$($a$)[5$ marks$]$ Suppose that you have won the UK National Lottery. You will receive

the prize money in $20$ installments. The first installment is an immediate payment

of $100,000,$ and then you will receive $19$ yearly payments for that same amount

starting one year after the initial payment.

i$.$ The operator of the National Lottery buys an annuity to cover this. If the

effective annual interest rate is $4\%,$ how much does the operator pay for this

annuity?

ii$.$ If you pay $40\%$ tax on the lottery payments, and inflation averages $6\%$ per

year, then what is the purchasing power of the final payment, after taxes, in

today's pounds?

$($b$)[6$ marks$]$ A bank offers a nominal interest rate of $5\%$ p$.$a$.$ compounded monthly.

One of its clients is planning to retire in $10$ years and has decided to put a fixed

amount $D$ in the bank at the beginning of each of the months before retirement,

so that the client is able to withdraw $1000$ at the beginning of each month for

the following $20$ years. At the time when the client starts to receive the monthly

payments, the nominal interest rate compounded monthly increases to $10\%$ p$.$a$.$

How large does $D$ need to be$?$