$7\mathrm{.}2\mathrm{.}1$ Liabilities of $1$ each are due at the ends of periods $1$ and $2.$ There are three securities available to produce asset income to cover these liabilities, as follows:

$($i$)$ A bond due at the end of period $1$ with coupon at rate $1\%$ per period, valued at a periodic yield of $14\%$;

$($ii$)$ A bond due at the end of period $2$ with coupon rate $2\%$ per period, valued at a periodic yield of $15\%$;

$($iii$)$ A bond due at the end of period $2$ with coupon rate $20\%$ per period, valued at a periodic yield of $14\mathrm{.}95\%.$

Determine the cost of the portfolio that exactly matches asset income to li$$bilities due using

$($a$)$ bonds $($i$)$ and $($ii$)$ only.

$($b$)$ bonds $($i$)$ and $($iii$)$ only

$($c$)$ Show that the combination of securities in $($b$)$ minimizes the cost of all exact$-$matching portfolios made up of a combination of the three securities$.$ Note that the minimum cost exactmatching portfolio does not use the highest yielding security in this case.