A fixed interest security was issued on $1$ January in a given year. The security pays half$-$yearly

coupons of $4\%$ per annum. The security is redeemable at $110\%20$ years after issue. An investor

who pays both income tax and capital gains tax at a rate of $25\%$ buys the security on the date of

issue. Income tax is paid on coupons at the end of the calendar year in which the coupon is received.

Capital gains tax is paid immediately on sale or redemption.

$($i$)$ Calculate the price paid by the investor to give a net rate of return of $6\%$ per annum

effective.

$($ii$)$ Calculate the duration of the net payments from the fixed interest security for an

investor who pays income tax as described above but who does not pay capital gains

tax, at a rate of interest of $6\%$ per annum effective.