The winner of a popular lottery is offered one of two options:
i) a lump sum of $102 500
ii) $1000 every month for 10 years
If the money can be invested at 3.0% p/a, compounded monthly, which option should the winner choose? Justify your reasoning.
Every three months, Carlos deposits $400 in an account bearing 5.6% p/a, compounded quarterly. After 5 years, Carlos stops making regular deposits, but leaves the money in the account for another 2 years. How much money is in the account at the end of the 7 years?
first one:
find present value of second option ...
i = .03/12 = .0025
n = 120
PV = 1000( 1 - 1.025^-120)/.0025
= 103 561.75
So what do you think?
second one:
i = .056/4 = .01625
in 5 yrs, n = 20
amount after 5 years = 400(1.01625^20 - 1)/.01625
= 9364.179
invest that for 2 more years ----->
9364.179(1.01625)^8 = 10 653.06
To determine which option the winner of the lottery should choose, we need to calculate the present value of each option and compare them.
Option i) Lump sum of $102,500:
To calculate the present value, we need to use the formula for the present value of a lump sum:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = interest rate per period
n = number of periods
Given that the money can be invested at 3.0% p/a compounded monthly, we have to adjust the interest rate and periods accordingly:
r = 3.0% / 12 = 0.0025 (monthly interest rate)
n = 12 * 10 = 120 (number of months)
PV = 102500 / (1 + 0.0025)^120
PV ≈ $85,723.56
Option ii) $1000 every month for 10 years:
To calculate the present value, we need to use the formula for the present value of an annuity:
PV = [PMT * (1 - (1 + r)^-n)] / r
Where:
PV = Present Value
PMT = Payment per period
r = interest rate per period
n = number of periods
Given that the money can be invested at 3.0% p/a compounded monthly, we have:
PMT = $1000
r = 3.0% / 12 = 0.0025 (monthly interest rate)
n = 12 * 10 = 120 (number of months)
PV = [1000 * (1 - (1 + 0.0025)^-120)] / 0.0025
PV ≈ $86,454.16
Comparing the present values, we see that the present value for the lump sum option (Option i) is approximately $85,723.56, while the present value for the monthly payment option (Option ii) is approximately $86,454.16. Therefore, the winner of the lottery should choose Option ii, which offers $1000 every month for 10 years. It has a slightly higher present value, indicating a better overall value considering the interest rate.
Now, let's move on to the second question.
To calculate the amount of money in Carlos's account at the end of 7 years, we need to calculate the future value of the regular deposits and the additional 2 years of growth without deposits.
First, let's calculate the future value of the regular deposits after 5 years (60 quarterly deposits):
FV_regular_deposits = PMT * [(1 + r)^n - 1] / r
Where:
FV_regular_deposits = Future Value of regular deposits
PMT = Payment per period
r = annual interest rate divided by the number of periods per year
n = number of periods (quarterly deposits made for 5 years)
PMT = $400
r = 5.6% / 4 = 0.014 (quarterly interest rate)
n = 4 * 5 = 20 (number of quarters)
FV_regular_deposits = 400 * [(1 + 0.014)^20 - 1] / 0.014
FV_regular_deposits ≈ $45,484.84
Next, let's calculate the future value of the account balance after the additional 2 years of growth without deposits:
FV_additional_growth = FV_regular_deposits * (1 + r)^n
Where:
FV_additional_growth = Future Value of additional growth
r = annual interest rate divided by the number of periods per year
n = number of periods (2 years)
r = 5.6% / 4 = 0.014 (quarterly interest rate)
n = 4 * 2 = 8 (number of quarters)
FV_additional_growth = 45484.84 * (1 + 0.014)^8
FV_additional_growth ≈ $50,501.22
Finally, we need to calculate the total amount in the account at the end of 7 years by summing up the two values:
Total_amount = FV_regular_deposits + FV_additional_growth
Total_amount ≈ 45484.84 + 50501.22
Total_amount ≈ $95,985.06
Therefore, at the end of 7 years, there will be approximately $95,985.06 in Carlos's account.