# 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!

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.

• I know it is calculated by weighted average, totally fine with the calculation... my main concern is why 2.74 year the time when price of bond is REPAID BY ITS INTERNAL CASH FLOW. I dont understand why it is totally repaid at that time Feb 3 '17 at 15:27
• In my opinion it is just an imprecise English language way of saying the same thing, that is is a weighted average. Look at the mathematical definition and not the "easy to remember" explanation using imprecise words like "internal cash flow". Already it would be an improvement to refer to "present value" and not "cash flow" (I don't know what "internal cash flow" is). Feb 3 '17 at 15:39
• Yea agree that expression is imprecise and kind of misleading... thanks Feb 3 '17 at 15:44

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)