Question

In 10 years you’d like to buy a boat that will cost $500,000 at that time. If your savings can earn 4% , how much will you need to save each year starting at the end of this year, in order to reach your goal? Round to the nearest cent. ​​[Hint: We are trying to solve for the cash flows of an annuity. The formula we need to use is dictated by whether we are given the PV or the FV of that annuity.]

Answer 1

Let the amount deposited at the end of each year =x

Future value of Amount deposited = X*(1+Interest rate)^time left after first deposit+X*(1+Interest rate)^time left after Second deposit+X*(1+Interest rate)^time left after third deposit+X*(1+Interest rate)^time left after fourth deposit+.........+X*(1+Interest rate)^time left after tenth deposit

=x*(1+4%)^9+x*(1+4%)^8+x*(1+4%)^7+......+x*(1+4%)^2+x*(1+4%)^1+x*(1+4%)^0

=x*(1+4%)^9+x*(1+4%)^8+x*(1+4%)^7+......+x*(1+4%)^2+x*(1+4%)^1+x

=x*Future value Annuity factor (4%,9 years)+x

=11.00611x+x

=12.00611x

Given the future value = 500000

therefore, 12.00611x=500000

=41645.47

Hence amount deposited should be 41645.47 each year starting at the end year 1

(Note: Since the first amount will deposited at the end of year 1 hence the time left after first deposit till the 10th year will be 9 years hence the same is used for compunding)