A man turns 40 today and wishes to provide supplemental lifetime retirement income of 3,000 at the beginning of each month starting on his 65th birthday. Starting today, he makes monthly contribution of X to a fund for 25 years. The fund earns a nominal rate of 8% compounded monthly. Every 9.65 of lifetime income paid at the beginning of each month starting at age 65 will cost 1,000 to purchase. Calculate x.

To get an income of $1, the man needs [tex]\frac{1000}{9.65}[/tex], therefore to get an income of $3000, the man needs [tex]\frac{1000*3000}{9.65}=310880.83[/tex].

Interest (i)= 8%/12 = 0.08/12 = 0.00667

Number of periods (N) = 12 months/year × 25 years = 300

Using actuarial notation:

[tex]Xs_{300/0.006667}=310880.83\\Where:\\s_{300/0.006667}=(1+0.006667)\frac{(1+0.006667)^{300}-1}{0.00667} =957.366[/tex]

Therefore:

[tex]957.366X=310880.83\\X=\frac{310880.83}{957.366} =324.72[/tex]