Part $3$ Bond pricing $(25\%)$

Assume that the time of the analysis is $t=0$ and consider a bond market with

four bonds with the following payments and prices, where time is measured in

years:

a$)$ Determine the zero$-$coupon interest rates in the market from the payments

and prices above. Also, price Bond $5,$ with the following payments:

Assume in the parts that follow that the zero$-$coupon term structure can be

approximated by the following specification:

$n_t=\sqrt[\phantom{2}]{t}$

b$)$ Use an initial guess of $=0\mathrm{.}02$ and determine the value of $$ that

minimizes the squared pricing errors of the four original bonds. Price

Bond $5$ with the resulting term structure.

Regardless of the $-$value estimated in part b$),$ use $=0\mathrm{.}03$ and the

specification above, for the term structure below.

c$)$ Plot the term structure $n_t$ for $t=\{1,2,3,4\}$ and add the forward rate term

structure to the plot.

d$)$ Assume you want to invest in Bond $3,$ but that Bond $3$ is very illiquid.

The other three bonds, however, are very often traded, and you decide to

construct a portfolio from the other bonds that mimics Bond $3.$

Form a portfolio of Bond $1,$ Bond $2,$ and Bond $4,$ which have the same

present value, duration and convexity as Bond $3.$

$($not on picture$)$ e$)$ Assume that immediately after you have formed your portfolio part d$),$

the $\backslash$alpha $-$parameter might take values anywhere between $0\mathrm{.}005$ and $0\mathrm{.}06.$

Within this range, determine the highest possible deviation between the

value of the portfolio you created in part d$)$ and the value of Bond $3.$