Question

What sum deposited today at 6​% compounded annually for 12 years will provide the same amount as ​$1300 deposited at the end of each year for 12 years at 8​% compounded​ annually? 1.What sum would have to be deposited today at 6​% interest compounded​ annually? Round to the nearest cent

Answer 1

First we will calculate the future value of $1300 deposited at the end of each year for 12 years at 8% compounded annually as below:

Here, the deposits will be same every year, so it is an annuity. We will use the future value of annuity formula as per below:

FVA = P * ((1 + r)n  - 1 / r)

where, , P is the periodical amount = $1300, r is the rate of interest = 8% and n is the time period = 12

Now, putting these values in the above formula, we get,

FVA = $1300 * ((1 + 8%)12 - 1 / 8%)

FVA = $1300 * ((1 + 0.08)12 - 1 / 0.08)

FVA = $1300 * ((1.08)12 - 1 / 0.08)

FVA = $1300 * ((2.51817011682 - 1 / 0.08)

FVA = $1300 * (1.51817011682 / 0.08)

FVA = $1300 * 18.9771264602

FVA = $24670.26

Calculation of sum to be deposited today at 6% compounded annually:

Here we will use the following formula:

PV = FV / (1 + r%)n

where, FV = Future value = $24670.26, PV = Present value, r = rate of interest = 6%, n= time period = 12

now, putting theses values in the above equation, we get,

PV = $24670.26 / (1 + 6%)12

PV = $24670.26 / (1 + 0.06)12

PV = $24670.26 / (1.06)12

PV = $24670.26 / 2.01219647184

PV = $12260.36

So, $12260.36 is the required amount to be deposited today.