25 CONTINUOUS ANNUITIES 231

Practice Problems

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%.

Problem 25.2

Find the force of interest at which s20

= 3s10.

Problem 25.3

If an = 4 and sn = 12, find .

Problem 25.4

There is $40,000 in a fund which is accumulating at 4% per annum convertible continuously. Ifmoney is withdrawn continuously at the rate of $2,400 per annum, how long will the fund last?

Problem 25.5

Annuity A offers to pay you $100 per annum, convertible continuously, for the next five years.Annuity B offers you $ X at the end of each year for ten years. The annual effective interest rate iis 8%. Find X such that you are indifferent between the two annuities.

Problem 25.6

Given = 0.1. Evaluatea(12)

10a10

.

Problem 25.7

You are given ddtst= (1.02)2t. Calculate .

Problem 25.8

Given an = n 4 and = 10%, find n0atdt.

Problem 25.9

Payments will be made to you at a continuous rate of $100 per year for the next five years. Theeffective rate of interest over this period is i = 0.06. Calculate the present value of this paymentstream.

Problem 25.10

Show that ddtat =

t

s1.

232 THE BASICS OF ANNUITY THEORY

Problem 25.11

Show that an < a(m)n < an < a

(m)n < an . Hint: See Example 10.15.

Problem 25.12

Find an expression for t, 0 < t < 1, such that 1 paid at time t is equivalent to 1 paid continuouslybetween 0 and 1.

Problem 25.13

A bank makes payments continuously at a rate of $400 a year. The payments are made between5 and 7 years. Find the current value of these payments at time 2 years using an annual rate ofdiscount of 4%.

Problem 25.14

A company makes payments continuously at a rate of $200 per year. The payments are madebetween 2 and 7 years. Find the accumulated value of these payments at time 10 years using anannual rate of interest of 6.5%.

Problem 25.15

Lauren is being paid a continuous perpetuity payable at a rate of 1000 per year. Calculate thepresent value of the perpetuity assuming d(12) = 0.12.

Problem 25.16

If i = 0.04, calculate the accumulated value of a continuous annuity payable at a rate of 100 peryear for 10 years.

Problem 25.17

If = 0.06, calculate the present value of a continuous annuity of 1 payable for 20 years.

Problem 25.18

You are given n0atdt = 100. Calculate an .

Problem 25.19

An nyear continuous annuity that pays at a rate of $748 per year has a present value of $10,000when using an interest rate of 6%, compounded continuously. Determine n.

Problem 25.20

Which of the following are true?

(I)(an

d

)(1 + i) = an1 .

25 CONTINUOUS ANNUITIES 233

(II) The present value of a 10 year annuity immediate paying 10 per month for the first eight

months of each year is 120a10a(12)

8/12.

(III) The present value of a perpetuity paying one at the end of each year, except paying nothingevery fourth year, is

s3is4

.

Problem 25.21

Payments of $3650 per year are made continuously over a five-year period. Find the present valueof this continuous annuity two years prior to the first payment. Given the nominal rate i(365) = 8%.