# Annuities and Perpetuities/AN

PROBLEM SET 3.1
A. (3pts) Interpret the following identities relating annuity-immediate and annuity-due
1. 𝑠𝑛|
̅̅̅ = 𝑎𝑛|
̅̅̅(1 + 𝑖)
𝑛
2. 𝑠̈𝑛|
̅̅̅ = 𝑎̈𝑛|
̅̅̅(1 + 𝑖)
𝑛
3. 𝑎̈𝑛|
̅̅̅ = 𝑎𝑛|
̅̅̅(1 + 𝑖)
4. 𝑠̈𝑛|
̅̅̅ = 𝑠𝑛|
̅̅̅(1 + 𝑖)
5. 𝑎̈𝑛|
̅̅̅ = 1 + 𝑎𝑛−1|
̅̅̅̅̅̅̅
6. 𝑠̈𝑛|
̅̅̅ = 𝑠𝑛+1|
̅̅̅̅̅̅̅ − 1
B. (12 pts) Using the formula of the present and accumulated values of the annuities presented in
class, derive the right-hand side of each of the above formulas.
PROBLEM SET 3.2
Show the solution and answer of the following items clearly, completely, and orderly.
1. A loan for PhP 500,000 must be repaid with 6 year end payments at an annual rate of 11%. What
is the annual payment?
2. An annuity immediate has quarterly payments of PhP 200 for 10 years at a rate of 12%
convertible monthly. Find its present value.
3. You lend a relative PhP 100,000 and he agrees to pay you 6% interest on the original PhP
100,000 at the end of every year for 5 years and then return the PhP 100,000. You can reinvest
the interest payments at 5%. How much will you have in total in 5 years?
4. An annuity pays PhP5,000 at the end of each of the next 5 years and PhP3,000 at the end of
each of 5 following years. If 𝑖 = 5%, find the present value of the annuity.
5. An annuity provides a payment of 𝑛 at the end of each year for 𝑛 years. The annual effective
rate of interest is 1/𝑛. What is the present value of the annuity?
6. Show that
a. 𝑎𝑚+𝑛|
̅̅̅̅̅̅̅̅ = 𝑎𝑚|
̅̅̅̅ + 𝑣
𝑚𝑎𝑛|
̅̅̅ = 𝑣
𝑛𝑎𝑚|
̅̅̅̅ + 𝑎𝑛|
̅̅̅
b. 𝑠𝑚+𝑛|
̅̅̅̅̅̅̅̅ = 𝑠𝑚|
̅̅̅̅ + (1 + 𝑖)
𝑚𝑠𝑛|
̅̅̅ = (1 + 𝑖)
𝑛
𝑠𝑚|
̅̅̅̅ + 𝑠𝑛|
̅̅̅
c.
1
1−𝑣
10 =
1
𝑠10|
̅̅̅̅̅
(𝑠10|
̅̅̅̅̅ +
1
𝑖
)

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