# Why different compounding in interest rates

This is more a philosophicalquestion than a financial question, let me explain.

There exist different types of interest rate (Annual Interest rate, Semi-annual interest rate, monthly interest rate, continuously compounded interest rate...)

The thing is that there is a relation between all this compondings (because at the end is a constant coefficient "a" to the power of time):

Annually compounded interest rate: (1+r)^t -> a^t where a = (1+r)

Semianual compounded interest rate: (1+r/m)^(mt)-> a^t where a = (1+r/m)^m

Continioulsy compounded interest rate: e^(rt) -> a^t where a = e^r

I'm coming from a scientific background (I hold a B.Sc. in Physics) and I don't understand why they are different if at the end is the same Because I think that this could lead to large confusions. One could be when someone is writing an article, or publishing some "interest rate" information on to a market data platform (Bloomberg, Reuters...) without adding the (compounding) information I could understand for example, that is continious compounded insted of annualy compounded that the publisher thinks it is.

As there is term structure of interest rates (so that, for each maturity we have an interest rate), why don't we create a discount curve instead of a yield curve. And create an international convention for that (that could help "street people" to understand better the financial product the bank is offering)

Edit: For example a good "international" choise could be the net return that is the annually compounded interest rate

• I'll make it a comment rather than an answer because it's entirely my opinion/guess. 1 conspiracy theory - sell-side banks have been promoting very confusing quoting conventions in hopes that some else will make a mistake and the bank would profit 2 misdirected creativity - especially in emerging markets, a committee of bureaucrats are tasked with designing fixed income market. They don't know what they're doing, but being different from all other countries is more important than clarifty/transparency of the quotes. – Dimitri Vulis Sep 27 '20 at 14:24

They are different because there are different conventions in different places.

Whilst it would make the maths more consistent to use discount (or accrual) factors to describe interest, the foremost concern historically has been absolute clarity on amounts of money, so having a simple way to calculate the coupon payment on a given bond. So a UK Gilt paying 5% on £100,000 pays exactly £2,500 every six months.

If you have an Interest Rate Swap which you are using to swap fixed coupons from a bond for floating coupons against an index, then it would be best if the payments for those two instruments lined up, at least in payment dates and ideally in accrual. So many swaps are quoted with bond basis, even though they are traded between sophisticated counterparties.

Interest rate and foreign exchange markets grew in a largely ad hoc, over the counter manner, not in some centralised system, so the conventions vary as different people did things their own way.

Especially Brazil.

I grappled with different compounding conventions for a while when I first joined the world of finance. After some time, I came to the conclusion that the best analogy is that of different units of speed in different countries: i.e. in the UK, the unit of speed is miles, whilst in continental Europe, it is kilometers. A car travelling at 100mph is travelling at 160km: the different units are just a different way of expressing the same quantity. The actual speed is the same.

When it comes to interest rates, consider an effective simple annual interest rate of 10%.

Using continuous compounding, we could write: $$e^{0.0953}=1.1$$: in other words, the rate of 9.53% continuously compounded for 1 year is equivalent to a simple rate of 10% per year (just like 160kmh is equivalent to 100mph)

Using semiannual compounding, we could write: $$(1+\frac{0.0976}{2})^2=1.1$$: again, the semiannual rate (annualized) of 9.76%, compounded semi-annually, is equivalent to 10% simple rate per year (just like 160kmh is equivalent to 100mph).

It is worth noting that the markets are very, very efficient at arbitraging any "opportunities" presented by different conventions: so that is to say, you cannot effectively make money by trying to exploit different conventions in various contracts (in the retail world though, some institutions have had trouble with regulators by playing around with conventions, so it's good to keep an eye on that if you ever sign any retail contracts bearing interest rates).