# Continuously Compounded rate less than a discretely compounded rate [closed]

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?

## closed as off-topic by byouness, Helin, Bob Jansen♦May 12 at 17:46

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• "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – byouness, Helin, Bob Jansen
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• To get the same discount factor or the same interest for a given period, the continuously compounded interest will of course be lower than the discreetly compounded interest rate. This is intuitive. If I pay you a given rate continuously vs discreetly over the same period, the continuous case will give a higher interest... so, if you want the same interest you have to use a lower continuously compounded rate. – byouness May 11 at 11:44
• Right I think I was getting confused because it looks like, the Rate reduces as compounding frequency increases, but the Future value actually increases. Is that right? – user6046760 May 11 at 13:16
• Yes, exactly :) – byouness May 12 at 13:42

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.

• Ok we go from the given rate of return (10.25%) backwards to get tis intrinsic CC rate (9.758). I was getting confused because e^.1=(10.52%), but this is the CC rate if 10% is compounded continuously - but this is not the same as the question asked. I think I understand this now thanks! – user6046760 May 11 at 13:24