Is the term "annual interest rate" misleading regarding compound interest?

Question

Compound interest can be computed by considering the recurrence relation $$a_{n+1}=a_n+sa_n=(1+s)a_n,$$ where $s>0$ is the growth rate, $n$ is the number of compounding cycles that have elapsed, $a_n$ is the current balance, and $a_{n+1}$ is the new balance after one "compounding cycle." This gives $$a_n=(1+s)^n a_0\\=a_n=(1+s)^{mt}a_0,$$ where $t$ is years and $m$ is number of compounding cycles per year. Typically, the compound interest equation is expressed as $$a_n=\left(1+\frac{r}{m}\right)^{mt}a_0,$$ where $r/m=s$, the compounding rate.

Does $r$ have any meaning other than the compounding rate scaled by $m$? What I mean is that calling $r$ the "annual interest rate" seems very misleading. I would expect the annual interest rate to be the interest rate such that, were I to be compounding annually, the growth would match my original $a_n$ equation. As it stands, if I were to compound annually with the "annual interest rate" for $a_n$ I would get a completely different growth than $a_n$; that is, $$a_n=\left(1+\frac{r}{m}\right)^{mt} \neq (1+r)^t.$$

How can this be reconciled? I suppose an answer would address why is $r$ called an annual rate, and whether $r$ has any meaning other than its seemingly arbitrary definition.

Answer 1

In the context of savings accounts,

• the nominal interest rate doesn't factor in compounding and is usually referred to as the stated interest rate or just the interest rate,
• whereas the effective interest rate accounts for compounding and is called the Annual Percentage Yield (APY) or the Annual Equivalent Rate (AER).

(For daily compounding, a $2.88\%$ p.a. nominal interest rate effectively corresponds to about $2.921756 \%\text{ p.a.})$ Let $n$ is the number of interest computations per year, $\$B$ be the account balance at the point of interest calculation, and $\$I$ be the corresponding interest payment. Then $$I=B\times\frac{i_\text{nominal}}n\tag1$$$$I=B\left((1+i_\text{effective})^{\frac1n}-1\right).\tag2$$

Contracts specify $i_\text{nominal}$ rather than $i_\text{effective};$ the first formula is more straightforward and computationally efficient than the second formula.

But yes, the standard practice of omitting the adjective ‘nominal’ is ambiguous, leading to nominal and effective interest rates being conflated and the misconception that $\$P$ earns exactly $\$P\times i_\text{nominal}$ in total interest in a year.

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For a loan with recurring repayment $\$A$ made $m$ times per year over $t$ years, neither of these two interest rates factors in compounding:

• the nominal interest rate or the stated interest rate or just the interest rate, given by $$\text{loan principal}=\frac{mA}{i_\text{nominal}}\left(1-\frac1{\left(1+\frac{i_\text{nominal}}m\right)^{tm}}\right),$$ disregards upfront and exit fees,
• whereas the Annual Percentage Rate (APR) is given by $$\text{loan principal $-$ upfront fees $-$ present value of recurring and exit fees}\\=\frac{mA}{\textbf{APR}}\left(1-\frac1{\left(1+\frac{\textbf{APR}}m\right)^{tm}}\right).$$

However, if the loan is compounded daily—as is typically the case—rather than at the repayment frequency, then for greater accuracy (while still disregarding leap-year variations), $\text‘i_\text{nominal}\text’,$ and likewise ‘APR’, in the above formulae should be replaced with $$m\left(\left(1+\frac{i_\text{nominal}}{365}\right)^{\frac{365}m}-1\right).$$

P.S. The commonly-quoted formula $$\textbf{APR}=\frac{\text{total interest payable $+$ total fees}}{\text{loan principal}}\times\frac{365}{\text{loan term in days}}$$ is incompatible with the above, for it actively assumes a simple-interest model, consistently underestimating the true cost of borrowing. For example, for a fee-free $30$-year $\$500000$ loan with a $\$4000$ monthly repayment schedule, the APR is $8.94\%$ according to the first formula and $6.27\%$ according to the deceptive second formula.