Question

What lump sum deposited today at 12​% compounded quarterly for 5 years will yield the same final amount as deposits of $6000 at the end of each​ 6-month period for 5 years at 4​% compounded​ semiannually?

What is the value of the lump sum?

Answer 1

Future value of both the option should be same. So first find the future value

Future value = P× [(1 + r)ⁿ - 1]/r

P = periodic payment =$6,000

r = rate of interest = 4%

But question is of semi-annual so r = 4%/2 = 2%

n = number of periods

n = 5 ×2 =10

Future value = 6,000 × [(1 + 0.02)¹⁰ - 1 ]/0.02

=6,000 × ( 1.21899442 - 1)/0.02

=6,000 × 0.21899442/0.02

=65,698.325998427

=$65,698.33

Now this should be equal to the future value at the end of year 5 of lumpsum deposit, so we will calculate the present value of $ 65,698.33 i.e.

Present value = A/(1 + r)ⁿ

^{A= Future value}

A = 65,698.33

r = 12%/4 = 3% (compounded quarterly)

n = 5×4 =20

Present value = 65,698.33/(1 + 3%)²⁰

= 65,698.33/ 1.8061112347

= 36,375.572410916

The value of lumpsum will be $ 36,375.57 .

Note :

The final figure is rounded to 2 decimals as nothing mentioned in the question for rounding off.