You are managing a portfolio of $$1$ million. Your target duration is $10$ years, and you can invest in two bonds, a zero$-$coupon bond with

maturity of five years and a perpetuity, each currently yielding $5\mathrm{.}2\%.$

Note from Ville: The duration of a perpetual bond $($perpetuity$)$ that pays the same coupon every year into infinity is $D=$

$(1+$YTM$)/$YTM$.$ Next year the duration of the zero$-$coupon bond will be shorter, but the duration of the perpetuity will remain the

same. The idea of this problem is to practice immunization. You can solve the problem like this:

Calculate the duration of a perpetuity.

It is $(1+$YTM$)/$YTM

Write down the following equation:

$w^{**}$ duration of the zero$-$coupon bond $+(1-w)**$ duration of the perpetuity $=$ target duration

Solve what $w$ is$.$

Calculate the percentages $($weights$)$ using the $w$ you just solved:

Then $w=$ the percentage you invest in the zero$-$coupon bond and $1-w=$ the percentage you invest in the perpetuity.

Same procedure works for $a.$ and $b.$

In b$.$ you just need to change the target duration and the duration of the zero$-$coupond bond.

Required:

a$.$ What weight of each bond will you hold to immunize your portfolio?

b$.$ How will these weights change next year if target duration is now nine years?