# Duration of portfolio equals to zero

I am solving the following problem:

Consider a 2000 dollars bond with maturity of 5 years and a half-year coupon of 25 dollars at a nominal interest rate of 8% p.a and a consolidation bond (eternal annuity) with a half-year coupon of 50 dollars and the same nominal interest rate. Create a portfolio of these two bonds, which will have a zero duration.

So I calculated the price of the first and the second bond and I know that the formula is: $$0=w_{1}D_{1}*w_{2}D_{2}$$

But how should I calculate weights so that it would be equal to zero?

• Do you want the portfolio duration or price to be zero? Also, consider that one of your weights might be negative. – Bob Jansen Dec 23 '19 at 13:44
• @BobJansen I want the portfolio duration to be zero – Daniel Dec 23 '19 at 13:52
• In your formula, does $P$ denote duration or price? – Bob Jansen Dec 23 '19 at 14:01
• @BobJansen P denotes duration I update it – Daniel Dec 23 '19 at 14:26

The equation to be solved should be $$w_1 D_1 + w_2 D_2=0$$ where $$D_1$$ and $$D_2$$ are the respective durations of the two bonds. However you need an investment constraint to fix the values of $$w_1$$ and $$w_2$$. Hence you also need $$w_1 P_1 + w_2 P_2 = \Pi$$ where $$\Pi$$ is the amount invested.
You can then subsitute $$w_1 = - w_2 D_2 /D_1$$ into the second equation to have a solution for both $$w_1$$ and $$w_2$$.
• So I wil got: $-w_{2}\frac{D_{2}}{D_{1}}P_{1}+w_{2}P_{2}=\prod$ and when I calculated P: $-w_{2}\frac{D_{2}}{D_{1}}1565,148+w_{2}1275,51=\prod$ but how would it help me? – Daniel Dec 23 '19 at 14:37
• Just decide on a value for $\Pi$ and solve for $w_2$. I assume you know how to calculate $D_1$ and $D_2$ – Dom Dec 23 '19 at 18:55