Tricky Problem

[bobjsure]

Hi guys,

I've been working on a practice problem the last couple hours but for some reason can't decide how best to solve it. The problem is as follows:

"you are given the following two portfolios:

Portfolio A consists of a 1000 par-value 4 year bond with 7% annual coupons and a 5 year zero coupon bond with a par value of X. Both bonds redeem at par."

Portfolio B consists of a single 4 year zero coupon bond with maturity value of 10000

All bonds yield an annual effective rate of 7%. The portfolios both have the same volatility. Find X."



Currently, I've done
(10,000/1.07^4)-1000 = 6628.95
But I'm not even sure I'm headed in the right direction.

Any help is appreciated. Thanks!

[Iñaki Viggers]

This being a thread from several months ago, I presume you found advice elsewhere. But here it goes in the unlikely event that you haven't.

I assume that by volatility you mean duration. Otherwise, please clarify.

The problem states that both portfolios have the same volatility (or, duration). Therefore, each portfolio will be on each side of the equation.

Duration of Portfolio B equals four years because it only consists of a four-year, zero-coupon bond. Check the formula in the aforementioned link to see why: t=4 is factored from the numerator; that leaves the rest of the numerator equal to the denominator, and thus they "cancel".

Applying the Duration formula to Porftolio A leads to the equation:
(70*v+2*70*v²+3*70*v³+4*1070*v⁴+5*X*v⁵ ) / (70*v+70*v²+70*v³+1070*v⁴+X*v⁵)= 4

Solving the equation for X gives the answer.