why is continously compounded return lower than simple return when there is coninously compounding ?
according to my text book "by contrast, a continously compounded return is a return that assumes continous compounding, meaning that interest is paid for very short sub-periods and that the interest is again reinvested at the same return.
I think they mean the continuously compounded rate is lower than its corresponding EAR. The sooner you have an interest compounding, the sooner the magic of “interest on interest” kicks in. Thus, the higher the compounding frequency, then the lower the nominal rate can be.
For example, assume an effective annual rate of 4%. The corresponding continuously compounded rate is 3.922% = ln(1.04) - 1. As a comparison, the nominal semi-annual rate is 3.9605% and the nominal quarterly rate is 3.9414%.