Hello David,
As I’m reviewing for the compounding frequencies from continuous to discrete and vice versa, I am wondering if there is such a thing as to convert from a discrete rate to a discrete rate of a different frequency (for example, from a semi-annual rate to quarterly compound rate). I know whenever we’re given a rate, say 10%, compounded semi-annually, we simply divide 10% by 2 to get the semi-annual rate (I might be wrong here), but wouldn’t this actually give us an annual effective rate greater than 10%? I think it is very straight forward for annual rates, but I always get confused whenever it comes to semi-annual and quarterly rates.

I think a 10% annual rate > 10% rate compounded semi-annually > 10% rate compounded quarterly, but I’m not sure how the calculation can be done. I would guess that the calculation would be similar to that of deriving forward rate from spot rates?

Thank you.

Hi Jack,

Although maybe not a common use, we could solve for the discrete by, given (m) for one and (q) for another:

(1+Rm/m)^(mn) = (1+Rq/q)^(qn), such that
(1+Rm/m)^m = (1+Rq/q)^q
Rm = [(1+Rq/q)^(q/m)-1]*m

for example, converting from semiannual (q=2) to monthly (m=12)
R(12) = [(1+Rq/2)^(2/12)-1]*12

Re “I know whenever we’re given a rate, say 10%, compounded semi-annually, we simply divide 10% by 2 to get the semi-annual rate (I might be wrong here), but wouldn’t this actually give us an annual effective rate greater than 10%?”

Yes, your effective annual rate *always* exceeds the stated rate; in a way, the effective annual rate is the rate that the converted rates all have in common.
8% continuous = EXP(8%) = EAR, continous = 8.162% semiannual = 2*(EXP[8%/2]-1)

So it is: Effective annual rate [10% annual rate] < EAR[10% rate compounded semi-annually] < EAR[10% rate compounded quarterly] because the 10% is “merely stated”

Hope helps, David