## 8 AM Class Group 3

Math Project Group 3 Part 1: Chase Chemero, Daniel Hanley, Sonja Henst, Luis Macancela, Maxim Shorey, Thanh Tran Formula Used: Amount in savings not including interest +(( Annuity amount + Interest earned) ˆ (( Annuity amount + Interest earned) * Tax percentage))) = New earned because must per year last year per year last yea Savings amount be taxed New savings amount + (New Savings amount * Interest percentage) = End of year total Justification for formula used: I used the above formula to get my answer because as stated in the instructions you will receive for 20 years a $50000 annuity and receive and interest of 6 % for what has been invest also at the beginning of each year you will be taxed 30 % on not only the annuity payment but also the interest you have accumulated over the past year. So as each step is followed (with the exception to the first) the interest earned over the year is taken out of the savings added to the annuity and then taxed, followed by being added back into saving to be reinvested for a years more of interest this is repeated for 20 years until the 21rst year where you receive no annuity payment and only earn and taxed on your interest as stated in the instructions. Work 50000-(50000*.3)=35000 35000*.3=2100 35000+2100 = 37100 End year 1 total 35000+((50000+2100)-((50000+2100)*.3)))=71470 52100 - 15630 =36470 71470+(71470*.06)=75758.20 End year 2 total 4288.2 71470+((50000+4288.2)-((50000+4288.2)*.3)))=109471.74 54288.2 - 16286.46 = 38001.74 109471.74+(109471.74*.06)=116040.04 End year 3 total 6568.30 109471.74+((50000+6568.30)-((50000+6568.30)*.3)))=149069.55 56568.30 - 16970.49 = 39597.81 149069.55+(149069.55*.06)= 158013.72 End year 4 total 8944.17 149069.55+((50000+8944.17)-((50000+8944.17)*.3)))= 190330.47 58944.17 - 17683.25 = 41260.92 190330.47+(190330.47*.06)= 201750.30 End year 5 total 11419.83 190330.47+((50000+11419.83)-((50000+11419.83)*.3)))=233324.35 61419.83 - 18425.95 = 42993.88 233324.35+(233324.35*.06)= 247323.81 End year 6 total 13999.46 233324.35+((50000+1399.46)-((50000+13999.46)*.3)))= 278123.97 6399.46 - 19199.84 = 44799.62 278123.97+(278123.97*.06) = 294811.41 End year 7 total 16687.44 278123.97+((50000+16687.44)-((50000+16687.44)*.3))) = 324805.18 66687.44 - 20006.23 = 46681.21 324805.18+(324805.18*.06) = 344293.49 End year 8 total 19488.31 324805.18+((50000+19488.31)-((5000+19488.31)*.3))) = 373447 69488.31 - 20846.49 = 48641.82 373447+(373447*.06) = 395853.82 End year 9 total 22406.82 373447+((50000+22406.82)-((50000+22406.82)*.3))) = 424131.77 72406.82 - 21722.05 = 50684.77 424131.77+(424131.77*.06) =449579.68 End year 10 total 25447.91 424131.77+((50000+25447.91)-((50000+25447.91)*.3))) = 476945.31 75447.91 - 22634.37 = 52813.54 476945.31+(476945.31*.06) = 505562.03 End year 11 total 28616.72 476945.31+((5000+28616.72)-((50000+28616.72)*.3))) =531977.01 78616.72 - 23585.02 = 55031.70 531977.01+(531977.01*.06) = 563895.63 End year 12 total 31918.62 531977.01+((50000+31919.62)-((50000+31918.62)*.3)))= 589320.04 81918.62 - 24575.59 = 57343.03 589320.04 + (589320.04*.06) = 624679.24 End year 13 total 35359.20 589320.04 +((50000+35359.20)-((50000+35359.20)*.3))) = 649071.48 65359.20 - 25607.76 = 59751.44 649071.48+(649071.48*.06) = 688015.77 End year 14 total 38944.29 649071.48+((50000+38944.29)-((50000+38944.29)*.3))) = 711332.48 88944.29 - 26683.29 = 62261 711332.48+(711332.48*.06) = 754012.43 End year 15 total 42679.95 711332.48+((50000+42679.95)-((50000+42679.95)*.3))) = 776208.44 92679.95 - 27803.99 = 64875.96 776208.44+(776208.44*.06) = 822780.95 End year 16 total 46572.51 776208.44+((50000+46572.51) ˆ((50000+46572.51)*.3))) = 843809.20 96572.51 - 28971.75 = 67600.76 843809.20+(843809.20*.06) = 894437.75 End year 17 total 50628.55 843809.20+((50000+50628.55)-((50000+50628.55)*.3))) 