How much is 6% a year in months?

Question

I am currently in high school where we are learning about present value.

I struggle with task like these: Say you get 6% interest each year, how much interest would that be each month?

Answer 1

In order to compute the monthly interest, you know that in a year (which has 12 months) you get $6\%$ interest. Call $a$ the (decimal) monthly interest. Thus, $$ (1+a)^{12}=1.06. $$ Then you just solve the equation for $a$.

Answer 2

Almost as Alberto Debernardi answered, let us note $y$ the interest per year and $m$ the interest per month. So, the equation which relates them is $$(1+m)^{12}=1+y$$ So $$m=(1+y)^{1/12}-1$$ To approximate it, you can use the binomial expansion and get $$m\approx \frac{1}{12}y-\frac{11 }{288}y^2=\frac{1}{288} (24-11 y) y$$ Applied to the case where $y=0.06$ ($6$% per year), this gives $m\approx 0.0048625$ (that is to say $0.486$% per month) while the exact solution would be $m\approx 0.0048676$. Almost no need of any calculator !

Answer 3

This question has two possible answers. It depends on whether you are using compound or simple interest.

Debernardi's answer is correct, but I suspect its not the answer they want. The answer they want is 0.5% per month. I'll explain why.

First you should know the difference. Say you earned 10% interest each 6 months. How much would you earn in a year if you invested $100?

If you use "simple interest", you get two lots of 10% which is 20%. But you could also do it like this. After 6 months you have 10% extra, which is 110. Then after another 6 months this has increased by 10%, and $110 * 1.1 = 121. The difference being the interest you earned on your interest. The large majority of bank accounts use compound interest, because the interest you earn stays in the account.

Debernardi's answer assumes this form of interest.

Instead, I think they want you to simply say that 6% per year is equivalent to 6%pa /12 = 0.5% per month. Which is a very slightly different number to what you get the other way.

I think they want you to do it this way because:

a) 6% per annum means it works out at exactly 0.5% per month, why pick a number which is very easy to work out for simple interest unless it is simple interest?

b) Compound interest is taught later, and if that's what they wanted they would probably have explicitly said this (to remove ambiguity).

So the answer thewy want is almost certainly 0.5% per month, being one twelfth of 6% per year.