## 8 AM Class Group 1

Lindsay Bissonnette James Gerosa Alyssa Huddleston Jennifer Rosa Amy Tran Math 1070 8:00 6 October 2008 Lottery Annuities There are numerous games in the Connecticut lottery, each of which would be considered a scam if run by a private business. They all do pay out prize money, but the vast majority of bettors wind up losing. Winners of large prizes are given the option of being paid their winnings as a lump sum or in the form of an annuity. The lump sum is considerably less than the advertised winnings; the annuity consists of payments made annually until the advertised amount won is distributed. Consider a nominal prize of one million dollars, to be paid in twenty annual payments of $50,000. Assume when each payment is made, the recipient immediately pays Federal income tax at a 25% rate, Connecticut state income tax at a 5% rate and invests the rest. Assume the funds invested earn 6% per year, with both Federal and Connecticut state income taxes paid from that income and the remainder reinvested. 1. The total amount the lottery recipient will have at the end of twenty-one years being paid $50,000 annual payments for twenty years, making 6% interest and 30% taxes deducted is $1,155,392.82. To find this value, we used a spread sheet for our calculations to come to our answer. The first year payment, as seen in row one, of $50,000.00 gets 30% taxes taken out. By multiplying $50,000.00 by .70, .70 comes from 100% minus 30% to get the remaining amount of money that will be left over once the taxes are taken out, which is $35,000.00. The $35,000.00 then gets multiplied by .06 for the interest that is made from the money in the account. Then to find the taxes taken from the interest made you multiply the amount of interest made, $2,100.00, by .70 to find the net interest of the first payment is $1,470.00. Then to find the Net total of dollars for the end of year one, the total net interest is added to the net payment to get $36,470.00. This amount then gets added in to the net payment of $35,000.00 for year 2. This accumulated amount then gets 30% taxes taken out and then receives 6% interest. The 6% interest made is then taxed by 30%. The remaining amount of interest after tax then gets added into the total dollars. The total dollars then gets accumulated. Then once the next payment is received the cycle continues. After the 20 years the amount is $1,108,822.28. When this amount is untouched for another year, after that year the money grows to $1,115,392.82. See Spread Sheet. 1 Years ACCUMULATOR Payments AFTERTX NET PAYMENT TOTAL DOLLARS INTEREST NET INT. NET 1 $50,000.00 $ 0.70 $ 35,000.00 $ 35,000.00 $ 2,100.00 $ 1,470.00 $ 36,470.00 2 $ 36,470.00 $50,000.00 $ 0.70 $ 35,000.00 $ 71,470.00 $ 4,288.20 $ 3,001.74 $ 74,471.74 3 $ 74,471.74 $50,000.00 $ 0.70 $ 35,000.00 $ 109,471.74 $ 6,568.30 $ 4,597.81 $ 114,069.55 4 $ 114,069.55 $50,000.00 $ 0.70 $ 35,000.00 $ 149,069.55 $ 8,944.17 $ 6,260.92 $ 155,330.47 5 $ 155,330.47 $50,000.00 $ 0.70 $ 35,000.00 $ 190,330.47 $ 11,419.83 $ 7,993.88 $ 198,324.35 6 $ 198,324.35 $50,000.00 $ 0.70 $ 35,000.00 $ 233,324.35 $ 13,999.46 $ 9,799.62 $ 243,123.98 7 $ 243,123.98 $50,000.00 $ 0.70 $ 35,000.00 $ 278,123.98 $ 16,687.44 $ 11,681.21 $ 289,805.18 8 $ 289,805.18 $50,000.00 $ 0.70 $ 35,000.00 $ 324,805.18 $ 19,488.31 $ 13,641.82 $ 338,447.00 9 $ 338,447.00 $50,000.00 $ 0.70 $ 35,000.00 $ 373,447.00 $ 22,406.82 $ 15,684.77 $ 389,131.78 10 $ 389,131.78 $50,000.00 $ 0.70 $ 35,000.00 $ 424,131.78 $ 25,447.91 $ 17,813.53 $ 441,945.31 11 $ 441,945.31 $50,000.00 $ 0.70 $ 35,000.00 $ 476,945.31 $ 28,616.72 $ 20,031.70 $ 496,977.01 12 $ 496,977.01 $50,000.00 $ 0.70 $ 35,000.00 $ 531,977.01 $ 31,918.62 $ 22,343.03 $ 554,320.05 13 $ 554,320.05 $50,000.00 $ 0.70 $ 35,000.00 $ 589,320.05 $ 35,359.20 $ 24,751.44 $ 614,071.49 14 $ 614,071.49 $50,000.00 $ 0.70 $ 35,000.00 $ 649,071.49 $ 38,944.29 $ 27,261.00 $ 676,332.49 15 $ 676,332.49 $50,000.00 $ 0.70 $ 35,000.00 $ 711,332.49 $ 42,679.95 $ 29,875.96 $ 741,208.46 16 $ 741,208.46 $50,000.00 $ 0.70 $ 35,000.00 $ 776,208.46 $ 46,572.51 $ 32,600.76 $ 808,809.21 17 $ 808,809.21 $50,000.00 $ 0.70 $ 35,000.00 $ 843,809.21 $ 50,628.55 $ 35,439.99 $ 879,249.20 18 $ 879,249.20 $50,000.00 $ 0.70 $ 35,000.00 $ 914,249.20 $ 54,854.95 $ 38,398.47 $ 952,647.67 19 $ 952,647.67 $50,000.00 $ 0.70 $ 35,000.00 $ 987,647.67 $ 59,258.86 $ 41,481.20 $1,029,128.87 20 $ 1,029,128.87 $50,000.00 $ 0.70 $ 35,000.00 $ 1,064,128.87 $ 63,847.73 $ 44,693.41 $1,108,822.28 21 $ 1,108,822.28 $ 1,108,822.28 $ 66,529.34 $ 46,570.54 $1,155,392.82 Totals $ 650,561.17 $455,392.82 2. The amount the winner will have at the end of twenty years when the winner receives the whole $1,000,000.00 in one payment is $1,593,868.25. To find this we used a spread sheet to organize our calculations of 30% tax and 6% interest on the money every year for 20 years. The net dollars after the original tax on the million is $700,000.00. This then makes 6% interest which is $42,000.00 in the first year. The tax then is taken out of the $42,000.00 by multiplying by .70, the inverse, to get the $29,400.00 net interest. This is then added to the $700,000.00 to get $729,400.00. This then gets accumulated into net dollars and interest is made and the taxes are deducted for the next 20 years to get the final total of dollars to be $1,593,868.25. See Spread Sheet 2 Payment After Tax Net Payment Net Dollars Interest Net Interest Net $1,000,000.00 0.7 $ 700,000.00 $ 700,000.00 $ 42,000.00 $ 29,400.00 $ 729,400.00 $ - 0.7 $ - $ 729,400.00 $ 43,764.00 $ 30,634.80 $ 760,034.80 $ - 0.7 $ - $ 760,034.80 $ 45,602.09 $ 31,921.46 $ 791,956.26 $ - 0.7 $ - $ 791,956.26 $ 47,517.38 $ 33,262.16 $ 825,218.42 $ - 0.7 $ - $ 825,218.42 $ 49,513.11 $ 34,659.17 $ 859,877.60 $ - 0.7 $ - $ 859,877.60 $ 51,592.66 $ 36,114.86 $ 895,992.46 $ - 0.7 $ - $ 895,992.46 $ 53,759.55 $ 37,631.68 $ 933,624.14 $ - 0.7 $ - $ 933,624.14 $ 56,017.45 $ 39,212.21 $ 972,836.35 $ - 0.7 $ - $ 972,836.35 $ 58,370.18 $ 40,859.13 $ 1,013,695.48 $ - 0.7 $ - $ 1,013,695.48 $ 60,821.73 $ 42,575.21 $ 1,056,270.69 $ - 0.7 $ - $ 1,056,270.69 $ 63,376.24 $ 44,363.37 $ 1,100,634.06 $ - 0.7 $ - $ 1,100,634.06 $ 66,038.04 $ 46,226.63 $ 1,146,860.69 $ - 0.7 $ - $ 1,146,860.69 $ 68,811.64 $ 48,168.15 $ 1,195,028.84 $ - 0.7 $ - $ 1,195,028.84 $ 71,701.73 $ 50,191.21 $ 1,245,220.05 $ - 0.7 $ - $ 1,245,220.05 $ 74,713.20 $ 52,299.24 $ 1,297,519.29 $ - 0.7 $ - $ 1,297,519.29 $ 77,851.16 $ 54,495.81 $ 1,352,015.10 $ - 0.7 $ - $ 1,352,015.10 $ 81,120.91 $ 56,784.63 $ 1,408,799.74 $ - 0.7 $ - $ 1,408,799.74 $ 84,527.98 $ 59,169.59 $ 1,467,969.33 $ - 0.7 $ - $ 1,467,969.33 $ 88,078.16 $ 61,654.71 $ 1,529,624.04 $ - 0.7 $ - $ 1,529,624.04 $ 91,777.44 $ 64,244.21 $ 1,593,868.25 Totals $1,276,954.64 $ 893,868.25 3. By doing a trial and error process, I found that the lump sum payment that will receive almost exactly the same amount of money at the end of 20 years as the annuity after twenty years is $724,898.58. This was done by guessing and checking the answer through the calculations on the excel sheet to find interest additions and tax deductions to lead to the lump sum payment amount. See Spread Sheet 3 Lump Sum Payment After Tax Net Payment Interest Interest Taxes Net Interest Net $ 724,898.58 0.7 $ 507,429.01 $ 0.06 $30,445.74 0.70 $ 21,312.02 $ 528,741.02 $ - 0.7 $ 528,741.02 $ 0.06 $31,724.46 0.70 $ 22,207.12 $ 550,948.15 $ - 0.7 $ 550,948.15 $ 0.06 $33,056.89 0.70 $ 23,139.82 $ 574,087.97 $ - 0.7 $ 574,087.97 $ 0.06 $34,445.28 0.70 $ 24,111.69 $ 598,199.66 $ - 0.7 $ 598,199.66 $ 0.06 $35,891.98 0.70 $ 25,124.39 $ 623,324.05 $ - 0.7 $ 623,324.05 $ 0.06 $37,399.44 0.70 $ 26,179.61 $ 649,503.66 $ - 0.7 $ 649,503.66 $ 0.06 $38,970.22 0.70 $ 27,279.15 $ 676,782.81 $ - 0.7 $ 676,782.81 $ 0.06 $40,606.97 0.70 $ 28,424.88 $ 705,207.69 $ - 0.7 $ 705,207.69 $ 0.06 $42,312.46 0.70 $ 29,618.72 $ 734,826.42 $ - 0.7 $ 734,826.42 $ 0.06 $44,089.58 0.70 $ 30,862.71 $ 765,689.12 $ - 0.7 $ 765,689.12 $ 0.06 $45,941.35 0.70 $ 32,158.94 $ 797,848.07 $ - 0.7 $ 797,848.07 $ 0.06 $47,870.88 0.70 $ 33,509.62 $ 831,357.69 $ - 0.7 $ 831,357.69 $ 0.06 $49,881.46 0.70 $ 34,917.02 $ 866,274.71 $ - 0.7 $ 866,274.71 $ 0.06 $51,976.48 0.70 $ 36,383.54 $ 902,658.25 $ - 0.7 $ 902,658.25 $ 0.06 $54,159.49 0.70 $ 37,911.65 $ 940,569.89 $ - 0.7 $ 940,569.89 $ 0.06 $56,434.19 0.70 $ 39,503.94 $ 980,073.83 $ - 0.7 $ 980,073.83 $ 0.06 $58,804.43 0.70 $ 41,163.10 $ 1,021,236.93 $ - 0.7 $1,021,236.93 $ 0.06 $61,274.22 0.70 $ 42,891.95 $ 1,064,128.88 $ - 0.7 $1,064,128.88 $ 0.06 $63,847.73 0.70 $ 44,693.41 $ 1,108,822.29 $ - 0.7 $1,108,822.29 $ 0.06 $66,529.34 0.70 $ 46,570.54 $ 1,155,392.83 4. Lump Sum Payments- Pros: The initial payment is more. You accrue more interest. You end up with more money in the end. Cons: More taxes are taken out. Annuities- Pros: You don‚t spend all your money at once. You don‚t accrue as much interest. Cons:Make less money in the end of the 20 years compared to the lump sum payment. 5. Some assumptions that we made were that this person did not spend any money over the 20 year period. Also we didn‚t take into account inflation. Another assumption we made was that the person used only one bank account. They could have put the money into different accounts with different interest rates. We also made the assumption that the interest rates didn‚t change during the 20 years. We also assume for simplistic purposes that you receive the payment on January first of each year and the interest is totaled and collected on December thirty-first.