# Calculating the daily continuously compounded return from index values

Given I have 3 index values at time $$t = 0, 1 , 2$$, how would I go about calculating the daily continuously compounded return?

Time: $$0, 1, 2$$

Index Values: $$4000, 4086, 4114$$

Any help would be much appreciated. Thanks.

In continuous compounding, a nominal (or an index value) in time $$t$$ is given by formula

$$N_t = N_0\mathrm{e}^{rt},$$

where $$r$$ is return (or interest) rate per annum.

Based on the equation above, the $$r$$ can be calculated as

$$r = \frac{1}{t}\ln\frac{N_t}{N_0}.$$

So, for $$t = 1$$ we have the annualized return: $$r_{t=1} = \frac{1}{1}\ln\frac{4086}{4000} = 2.1272\, \%.$$

And for $$t = 2$$ we have the annualized return:
$$r_{t=2} = \frac{1}{2}\ln\frac{4114}{4000} = 1.4051\, \%.$$

• Hi there, Just wanted to ask a quick question given some details that I may have skimmed over, and whether this would change the answer. Firstly, if instead of time being $t = 1,2,...$, it was given as dates starting from, say, 16/07/20 and moving up in increments of single days, would this change the daily continuously compounded returns? I am also told to assume that daily market returns are independent. Does this change anything? Thanks for your help again. May 12 '20 at 20:45
• @DPJDPJ: Hi, firstly you can always convert date to integer 1,2, etc. where 1 is assigned to the first date. Then the compounding will be the same as above. Secondly, no as the returns in the answer can be considered as average return over period from 0 to some $t$. May 13 '20 at 7:14