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Macaulay and Fischer-Weil Duration with Different Discount Rates

Article Quant Q&A · Author: Amrit Prasad
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Summary

The document explains why the choice of discount rates changes a bond duration measure. Traditional Macaulay duration discounts each cash flow using the bond’s single yield to maturity, then weights its payment time by the present value of that cash flow relative to the bond’s dirty price. A yield calibrated to the bond price can produce the same total present value as discounting with zero rates, while producing different time-weighted cash flows.

The answer identifies duration calculated from a flat yield as traditional Macaulay duration and says that using zero rates instead gives Fischer-Weil duration. It notes that Fischer-Weil duration is commonly used when the term structure varies substantially across maturities and cash flows are long dated. The text offers this distinction but does not derive the measure or discuss its assumptions and limitations in detail.

Key ideas

  • Traditional Macaulay duration discounts cash flows using a single yield to maturity.
  • Discounting with zero rates can preserve the bond price while changing the duration weights.
  • Duration based on zero-rate discounting is called Fischer-Weil duration.
  • Fischer-Weil duration is used for bonds with long cash flows when rates vary across maturities.

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Fixed incomeRisk managementStatistics
Full text
# Macaulay's Duration with Zero Rates


# Macaulay's Duration with Zero Rates












The definition of Macaulay's Duration is the weighted average maturity of cash flows and is calculated as-

$$D_{mac}=\frac{\sum_ttPV(C_t)}{V}$$

where $PV(C_t)$ is the present value of the cash flow at time t of the bond in question while $V$ is the current dirty price of the bond.

The $PV(C_t)$ is calculated as $e^{-yt}C_t$ where $y$ is the Yield to Maturity of the bond.

My question is regarding this choice of discounting. Could we use the zero rates instead of the flat yield to calculate the $PV(C_t)$? In the case of a bond price it wouldn't make a difference since the yield to maturity is calculated assuming that it's the flat rate that allows your discounted net present value to equal the one that you got using the zero rates.

$$\sum_tC_tZ(0,t)=\sum_tC_te^{-yt}$$

But in the case of time weighted cash flows the two sums would be different.

## Answer by Amrit Prasad (score 1)

https://quant.stackexchange.com/a/40495

If the discounting is done using the flat yield, then we get the traditional measure of Macaulay's and Modified durations. Discounting using zero rates while calculating durations results in what is known as the Fischer-Weil duration. This measure is usually used when the yield term structure varies a lot for different tenors and the cash flows are longer term.

Shown 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.