```9:20 AM Class Group 4

Group 4: David Sanchez, Leo Carey,

Eric Demmons, Ryan Mcnally

Math Report

We were asked to determine the total amount of two forms of payments made by the Connecticut Lottery to winners of one million dollars. The first payment was an annuity which allows the winner to receive payments of \$50,000 annually for 20 years. Noting that all Federal and Connecticut state income taxes were to be deducting from this prize money yearly; after all a prize is a form of income. Following this, once you have received your payment it was to be invested at an earnings rate of 6% annually, which also has to be taxed. The second form of payment is a Lump Sum payment which allows the winner to claim all the money at one time. Similarly, you must deduct all Federal and Connecticut state income taxes immediately upon receiving the payment and investing it at a 6% earnings rate for 20 years. By splitting these two forms of payments up between our group members, it was concluded that the form of payment that received the most amount in the end of the 20 years was the Lump Sum payment.

Annuity

1.         Starting off with the winning Lottery prize of one million dollars, this will be divided into 20 yearly payments of \$50,000. Again we must remember to deduct all taxes from our prize. After subtracting our Federal income taxes at 25% and Connecticut state income tax at 5% which is a total of \$15,000.00 to taxes. Respectively, once the \$15,000.00 is deducted from the original \$50,000 you are left with only \$35,000.00 payments received to you every year. The money you receive is then invested at a rate of 6% yearly for a period 20 years. By using the future value of annuity formula which is shown below the total value of the annuity comes to \$1,287,495.69.

The future value of an annuity formula: notation used (FVA). It has four variables, each of which can be solved as the fallowing:

1.	FV(A), the value of the annuity at time = n
2.	A, the value of the individual payments in each compounding period

* It would A= \$35,000 with all the taxes deducted

3.	i, the interest rate that would be compounded for each period of time.

* i=1.06. → (being that the .06 is the 6% investment intrest)

4.    n, the number of payment periods.

*In our case it would be for n=20 years

Future Value of Annuity = 35,000 (1.06) ^20 – 1

I

Equation simplified to equal: 35,000 ・ (36.7855912) =

1,287,495.69

Lump Sum

2.         For the lump sum payment the winner received the total amount of the prize money at one time. After deducting our Federal income taxes at 25%, equaling \$250,000 and Connecticut state income tax at 5%, equaling \$50,000 leaves us with a total of \$300,000 worth of taxes paid immediately upon receiving the payment. So our acquisition is as fallows:

Payment

Federal Income 25%

CT State Tax 5%

Total Taxes

Income

Capital

\$ 1,000,000

\$250,000

\$50,000

\$300,000

\$ 700,000

\$700,000

Seeing that our total income is \$700,000 we will multipy this by the 6% investment that will be earned the first year shown below:

700,000 ・ .06= 42,000

Interest 6%

Federal Income 25%

CT State Tax 5%

Total Taxes

Income

Capital

\$42,000

\$10,500

\$2,100

\$12,600

\$29,400

\$729,400

You would add the  income to the \$700,000 and get the Capital of \$729,400. this same process would continue until you have reached the 20th  year. The final numbers would look like this after 20 years:

Interest 6%

Federal Income 25%

CT State Tax 5%

Total Taxes

Income

Capital

\$91,777.44

\$22,944.36

\$4,588.87

\$27,533.23

\$64,244.21

\$1,593,868.25

It is concluded here that the final amount of  the Lump Sum is \$1,593,868.25. Being that the this amount is more then the anuity (\$1,287,495.69 ), makes it clear that the lump sum is the  choice with  the most money in the end.

3.         To further our research on the best choice of reciving the prize money we were also asked to determine the amount for a lump sum payment that would leave the winner with the same amount as the annuity, \$1,287,495.69  after twenty years. This problem was similarly solved the way that we did the annuity formula. In order to calculate the lump sum that will yield a future value of 1,287,495.00, the formula for the future value of a lump sum is re-arranged so that it can be solved for the present value at a rate of 6% over a twenty year period. Therefore, the present value of \$1,287,495.69 is \$401,447.24 after taxes are deducted. In order to find the pre-tax amount, divide the PV by 70%.

