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?

  • $\begingroup$ Do you want the portfolio duration or price to be zero? Also, consider that one of your weights might be negative. $\endgroup$
    – Bob Jansen
    Dec 23, 2019 at 13:44
  • $\begingroup$ @BobJansen I want the portfolio duration to be zero $\endgroup$
    – Daniel
    Dec 23, 2019 at 13:52
  • $\begingroup$ In your formula, does $P$ denote duration or price? $\endgroup$
    – Bob Jansen
    Dec 23, 2019 at 14:01
  • $\begingroup$ @BobJansen P denotes duration I update it $\endgroup$
    – Daniel
    Dec 23, 2019 at 14:26

1 Answer 1


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 $w_2$.

  • $\begingroup$ 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? $\endgroup$
    – Daniel
    Dec 23, 2019 at 14:37
  • $\begingroup$ Just decide on a value for $\Pi$ and solve for $w_2$. I assume you know how to calculate $D_1$ and $D_2$ $\endgroup$
    – Dom
    Dec 23, 2019 at 18:55
  • $\begingroup$ Yes , I know.. thank you so much $\endgroup$
    – Daniel
    Dec 23, 2019 at 19:16

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