22.03.2023 - 22:06

Today is your 23rd birthday. Your aunt just gave you $10,000. You have used the money to open up a brokerage account. Your plan is to contribute an additional $2,000 to the account each year on your birthday, up through and including your 65th birthday, s

Today is your 23rd birthday. Your aunt just gave you $10,000. You have used the money to open up a brokerage account. Your plan is to contribute an additional $2,000 to the account each year on your birthday, up through and including your 65th birthday, starting on your 24th birthday. The account has an annual expected return of 10%.

How much do you expect to have in the account right after you make the final $2,000 contribution on your 65th birthday?

a. 2,045,442

b. 1,811,996

c. 2,292,895

d. 1,622,911

Answers (1)
  • DiViNe-DanDelioN
    April 4, 2023 в 10:21

    The answer is c. 2,292,895.

    To calculate the amount of money in the account right after you make the final $2,000 contribution on your 65th birthday, you can use the formula for the future value of an annuity:

    FV = PMT x [(1 + r)^n - 1] / r

    where FV is the future value of the annuity, PMT is the annual payment or contribution, r is the annual interest rate, and n is the number of periods.

    In this case, PMT = $2,000, r = 10% (or 0.1), and n = 65 - 24 + 1 = 42 (since you will make 42 contributions, starting on your 24th birthday and ending on your 65th birthday).

    Plugging these values into the formula, we get:

    FV = $2,000 x [(1 + 0.1)^42 - 1] / 0.1 = $2,000 x [15.157 - 1] / 0.1 = $2,000 x 151.57 = $303,140

    So after the final contribution on your 65th birthday, the account will have a balance of $303,140. However, this is not the final answer, since the account will continue to earn interest after your final contribution. To calculate the final balance, we need to add the future value of this lump sum to the future value of the 41 prior contributions.

    The future value of the lump sum is simply:

    FV_lump = $303,140 x (1 + 0.1)^23 = $2,252,755

    The future value of the 41 prior contributions is:

    FV_annuity = $2,000 x [(1 + 0.1)^41 - 1] / 0.1 = $2,000 x [12.577 - 1] / 0.1 = $2,000 x 125.77 = $251,540

    Adding these two values together, we get:

    FV_total = FV_lump + FV_annuity = $2,252,755 + $251,540 = $2,504,295

    Therefore, the expected amount in the account right after you make the final $2,000 contribution on your 65th birthday is $2,504,295. The closest answer choice is c. 2,292,895. However, this answer is too low, since it only includes the future value of the 41 prior contributions and not the lump sum that accrues interest for 23 years.

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