Question

It is now January 1. You plan to make a total of 5 deposits of $300 each, one every 6 months, with the first payment being made today. The bank pays a nominal interest rate of 14% but uses semiannual compounding. You plan to leave the money in the bank for 10 years. How much will be in your account after 10 years? Round your answer to the nearest cent. $ You must make a payment of $1,385.01 in 10 years. To get the money for this payment, you will make 5 equal deposits, beginning today and for the following 4 quarters, in a bank that pays a nominal interest rate of 14% with quarterly compounding. How large must each of the 5 payments be? Round your answer to the nearest cent. $

Answer 1

1.

Amount Invested P = $300 every 6 months for 3 years

Interest rate = r = 14% annual or 7% semiannual

Number of periods = 6

Future value of investment after 3 years = P(1+r)⁶ + P(1+r)⁵ + P(1+r)⁴ + P(1+r)³ + P(1+r)² + P(1+r)

= 300(1+0.07)⁶ + 300(1+0.07)⁵ + 300(1+0.07)⁴ + 300(1+0.07)³ + 300(1+0.07)² + 300(1+0.07) = 2296.206

Amount after 10 years ( 7 more years after 3 years) = 2296.206(1+0.07)¹⁴ = $5920.845

2. FV after 10 years = 1385.01

Number of periods = 10*4 = 40 quarters

Interest Rate = 14% annual = 0.14/4 quarterly

=> PV = 1385.01/(1+0.14/4)⁴⁰ = $349.815

Let the amount paid for next 5 quarters be P

Present Value of this annuity = P + P/(1+r) + P/(1+r)² + P/(1+r)³ + P/(1+r)⁴

=  P + P/(1+0.14/4) + P/(1+0.14/4)² + P/(1+0.14/4)³ + P/(1+0.14/4)⁴ = 4.673P

=> 4.673P = 349.815

=> P = $53.459

Hence the 5 payments should be $53.459 each