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).
To determine which option, annual payments or a lump sum, is better for a lottery winner, we need to compare the present value of the annuity payments over 25 years to the lump sum cash-out value.
Let's start by calculating the present value (PV) of the annuity payments. We'll use the formula a1 = (1 - r^n) / (1 - r), where a1 is the original amount (30 million or 18.2 million), r is the rate (0.06), and n is the number of years (25).
For a $30 million annuity:
a1 = (1 - 0.06^25) / (1 - 0.06)
a1 ≈ 12.4178
PV = a1 * 30 million
PV ≈ 12.4178 * 30 million
PV ≈ 372,534,000
For an $18.2 million annuity:
a1 = (1 - 0.06^25) / (1 - 0.06)
a1 ≈ 12.4178
PV = a1 * 18.2 million
PV ≈ 12.4178 * 18.2 million
PV ≈ 224,341,960
The present value (PV) of the annuity payments for $30 million is approximately $372,534,000, and for $18.2 million is approximately $224,341,960.
Next, let's calculate the future value (FV) of the lump sum cash-out value using the formula FV = PV(1 + i/n)^(n*t), where PV is the present value (18.2 million), i is the rate (0.06), n is the number of compounds per year (1), and t is the time (25).
FV = 18.2 million * (1 + 0.06/1)^(1*25)
FV ≈ 18.2 million * (1.06)^25
FV ≈ 18.2 million * 2.9931791
FV ≈ 54,501,564
The future value (FV) of the lump sum cash-out value is approximately $54,501,564.
Comparing the present value (PV) of the annuity payments to the future value (FV) of the lump sum cash-out value:
- If the annuity is $30 million, PV ($372,534,000) is greater than FV ($54,501,564). Therefore, taking the lump sum cash-out option would be financially better.
- If the annuity is $18.2 million, PV ($224,341,960) is also greater than FV ($54,501,564). Thus, the winner should still choose the lump sum cash-out option for the same reasons.
In conclusion, regardless of the original amount, it is better for the lottery winner to choose the lump sum cash-out option ($18.2 million) as it results in a higher overall value compared to the annuity payments over 25 years.