Calculate the price at each call date an d the maturity date and …jbeckley/WD/MA 373/F17/Math... · 2018-05-01 · Tom purchases a 2 year bond which matures for 20,000. The bond

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

MATH 373

Test 3

Fall 2017 November 16, 2017

1. Jackson purchases a callable bond. The bond matures at the end of 20 years for 52,000. The

bond pays semi-annual coupons of 1300.

The bond can be called at the end of 14 years. The call value is 54,925.



Jackson buys the bond to yield 4% convertible semi-annually.

Determine the price of the bond.

Solution:

Calculate the price at each call date and the maturity date and pick the lowest price.

/ 4% / 2 2; 1300

28 54,925 59,123.20

32 53,950 59,136.50

36 52 59,105.,975

40 52,000 59,112.

07

42

I Y PMT

N FV PV


2. The common stock of Zhang Corporation pays a quarterly dividend. The next dividend of 5.00

will be paid in one month. Future dividends are expected to increase such that each dividend is

2% greater than the prior dividend. In other words, a dividend of 5.00 will be paid at the end of

one month. A dividend of 5.00(1.02) will be paid at the end of 4 months. A dividend of

25.00(1.02) will be paid at the end of 7 months, etc.

Using the dividend discount method, determine the price that Summer should pay in order to

have an annual effective return of 12%.

Solution:

1/12 4/12

1/12

3/12

5(1.12) 5(1.02)(1.12) ...

5(1.12)583.17

1 (1.02)(1.12)

PV


3. Wendy is the beneficiary of an annuity due which makes monthly payments for 15 years. Each

monthly payment in the first year are 1000. Each monthly payment in the second year is 2000.

The payments continue to increase until each monthly payment in the 15th year is 15,000.

Calculate the present value of Wendy’s annuity at an interest rate of 9% compounded monthly.

Solution:

15

15

(12

We have to use the formula that does not follow the rules since the payments are

level during each year but increase year to year. We also note that this is an annuity

due.

15(1 )1000 i

a iPV

i

(12)

)

(12)12

1515

112

12

0.090.0075 and (1.0075) 1 0.093806898

12 12

1 (1.093806898)(1.093806898) 15(1.093806898)

0.0938068981000 1.0075

0.0075

633,233.59

i

ii

PV


4. A continuous perpetuity that pays at a rate of 1000t at time t has a present value of 25,000

when calculated at a force of interest of .

Chengjia is receiving a continuous 20 year annuity that pays at a rate of 500t at time t .

Calculate the present value of Chengjia’s annuity using a force of interest equal to 0.5 which is

one half the force of interest used to calculate the present value of the perpetuity.

Solution:

2

2

20(0.1)20(0.1)

20( )

20

We will use the perpetuity to find .

1000 125,000 0.2

25

Now we will find the present value of the annuity at 0.5 0.10

12020 0.1(500) (500)

0.1

eea e

PV

29,699.71


5. Tom purchases a 2 year bond which matures for 20,000. The bond has semi-annual coupons.

The coupons are not level. The first two coupons are each equal to 1000. The second two

coupons are each equal to 2000.

The bond is bought to yield 13% convertible semi-annually.

Complete the following amortization table for Tom’s bond. Show formulas if you want full

credit.

Time k Coupon Interest in Coupon Principal in Coupon Book Value

0 --- --- ---

Present Value of Cash Flows =

1000v+1000v2+2000v3 +(2000+20000)v4

=20,577.43

1 1000 (20,577.43)(0.065)

= 1337.53 1000 – 1337.53

= -337.53 20,577.43 – (-337.53)

=20,914.96

2 1000 (20,914.96)(0.065)

= 1359.79 1000 – 1359.47

= -359.47 20,914.96 – (-359.47)

=21,277.43

3 2000 (21,277.43)(0.065)

= 1382.84 2000 – 1382.84

= 617.16 21,277.43 – 617.16

= 20,657.27

4 2000 (20,657.27)(0.065)

= 1342.72 2000 – 1342.72

= 657.28 20,657.27 – 657.28

= 20,000


6. A 20 year loan is being repaid with 20 annual non-level payments. The first payment is 25,000.

The second payment is 24,000. The payments continue to decrease until the last payment of

6000 is paid. The interest rate on the loan is an annual effective rate of 6%.

Calculate the principal in the 11th payment.

Solution:

10

10 10 10

11 10

11

We want to find the outstanding loan balance at time 10.

1000OLB (15,000) 10(1.06) 80,798.98

0.06

(0.06) (80,798.98)(0.06) 4847.94

15,000 4847.94 10,152.06

a a

I OLB

P


7. Kanishk can purchase either of the following two bonds:

a. Bond A has a par value of 25,000 and semi-annual coupons. The bond sells for 30,000.

The coupon rate is 6% convertible semi-annually. The amount of principal in the first

coupon is 71.70.

b. Bond B is a 20 year zero coupon bond. This bond also has a price of 30,000.

