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!