Question number 9, page 51, volume 4 curiculum 840, its, understand, product, number

Darien

Can any explain for me why to get par value of the 10-year bond, we take dellar duration of the 5-year ($454,840) divide by the product of the duration of the 10-year and its quoted price and 0.01 to get the par value of the 10-year.
I don’t not really understand the logic behid doing this.
Thanks,

Sunshine4ever

dollar duration = duration * market value* 0.01
portfolio dollar duration is the sum of dollar duration of all the bonds in it.
If you sell the 5 year bonds and invest in 10 years yet maintain the dollar duration of the portfolio, then :
dollar duration of the 10 year bonds purchased = dollar duration of the 5 year bonds sold.
The market value of 5 year bonds in the portfolio = price * par value = 10,040,625
So the dollar duration of 5 year bonds = duration * market value * 0.01 = 4.53 * 100,406.25=454,840
Now to get par value for 10 year bonds to be purchased , reverse the calculation with new price and duration for 10 year bonds
duration * market Value * 0.01 = 454,840 = 8.22 * Par Value * 1.0909375 * 0.01
Par value = notional/duration/Price/0.01
so Par value = 453,840 / 8.22 /  1.0909375 / 0.01 = 5,072,090