# 1 calculate the principal in the coupon paid at the end

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MATH 373 Test 2

Fall 2018

November 1, 2018

1. A 20 year bond has a par value of 1000 and a maturity value of 1300. The semi-annual coupon rate for the bond is 7.5% convertible semi-annually. The bond is purchased to yield 9% convertible semi-annually. Calculate the principal in the coupon paid at the end of the 12th year. Solution:

This is a question that can be done on your calculator. FV 1300; PMT (1000)(0.075 / 2) 37.50; N (20)(2) 40; I / Y 9 / 2 4.5 CPT PV 2nd Amort P1 P2 (12)(2) 24 PRN 9.94

April 2, 2021 Copyright Jeffrey Beckley 2018, 2019, 2020, 2021

2. Haozhe has 420,000 that he wants to invest. He has the following possible investments:

a. Purchase a perpetuity immediate with annual payments for 420,000. The perpetuity will pay 1000 at the end of the first year, 2000 at the end of the second year, 3000 at the end of the third year,etc.

b. Make a loan of 420,000 to Jun. Jun will repay the loan with level annual payments of 30,000 followed by a drop payment.

The annual effective interest rate for both investments is the same.

Calculate the amount of the drop payment.

Solution:

First determine the interest rate to be sued using Part a.

1000 1000 420, 000 1000i 1000 420, 000i2 i 1 420i2

i

i2

420i2 i 1 0 i (1)

(1)2 (4)(420)(1) 0.05

2(420)

Now we find the drop payment:

PV 420, 000; I / Y 5; PMT 30, 000; CPT N 24.676

2nd Amort P1 1; P2 24; Bal 19, 482.01 Drop (19, 482.01)(1.05) 20, 456.11

April 2, 2021 Copyright Jeffrey Beckley 2018, 2019, 2020, 2021

3. Kimberly purchases an 18 year continuous annuity that pays at a rate of 1300t at time t . Calculate the present value of this annuity using a force of interest of 0.0625.

Solution:

1300(Ia) 18

1 e18(0.0625)

1300

0.0625

18e 18( 0.0625)

103, 205.78

0.0625

April 2, 2021 Copyright Jeffrey Beckley 2018, 2019, 2020, 2021

4. Tomas is receiving an annuity due with monthly payments for the next five years. The first

payment is 500 at the start of the first month. The second payment is 500(1.08) at the beginning of the second month. The third payment is 500(1.08)2 at the beginning of the third

month. The payments continue to increase in the same pattern.

Calculate the present value of this annuity using an interest rate of 12% compounded monthly.

Solution:

i (12 )

0.12

Since payments ar monthly, we need which is 0.01

12

12

PV 500 500(1.08)(1.01)1 ... 500(1.08)59 (1.01)59

500 500(1.08)60 (1.01)60 1 (1.08)(1.01)1

394,887.72

April 2, 2021 Copyright Jeffrey Beckley 2018, 2019, 2020, 2021

5. Ram buys an annuity immediate for his Mom. The annuity will make quarterly payments to his Mom for 20 years. The payments are 1000 each quarter in the first year. The payments are 1100 each quarter of the second year. The payments continue to increase in the same pattern until payments of 2900 are paid each quarter of the 20th year.

Using an interest rate of 8% compounded quarterly, calculate the price that Ram paid for this annuity. (The price is the present value of the payments.)

Solutions:

To solve this problem, we need to use the formula that does not follow the rules. However, since the first payment does not equal the amount of the increase, we must split the annuity into level payments of 900 and payments that are 100 the first year, 200 the second year, etc.

To use the Formula that does not follow the rules, we need both i(4) and i. 4

i(4) 0.08 0.02 44

i

1

i(4) 4

4

1

(1.02)4

1

0.08243216

a

20(1.08243216)20

PV

900a 80 0.02

100

20 0.08243216

0.02

900

1

(1.02)80 0.02

100

1

(1.08243216)20 0.08243216

(1.08243216) 0.02

20(1.08243216)20

67, 448.36

April 2, 2021 Copyright Jeffrey Beckley 2018, 2019, 2020, 2021

6. The Huang Company invests 10,000 at the end of each year with DeWitt Bank. DeWitt Bank pays an annual effective interest rate of 6%.

At the end of each year, Huang withdraws the interest earned from DeWitt and reinvests it in the Carvajal Fund which pays an annual effective interest rate of 8.5%.

Determine the total amount that Huang has at the end of 15 years. Solution: The first year, there is no interest transfered to Carvajal as there is no money invested in DeWitt the first year. At the end of the second year, 600 is withdrawn from DeWitt and invested in Carvajal. At the end of the third year, 1200 is withdrawn from DeWittt and invested in Carvajal. This amount contines to increase each year.

At the end of 15 years, there will be (15)(10, 000) 150, 000 in the DeWitt Bank.

At the end of 15 years, there will be

600a14

600 i

a 14(1.085)14 14

(1.085)14

93, 404.25

Total 150, 000 93, 403.97 243, 404.25

April 2, 2021 Copyright Jeffrey Beckley 2018, 2019, 2020, 2021