Future Value = PV (1+i) ^n

(1+i)^2            (1+i) ^n

Present Value = Future Value = \$1,287,495.69

(1+i)^n              (1.06)^20

Present Value: \$401,447.24

(X)(0.70) = 401,447.24

0.70      0.70

X= 573,496.06 before Taxes

Therefore the present value would be \$401,447.24

Pros and Cons

4.          The first method for receiving the payment was an annuity which allows the winner to receive payments of \$50,000 annually for 20 years. After both the federal and state taxes, and by using the formula for the future value of annuity, the total value of the annuity after 20 years comes to \$1,287,495.69. The other method that can be used for receiving the money is a Lump Sum payment. After, again, both Federal and State taxes and investing it at a 6% earnings rate, the total after 20 years comes to \$1,593,868.25. One pro for the annuity is that it is better for people who like to save their money and plan for their future. It also may prevent people from spending large amounts of their money on certain things. People who choose the Lump Sum can make this mistake and may make some bad investments. But, at the same time, this can actually be a pro for the Lump Sum and a con for the annuity because you may actually want to invest in something that requires a large amount of money all at once and you may not be able to afford it at that time with only the \$50,000 a year. But of all the pros and cons of both the annuity and the Lump Sum, the most important one is the fact that the total for the Lump Sum payment after 20 years is over \$300,000 more than the annuity payments after 20 years.

5.         Some of the ways that our group used to simplify this assignment was by recognizing the money that was being earned in the investment, was not being used during the duration of the 20 years. Also, by using the formulas of the future value of an annuity along with the present value helped us save time and gave us more accuracy on our conclusion.

Lump Sum payment of \$1,000,000 Payment Federal Income 25% CT State Tax 5% Total Taxes Income Capital Acquisition   \$1,000,000.00 \$250,000.00 \$50,000.00 \$300,000.00  \$  700,000.00 \$700,000.00 Interest 6% Federal Income 25% CT State Tax 5% Total Taxes Income Capital Year 1 \$42,000.00 \$10,500.00 \$2,100.00 \$12,600.00 \$29,400.00 \$729,400.00 Year 2 \$43,764.00 \$10,941.00 \$2,188.20 \$13,129.20 \$30,634.80 \$760,034.80 Year 3 \$45,602.09 \$11,400.52 \$2,280.10 \$13,680.63 \$31,921.46 \$791,956.26 Year 4 \$47,517.38 \$11,879.34 \$2,375.87 \$14,255.21 \$33,262.16 \$825,218.42 Year 5 \$49,513.11 \$12,378.28 \$2,475.66 \$14,853.93 \$34,659.17 \$859,877.60 Year 6 \$51,592.66 \$12,898.16 \$2,579.63 \$15,477.80 \$36,114.86 \$895,992.46 Year 7 \$53,759.55 \$13,439.89 \$2,687.98 \$16,127.86 \$37,631.68 \$933,624.14 Year 8 \$56,017.45 \$14,004.36 \$2,800.87 \$16,805.23 \$39,212.21 \$972,836.35 Year 9 \$58,370.18 \$14,592.55 \$2,918.51 \$17,511.05 \$40,859.13 \$1,013,695.48 Year 10 \$60,821.73 \$15,205.43 \$3,041.09 \$18,246.52 \$42,575.21 \$1,056,270.69 Year 11 \$63,376.24 \$15,844.06 \$3,168.81 \$19,012.87 \$44,363.37 \$1,100,634.06 Year 12 \$66,038.04 \$16,509.51 \$3,301.90 \$19,811.41 \$46,226.63 \$1,146,860.69 Year 13 \$68,811.64 \$17,202.91 \$3,440.58 \$20,643.49 \$48,168.15 \$1,195,028.84 Year 14 \$71,701.73 \$17,925.43 \$3,585.09 \$21,510.52 \$50,191.21 \$1,245,220.05 Year 15 \$74,713.20 \$18,678.30 \$3,735.66 \$22,413.96 \$52,299.24 \$1,297,519.29 Year 16 \$77,851.16 \$19,462.79 \$3,892.56 \$23,355.35 \$54,495.81 \$1,352,015.10 Year 17 \$81,120.91 \$20,280.23 \$4,056.05 \$24,336.27 \$56,784.63 \$1,408,799.74 Year 18 \$84,527.98 \$21,132.00 \$4,226.40 \$25,358.40 \$59,169.59 \$1,467,969.33 Year 19 \$88,078.16 \$22,019.54 \$4,403.91 \$26,423.45 \$61,654.71 \$1,529,624.04 Year 20 \$91,777.44 \$22,944.36 \$4,588.87 \$27,533.23 \$64,244.21 \$1,593,868.25

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