lottery offers two options for the prize. (7)
Option A: $1000 a week for life.
Option B: $600 000 in one lump sum.
If you choose Option B, you have the opportunity to place the winnings into an investment that also makes regular payments, at a rate of 3%/a, compounded monthly.
Which option would the winner choose if s/he expects to live for another 50 years?
At what point in time is Option A better than Option B?
I assume we have to ignore taxes, which would be a silly assumption, but anyway ....
We have to compare $600,000 with the present value of the annuity
PV of annuity = 1000( 1 - 1.0025^-600)/.0025
= $310,580.71
what would be your choice?
To determine which option is better, we need to compare the total amount of money received over 50 years for each option.
Option A:
- The winner will receive $1000 per week for life. Since there are 52 weeks in a year, this means they will receive $52,000 per year.
- Over 50 years, the total amount received will be $52,000 x 50 = $2,600,000.
Option B:
- The winner will receive a lump sum of $600,000 initially.
- If they choose to invest the money at a rate of 3% compounded monthly, we can calculate the future value of the investment after 50 years.
- Using the compound interest formula: A = P(1 + r/n)^(nt), where A is the future value, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years, we can calculate the future value.
A = $600,000(1 + 0.03/12)^(12*50)
A ≈ $600,000(1.0025)^600
A ≈ $600,000(4.41759)
A ≈ $2,650,554
Even though Option A guarantees a lower initial amount, it ends up being better in the long run. The winner would choose Option A if they expect to live for another 50 years.
Option A becomes better than Option B when the amount received from Option A exceeds the future value of Option B's investment. In this case, Option A becomes better after 50 years.
To determine which option would be better, we need to compare the total value of Option A over 50 years with the total value of Option B, considering the investment interest rate of 3% compounded monthly.
For Option A, the winner would receive $1000 per week for life, which we assume to be 50 years. Since there are 52 weeks in a year, the total number of weeks is 52 * 50 = 2600 weeks. Therefore, Option A would yield a total value of 2600 * $1000 = $2,600,000 over 50 years.
For Option B, the winner would receive a lump sum of $600,000. If they invest it at a 3% interest rate compounded monthly, they would earn interest on the initial amount over time. To calculate the future value of the investment, we can use the formula for compound interest:
Future Value = Present Value * (1 + (interest rate / number of compounding periods))^(number of compounding periods * number of years)
Let's calculate the future value of the investment in Option B after 50 years:
Future Value = $600,000 * (1 + (0.03 / 12))^(12 * 50)
Using a calculator or spreadsheet, the future value comes out to be approximately $4,458,528.81.
Comparing the total values, we can see that Option B has a higher value than Option A over 50 years - $4,458,528.81 vs. $2,600,000.
Therefore, if the winner expects to live for another 50 years, they would choose Option B as it has a higher total value.
To find the point in time when Option A becomes better than Option B, we need to compare the total values over different time periods. We can use the same calculations as above for various time frames and compare the values. The point at which Option A becomes better than Option B is when the total value of Option A exceeds the total value of Option B.
This calculation depends on the assumptions of the investment interest rate and the time frame to compare. If you provide the specific time frame and interest rate, I can help you calculate the point at which Option A becomes better than Option B.