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If a student makes monthly deposits of 1,200 into an account with a nominal annual interest rate of 4.5% compounded monthly, will he have enough after 5 years to purchase a $105,000 property in cash?

I already have the solution.

I just want to understand why did he use the following to find the annual effective interest rate

i = 4.5% / 12 = 3.75%

and the Number of compounding periods he used is 60

Why he didn't use the following standard formula:

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OMAR
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  • I don't know about the formula, but I would calculate it as $$ (1.045)^{1/12} \approx 1.0036748 $$ So that's $0.36748~%$ per month. Slightly different from $0.375~%$ (after typo correction) – Matti P. Aug 13 '21 at 09:04
  • What was the calculation in whole? I'm not sure where your problem is. – callculus42 Aug 13 '21 at 11:55

1 Answers1

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If the nominal annual interest rate is 4.5%, then 4.5%/12 represents the monthly rate. The annual effective rate (compounding monthly) as per your formula is

$$(1+0.045/12)^{12}-1.$$

Now, as per the problem you mention, the timings are unclear, but let me give you a general formula (you can adjust accordingly to deal with different timings). If you make deposits of $x$ at the beginning of every period (starting immediately) and the per-period interest rate is $r$, observe that at the end of $t$ periods, your balance will be

$$x(1+r)^t+x(1+r)^{t-1}+...+x(1+r)=x(1+r)\frac{(1+r)^t-1}{r}.$$

Your setup is the case $x=1200,t=5*12=60,r=0.045/12.$

Golden_Ratio
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