Question

What lump sum deposited today at 8​% compounded quarterly for 15 years will yield the same final amount as deposits of $4000 at the end of each​ 6-month period for 15 years at 8​% compounded​ semiannually?

Answer 1

First calculating the future value of deposits of $4000 at the end of each​ 6-month period for 15 years at 8​% compounded​semiannually.

So, FV of this annuity is calculated using formula

FV = PMT*((1+r/n)^(n*t) - 1)/(r/n)

Where PMT = deposit at the end of every six months = $4000

r = interest rate = 8% compounded semiannually

n = compounding frequency = 2

t = time period = 15 years

So, Future value of deposits = 4000*((1+0.08/2)^(2*15) - 1)/(0.08/2) = $224339.75

An equivalent lump sump deposit today at 8% compounded quarterly for 15 years with same FV is calculated using formula

PV = FV/(1 + r/n)^(n*t)

here n = 4 quarters in a year

So, Lump sum deposited today = 224339.75/(1 + 0.08/4)^(4*15) = $21325.83