Calculating Continuously Compounded Returns from Index Values

Summary

The note explains how to calculate a continuously compounded return from index levels observed at different times. It starts from the exponential growth relationship between an initial value and a later value, then rearranges it to express the annualized rate as the logarithm of their ratio divided by elapsed time. The example uses index values of 4,000, 4,086, and 4,114 at times 0, 1, and 2, and reports annualized rates for the first and second observations relative to the initial level.

This method gives a continuously compounded rate over the specified interval; the annualized interpretation depends on time being measured in years. It does not separately calculate each one-period daily return from consecutive index values, so readers seeking daily returns should apply the log ratio to adjacent observations and account for the time unit. The example is illustrative and provides no discussion of fees, dividends, or other adjustments to index levels.

Key ideas

• A continuously compounded rate follows from the logarithm of the ratio between later and initial values.
• Divide that log ratio by elapsed time to express the rate per time unit.
• The example calculates annualized rates from the initial index value to later observations.
• Adjacent index values can be compared to calculate returns over individual intervals.

Tags

# Calculating the daily continuously compounded return from index values


# 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.

## Answer by Martin Vesely (score 2, accepted)

https://quant.stackexchange.com/a/53865

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: $$ r_{t=2} = \frac{1}{2}\ln\frac{4114}{4000} = 1.4051\, \%. $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.