You just won a $12,000 prize at the local talent show. You can receive the money

Question

You just won a $12,000 prize at the local talent show. You can receive the money all up front (t=0), or choose to get 24 monthly payments, with the first payment starting 6 months from now. The annual percentage rate (APR) is 5%.

a. Calculate the monthly payment that makes you indifferent between receiving the money all up front or in monthly installments as described above.

b. The talent show organizers propose another method of payment, where you receive 8 semi-annual payments of $975 each, starting one year from now and 25 weekly payments of $200 starting immediately (t0). Calculate the value of this choice and decide whether this is preferred or not

Explanation / Answer

a APR =5% Half Yearly Rate =2.5% Monthly Rate =5/12=0.4167% PV of Annuity Formula= =A*[(1+k)^n-1)/k(1+k)^n A= Monthly Istallment   n=24 k=0.4167% PV = A*(1.004167^24-1)/0.004167*1.004167^24 PV =22.794 A So PV of the Annuity afte 6 months= 22.794*A Current PV= 22.794*A/1.025= 22.238*A At Indifferent condition : 12000=22.238*A A= 539.62 So Indifferent monthly payment= $      539.62 b Option 1 PV of Annuity after 1 year = =A*[(1+k)^n-1)/k(1+k)^n A =975 per half year n=8 k=2.5% per half year PV =975*(1.025^8-1)/0.025*1.025^8 PV =$6991 So PV after 1 year =6991 Current PV =6991/1.05= $ 6,658.10 So Current PV of the option 1= $ 6,658.10 Option 2 .Weekly Interest rate =5/52= 0.0962% Pv OF Annuity = =A*[(1+k)^n-1)/k(1+k)^n A=200 per week n=25 k=0.0962% per week PV =200*(1.000962^25-1)/0.000962*1.000962^25 PV =4938 So PV of Option 2 =$4938 So the semi annual payment option is better than weekly payment   option, but PV of both Are much   lower Than $12000 So On the spot $12000 is the best   option along with monthly   annuity of $539.62 option for 24 months

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