Chuck needs to purchase an item in 10 years The item costs 2

Chuck needs to purchase an item in 10 years. The item costs 200 today, but its price increases by 4% per year. To finance the purchase, Chuck deposits 20 into an account at the beginning of each year for 6 years. He deposits an additional X at the beginning of years 4, 5, and 6 to meet his goal. The annual effective interest rate is 10%. Calculate X.

Solution

The fund will equal the accumulated value of the 200 at inflation of 4:

Sum from i = 5 to 10 of 20(1.1ⁱ) + Sum from j = 5 to 7 of x(1.1)^j = 200(1.04¹⁰)

So we get

x = (200(1.04¹⁰) =296.05

Check deposts 20 into an account per year at the beginning of each year for 6 years. This is an annuity due @ 10% for 6 years. There is also a 3 year annuity due at 10% of X.

296.05 = Cost in year 10

296.05 = 248.52 + x(1.1)⁷ + x(1.1)⁶ + x(1.1)⁵

47.53 = x(5.33079)

x = 8.912