Question

How long will it take to save $14,500 by making deposits of $110 at the end of every month into an account earning interest at 10.9​% compounded semi-annually​?

State your answer in years and months​ (from 0 to 11​ months).

It will take___​year(s) and ___​month(s) to save $14,500.

Answer 1

Amount to be saved (Future Value) = $14,500

Periodic Deposits at the end of every month = $110

Interest rate = 10.9% compounded semi-annually​

Calculating its EAR:-

where, r = interest rate = 10.9%

m = no of times compounding in a year = 2

EAR = 11.1970%

Now, usig EAR to calculating Nominla Rate for mnthly compounding as assuming EAR of both monthly compounding and semi-annual compoundings are same.

where, r = interest rate

m = no of times compounding in a year = 12

EAR = 0.111970

Taking 12-root on both sides..

1.00888366 = (1+ r/12)

0.00888366 = r/12

r = 10.66%

Now, Calculating the no of period it will take to save $14,500:-

Where, C= Periodic Payments = $110

r = Periodic Interest rate = 10.66%/12 = 0.888333%

n= no of periods

Future value = $14,500

1.17098484804 = (1+0.00888333)^n - 1

2.17098484804 = (1+0.00888333)^n

Taking log on both sides,

Log(2.17098484804) =n*Log(1+0.00888333)

0.33665679 = n(0.0038409476)

n = 87.65 months

So, it would take 7 years & 4 months.

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