Calculate the price at each call date an d the maturity ...

[Pages:12]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. The bond can be called at the end of 16 years. The call value is 53,950. The bond can be called at the end of 18 years. The call value is 52,975.

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.

I / Y 4% / 2 2; PMT 1300

N FV

PV

28 54,925 59,123.20

32 53,950 59,136.50

36 52,975 59,105.07 40 52, 000 59,112.42

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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 5.00(1.02)2 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:

PV 5(1.12)1/12 5(1.02)(1.12)4/12 ...

5(1.12)1/12

583.17

1 (1.02)(1.12)3/12

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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:

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.

PV

1000

a 15 i

15(1 i)15 i (12 ) 12

1

i (12 ) 12

i(12) 0.09 0.0075

and

i (1.0075)12 1 0.093806898

12 12

PV

1000

1 (1.093806898)15

0.093806898

(1.093806898)

0.0075

15(1.093806898)15

1.0075

633, 233.59

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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:

We will use the perpetuity to find .

25, 000

1000 2

2

1 25

0.2

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

PV

(500)

a 20

20e20( )

(500)

1 e20(0.1) 20e20(0.1) 0.1 0.1

29, 699.71

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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:

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

OLB10

(15, 000)a 10

1000 0.06

a 10(1.06)10 10

80, 798.98

I11 OLB10 (0.06) (80, 798.98)(0.06) 4847.94

P11 15, 000 4847.94 10,152.06

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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: BV0 Price 30, 000 Coupon (25, 000)(0.06 / 2) 750 I1 Coupon P1 750 71.70 678.30 but I1 (BV0 )r (30, 000)(r) r 678.30 0.02261

30, 000 Price of Bond B = (Maturity Value)(1 r)20(2) since r is for a six month period. Maturity Value (30, 000)(1.02261)40 73,370.70

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

8. Connor buys a 12 year bond with a par value of F . The bond matures for F 500 . 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: C P 400 P C 400

C F 500 P F 500 400 F 100

P Fra Cv24 24

F

100

(F

)(0.07

/

2)

1

(1.045)24 0.045

(F

500)(1.045)24

F

(500)(1.045)24 100

73.85173676 509.48

1

(0.07

/

2)

1

(1.045)24 0.045

(1.045)24

0.144954784

May 1, 2018 Copyright Jeffrey Beckley 2017, 2018

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