Financial Math

Homework 6

October 18, 2022

1. (Problem 18.2) Which of the following are equal to a3 (check all of them, as there may be more than 1 (or none!)).

(i) ν + ν2 + ν3

(ii) ä4 − 1

(iii) a∞ (1 − ν3)

2. (Problem 18.5) A perpetuity-immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. Calculate K.

3. (Problem 19.11) An annuity pays 1 at the end of each year for n years. Using an annual effective interest rate of i, the accumulated value of the annuity at time (n+ 1) is 13.776. It is also known that (1 + i)n = 2.476. Calculate n.

4. (Problem 21.1) James deposits 1,000 into an account at the end of each year for the next 6 years. The account earns 5% interest. James also deposits 1,000 at the end of each of years 7 through 10 into another account earning 4%. Calculate the total amount James will have in both accounts at the end of ten years.

5. (Problem 21.8) Find the present value of an annuity-immediate which pays 1 at the end of each half-year for five years, if the rate of interest is 8% convertible semiannually for the first three years and 7% convertible semiannually for the last two years.

6. (Problem 25.1) Calculate the present value of a continuous annuity of 1,000 per annum for 8 years at:

(a) An annual effective interest rate of 4%;

(b) A constant force of interest of 4%.

7. (Problem 25.14) A company makes payments continuously at a rate of $200 per year. The payments are made between 2 and 7 years. Find the accumulated value of these payments at time 10 years using an annual rate of interest of 6.5%.

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