# Why does compound interest exist?

## Background

My understanding is that compound interest arises in the following way:

The bank offers its clients some interest rate $$r$$ on an account with principal $$P$$ that yields $$rP$$ after some time $$t_0$$. But clients, not wanting to wait for $$t_0$$ to pass before seeing any returns, ask if they can instead have some fraction of the full return every time interval $$t_0 / n$$, where $$n$$ is some natural number. Reasonably, the bank agrees to pay out $$r/n$$ times the current balance every $$t_0 / n$$. The amount of money $$A(t)$$ in an account with principal $$P$$ after time $$t$$ is then the usual compound interest formula $$$$A(t) = P\left(1 + \frac{r}{n}\right)^{\frac{nt}{t_0}} \label{comp-int}\tag{1}$$$$

## The Problem?

While the deal that the bank offers may seem reasonable for the bank, it results in the interest earned after time $$t_0$$ exceeding the original offer. Orignally, after $$t_0$$, the client would have a total balance of $$(1 + r)P$$. With compound interest, after $$t_0$$, the client would have a total balance of $$A(t_0) = P(1 + r/n)^n$$. The bank overpays (compared to its original promise) by a ratio of $$\frac{\left(1 + \frac{r}{n}\right)^n}{1 + r}$$ which is bounded by $$1 \leq \frac{\left(1 + \frac{r}{n}\right)^n}{1 + r} \leq \frac{e^r}{1 + r} \label{bound}\tag{2}$$ The problem, it seems to me, is that the offer is all downside for the bank. They give their clients money more often and end up giving them more money than the original offer. Why would a bank ever offer this option?

## An Alternative?

If there were no other reasonable way to compute a partial return on an investment, then of course client demand would eventually force banks to make the compound interest offer.

But there is a readily available and (to me) perfectly reasonable alternative. Why not just compute the current balance as $$A(t) = P(1 + r)^\frac{t}{t_0} \label{alt}\tag{3}$$ This has a few advantages. It doesn't require a new parameter $$n$$. It can be calculated continuously, but unlike the typical limit as $$n$$ goes to infinity (i.e. $$Pe^{rt/t_0}$$) it doesn't result in the bank overpaying. That is, $$A(t_0) = P(1 + r)$$, which was the original offer.

## Speculation

I would speculate that the reason interest isn't calculated as in \ref{alt} is because it doesn't make sense to the mathematical layman. To the layman, the fair amount to be paid after time $$t_0/n$$ is $$r/n$$ times the current balance so that the new balance is $$1 + r/n$$ times the old balance — you just divide both by $$n$$. But the mathematically correct (I claim) way to evenly divide the payments would be to multiply the current balance by $$(1 + r)^\frac{1}{n}$$ every $$t_0/n$$, which seems more complicated and results in an effective interest rate less than $$r/n$$ per $$t_0/n$$. I can imagine that would be a tough sell to someone who doesn't understand the math. Instead the bank opts for the more approachable $$1 + r/n$$ factor, with the understanding that the deviation from the original offer is bounded (as \ref{bound} shows) and so an acceptable loss if clients are more likely to accept the offer.

Aside: musicians (or at least piano tuners) have to understand this math. The correct ratio between half-steps for equal temperament is $$2^{1/12}$$, not $$1 + \frac{1}{12}$$. While the latter would be somewhat close to the correct tuning, it would lead to an octave that is about $$31\%$$ sharp.

## TL;DR

Why is compound interest calculated with \ref{comp-int} and not the simpler and (I argue) more mathematically correct \ref{alt}?

• This seems to me to be less a math question and more like a business model/philosophical question.
– Mike
Apr 7, 2022 at 20:42
• I included the piano tuner comment in the hopes of somewhat clarifying my question. It would be simply wrong to tune a piano by a ratio of $1 + 1/12$ and not $2^{1/12}$. Is there a mathematical reason that the compound interest formula is not simply wrong in the same way? If not, why is it still used / taught? Apr 7, 2022 at 20:50
• Business-model-wise, what doesn't seem to be taken into consideration here is what the bank is doing with your money between interest payments. It's using your money —aggregated with that of hundreds/thousands/millions of other clients— to make investments that earn dividends, provide loans that earn their own interest, etc, etc, etc. In short: Banks make money off of your money! And (presumably) they make enough money off of your money that (1) they can afford compound interest and/or (2) simple interest would be considered exploitative.
– Blue
Apr 7, 2022 at 20:53
• I doubt the bank "over pays". The bank doesn't offer an interest rate, then let the customer decide how often it is compounded. The bank factors the compounding into their decision on the interest rate to offer, and may even advertise the interest rate in terms of APY instead of APR. As for why this equation became the one to be used, maybe your post would be better on hsm.stackexchange.com
– Joe
Apr 7, 2022 at 20:57
• Simply put: yes, what you describe results in the bank paying out extra to their savings accounts. But it also results in them receiving extra from people they have lent to--which is always at a higher interest rate. Using Eq (3) would be a fantastic loss for the bank. Apr 7, 2022 at 21:00

One advantage of the current reporting method is that it allows for comparability across timelines.

