Rate rt

1 0.060

2 0.055

3 0.050

4 0.048

5 0.046

The notional amount for the swap changes each year. The notional amount during the first year

is 500,000. The notional amount during the second year is 300,000. The notional amount

during the third year is 100,000.

Calculate the swap rate for this swap.

Solution:

* * *

1 [0,1] 1 2 [1,2] 2 3 [2,3] 3

1 1 2 2 3 3

[0,1] 1

2

[1,2]

3

[2,3] 2

1 2

0.06

(1.055)1 0.050023585

1.06

(1.05)1 0.040070978

(1.055)

(500,000)(0.06)(1.06) (300,000)(0.050023585)(1.055)

Q f P Q f P Q f PR

Q P Q P Q P

f r

f

f

3

1 2 3

(100,000)(0.040070978)(1.05)

(500,000)(1.06) (300,000)(1.055) (100,000)(1.05)

0.054671

MATH 373 Test 4 Fall 2017jbeckley/WD/MA 373/F17/Math...Cv v §· ¨¸ ©¹ §· ¨¸ ©¹ ¦ ¦ 1 2...

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