Question

Find the present value of payments of $250 every six months starting immediately and continuing through 6 years from the present, and $200 every six months thereafter through 12 years from the present. if i^(2) = 4%.

Answer 1

i^2=4%

or, i=4%^0.5 =2%

Since i =2% is the effective annual rate we need the annual rate with semi annual compounding

Hence, 2% = (1+i/2)^2-1

or, (1+i/2)^2=1+0.02

or, 1+i/2 = 1.02^0.5

or, i/2 = 1.0099 -1 = 0.00995

or, i=0.00995*2

or, i=0.0199 or 1.99%

Hence Semi annual rate=1.99%/2=0.995%

Amount of Payment first 6 years =$250

No of payments after year 0 to year 6=12

Total no of Payment =13 (including first payment)

Hence PV of last 12 payments = A*(1-(1+r)^-n)/r

=250*(1-(1+0.995%)^-12)/0.995%

=250*(1-1.00995^-12)/0.00995

=250*(1-0.888)/0.00995

=250*0.112/0.00995

=$2814.66

Hence Total PV including first payment= 2814.65+250 = $3064.66

No of payments after year 6 to year 12=12

Hence PV of 12 payments after year 6 to year 12 at the end of year 6 = A*(1-(1+r)^-n)/r

=200*(1-(1+0.995%)^-12)/0.995%

=200*(1-1.00995^-12)/0.00995

=200*(1-0.888)/0.00995

=200*0.112/0.00995

=$2251.73

Hence PV of $2251.73 at the end of year 0 =2251.73/(1+2%)^6 = 2251.73/1.02^6=2251.73/1.1262=$1999.47

(here 2% rate is used as it is the effective annual rate)

Hence Total PV = 3064.66+1999.47=$5064.13