Convex combinaton of bonds with different maturity

Question

I am given two bonds, Bond 1 with maturity 30 years and Bond 2 with maturity 2 years. How to find an $\alpha$ such that portfolio $(\alpha$, $1-\alpha)$ of bond a and bond 2 has a duration of 10 years. Here $\alpha$ lies in $(0,1)$. I am confused that as $\alpha$ is given positive and less than 1, how it is possible to cancel out the cashflow of 30 year bond.

Answer 1

While the concept of duration in text books is taught by using the timing of cashflows, it is really best to understand it in mathematical terms. The duration is the sensitivity of the bond price with respect to the change in yield. In mathematical terms, it is the first derivative of the bond price equation with respect to yield.

The portfolio duration is just the weighted average of the duration of the respective bonds in the portfolio. In your case, you have a two bond portfolio thus the duration of your portfolio is:

$$D_{Portfolio} = \alpha * D_{Bond1} + (1-\alpha) * D_{Bond 2} = 10$$

You can easily solve for the weights of Bond1 and Bond2.

The constraints in your case mean that there is no short selling and no leverage.