Compound interest: why does everyone get it wrong?

Question

The compound interest formula is: $$A_t=A_0(1+r)^t$$ There is a simple derivation for this which works by starting with $A_1$ and then considering $A_2$ and then extrapolating. The above formula can be manipulated to solve any related problem we have. Rates are almost always stated as APR's, which once stated are a legal obligation so daily or monthly rates must be derived from the APR. This can be done thus: $$(\frac{A_t}{A_0})^{1/t}-1=r$$ So, for example, if the APR is 4 and we want the monthly rate we can say: $$1.04^{1/12}-1=0.003274$$ since we know that $A_t/A_0=1.04$ and if we want to work out the amount after five months from a deposit of $300$ we can say: $$\frac{A_t}{A_0}A_0=(r+1)^tA_0=(1+0.04)^{5/12}\times300=304.942867$$ Crucially, this is less than $305.033445$ which is what we would get if we set the monthly rate at $0.04/12$ which is how it's done in the formula: $$A_t=A_0(1+\frac{r}{n})^t$$ This formula, which is most usually given in books and websites, gives a total of $312.222463$ at $t=12$ rather than $312$. The commonly used formula can therefore not possibly be correct! Yet no one seems too bothered. Everyone just goes on using the wrong formula getting misleading results. Why? What can be done to change the situation?

Answer 1

I will just point out that $$(1+r)^{1/n}=\sum_{k=0}^{\infty}\binom{1/n}{k}r^k=1+\frac{r}{n}+\cdots$$ by using the Taylor series, where $\binom{x}{k}=\frac{x(x-1)\cdots (x-k+1)}{k!}$ when $x$ is real.

So letting $r_n$ being the rate of the $n$-th part of some unit of time rate $r$, we have $$r_n=(1+r)^{1/n}-1\sim\frac{r}{n}$$ by ignoring terms of higher order. This gives a decent approximation for $r$ small (as I think it usually is).

Ignorant of the history of the compound interest formula, this approximation might have been preferred in an era where computing the $n$-th root of something was really difficult due to the absence of the computational resources we have today (and where many calculators only performed addition, substraction, multiplication and division).

Answer 2

If you have a yearly interest rate at $0.04$, then that in itself, a priori, is not a compounded interest rate. It means that you have $\$100$ in a bank acccount, and after a year, the bank thanks you for choosing them to keep your money by adding $\$4$ to it. It then stands to reason that if you only have that money in the bank for a month, they would pay you $\$\frac13$ for that.

Answer 3

You are right. Let the (yearly) interest rate denoted as $i$. Then the annual equivalent interest rate (AER), compunded m times a year, is $i^e_m=\sqrt[m]{1+i}-1$. You are right that there is a difference to the annual percentage year rate (APR) which is $i^p_m=\frac{i}{m}$. If the bank pays interest more than once in a year ($m>1$) then it is an advantage for the client if the bank uses APR and the account is positive. But if $m=1$ then $AER=APR$

In general you have to look at the general terms and conditions of your bank. "By law, credit card companies and loan issuers must show customers the APR to facilitate a clear understanding of the actual rates applicable to their agreements. Credit card companies are allowed to advertise interest rates on a monthly basis (e.g. 2% per month), but are also required to clearly state the APR to customers before any agreement is signed."(http://www.investopedia.com/terms/a/apr.asp)

It is up to the client to investigate which kind of interest rate the bank uses and how often the intrest rates a payed. I think it is not necessary to change the situation. But the goverments or other institutions should make a campaign about that issue. Then the clients are better informed and can make their own (rational) decisions.

Answer 4

In the formula $A_t=A_0(1+r)^t$, correctly interpreted, $r$ is the interest rate per compounding period and $t$ is the number of periods. Nowhere in the formula is there any definition of what the compounding period is--it could be an entire year, or a month, or an hour.

People sometimes write $A_t=A_0\left(1+\frac rn\right)^t$ instead when there are $n$ compounding periods per year and they would like $r$ to be seen as an "annual" interest rate. That is, if an bank says your certificate of deposit is earning $3\%$ interest compounded monthly, it means precisely that every month they credit you with interest equal to $\frac 14\%$ of the balance in that account. But this basically comes down to a semantic question of which number you want to use for $r$; whichever formula you use, if you are using it correctly, you will end up with the same numeric value added to $1$ inside the parentheses; the only difference is whether you wrote $r$ or $\frac rn$ to "name" that numeric value.

There are a great number of ways that financial deals involving interest can be structured. Not only can one use different rates, but different compounding periods, and one can have additional payments such as up-front fees. The idea of the "APR" is that it takes all of these things and rolls them into one number that will allow you to compare one financial deal against another--it is in some sense the rate that the financial institution would have to set in order to have an "equivalent" deal using a certain standard set of assumptions about the payments--but it is not necessarily equal to any of the numbers that is used during the actual calculation of interest, because the contract may be based on different assumptions about payments.