An investor who is not liable to tax had the choice of purchasing two investments made on $1$ Apr $2018.$

$($A$)$ A $10-$year bond of a nominal of B$100,$ bearing a half$-$yearly coupon of $9\%$ per annum and redeemable at par. The issue price was at $110\%.$

$($B$)$ A $10-$year index linked bond at a price of B$135$ per B$100$ nominal, bearing a half$-$yearly coupon of $4\%$ per annum and redeemable at par. The CPI base figure for indexing is $100\mathrm{.}24$ and the CPI figure applicable to the next coupon $($payable on $1$ Oct $2018)$ is $145\mathrm{.}68.($Here $145\mathrm{.}68$ is the CPI index on $1$ Apr $2018).$

Assume that CPI will grow at a rate of $2\mathrm{.}5\%$ per annum from its latest known value of $145\mathrm{.}68$ on $1$ Apr $2018.$

$($a$)$ Calculate the real rate of return $($yield$)$ per annum earned on both investments A and B$.$

$($b$)$ Determine which of the two investments yielded the highest real rate of return per annum.