Fixed Income - question related to Bond Duration understand, continuous, investment

Bluetick1010

Bond duration is defined as a measure of how long on average the holder of the bond has to wait before receiving cash payments. For a coupon paying bond, I understand this as at what point in entire tenure will I recover my investment. I referred to examples given in books and according to one of them:
consider a 3 Year 10% coupon bond with face value of $100. Suppose that the yeild is 12% per annum with continuous compounding. On computing using Macaulay’s method, I can see that the duration for this bond is 2.653 years. The bond price (computed by summing all cash flows discounted using the yield %) is $94.213.
Now my question is I am not able to prove it mathematically that in 2.653 years I am able to recover 94.213 given all above conditions.
can someone please help?

atate2007

I recommend not wasting time and going too deep. From what ive read and seen, you should view and define bond duration as price sensitivity rather than maturity for simplicity.

wizofoz

roy_cfa wrote:
Now my question is I am not able to prove it mathematically that in 2.653 years I am able to recover 94.213 given all above conditions.
Good, because you shouldn’t be able to. Macaulay’s duration is the weighted average maturity of cash flows. No one said that you’re gonna recover the PV (or FV) of all of your CFs in years.
A brief look at the formula might help you gain some appreciation of what’s going on. You should realize that Macaulay’s is just the sum of the (present values of each cash flow * time (in years) when it is received) scaled by the present value of the bond. It’s like the average age of your CFs.
Notice that you WILL recover everything in years if you’re dealing with a zero-coupon bond. This is because the PV of the single CF will be equal to the PV of the bond (so the ratio=1); multiply that by t, and you get duration=t. Notice, however, that this is just a special case, not the definition of duration.

adehbone

Thank you. This is what I was looking for.