914249.18 100628.55 - 30188.57 = 70439.98 914249.18+(914249.18*.06) = 969104.13 End year 18 total 54854.95 914249.18+((50000+54854.95)-((50000+5485.95)*.3))) = 987647.64 104854.95 - 31456.49 = 73398.46 987647.64+( 987647.64*.06) = 1046906.50 End year 19 total 54258.86 987647.64+((50000+59258.86)- ((50000+59258.86)*.3))) = 1064128.84 109258.86 - 3277.66 = 76481.20 1064128.84+(1064128.84*.06) = 1127976.57 End year 20 total 63847.73 1064128.84+(63847.73-(63847.73*.3))= 1108822.25 63847.73-19154.32 =44693.41 1108822.25+(1108822*.06)= 1175351.59 End year 21 66529.34 Part 2 Formula used to obtain answer: (Amount in savings + (interest amount - (Interest amount * .03)) = New savings Amount with out interest) from previous year) from previous year) New savings amount+( New savings amount *.06 ) = End of year total Justification for formula used: I used this process for figuring out the lump sum payment under the same conditions as part 1 because your only income is the 1rst and only initial lump sum payment you must pay a huge tax on it which is one in the first part and proceeding after that is that as stated in the instructions after each year you get interest on what you have in your savings account it is only until the first day of the next year are you taxed on you interest ( your only other source of income and there fore must be taxed) But you aren‚t taxed on your interest until the first day of the new year, after which you get to keep the rest of your interest for further investment to gather another years interest to be added to your savings Work for lump sum 1000000-(1000000*.3)=700000 300000 700000+(700000*.06)=742000 End year 1 700000+(42000-(42000*.3))=729400 42000-12600=29400 729400+(729400*.06)=773164 End year 2 43764 729400+(43764-(43764*.3))= 760034.8 43764-13129.2=30634.80 760034.80+(760034.80*.06)= 805636.89 End year 3 45602.09 760034.8+(45602.09-(45602.09*.3))= 791956.26 45602.09-13680.63=31921.46 791956.26+(791956.26*.06)= 839473.64 End year 4 47517.38 791956.26+(47517.38-(47517.38*.3))= 825218.43 47517.38-14255.21=33262.17 825218.43+(825218.43*.06)=874731.54 end year 5 49513.11 825218.43+(49513.11-(49513.11*.3))=859877.61 49513.11-14853.93=34659.11 859877.61+(859877.61*.06)=808284.95 end year 6 51592.66 859877+(51592.66-(51592.66*.3))= 895992.47 51592.66 ˆ 15477.8=36114.86 895992.47+(895992.47*.06) = 949752.02 End year 7 53759.55 895992.47+(53759.55-(5375955 *.3)) =933624.15 53759.55-16127.87=37631.68 933624.15+(933624.15*.06)=989641.60 End year 8 56017.45 933624.50+(56017.45-(56017.45*.3))= 972836.71 56017.45-16805.24= 39212.21 972836.71+(972836.71*.06)= 1031206.91 End year 9 58370.20 972836.71+(58370.20-(58370.20*.3))= 1013695.85 58370.20-17511.06=40859.14 1013695.85+(1013695.85*.06)= 1074517.60 End year 10 60821.75 1013695.85+(60821.75-(60821.75*.3)) = 1056271.08 60821.75-18246.53=42575.23 1056271.08+(1056271.08*.06)=1119647.34 End year 11 63376.26 1056271.08+(63376.26-(63376.26*.3)) = 1100634.46 63376.26-19012.88= 44363.38 1100634.46+(1100634.46*.06) = 1166672.53 End year 12 66038.07 1100634.46+(66038.07-(66038.07*.3)) = 1146861.11 66038.07- 19811.42=46226.65 1146861.11+(114681.11*.06) = 1215672.78 End year 13 68811.67 1146861.11+(68811.67-(68811.67*.3))= 1195029.28 68811.67- 20643.50= 48168.17 1195029.28+(1195029.28*.06) = 1266731.04 End year 14 71701.76 1195029.28+(71701.76-(71701.76*.3)) = 1245220.51 71701.76-21510.53=50191.23 1245220.51+(1245220.51*.06) = 1319933.74 End year 15 74713.23 1245220.51+(74713.23-(74713.23*.3))= 1297519.77 74713.23- 22413.97= 52299.26 1297519.77+(1297519.77*.06)= 1375370.96 End year 16 77851.18 1297519.77+(77851.19-(77851.19*.3))= 1382015.60 77851.16- 23355.36= 54495.83 1382015.60+(1382015.60*.06) = 14631636.54 End year 17 81120.94 1382015.60+(81120.94-(81120.94*.3)) = 1438800.26 81120.94-24336.28= 56784.66 