An employer establishes a DC pension plan. Consider a male new entrant aged

$30.$ Suppose that this member will survive to age $65$ and retire at that age. The contribution

rate is set using the following assumptions:

$$ At retirement, the employee will use the accumulated value to purchase a life annuitydue.

$$ The target replacement ratio is $60\%.$

$$ Contributions are payable at the end of each year at a fixed percentage of his salary

earned during the year.

$$ Salaries increase by $2\mathrm{.}25\%$ at the beginning of each year.

$$ Contributions earn investment returns of $8\%$ per year.

$$ Annuities $($life annuity$-$due$)$ purchased at retirement are priced assuming an interest

rate of $5\%$ per year and based on $2001$ CSO Mortality Table. You are given that

a$65=11\mathrm{.}21147.$

$($a$)$ Calculate the contribution rate required to meet the target replacement ratio for this

member.

$($b$)$ Calculate the contribution rate required to meet the target replacement ratio for this

member, if the annuity purchased at retirement is a life annuity$-$due with a $10-$year

guarantee period. You are given that a$75=7\mathrm{.}99324$