Problem $3(25$ points$)$

Suppose an IROR curve is constructed from the internal rates of return $r_i(i=1,$dots,$30)$ of

$30$ bonds not necessarily priced at par; bond i pays a fixed positive coupon $c_i$ each year and

matures in year $i,$ paying back $100$ in addition to the coupon at maturity. Prove that when

the IROR curve is upward$-$sloping and positive, the zero curve bootstrapped from this IROR

curve sits above the IROR curve, i$.$e $z_i{r}_i$ where $z_i$ is the bootstrapped zero rate for year $i.$

Use discrete compounding. Hint: use induction on the zero curve.