Problem $3(10pt)$

A treasury bond has maturity $T=7Y$ and pays at the end of each year $$6\mathrm{.}00$ on a face value of $$100.$ The risk$-$free interest rate for all maturities is $5\mathrm{.}0\%$ with continuous compounding. The bond face value $($notional$)$ is $$100.$

i$)$ compute the price of the bond.

ii$)$ compute the duration and convexity of the bond.

iii$)$ Suppose that the risk$-$free interest rate increases to $6\mathrm{.}0\%.$ Recalculate the bond price and compare with the result of the duration$-$convexity approximation

$P(r+r)=P(r)(1-D(r)*r+\frac{1}{2}C(r)(r{)}^2)$

Compare the bond price obtained from this approximation with the exact bond price.