利息理论习题

1.1

1. Sally has two IRAs. IRA 1 earns interest at 8% effective annually and IRA 2 earns interest at 10% effective annually. She has not made any contributions since January 1, 1985, when the amount in IRA 1 was twice the amount in IRA

2.The sum of the two accounts on January 1, 1993 was $75000. Determine how much was in IRA 2 on January 1, 1985?（Individual Retirement Account）

2. Suppose we are given that the effective rate of interest is 5% in the first year and 6% in the second year .We invest $1 at time 0. How much is in the fund at the end of two years?

3. An investor puts 100 into Fund X and 100 into Fund Y. Fund Y earns compound interest at the annual rate of j, and Fund X earns simple interest at the annual rate of 1.05j . At the end of 2 years, the amount in Fund Y is equal to the amount in Fund X. Calculate the amount in Fund Y at the end of 5 years?

4. Eric deposits X into a savings account at time 0, which pays interest at

a nominal rate of i , compounded semiannually. Mike deposits 2X into a different savings account at time 0, which pays simple interest at an annual rate of i . Eric and Mike earn the same amount of interest during

the last 6 months of the 8th year. Calculate i.

5. John invests 1000 in a fund which earns interest during the first year at a nominal rate of K convertible quarterly. During the 2nd year the fund earns interest at a nominal discount rate of K convertible quarterly. At the end of the 2nd year, the fund has accumulated to 1173.54. Calculate K.

6. A deposit of X is made into a fund which pays an annual effective interest rate of 6% for 10 years. At the same time, X/2 is deposited into another fund which pays an annual effective rate of discount of d for 10 years. The amounts of interest earned over the 10 years are equal for both funds. Calculate d.

7. You are given: 2()A t Kt Lt M =++ for 02t ≤≤

(0)100,(1)110,(2)136A A A === Determine the force of interest at time 12

t =

. 8. At time 0, 100 is deposited into Fund X and also into Fund Y. Fund X accumulates at a force of interest ()20.51t t δ-=+. Fund Y accumulates at an annual effective interest rate of i . At the end of 9 years, the accumulated value of Fund X equals the accumulated value of Fund Y. Determine i .

1.2

1. At an effective annual interest rate of ,0

i i>, each of the following two sets of payments has present value K:

1) A payment of 121 immediately and another payment of 121 at

the end of one year.

2) A payment of 144 at the end of two years and another payment

of 144 at the end of three years. Calculate K.

2. You are given:

1)The sum of the present values of a payment of X at the end of 10

years and a payment of Y at the end of 20 years is equal to the

present value of a payment of X+Y at the end of 15 years.

2)X+Y=100

3)5%

i=. Calculate X.

3.A customer is offered an investment where interest is calculated

according to the following force of interest :

0.0203

0.0453 t

t t

t

δ

≤≤

⎧

=⎨

>

⎩

The customer invests 1000 at time 0. What nominal rate of interest , compounded quarterly, is earned over the first four-year period?

4. Payments of 300,500 and 700 are made at the end of years five, six