Compound interest: how to use the textbook formula?

Question

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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Answer 1

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.

Answer 2

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.