An investor who pays income tax at $20\%,$ but no CGT buys a $15-$year bond to be redeemed at $105\%,$ bearing semi$-$annual coupons of $10\%pa.$

$($a$)$ Calculate the price pe $$100$ nominal to give a yield of $9\%$ pa effective.

$($b$)$ Just after the $20$th coupon payment, the income tax rate changes to $15\%.$ If the investor holds the bond to redemption, calculate the realised yield on the whole transaction.