$6188.88 is the value today of Social Security's promise
Solution:
This is a deferred annuity, since the payments begin in year 45. The value of the annuity in
year 44 is:
[tex]PV_{44}[/tex] = [tex]\frac{c}{r}[/tex] ( 1 - [tex]\frac{1}{(1+r)^{2} }[/tex] )
= [tex]\frac{50,000}{0.10}[/tex] ( 1 - [tex]\frac{1}{(1+0.10)^{18} }[/tex] )
= 500,000 ( 1 - [tex]\frac{1}{5.5599}[/tex] )
= 500,000 ( 0.8202 )
= 410,100
Since this is the value of the annuity in year 44, we need to discount this back 44 years to the
present:
[tex]PV_{0}[/tex] = [tex]\frac{PV_{44}}{(1+r)^{44} }[/tex]
= [tex]\frac{410,100}{(1+0.10)^{44} }[/tex]
= [tex]\frac{410,100}{66.2640}[/tex]
= $6188.88
