Period returns

morning, ive memorised a lot of formulas etc but now trying to get a better understanding.

period returns, using the continuously compounded the annulised formula formula is 1/N*LN(S1/S0).
when calculating the annualised return of a bond [(FV+Coupons+interest earned)/Price Paid]^(1/N).

why do you divide by N when calculating the CC return but take the power when calculating normal annualised return over a period?

many thanks

There are 2 different equations for continuous compounding vs non-continuous compounding:

Continuous case (solve for r using LN):

S1 = S0 * (e^r)^N

Non-continuous case (solve for r by taking Nth root):

S1 = S0 * (1 +r)^ N

legend, very much appreciated