Banks offer time deposits for periods of 1 day, 1 week, 1 month, 3 months, 6 months, 1 year and 5 years.

We could report different rates such that each would report the total return over each holding period, and the calculation would be what the OP describes.

But, it is not easy to compare rates across time periods. Under this scheme, the overnight rate might be $$0.0086\%$$ the one week rate $$0.0615\%,$$ and the three month rate $$0.800\%.$$

Instead we say the overnight rate is $$3.096\%$$ the one week rate is $$3.198\%$$ and the three month rate is $$3.20\%.$$

The reported interest rates are simple interest rates for short periods. Your expected total return, if you rolled over these time deposits for a year would be higher than that due to compounding. The appropriate compounding frequency would be in line with the length of the time deposit.

Longer-term loans and time deposits are reported on a semi-annual or annual compounding frequency.

Debts are calculated in a similar way. If you don't pay off your credit card for a month you will be charged a monthly financing fee. The rate is quoted on an annualized basis, and if you didn't pay your bill for a year, (or charge more debts on that card) at the end of a year your bill would be $$(1+\frac {r}{12})^{12}$$ times your starting debt.

Your suggested formula behaves identically to ordinary compound interest in the case $$t$$ is an integer multiple of $$t_0$$ (with $$n=1,$$ but that’s irrelevant.) Thus there is really no question here: banks banks do not in fact need to calculate your account balance continuously. Instead, they calculate it every $$t_0$$, where $$t_0$$ may be a month, a quarter, a year, or some more exotic amount of time.

As for the apparent overpaying if $$t_0$$ is not one year but $$r$$ is reported over one year, in fact the financial industry has a solution for that: report APY, the equivalent non-compounding interest rate, rather than $$r$$, which is called the APR. Of course the difference is usually exquisitely small if you’re saving, though it may be nontrivial if you’re borrowing! So no matter what, the bank always wins.

• I disagree with the first paragraph. The formulas are only identical if $n = 1$, i.e. if there is no compounding. My formula still allows for discrete calculations. You would have $A(k t_0 / n) = P(1 + r)^{k/n}$ for integer $k$, which is different from the typical compound interest formula $A(k t_0 / n) = P(1 + r/n)^{k}$. Apr 7, 2022 at 21:28
• @charlesHudgins Yes, if you assume $n$, $r$, and $t_0$ are all chosen separately and ahead of time, then there is a difference. But this is not how it works. If you want quarterly compounding with your formula, just set $t_0$ to be one quarter and reset $r$ to $r/4.$ The models are totally equivalent under this process. Apr 7, 2022 at 21:32
• As for the distinction between APY and APR, that is essentially my question. Why not only report one rate and have that rate actually pay out the promised amount in the promised time? It seems like (and my question is whether this is so) APY and APR are just a shell game banks play on mathematically unsophisticated clients. Evidence for this is that APY and APR are basically synonymous in what they stand for. Apr 7, 2022 at 21:32
• I still think I disagree. The distinction is between a growth rate of $(1 + r)^{1/n}$ and $1 + r/n$. The former I'm arguing is the correct growth rate because it produces the correct final balance after $n$ iterations, i.e. $(1 + r)P$, whereas the latter yields $(1 + r/n)^n P$, which is larger for $n > 2$. The transformation between the two is not as simple as you make it seem. $1 + r$ and $(1 + r'/n)^n$ are the same only if $r = (1 + r'/n)^n - 1$ or, equivalently, $r' = n[(1 + r)^{1/n} - 1]$. Apr 7, 2022 at 21:40

No, that's not how banks compute interest payments rates per my knowledge. Suppose an interest rate $$r_{T}$$ is offered at maturity $$T$$. If a client wants $$n$$ infra-periodic payments that will result in the same $$(1+r_{T})$$ at time $$T$$, the computation of the rate $$r_{T,n}$$ is given by the equation $$(1+r_{T,n})^n=(1+r_{T})$$ which implies $$r_{T,n}=(1+r_{T})^{1/n}-1$$. Furthermore, the continuous time cash value in $$t \in [0,T]$$ of the periodic payment account $$A_n(t)$$ is given by $$A_n(t)=(1+r_{T,n})^{\lfloor nt/T\rfloor}$$ obviously if no withdrawals occur.

However, if you compute $$(1+r_T)^{t/T}$$ you rightly obtain a fair value of the account. That is, the value is given by the current cash held $$A_n(t)$$ plus the accrued interest $$c(t)$$ of the next payment which can be given by $$c(t)=(1+r_T)^{t/T}-A_n(t)$$.

