Yumi's grandparents presented her with a gift of $12,000 when she was 10 yr old to be used for her college education. Over the next 7 yr, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 5.5%/year compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 yr, starting at age 18. If the college fund is expected to earn interest at the rate of 6%/year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.)
To find the size of each installment, we need to calculate the future value of the $12,000 Yumi's grandparents gave her, with interest, until she turned 17. Then we will calculate the equal installments using the future value.
Step 1: Calculate the future value of the investment until Yumi turns 17.
Since the interest is compounded monthly, we need to find the future value using the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = future value
P = principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, P = $12,000, r = 5.5% = 0.055 (converted to decimal), n = 12 (compounded monthly), and t = 7 years.
A = 12000(1 + 0.055/12)^(12*7)
A = 12000(1.0045833)^84
A ≈ 12000 * 1.522999835
A ≈ $18,275.99 (rounded to the nearest cent)
Therefore, the future value of the investment when Yumi turns 17 is approximately $18,275.99.
Step 2: Calculate the equal annual installments using the future value.
To find the equal annual installments, we will use the future value and the interest rate she expects to earn after turning 17.
Now, we will use the future value, A = $18,275.99, interest rate (compounded annually) = 6% = 0.06 (in decimal form), and the number of years, t = 4, to calculate the equal annual installments.
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = present value (future value)
PMT = the equal annual installments
r = annual interest rate (in decimal form)
n = number of payments (years)
Substituting the given values:
$18,275.99 = PMT * (1 - (1 + 0.06)^(-4)) / 0.06
Simplifying this equation:
$18,275.99 = PMT * (1 - (1.06)^(-4)) / 0.06
$18,275.99 = PMT * (1 - 0.792094)
Isolating PMT:
PMT = $18,275.99 / (1 - 0.792094)
PMT ≈ $18,275.99 / 0.207906
PMT ≈ $87,860.74 (rounded to the nearest cent)
Therefore, the size of each installment will be approximately $87,860.74.