To derive a general compound interest formula we can say: $$A_1=A_0 + rA_0=A_0(1 + r)$$ $$A_2=A_0 + rA_0 + r(A_0 + rA_0)=A_0 + 2rA_0 + r^2A_0=A_0(1 + r)^2$$ and so on. In general: $$A_t=A_0(1 + r)^t$$ But the formula sometimes given in textbooks is: $$A_t=A_0(1 + \frac{r}{n})^{nt}$$ where $r$ is the annual rate and $n$ is the number of compoundings per year. However, the two formulas give different results; here is how to use them if the interest is compounded monthly but we are given an APR. We use a principal of $300$, a period of five months and an APR of $5$ i.e. $0.05$: $$A_t=300(1.05)^{5/12}=306.16118441572989$$ Using the 'textbook' formula we have: $$A_t=300(1+\frac{0.05}{12})^{12\times5/12}=306.302300799711251$$ Which is correct?! (NB $t=5/12$ is 'in' years not months.)
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$\begingroup$ The formula $A(t) = A_0(1 + r)^t$ is used when interest is compounded annually. For monthly compounding, you must use the formula $$A(t) = A_0\left(1 + \frac{r}{n}\right)^{nt}$$ with $n = 12$. Notice that when $n = 1$, the second formula agrees with the first, as we would expect since annually means once per year. $\endgroup$– N. F. TaussigJun 5, 2016 at 23:36
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$\begingroup$ Whether that's true depends on whether the reference rate is the annual rate $r$ or $r/n$. $\endgroup$– user301988Jun 5, 2016 at 23:51
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$\begingroup$ Normally, the nominal interest rate is stated, which obviously can be a source of confusion. $\endgroup$– N. F. TaussigJun 6, 2016 at 1:15
2 Answers
Refine your initial formula in the following manner.
$$A_1=A_0+\frac rnA_0+r(A_0+\frac rnA_0)+\frac rn(A_0+\frac rnA_0+\frac rn(A_0+\frac rnA_0))+\dots$$
where we repeat this $n$ times. This is interest compounded $n$ times in one year.
In words, to make it less confusing, we start with $A_0$.
$$A_0$$
We add on $\frac rn$ times this initial amount after $1/n$ of the year.
$$A_0+\frac rnA_0$$
After the next $1/n$ of the year, we add on $\frac rn$ times what we currently have.
$$A_0+\frac rnA_0+\frac rn(\color{red}{A_0+\frac rnA_0})$$
The red part is what you have before we apply the additional interest, which now includes the red part.
We repeat this $n$ times to reach a full year, and then we derive the formula like yours, but with the $n$ in it.
Note that compounding is basically applying interest, and sometimes we apply interest multiple times during the year, making us have to use this formula.
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$\begingroup$ What they're doing is defining the monthly interest as a fraction of the annual interest, which works if it's made clear. $\endgroup$– user301988Jun 5, 2016 at 23:48
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$\begingroup$ @selfawareuser That's exactly what I'm doing as well. The $\frac rn$ is the "fraction of the annual interest" that you've noted, which is usually understood to be if interest isn't compounded annually. $\endgroup$ Jun 5, 2016 at 23:49
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$\begingroup$ @selfawareuser I do think the actual correct correct formula is: $$A_t=A_0\left(1+\frac rn\right)^{\lfloor nt\rfloor}$$If interest is compounded only at $n$ times during the year, and $t$ is measured in years. $\lfloor\dots\rfloor$ is the floor function. $\endgroup$ Jun 6, 2016 at 22:20
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$\begingroup$ I think just $nt$ is enough e.g. $15$ months would be $1.25$, or $15/12$ in which case we cancel and get $15$. But $r/n$ must be the reference rate (i.e. not $r$ on its own) meaning the rate the bank have committed to. That's why I thought it was strange - banks usually give an APR and derive a daily rate which I've seen as a very long decimal number. If the daily rate was actually the consciously set rate it would presumably be a short number. I assumed they had done this: $r=(A_t/A_0)^{1/t}-1$. $\endgroup$– user301988Jun 6, 2016 at 23:20
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1$\begingroup$ @selfawareuser sounds like a good plan. And if it comes out wrong, you should ask them how they calculate their rates and post it here lol. $\endgroup$ Jun 6, 2016 at 23:26
The first formula you gave is simplified and assumes that the interest is compounded once per period. The bottom formula is more general and does not make this assumption. They are both correct though, just different assumptions.
