father wants to set up a fund for his newborn babys college

father wants to set up a fund for his newborn baby’s college education. To do this, he estimates that he will need $120,000 when the child turns 18. The account pays 7% interest per year compounded monthly. (a) How much money should he invest now so that the account has the value $120,000 in 18 years. (b) Alternatively, he can make monthly payments for the next 18 years. What is the size of the monthly payment?

Solution

a) The formula for compound interest is F = P(1+r)ⁿ , where P is the initial investment/principal, F is the future value of the investment , r is the rate of interest per period in decimals and n is the number of periods. Here, F = $ 120000, r = (7/100)*1/12 = 7/1200 and n = 18*12 = 216. Therefore, 120000= P( 1 + 7/1200)²¹⁶ = 3.512539319 P . Then P = 120000/3.512539319 = $ 34163.32 ( on rounding off to the nearest cent). Thus, the father should invest $ 34163.32 now so that the account has the value $120,000 in 18 years.

b) The formula for the future value (F) of an ordinary annuity is F = P [(1 + r)ⁿ – 1] / r]

Where, P is the amount of each annuity payment, r is the interest rate per period in decimals, and n is the number of periods over which payments are made. Here, F = $ 120000, r = (7/100)*1/12 = 7/1200 and n = 18*12 = 216. Therefore 120000= P [ {( 1 + 7/1200)²¹⁶ -1}/ 7/1200] = (1200P/7) [ 1207/1200)²¹⁶ -1] = (1200P/7 )* 3.512539319. Then, P = (120000*7 )/ (1200 * 3.512539319)= 840000/4215.047183 = $ 199.29 ( on rounding off to the nearest cent). Thus, the size of the monthly payment is $ 199.29.