Jim, a current retiree can purchase an eleven$-$year indexed annuity payable monthly at an effective annual yield of $6\%$ p$.$a$.$ through Challenger Financial Services. Suppose Jim has a superannuation lump sum of $$800,000$ with which to implement an MVE strategy.

$($i$)$ In the first part of the strategy, he purchases an eleven$-$year annuity which pays $$40,000$ p$.$a$.$ payable monthly and indexed at $3\%$ p$.$a$.$ Find the annuity unit cost and hence the total purchase price of this annuity.

$($ii$)$ In the second part of the strategy, assume the net annual return on the Vanguard Accumulation Index fund $($in which the remainder of the lump sum is invested$)$ is $9\mathrm{.}5\%$ p$.$a$.$ compound. Find the expected accumulation of the remainder of the lump sum at the end of the guaranteed income years.