# Deriving a simpler formula for bond duration.

• Oct 9th 2012, 09:11 AM
ffezz
Deriving a simpler formula for bond duration.
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 \$\displaystyle ((((CFn/(1+i)^n)/Price*100)*n)/100)\$
\$\displaystyle ((((100/1.1)/1000*100)*1)/100)+((((100/1.1^2)/1000*100)*2)/100) etc\$

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 \$\displaystyle (1-(1/(1+i)^n)/i\$
Is it even possible?
• Oct 9th 2012, 06:47 PM
chiro
Re: Deriving a simpler formula for bond duration.
Hey ffezz.

I recall when I did a first year applied math course that they derived the annuity equations for an nth time period using difference equations like this:

http://www.ziegenbalg.ph-karlsruhe.d...ClosedForm.pdf

Can you outline your model in terms of a difference equation for the value of the annuity at x_(t+1) given the annuities value at x_t along with the other information? This will make it a lot easier to give a more specific answer.
• Nov 4th 2012, 08:19 PM
akilina123
Re: Deriving a simpler formula for bond duration.