1438800.26+(1438800.26*.06) = 1525128.28 End year 18 86328.01 1438800.26+(86328.01-(86328.01*.3))= 1499229.87 86328.01- 25898.40= 60429.61 1499229.87+( 1499229.87*.06)= 1589183.66 End year 19 89953.79 1499229.87+(89953.79-(89953.79*.3)) 1562197.52 89953.79-26986.14=62967.65 1562197.52+(1562197.52*.06) = 1655929.37 End year 20 93731.85 1562197.52+(93731.85-(93731.85*.3)= 1627809.81 93731.85-28119.56 = 65612.29 1627809.81+( 1627809.81*.06)= 1725478.40 End year 21 97668.59 Part 3 Formula 1000000 X ------------ = -------- Part 2 answer Part 1 answer X: Represents the answer for part 3 which is what the amount for a lump sum payment that would leave the winner with the same amount as the annuity after twenty years. This is set up as proportion to be set up as X = (Part 1 answer) * (1000000) -------------------------------------- (Part 2 answer) This works because the $1000000 relates to the Part 2 answers by $1 million being a lump sum and the Part 2 answer by being a lump sum that is being taxed with interest. The Part 3 answer will relate to the Part 1 answer by what lump sum amount will get you the same answer with taxes and interest as the Part 1 answer for annuity payments. With all the numbers put in it looks like 1000000 X ------------ = ------------- 1725478.40 1127976.57 In the correct format to solve for X is X= (1127976.57) * (1000000) ------------------------------------- 1725478.40 X= 653718.16 X is the lump sum payment that under the same conditions as Part 1 will get you the same amount as Part 1 Part 4 Lump Sum: Pros Receive a larger amount of money at once Higher interest rates per year Greater sum in the end Cons: Overall higher tax payments per year If incorporating a human-psychological factor: with so much money given at once the lower the chances are for smart investments and savings Annuity: Pros Overall lower tax payments per year If incorporating a human-psychological factor a steady increase in money will increase the chances for smart investing and savings Cons: Over all lower savings amount Less money overall per year and in the end Overall lower interest payments per year Common Similarities that many be over looked by the normal person: With out incorporating taxes and interest, you essentially getting the same amount of money whether it is $50,000 for 20 years ( 50,000 * 20 = 700,000) or $700,000 at once The only reason why the lump some payment is larger in the end over the annuity ( regardless of the huge initial tax payment) is because the money left over from the taxes is so large that the interest per year grows faster than with the annuity payments thus resulting in a larger sum. Part 5 There are multiple assumptions made in this project to make the problem simpler to solve The assumptions are as follows: 1. There are no other expenditures ( such as money spent on housing, clothes, food essentially any durable and non-durable goods were not incorporated) made throughout each year 2. There is no other source of income (whether he/ she has another job with a income that can be added and taxed to this savings account other possible winnings from other contests ect.) 3. There are no other investments ( some people have many types of bank accounts in this particular case there is just one savings account, some people have checking account and other investment bank accounts) 4. The interest earned each year is the same ( In reality savings interest earned never stays the same nor is it ever that high) 5. Taxed percentage stays the same ( In reality taxed percentage are subject to change by either the states government and/or federal government) 6. There are only 2 taxes that have to be paid ( in reality there are many other taxes that are paid besides just a federal and states tax, which can include Medicare tax and Social security tax)