Question

It is now January 1. You plan to make a total of 5 deposits of $500 each, one every 6 months, with the first payment being made today. The bank pays a nominal interest rate of 8% 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 $1,438.94 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

Total 5 deposits of $500 each for semi-annual compounding

Therefore we consider 8%2 = 4% per compounding period

Timeline will be

$500 $500 $500 $500 $500

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Pay1 2 3 4 5

One approach is calculated to present value of the account till 5 periods and then we can check the future value of the account at 10 years (easiest approach)

Pay 1=$500 (current period on Jan 1)

Pay 2=500/1.04=480.77

Pay 3=500/(1.04)²⁼462.28

Pay 4=500/(1.04)³=444.5

Pay 5=500/(1.04)⁴=427.40

Total Present value of the account=$ 500+480.77+462.28+444.5+427.40=$2314.95

Value at end of 10 years semi-annually compounded @8% is

Note as semi-annually we have 20 periods at 4% rate

2314.95*(1.04)²⁰

=$5072.34

For question number 2

We need to pay $1438.94 after 10 years

therefore we can calculate how much to save, however, we pay an equated amount every quarter, therefore, we can use annuity formula, but firstly

if today is Jan 1 then

pay 1 today

pay2=april 1

and so on, therefore amount needs to be.

10 years=40 periods @3.5%

40-5 periods=35 periods

we calculate the present value of $1438.94 at 5periods

1438.94/(1.035)³⁵

=$431.64 need to be accumulated

now we use annuity formula

Deposit per month=[(431.64*0.035)*(1+0.035)⁵)] / (1.035⁵-1)

=15.10*1.187/0.188

$95.34 per quarter need to be deposited.