Calculating Macaulay Duration with Spot Rates
Summary
The question asks whether Macaulay duration can be calculated by discounting each cash flow with the spot rate for its payment date, rather than using one yield to maturity. It proposes weighting each cash flow’s time by its present value under the corresponding zero-coupon spot rate, then dividing by total present value.
The accepted response confirms this approach by referring to the duration definition, but provides no derivation, worked example, or discussion of assumptions. The note therefore supports the basic idea that cash flows along a non-flat curve can be discounted at their respective spot rates when forming a present-value-weighted average time. It does not explain how modified duration should be adapted, or address compounding conventions, curve construction, or rate sensitivities.
Key ideas
- Macaulay duration is a present-value-weighted average of cash-flow times.
- When using a spot curve, discount each cash flow at the spot rate corresponding to its maturity.
- The answer confirms the proposed approach but gives no derivation or implementation details.
Tags
Full text
# Calculation of duration based on spot rates
# 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 $t$.
## Answer by ZRH (score 0, accepted)
https://quant.stackexchange.com/a/44510
yes, definitely, see definition of Macaulay durationShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.