The following information relates to Questions $3$ and $4$

You are managing a portfolio of $$1$ million. Your target duration is $10$ years, and you can choose from two bonds: a zero$-$coupon bond with maturity $5$ years, and a perpetuity, each currently yielding $5\%.$

$3.$ How much of each bond will you hold in your portfolio?

$5-$year zero$-$coupon bond: $$

Perpetuity: $$

$4.$ How will these fractions change next year if target duration is now nine years?

$5-$year zero$-$coupon bond: $$

Perpetuity: $$