"Natural" Interest

[Amatuer Geek]

Hello

I've watched a few videos that talk (in a way I understand) about how splitting interest over different periods of time (eg 1 year, 1 month, 1 day) leads to the formula (1+r/n)^n and this tends to e^r as n tends to infinity. I get the maths here and anything that leads to an answer with e in it is probably tapping into some "natural truth", but my question is about why is splitting interest rates in this way "natural"?

To me, the "correct" way to split interest over different periods of time is to do it in such a way that it compounds up to be the same at the end of the period.

eg Getting 5% interest per year on £100 would result in £105 after 1 year if it was paid in 1 payment. If you were to get interest at 2 points during the year, it makes more sense to me that the interest paid at each point isn't 2.5% but rather 2.4695...% as this rate will compound back to £105 at the end of a year and the interest rate remains 5% after a year.

In general, if you are splitting an annual rate of r into n time periods, the interest applied at each period could be calculated as (r+1)^(1/n)-1 instead of r/n

Of course, everyone is free to choose any convention that they want and r/n is definitely simpler, but as r/n leads to a "natural" limit, is there some underlying truth that makes it the best method, or is the fancy limit just an interesting coincidence?

Thanks for considering this.

Amateur Geek

 

[Deleted member 4993]

Hello

I've watched a few videos that talk (in a way I understand) about how splitting interest over different periods of time (eg 1 year, 1 month, 1 day) leads to the formula (1+r/n)^n and this tends to e^r as n tends to infinity. I get the maths here and anything that leads to an answer with e in it is probably tapping into some "natural truth", but my question is about why is splitting interest rates in this way "natural"?

To me, the "correct" way to split interest over different periods of time is to do it in such a way that it compounds up to be the same at the end of the period.

eg Getting 5% interest per year on £100 would result in £105 after 1 year if it was paid in 1 payment. If you were to get interest at 2 points during the year, it makes more sense to me that the interest paid at each point isn't 2.5% but rather 2.4695...% as this rate will compound back to £105 at the end of a year and the interest rate remains 5% after a year.

In general, if you are splitting an annual rate of r into n time periods, the interest applied at each period could be calculated as (r+1)^(1/n)-1 instead of r/n

Of course, everyone is free to choose any convention that they want and r/n is definitely simpler, but as r/n leads to a "natural" limit, is there some underlying truth that makes it the best method, or is the fancy limit just an interesting coincidence?

Thanks for considering this.

Amateur Geek

Do you know the differences between

Annual percent rate (APR) and

Annual percent yield (APY)?

It might be interesting for you to Google those terms

 

[Amatuer Geek]

Thanks for replying.

I've looked into definitions and connections between APR and APY but they both seem to come from the initial assumption that the way to subdivide an interest rate r into n periods you should use interest for each period of r/n and then go on to show the maths that follows from this assumption, but that initial assumption leading to the end relationship seems circular to me.

If I've understood them correctly, for the explanations of APR and APY, neither addressed my question which is why (if at all) is splitting the interest rate this way a natural decision to pick rather than an arbitrary one (eg picking radians has a naturally compelling reason over degrees which are an arbitrary measurement).

Apologies if the reason is in properly understanding APY and APR and I've just missed it.

A-G

 

[HallsofIvy]

I think you are putting too much into the word "natural". I haven't seen this particular textbook so I can't say for sure but in general mathematics the logarithm "base e" is called the "natural logarithm" and I suspect that the word "natural" was taken from that.

And base e is "natural" for this reason: the derivative (rate of change) of the function \(\displaystyle f(x)= log_a(x)\) (the logarithm of x to base a) is a constant times \(\displaystyle \frac{1}{x}\). That is, \(\displaystyle \frac{d log_a(x)}{dx}= \frac{C_a}{x}\) where \(\displaystyle C_a\) depends upon a but not x. The "natural logarithm", ln(x), with base "e", has property that the constant is 1: \(\displaystyle C_e= 1\) so \(\displaystyle \frac{d ln(x)}{dx}= \frac{1}{x}\). That simple property, together with the fact that a logarithm to any base can be converted to the natural logarithm, makes the natural logarithm very important. That is the sense in which "natural" is being used here.

 

[JeffM]

By "natural" interest, do you mean how interest is calculated among kangaroos or rabbits? I cannot begin to understand this question until you can tell me the definition of "natural interest" and how it is distinguished from "unnatural interest."

The alternative approach is to forget the term "natural interest," which is never used in commercial practice or legal documents and look how "interest" is actually computed in practice. The rate is almost always quoted as an annual rate. But frequency of payment is a separate contractual term. So if the contract says that the rate is 8 per cent per annum to be paid quarterly, the actual interest payment due is the principal times 0.08 divided by 4. I have no idea whether that is natural, but it is simple enough that the calculations can be done by hand, which was an important consideration before the twentieth century.

