Why would Annual Percentage Interest Rate per period be calculated using division instead of n-th root?

Question

It happens on many places on the web. Even economics professors would divide the annual interest eg. by 12, instead of calculating 12-th root. For example interest rate for 3% APR target with montly payments (12 periods) is calculated as: 1 + 0.03/12 = APR(3.04%) instead of the proper (1.03)^1/12. I initially assumed it's a mental approximation but it's not as if modern society lacks calculators, so what am i missing?

Answer 1

It’s a vestigial convention left over from the time before calculators, to make things easier to calculate, they just set it as a rule that that’s how compounding things multiple times a year worked to avoid complication. now enough people do it that way that it’s just too hard to change

Answer 2

Because that is what we mean.

When we say $3\%$ annual interest we do NOT mean after a year your principal will grow by $3\%$.

We actually do mean $\frac {\text{percentage of growth in one period}}{\text {per time of period}}$ converted to a time length of a year. So, yes, if in actuality an interest payment is a rate of $1$ payment $0.25\%$ per month, that is precisely what we mean when we say $\frac {0.25\%}{\text {month}}\times \frac {12\text{ months}}{year} = 3\%$ per year.

If we wanted so say "You investment will grow by $3\%$ in a year" we would say something else.

Now you might wonder why we use the term annual interest to mean extrapolated/converted rate adjusted linearly to a annual time period when effective actual growth seems more useful and natural.

I imagine that is probably because interest payments are not linear and the effective interest with the same rate will have different growth based on time so it'd be impossible to give a meaningful rate. $3\%$ APR compounded monthly will be $1.0025^{12}-1$ or $3.042\%$ if you have the loan for $1$ year. But if you have it for any other period of time, say 2 years, it'd be $6.176\%$.

........

This comes from the idea that rates are intantaneous and extrapolated to a standard time unit and not that there is a time unit that must occur. If you travel to the store and $20$ miles per hour, that does not mean you actually must, by law, drive for one hour and in that hour you must drive exactly $20$ miles. You could be be driving $20$ miles per hour and in the next $36$ miles per hour. And that means is your tachametor is going so fast and if we extrapolate it will be $20$ miles in an hour.

Perhaps a better analogy is if you are accelerating at constant acceleration. We say you velocity at this moment is $20$ mph but we know damned well that in an hour you will have gone a darned lot more than $20$ miles because in the very next instant are velocity will be faster and in the next instant it will be faster yet.

Are we "lying"? Are we mistaken? No. We simply recognize when we velocity we do not mean hour far we travel in an hour. If we want to talk about how far we travel in an hour we'd ask a different question. We'd talk about total distance traveled.

For continuous compound interest (which I believe is not legal) this what be a perfect example as a rate of $3\%$ compounded every instant would be $(1.03)^t$ where $t$ is time express with years as our standard unit. How much you actually make would be $\int_0^t (1.03)^x dx - 1$. That is a completely different think.

Admittedly monthly interest would by analogous to, instead of having a car you have a transporter and every $12$ minutes the transporter instantly transports you $4$ miles. Thus thats $\frac {4miles}{12minutes} = 20$ mph.

Answer 3

It's just a quote convention. The market practice followed by banks and financial institutions is to quote the interest rate as an annual percentage rate (APR) along with the compounding frequency and the tenor. So, while a 12% annual percentage rate compounded quarterly $(1.04)^3$ is different from monthly compounding $(1.01)^{12}$, the annual rate and the effective rate are mathematically linked. In general, if you compound discretely, the effective rate is just -

$$\left(1+\frac{r}{m}\right)^{m}$$

where $m$ is the number of periods per year.

1. I suppose, it is also easier to work with. You could have different payment schedules e.g. a balloon principal payment at the end, retirement of principal in tranches, or diminishing balance. So, its easier to start with an annual rate and then apply it to the outstanding notional on each payment date, to derive both interest & principal repayment cash-flows.

2. It's not too often, that you move money from one market to another for example repos to bonds.

3. Unlike bonds which are quoted in terms of their yield to maturity, there is not a very deep liquid secondary market or exchange for loans unless they are sliced and packaged into bundles according to the risk appetite of a buyer. It's not an asset traded too often between banks.