Question

A 10-year annuity pays $2,100 per month at the end of each month. If the discount rate is 8 percent compounded monthly for the first seven years and 8 percent compounded monthly thereafter, what is the present value of the annuity?

Answer 1

PV for first 7 years

r=7%

Monthly discount rate=7%/12=0.5833%

n=7*12=84

PVat the end of year 7= A*(1-(1+r)^-n)/r

=2100*(1-(1+0.5833%)^-84)/0.5833%

=2100*(1-1.005833^-84)/0.005833

=2100*(1-0.6135)/0.005833

=2100*0.3865/0.005833

=$139142.1

PV for last 3 years

monthly rate=8%/12=0.67%

n=3*12=36

PV= A*(1-(1+r)^-n)/r

=2100*(1-(1+0.67%)^-36)/0.67%

=2100*(1-1.0067^-36)/0.0067

=2100*(1-0.7863)/0.0067

=2100*0.2137/0.0067

=$66975.35

PV at the end of year 0=66975.35*(1+0.5833%)^-84=66975.35*0.6135=$41090.46

Total PV=139142.1+41090.46= $180232.56