That's interesting, Steve (to me, because I write so many questions).

With regard to "An asset is quoted at 12% annually with continuous rate. Interest is paid quarterly." Note three timeframes are invoked:

- Interest paid quarterly (4 per year)
- The rate curve used to compound or discount (FV or PV more likely) should always be expressed "per annum" which is independent of compound frequency; i.e., even if the "annually" were omitted, we would assume the 12.0% is per annum
- Compounding frequency is continuous

A modern version of the question is more likely (imo) to rephrase, in a manner typical of Hull, as follows (eg):

**"An asset pays interest quarterly and the [spot | zero | discount | swap rate curve] is flat at 12.0% per annum with continuous compounding**"

... Note in a carefully phrased question, how we can easily see that purpose of the 12% is to discount to price (or compound forward to an expected future price)

So, as phrased above, the continuous would not be converted;

For example, assume a bond with a 12% quarterly coupon when the spot rate curve is flat at 12.0% per annum with continuous compounding.

It's price is slightly less than par:

($100*12%/4)*exp(-0.25*12%); first quarterly coupon discounted continuously

+ ($100*12%/4)*exp(-0.50*12%); i.e. coupon paid quarterly but discounted continuously

....

+ $100+($100*12%/4)*exp(-n*12%)

= slightly less than $100

which is why it's more typical to assume, or better yet to explicitly set the coupon frequency = compound frequency (but to keep in mind they are different things!):

"an asset pays pays a quarterly 12.0% coupon when the [spot] rate curve is flat at 12.0%"

*.... does not explicitly give us the compound frequency, so assume that it means:*
"an asset pays pays a quarterly 12.0% coupon when the [spot] rate curve is flat at 12.0% with quarterly compounding"

... and that will discount a bond to par, as we would expect

my point is that it helps to separate (conceptually) the curve being used to discount from the coupon (interest payments) which has its own per annum r% and compound frequency, often the same. (sorry for length). You asked i think a great question:

"When we say compounded monthly, are we assuming that the interest income is withdrawn each month too?"

My answer is:

**no we do not make this assumption,** unless we have no other information from which to assume.

On the other hand, while it is unnatural to infer from compound frequency --> coupon frequency, the reverse inference is quite natural:

"An asset pays interest monthly"

... while a good question explicates the compound frequency, in most cases, it is totally natural to infer from monthly coupon --> monthly compound frequency, as that describe the actual nature of reinvested cash flows.

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