Maggie wins the lottery and is awarded 500,000 at the beginning of each quarter

Question

Maggie wins the lottery and is awarded 500,000 at the beginning of each quarter for the next 20 years. The government takes 40% of her winnings and she spends 90% of what is left. The remainder is put into a special account with an insurance company that guarantees 7% compounded monthly for the first 10 years and 4% compounded 6 times a year for the following 10 years. At the end of 20 years the accumulated value in the account is used to purchase a perpetuity-immediate with semi-annual payments of X based on a force of interest of 6% per year. FInd X

Explanation / Answer

Actual Lottery 5,00,000 Tax 200000 Remaining 3,00,000 Expenses 270000 Avaibale for Investement 30,000 Number of periods (20 years, 4 quarters) 80 First 40 Payments put into 7% monthly compunding So effective annual rate 0.072290081 Quarterly rate 0.01807252 No of quarters 40 Pmt 30000 Accumulated Value $     17,38,238.48 Now this will give 4% compunded 6 times for the next 10 years Effective annual rate 0.040672622 Time in years 10 PV $     17,38,238.48 Accumulated Value of this first 40 payments after 20 years $     25,89,707.14 Next 40 payment that will give 4% compunded 6 times a year for 10 years Effective annual rate 0.040672622 Quarterly rate 0.010168156 No. of quarters 40 Pmt 30,000 Accumuated value $     14,71,686.58 So Total Accumulated Value $     40,61,393.72 Now there is a purchase of a perpetuity with the accumulated value PV of the perpetuity $     40,61,393.72 Rate 0.06 Hence Annual Payments $       2,43,683.62 Semi Annual Payments (X) will be $       1,21,841.81

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