Question

Tony just won the actuarial lottery. He is allowed to select one of three payout options in order to collect his winnings:(A) $575,000 payable six months from today.(B) $300,000 today plus an additional $300,000 payable two years from today.(C) $20,500 paid at the beginning of each six-month period for 40 years (with the first payment coming six months later).The nominal rate of interest, convertible semiannually, is 8%. Select the option (A, B, or C) that results in the highest present value for Tony. Evaluate all three options and show all work.

Answer 1

Given :
Interest rate =8% pa compounded semi annually.
Effecteive Annual Rate =(1+8%/2)^2-1=8.16% pa
Let us find the PV of the 3 options given to Tony.
Option A.
$575,000 receivable after 6 months (0.5 years)
PV of the payout=575000/1.0816^(0.5)=	$              552,884.62

Option B.
$300,000 receivable today , PV =$300,000
Plus : $300,000 receivable After 2 years.
PV of the second payment =$300,000/1.0816^2=	$              256,441.26
So Total PV of the two payouts=$300,000+$256,441.26=	$              556,441.26

Option C.
We need to find the PV of the $20,500 annuity
for 40 years (80 periods)
Semi annual interest =8.16%/2=4.08% semi annually.
Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]
PV = Present value of Annuity
A = semi annual amount =$20500
k=interest rate=4.08% semi annually
n=periods=80 half years.
PV =20,500*[(1.0408^80-1)/(4.08%*1.0408^80)]
PV =$481,953
So PV of the 40 year annuity of $20,500=$481,953

So option B results in the highest payout for Tony.