Bond A and Bond B have the same yield rate.

Calculate the maturity value of Bond B.

Solution:

0

1 1 1 0

20(2)

Price 30,000

(25,000)(0.06 / 2) 750

750 71.70 678.30 but ( ) (30,000)( )

678.300.02261

30,000

Price of Bond B = (Maturity Value)(1 ) since is for a six month perio

BV

Coupon

I Coupon P I BV r r

r

r r

40

d.

Maturity Value (30,000)(1.02261) 73,370.70


8. Connor buys a 12 year bond with a par value of F . The bond matures for 500F . The bond

has semi-annual coupons paid at a rate of 7% convertible semi-annually.

At yield rate of 9% compounded semi-annually, bond is bought at a discount of 400.

Determine F .

Solution:

24

24

2424

24

2424

400 400

500 500 400 100

1 (1.045)100 ( )(0.07 / 2) ( 500)(1.045)

0.045

(500)(1.045) 100 73.85173676

0.14495471 (1.045)1 (0.07 / 2) (1.045)

0.045

C P P C

C F P F F

P Fra Cv

F F F

F

509.4884


9. A 10 year bond pays semi-annual coupons that are increasing. The first coupon is 500. The

second coupon is 600. The third coupon is 700. The coupons continue to increase in the same

pattern. The bond has a maturity value of 13,000.

Calculate the price of the bond to yield 10% convertible semi-annually.

Solution:

20 20

20 20

20 2

The price is the present value of cash flows. Since each coupon is increasing, we must

use the P&Q formula.

100500 20(1.05) (13,000)(1.05)

0.05

1 (1.05) 100 1 (1.05)500

0.05 0.05

PV a a

020 2020(1.05) (13,000)(1.05)

0.05

20,979.51


10. Yuchen has a loan which is being repaid with level monthly payments of 500. The interest in the

30th payment is 113.74. The principal in the 36th payment is 402.77.

Determine the amount of the loan.

Solution:

30

1

66 6

30 36

1 30 29

1 30 1

1 0 0

Principal in 30th Payment 500 113.74 386.26

402.77(1 ) 386.26(1 ) 402.77 1 0.007

386.26

(1 ) 386.26(1.007) 315.52

500 315.52 184.48 ( ) (0.007)

A

P

P i P i i

P P i P

I OLB i OLB

0

184.48mount of Loan 26,354.29

0.007

If you carry more decimals or do the problem a different way, your answer

will be slightly different.

OLB


11. Chufan is considering the following two investments:

a. A preferred stock issued by Osborn LTD. The preferred stock pays quarterly dividends of

25 with the next dividend payable later today. The price of the stock is 1000.

b. A bond issued by Johnson Inc. The bond has a par value of 1000. The coupons are paid

semi-annually at a rate of 8.6% compounded semi-annually. The bond matures for 1500

at the end of 20 years.

The preferred stock of Osborn LTD and the bond of Johnsons Inc are expected to provide the

same annual effective interest rate.

Determine the price of the bond.

Solution:

(4) (4)

(4) (4)

4(4)

(2) (2)0.5 0.5

From the stock,

25 25 25Price=1000 1 25 0.025641

4 4 1000 25

4 4

1 1 0.1065774

But we need for the bond so (1 ) 1 (1.106577) 1 0.051939642 2

(1000)(0.0

i i

i i

ii

i ii

P

40

401 (1.05193964)86 / 2) 1500(1.05193964) 916.56

0.051939964

(1000)(0.043) 43; 1500; / 5.193964; 40

916.56

or

PMT FV I Y N

CPT PV


12. Lin has the choice of the following two loans:

a. An 10 year amortization loan for 100,000 from the Bank of Senese which requires

annual payments based on an annual effective interest rate 9.25%.

b. A sinking fund loan from Pitman Bank. The amount of the loan is 100,000 and must be

repaid over 10 years. At the end of each year, Lin would pay interest to Pitman Bank at

an annual effective interest rate of i . Additionally, Lin would have to make a deposit

into a sinking fund at the end of each year so that the amount in the sinking fund would

exactly repay the loan at the end of 10 years. The sinking fund will earn an annual

effective interest rate of 7%.

The annual payment under the loan from the Bank of Senese is equal to the total of the interest

payment and sinking fund deposit under the loan from Pitman Bank.

Determine i

Solution:

10

10

10

For the amortization loan, we find the payment as

100,000 100,00015,753.89

1 (1.0925)

0.0925

For the sinking fund loan, we have two payments ==> I and D

(100,000)

100,000 100,000

(1

Qa

I iL i

Ds

10

7237.75.07) 1

0.07

15,753.89 7237.7515,753.89 (100,000) 7237.75 0.0851614

100,000i i

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Calculate the price at each call date an d the maturity date and …jbeckley/WD/MA 373/F17/Math... · 2018-05-01 · Tom purchases a 2 year bond which matures for 20,000. The bond