# Calculation of duration based on spot rates

As I know, Macaulay and modified durations are defined in terms of yield to maturity (YTM), in other words, in order to calculate durations we use yield to maturity as a discount factor. Suppose, that instead of YTM we want to calculate durations based on spot rates. Should I calculate Macaulay duration in the following way:$$MacD=\frac{\sum_{t=1}^{n}tCF_t/(1+s_t)^t}{\sum_{t=1}^{n}CF_t/(1+s_t)^t},$$ where $$s_t$$ is spot rate (zero-coupon rate) for period $$t$$.

yes, definitely, see definition of Macaulay duration

• Thank you for your answer. I have considered the attached article about duration, but, unfortunately, I didn't find something about duration calculation based on spot rates.
– sane
Mar 11 '19 at 9:14
• The fact that the definition refers to the $PV_i$, i.e. the present values of each single cashflow, requires that you use spot rates. If you used yields, the $PV_i$ would not be correct
– ZRH
Mar 11 '19 at 9:17
• I see, thank you.
– sane
Mar 11 '19 at 9:24
• When you use the spot rates it is called the Fisher-Weil duration bondtutor.com/btchp6/topic6/topic6.htm Macaulay used ytm's in his work. Mar 11 '19 at 12:21
• @Alex C: Interestingly, Wikipedia seems to suggest otherwise
– ZRH
Mar 11 '19 at 20:09