Hello,

I have a question regarding bond duration and whether it is possible to achieve a simpler formula such as the one for PV of annuity.

We know that bond duration is based on the coupons that are received prior to maturity of the bond.

To focus my question; How can I sum some numbers that are raised to an exponent and then multiplied by increasing n value of the year?

Example:

We have a 10 year bond with 10% coupon with 1000 face value. Since coupon rate and i are the same, the bond is priced at par ($1000). Using this information, we can determine that the duration of the bond is 6.7585. We get this by adding all of the discounted cash flows using the formula

How can I derive a formula that can give me the duration of 1 dollar for the specified parameters and then multiply by the coupon to give me the duration of the coupons alone (similar to how you can price a bond using the annuity formula x coupon to get the PV of the coupon payments).

The annuity formula I am talking about is

Is it even possible?