Question

New parents wish to save for their newborn's education and wish to have $35,000 at the end of 18 years. How much should the parents place at the end of each year into a savings account that earns an annual rate of 5.4% compounded annually? (Round your answers to two decimal places.) How much interest would they earn over the life of the account? Determine the value of the fund after 12 years.

Answer 1

Answer :-

Amount needed at the end of 18 years i.e future value (FV) = $35,000.

Given,

Number of years of investment (n)=18 years.

Annual rate of interest (r) = 5.40%.

We have to find period annual payments need to be made at the end of each year in order to get $35,000 at the end of 18 years. This can be represented as follows

(i)Periodic Payment ???

Future value (FV) =periodic payments(P) * FVIFA(r%,n)------------ equation (1)

FV = $35,000 ; r = 5.4% ; n = 18 years.

Periodic payment = P .

Substituting values in equation (1)

$35,000 = P * FVIFA(5.4%,18)

P * FVIFA(5.4%,18) = $35,000

P * 29.2055 = $35,000. { FVIFA = [(1+r)ⁿ-1]÷r }

P = $35,000 ÷ 29.2055

Periodic payment = $1198.40

# Annaul payment to be made at the end of each year = $1198.40.

(ii) Total interest over the life of the account ??

Life of the account = 18 years i.e total years for maturity.

Interest they would earn = Maturity value - Periodic payments * No.of years of investment.

= $35,000 - $1198.40 * 18 years.

=$35,000 - $21,571.28( rounded off )

= $13,428.72

Total interest they would earn = $13,428.72.

(iii) Value of fund after 12 yeas =???

Annual periodic payment (P) = $1198.40 ;

Annual rate of interest (r) = 5.4% ;

Value of fund after 12 years (FV) = ??

This can be shown in the form of equation as follows

FV = Periodic payment ×FVIFA(r%,n)

FV = $1198.40 × FVIFA(5.4%,12)

FV = $1198.40 × 16.2906

FV = $19,522.70

Value of fund at the end of 12 years = $19,522.70

Notes :-

(1) FVIAF(5.4%,18) = {(1+5.4%)¹⁸ -1} / 5.4%

= (2.577 -1) / 0.054 = 29.2055.

(2) FVIAF (5.4%,18) = {(1+5.4%)¹²-1}/5.4%

= (1.8796-1)/0.054 = 16.2906.