Converting Semiannual Compounding to a Continuous Rate

Summary

The document explains how to convert a quoted annual rate with semiannual compounding into an equivalent continuously compounded annual rate. The example treats a 10% annual quote as 5% for each six-month period. Compounding those two periods gives a one-year growth factor of 1.1025, equivalent to an annual effective return of 10.25%.

To find the continuous rate, the answer equates the exponential growth factor over one year to that same accumulated value and solves for the rate, obtaining about 9.758%. The continuous rate is lower than the semiannually quoted nominal rate because the two rates use different compounding conventions while matching the same ending value. The explanation is limited to this one-year example and assumes the stated semiannual convention; it does not discuss day-count conventions, fees, or other rate quotations.

Key ideas

• A 10% annual nominal rate compounded semiannually applies 5% in each half-year period.
• Two semiannual periods produce a one-year accumulation factor of 1.1025.
• An equivalent continuously compounded rate is found by matching the same accumulated value.
• Different compounding conventions can yield different quoted rates for identical growth.

Tags

# Continuously Compounded rate less than a discretely compounded rate


# Continuously Compounded rate less than a discretely compounded rate












I'm looking at an example in a well known book and its saying

"consider an interest rate that is quoted as 10% per annum with semi annual compounding"

The book puts 10% as the semi-annual rate, then uses a formula which doesn't make sense and gets to the continuous compounded rate which is then 9.758?

How can the continuously compounded rate be smaller that the semi-annually compounded rate?

## Answer by rbm (score 1, accepted)

https://quant.stackexchange.com/a/45547

Well let's just do the math. 10% p.a. with semi annual means 10%/2 for 6 months, so you get

$1.05*1.05=1.1025$

That is, for 1 dollar you'll have 1.1025 in 1 year, i.e. 10.25% p.a. if it was annualy compounded.

What should be the rate for continuous compounding (annual)? Well:

$e^{r\times1}=1.1025$

gives

$r=0.0975803$

or 9.758% as stated in your book.

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.