Now if you are a banker, interest can be earned on interest payments received. I lend to X Corp for a year 10,000,000 pounds at 4% per annum payable semi-annually, Thus, I receive 200,000 pounds at the end of the first half year. I can lend that 200,000 pounds. On what terms? Let's assume that I can lend it to Y for half a year also at 4% per annum. Then at the end of the year, I get four payments: the principal of 10,000,000 plus interest of 200,000 from X and principal of 200,000 plus interest of 4,000 from Y. Thus, I get a total of 10,404,000. So the banker says to himself: I should be indifferent between lending at 4% per year payable semi-annually and lending at 4.04% per year payable annually. The whole math that you are learning grows out of this specific thought process.

So the math you are learning is partly a reflection of how interest payments are actually computed in commercial practice and partly a consequence of two economic assumptions, one being that the yield curve is currently flat and the other being that the yield curve will not shift up or down over time. Neither assumption is likely to be true, but the difficulty of computations using alternative assumptions is huge. In other words, the math actually is a compromise between established commercial practice and ease of computation; it has nothing to do with the world of nature, of flowers, rabbits, and kangaroos.

I now give a historical hypothesis (meaning I have done no special historical research) that current commercial practice itself arose in part from ease of computation by hand and in part from the legal interpretation of usury laws in early modern Europe.

 

[Amatuer Geek]

Firstly, thanks to everyone who has spent their valuable time thinking about this on my behalf, I really appreciate it.

I've clearly confused the issue by using the word natural here. What seemed to be an arbitrary convention of splitting interest (splitting the rate r over n periods and saying each period should be r/n) leads to a major mathematical constant popping up when calculating continuous interest (e^rt). To me, this constant appearing suggests that there is a deeper level of maths at play and some intuitive reason for picking r/n which escaped me.

I was always concerned as it didn't feel like the inverse of the process and it causes terms like APR and APY to appear (which are generally confusing to many) whereas agreeing on a system so that 8% remained 8% no matter how it divided up made more sense to me as it puts the complexity onto the banks and remains simple (and fairer sounding to me) from the consumer's point of view.

Jeff's explanation of the convention coming from the banker's point of view makes perfect sense to me - thank you so much.

I suppose the e^rt formula is just a happy coincidence (kind of like using Fibannacci Numbers to convert between miles and kilometres).

 

[HallsofIvy]

I talked about the "natural logarithn before. Since the natural logarithm is the inverse function to the "exponential", here is going the other way:

The general exponential, \(\displaystyle a^x\) has the property that its derivative (rate of change) is a constant times \(\displaystyle a^x\). (To see that, start with the change as you go from x to x+h, \(\displaystyle a^{x+h}- a^x= a^x(a^h)- a^x= a^x(a^h- 1)\), then calculate the average rate of change, \(\displaystyle \frac{a^{x+h}- a^x}{h}= a^x\frac{a^h- 1}{h}\), then finally get the instantaneous rate of change by taking the limit as h goes to 0, \(\displaystyle \lim_{h\to 0} a^x\frac{a^h- 1}{h}= a^x\left(\lim_{h\to 0}\frac{a^h- 1}{h}\right)= C_aa^x\). The constant is that \(\displaystyle C_a= \lim_{h\to 0}\frac{a^h- 1}{h}\). The base e is used because it is the number such that that rate of change is \(\displaystyle C_e= 1\). It is the function whose rate of change is equal to the function itself: \(\displaystyle \frac{de^x}{dx}= e^x\).

That's not a "coincidence" at all, that's why that particular number, e, is used.

 

[JeffM]

What you see as surprising seems to me to be the inevitable result of picking a convention that reduces to a fairly simple computation

[MATH](1 + i)^n, \text { where } i = \dfrac{r}{n}.[/MATH]
The only way e gets infolved is if we let n go to infinity, which never happens in practice. (Actually, in the late 1970's, some US banks used to advertise that they paid interest continuously, but of course they did not compute interest continuously. One of my first jobs at the bank where I ended my career as a director was figuring out how to make that advertising claim effective. I assure you my solution did not involve having computers calculate interest every microsecond on hundreds of thousands of accounts.)

 

[HallsofIvy]

I suppose the e^rt formula is just a happy coincidence (kind of like using Fibannacci Numbers to convert between miles and kilometres).

The "e^rt formula" is a much happier coincidence since this last is not true!
It, for example, asserts that 5 and 8 are consecutive numbers in the Fibonacci sequence, which is true, and that 5 miles is 8 km. No, 8 km is approximately 4.8 miles, not 5. It also asserts that 21 and 34 are consecutive numbers in the sequence (true) and that 34 km is 21 miles (NOT true) 34 km is approximately 20.4 miles. Those, we can grant, but that skips over other consecutive Fibonacci numbers, like 2 and 3 (3 km is 1.8 miles, not 2) ,

What you see as surprising seems to me to be the inevitable result of picking a convention that reduces to a fairly simple computation

[MATH](1 + i)^n, \text { where } i = \dfrac{r}{n}.[/MATH]
The only way e gets infolved is if we let n go to infinity, which never happens in practice. (Actually, in the late 1970's, some US banks used to advertise that they paid interest continuously, but of course they did not compute interest continuously. One of my first jobs at the bank where I ended my career as a director was figuring out how to make that advertising claim effective. I assure you my solution did not involve having computers calculate interest every microsecond on hundreds of thousands of accounts.)

Calculating interest continuously does NOT mean "calculate interest every microsecond". It means the exponential function which CAN be easily done.