The state lottery (which has a 6% lottery tax) offers to pay winnings in 25 annual payments or one lump sum,
sometimes called a cash-out option. This week’s lottery has a jackpot of $30 million and
a cash-out value of $18.2 million. Granted that the odds are highly unlikely one would
win, which option should a winner take—annual payments or a lump sum? Why?
Show work and formulas used.
These are the formulas I am choosing between:
annuity: a1=(1-r^n)/(1-r) where a1 is the original amount (30M or 18.2M), r is the rate (.06), and n is the number of years (25).
or future value: FV=PV(1+i/n)^nt where PV is the present value (30M or 18.2M), i is the rate (.06), n is the number of compounds per year, and t is the time (25).
I saw this question when you posted it yesterday. I did not answer it since it wasn't quite clearly stated. Is the 30 million split into 25 equal annual payments?? I will assume that.
In that case we are simply finding the present value of those payments
payment = 30 million/25 = 1200000
r = .06
n = 25
PV = 1200000( 1 - 1.06^-25)/.06
= $15,340,027.37
So I would take the cash-out value of 18.2 million
To determine which option, annual payments or a lump sum, the winner should choose, we need to compare the present value of the annuity payments to the lump sum cash-out value.
Let's calculate it step-by-step:
1. Present Value of Annuity Payments:
Using the formula for the present value of an annuity, a1=(1-r^n)/(1-r), where "a1" represents the original amount, "r" is the rate, and "n" is the number of years, we can calculate the present value of the annuity payments.
For the annual payments, a1=(1-0.06^25)/(1-0.06)
a1 ≈ 16.0152
Considering the original amount is $30 million:
Present value of annuity payments = 16.0152 × $30 million
Present value of annuity payments ≈ $480,456,480
2. Future Value of Lump Sum:
Using the formula for the future value of a lump sum, FV=PV(1+i/n)^nt, where "FV" is the future value, "PV" is the present value, "i" is the rate, "n" is the number of compounds per year, and "t" is the time, we can calculate the future value of the lump sum.
For the lump sum amount of $18.2 million:
Future value of lump sum = $18.2 million × (1 + 0.06/1)^(1 × 25)
Future value of lump sum ≈ $107,420,227.47
Now we can compare the present value of the annuity payments to the future value of the lump sum.
If the present value of the annuity payments is greater than the future value of the lump sum, the winner should choose annual payments. Otherwise, if the future value of the lump sum is greater than the present value of the annuity payments, the winner should choose the lump sum.
Based on the calculations above:
- Present value of annuity payments ≈ $480,456,480
- Future value of lump sum ≈ $107,420,227.47
Since the present value of the annuity payments is significantly greater than the future value of the lump sum, the winner should choose the annual payments option.
To determine which option a winner should choose - annual payments or a lump sum - we will compare the present value of the annuity payments with the cash-out value.
Let's start with calculating the present value of the annuity payments.
Using the formula for an annuity:
a1 = (1 - r^n) / (1 - r)
For the jackpot of $30 million, the original amount (a1) is $30 million, the rate (r) is 6% (0.06), and the number of years (n) is 25.
Plugging in these values, we get:
a1 = (1 - 0.06^25) / (1 - 0.06)
a1 ≈ 12.1981681
So the present value of the annuity payments is approximately $12.198 million.
Now let's calculate the future value of the cash-out option.
Using the formula for future value:
FV = PV(1 + i/n)^(n*t)
For the cash-out value of $18.2 million, the present value (PV) is $18.2 million, the interest rate (i) is 6% (0.06), the number of compounds per year (n) is 1, and the time (t) is 25 years.
Plugging in these values, we get:
FV = 18.2(1 + 0.06/1)^(1*25)
FV ≈ 88.9854099
So the future value of the cash-out option is approximately $88.985 million.
Comparing the present value of the annuity payments ($12.198 million) with the future value of the cash-out option ($88.985 million), it is clear that the cash-out option offers a higher value.
Therefore, if someone wins the lottery, they should choose the lump sum or cash-out option of $18.2 million because it provides a higher immediate value compared to the 25 annual payments.