• Internally, of course, the bank can calculate $r_{T, n}$, but that isn't what clients see. What clients see is an $r$, called APR, which in fact corresponds to a real interest rate of $(1 + r/n)^n - 1$, called APY. My question is (in part) if there is any mathematically sound reason for both of these rates to exist in the present other than to confuse a mathematically unsophisticated client. Apr 7, 2022 at 22:05
• That is not unsophistication, it is a different context. Always per my knowledge, if I recall correctly, APY is calculated over fictitious annual rates, i.e. conventions. That is, an 'annual rate' $v$ is given, but the real rate at which compound payments are made is $v/n$ where $n$ is the number of payments in the year. That 'annual rate', however, is inaccessible as annual interest. In my example, $r_T$ is on the contrary an accessible rate for an investment. Apr 7, 2022 at 22:11
• Unless I'm mistaken, if your APR is $r$, after time $T$, you will have an amount in your account as if you had only compounded once at a rate of $(1 + r/n)^n - 1$, which I believe is called the APY. Why would you ever want to know the APR if the amount of money that you realize after time $T$ is $(1 + APY)P$? Why would you ever want to know the APR? At best you might want to know APR/n, since that is the rate you'll see each compounding. But I can think of no use for APR itself except to make a loan seem like it's cheaper than it is. Apr 7, 2022 at 22:22
• Yes, I can confirm you that many conventions, such as presenting 'annual rates' which however are only used to compute a (effective) periodic rate, are made for marketing purposes and industry inertia. It is reasonably understandable, but not commendable, that institutions care less about maths and more about marketing when small clients are involved. Apr 7, 2022 at 22:32
• Thank you. I just realized I accidentally repeated myself in the previous comment, but alas the edit window has expired. Sorry if it caused confusion. Apr 7, 2022 at 22:40

There is a practical fact that we really should take into account: banks typically set their own interest rates. And different banks can choose different rates.

So let's say Bank $$1$$ offers the interest rate $$r_1$$ with traditional compound interest compounded $$n$$ times per period $$t_0$$. The balance in the account after time $$t$$ is

$$A_1(t) = P\left(1 + \frac{r_1}{n}\right)^{{nt}/{t_0}}. \tag1$$

And let's say Bank $$2$$ offers the interest rate $$r_2$$ with interest accrued according to your scheme, $$A_2(t) = P(1 + r_2)^{{t}/{t_0}}.$$

This implies that $$A_1(t) = P\alpha^t$$ where $$\alpha = \left(1 + \frac{r_1}{n}\right)^{{n}/{t_0}},$$ whereas $$A_2(t) = P\beta^t$$ where $$\beta = (1 + r_2)^{{1}/{t_0}}.$$

So both banks are compounding the depositors' interest in the same way: exponential growth. There are some possible questions about details such as what happens if a depositor at Bank $$1$$ withdraws their balance at, say, the time $$t = 2.5 t_0/n.$$ (Do they get only as much money as if they had withdrawn it at $$t = 2 t_0/n$$?) And we could ask how Bank $$2$$ rounds the exponent $$t/t_0$$ when they compute interest. But these details just amount to small local "wiggles" in a growth curve that is essentially the same shape as any other exponential growth curve.

Of course which bank pays out more at the end of a period $$t_0$$ depends on whether $$\alpha$$ or $$\beta$$ is larger. If the two banks use the same nominal interest rates, that is, if $$r_1 = r_2,$$ then $$\alpha$$ will be slightly larger than $$\beta$$. But if Bank $$1$$ doesn't want to pay more than Bank $$2,$$ all it has to do is to observe the rate $$r_2$$ that Bank $$2$$ has offered, and then set

$$r_1 = n(1 + r_2)^{1/n} - 1.$$

This will ensure that $$\alpha = \beta$$, and then Bank $$1$$ will not be "overpaying" its depositors.

In short, the entire argument in the question is just about two different ways to specify a parameter (and "interest rate") that describes the exponential growth of deposits in an account. And you may see both versions of the "interest rate" in a bank's advertisements, because while the bank is required by law to specify the interest rate in the traditional way, the second way allows them to advertise a higher numerical percentage in order to lure people to deposit money in the bank.

A historical note: there was, at least in New York in the 1970s, an exception to the observation in the first paragraph, namely, consumer banks were legally prohibited from offering an interest rate above a certain amount ($$5.25\%$$ if I remember correctly). But interest rates on loans were much, much higher at that time, and banks would have liked to be able to offer higher rates to compete for deposits that they could then lend out. Since they could not offer higher rates, instead they all offered $$5.25\%$$ with shorter and shorter compounding periods, with at least some banks advertising that their interest was compounded continuously. That is, their account balance as a function of time was

$$A(t) = Pe^{0.0525t},$$

as pure a form of exponential growth as you can get.