Quoted Annual Rates and Periodic Compounding

Summary

The document explains why a quoted annual interest rate is divided by the number of compounding periods per year. The annual quote is a convention; the rate applied at each period is the quoted rate divided by that frequency. Applying this periodic rate over the total number of periods gives the stated compound growth formula for an investment held over several years.

The explanation is conceptual rather than a derivation from continuous-time interest or an empirical analysis. It clarifies the meaning of the rate and how the formula follows from repeated compounding. The result depends on the stated convention that the annual rate is nominal and compounded at the specified frequency; it does not address effective annual rates, varying rates, or other day-count conventions.

Key ideas

• A quoted annual rate is divided by the compounding frequency to obtain the periodic rate.
• The periodic rate is applied once for each compounding interval.
• The total number of intervals is the compounding frequency multiplied by the investment duration in years.
• The formula describes nominal-rate compounding and does not cover alternative interest-rate conventions.

Tags

# Derivation of the formula for $m$ compounding periods per year: $(1+\frac{i}{m})^{mt}$


# Derivation of the formula for $m$ compounding periods per year: $(1+\frac{i}{m})^{mt}$












A dollar return with interest $i$ invested for $T$ years with compounding interest frequency of $m$ times each year is:

$$1*(1+\frac{i}{m})^{mt}.$$

My Question

- Why do we divide $i$ by $m$? Is this because $i$ represents annual interest rate, but it is compounded $m$ times a year, so we need to compute the effective interest rate at each compounding periods?

- How do we analytically derive this formula?

## Answer by Alex C (score 3, accepted)

https://quant.stackexchange.com/a/49010

This is about interest rate conventions and terminology. When people say "i percent a year compounded m times a year" it means the following: $i$ is called the quoted rate, which is not directly used in the calculation. Instead the first step is to calculate $\frac{i}{m}$ which is called the periodic rate and then apply this rate to every period. If there are t years, there are $mt$ periods and therefore the formula above follows.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.