Macaulay Duration as the Present-Value-Weighted Average Payment Time
Summary
The document examines the interpretation of Macaulay duration as the time-weighted average at which a bond’s present value is returned through its cash flows. It presents a bond with annual coupon and principal payments discounted at a constant yield, then calculates each payment’s present value, the bond’s total value, and duration by weighting payment dates by their shares of total present value.
The apparent puzzle is that discounting a single cash flow at the calculated duration does not reproduce the bond’s price. An answer clarifies that duration is an average over payment times, illustrated by treating each dollar of present value as a unit with a payment date; it is not a single repayment date. Another response suggests a reinvestment-based payback interpretation under a flat yield curve and parallel shifts, but offers no completed demonstration. The discussion is illustrative and does not establish that interpretation rigorously or cover other duration definitions and yield-curve changes.
Key ideas
- Macaulay duration weights each cash-flow date by that payment’s share of the bond’s present value.
- The weighted average date is not a date at which one discounted cash flow equals the full bond price.
- The example uses discounted coupon and principal payments to calculate duration.
- A reinvestment interpretation is proposed under restrictive assumptions, but the document does not prove it.
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Full text
# Bond duration and the mathematical proof of 'bond price recovery'
# Bond duration and the mathematical proof of 'bond price recovery'
> The term duration has a special meaning in the context of bonds. It is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows.
I have read this statement from the textbook and try to use the mathematical way to proof (the bolded statement) that is true. Thus, I have made up an example as follow:
```
Take the discount rate as 7% per annum
Term (yr) Cash Flow PV
1 100 93.45794393
2 100 87.34387283
3 1100 897.9276646
Fair value = 93.45794393 + 87.34387283 + 897.9276646 = 1078.729481
Duration = 1*93.45794393/1078.729481 + 2*87.34387283/1078.729481
3*897.9276646/1078.729481
= 2.745756684
```
Then I was getting stuck. When I try to add up the PV of cash flow at 2.7458 year, the result is not equal to the price of the bond (i.e. $1078.729481)
Can anyone explain (in mathematical sense) why duration is a measure that calculates the time it takes for the price of a bond to be repaid by its internal cash flows , by using the above example? Rigorous proof by formula is also appreciated. Thans!
## Answer by nbbo2 (score 1)
https://quant.stackexchange.com/a/32234
Some of the PV is paid back after 1 year, some after 2 years, and the rest after 3 years. The time weighted average of these three numbers gives you the duration:
Imagine a medical experiment done with rats. 8.66% of the rats lived for one year, 8.09% of the rats lived for 2 years and the rest (83.23%) survived for 3 years. What is the average survival of rats in this exeriment. The answer is 2.74 years. Duration is just the same except with "dollars of present value" instead of rats.
## Answer by Randor (score 0)
https://quant.stackexchange.com/a/32473
i have an excel that i made a while back, that i can neaten up and attach to show you exactly how one can see that it is like a payback period measure , ie , irrespective of what the price is at that future time , assuming you reinvested all coupons back in the bond , and yield curve is flat with only parallel shifts in yield curve , then your investment cost by that time should have been recouped. but , is it at all possible to post my excel here, i dont see how?!
it seems like value at future time @ current yield (y0) = value at future time @ any other yield (future time = current time + duration , value at future time = accumulation of coupons between now and then at y , and pv of coupons between then and maturity at y)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.