The Browns wish to accumulate at least $150,000 at the time of their last deposi

Question

The Browns wish to accumulate at least $150,000 at the time of their last deposit in a college fund for their daughter by contributing an amount A into the account at the end of each year for eighteen years. What is the smallest annual payment A that will sufce if the college fund earns a level annual effective interest rate of 5%? If at the end of ten years, it is announced that the annual effective interest rate will drop to 4.5%, how much must the Browns increase their payments in order to reach their accumulation goal? Assume that the Browns wish to continue to make level payments except for a slightly reduced nal payment. (No excel)

Explanation / Answer

1.

Amount required at the end of 18 years from now = $150,000

Annual interest rate = r = 5% = 0.05

Number of annual payments = n = 18

Future value of ordinary annuity = Annuity payments * {(1+r)n-1}/r

$150,000 = A*(1.0518-1)/0.05

$150,000 = A*28.13238

A = $150,000/28.13238 = $5,331.93

Hence, smallest annual payment = $5,331.93

2.

Amount accumulated at the end of 10 years = $5,331.93*(1.0510-1)/0.05 = $67,064.48

Value of this accumulated balance at the end of 18 years = $6,7064.48 * 1.0458 = $95,372.44

Remaining amount required in next 8 years = $150,000 - $95,372.44 = $54,627.56

Reduced interest rate = r = 4.5% = 0.045

Number of payments = n = 8

$54,627.56 = Annual payments*(1.0458-1)/0.045

$54,627.56 = Annual payments*93.38001

Annual payments = $54,627.56/93.38001 = $5,823.82

Increase in annual payments = $5,823.32 - $5,331.93 = $491.89

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