A discretely compounded rate of return is the return that is compounded for a discrete or defined interval of period (e.g. monthly, quarterly, semi-annually, etc.)
A continuously compounded return is compounded continuously such as the interval period for compounding is almost zero.
The continuously compounded return associated with a holding period is the natural logarithm of 1 plus that holding period return.
rt,t+1 = ln(1+ Rt,t+1) = ln(St+1/St) where Rt,t+1 is the holding period return between period t and t+1 and rt,r+1 is the continuously compounded return during that period. St+1 and St are the asset's values at periods t+1 and t respectively.
If the asset's continuously compounded return is normally distributed, then the future asset price is necessarily lognormally distributed.
If the returns between the holding periods are assumed to be independently and identically distributed with expected value of µ and variance of σ2, then the mean and variance of T holding period compounded continuously are µT and σ2T respectively.
Standard deviation for T days = Daily Standard deviation*√T
Effective annual rate (EAR) = exp(rcc) - 1
ln(1+EAR) = rcc
1+ EAR = (1+HPR)1/N
where rcc is continuously compounded annual rate of return and HPR is holding period return, and N is the time period in years
rcc = ln(1+EAR) = ln(1+HPR)1/N = (1/N)ln(1+HPR) = (1/N)ln(ST/S0) for a non-dividend paying stock
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Example 8: Calculating continuously compounded return
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A stock was bought at $15 and sold at $20 after two years. Compute its continuously compounded annual rate.
Solution:
Holding period return = (20/15)-1 = 0.3333
Continuously compounded annual rate of return = (1/2)ln(1+0.3333) = 0.1438 = 14.38 percent.
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