A man aged exactly $45$ has decided to invest some money to purchase a deferred annuity from an insurance company. The man plans to invest R $1500$ per annum for $20$ years. The first payment will be made now, with subsequent payments made annually if he is alive. In addition, he will have lump sums of R $3000$ to invest in $5$ and $10$ years time if he is then alive. The man has to choose between a standard whole life annuity and an annuity where the benefits are guaranteed to be paid for at least $10$ years and for life thereafter. Both annuities are level, commencing at age $65$

and are payable annually in arrears.

i$.$ Write down the equation of value for the standard whole life annuity and for the guaranteed annuity that can be bought.

ii$.$ Show that by choosing the guaranteed annuity option, the annuity payable is R $155$ per annum lower than under the standard option.

iii. The policyholder chooses to take the standard annuity option. He pays the premium and reaches age $65.$ Calculate the probability that the policyholder survives long enough so that he receives annuity payments whose nominal amount exceeds the premiums paid.

Basis: Mortality before age $65$: AM$92$ ultimate Mortality from age $65$: PMA$92$C$20$ Interest: $4\%$ p$.$a$.$ effective throughout.