Question

Seamus has paid ​$19,000 for a retirement annuity from which he will receive $1,938 at the end of every month. The payments are deferred for 12 years and interest is 9% compounded monthly.

​(a) How many payments will Seamus ​receive?

​(b) What is the size of the final​ payment?

​(c) How much will Seamus receive in​ total?

​(d) How much of what he receives will be​ interest?

Answer 1

a)

FV = PV x (1+r) ⁿ

r = Interest rate = 0.09/12 = 0.0075 p.m.

n = Number of periods = 12 years x 12 months = 144 periods

FV in 12 years = $ 19,000 x (1+0.0075)¹⁴⁴

                      = $ 19,000 x (1.0075)¹⁴⁴

                      = $ 19,000 x 2.9328367736409

                      = $ 55,723.8987

Fund of $ 55,723.8987 will facilitate retirement monthly cash flow of $ 1,938 for n numbers of periods, which can be computed using formula for PV of annuity.

PV = C x [1- (1+r) ^‑n/r]

Or

-n = log [1 – {(PV x r)/C}]/ log (1+r)

-n = log [1 – {($ 55,723.8987 x 0.0075)/ $ 1,938}]/ log (1+0.0075)

   = log [1 – ($ 417.92924025/ $ 1,938)]/ (log 1.0075)

= log [(1 – 0.215649762770898)]/ (log 1.0075)

= log 0.784350237229102/ log 1.0075

= -0.10548996776/0.00324505481

= -32.50791556

n = 32.50791556 or 33 periods

Seamus will receive 33 payments.

b)

PV of 32 payments =

Final payment = (32.50791556 – 32) x $ 1,938 = 0.50791556 x $ 1,938

= $ 984.3403606 or $ 984.34

c)

Total payment = Number of payments x Monthly payment

                        = 32.50791556 x $ 1,938 = $ 63,000.34035528 or $ 63,000.34

d)

Interest received = Total payment – Principal of deposit

                           = $ 63,000.34 - $ 19,000 = $ 44,000.34