Yumi's grandparents presented her with a gift of $22,000 when she was 9 years old to be used for her college education. Over the next 8 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 4.5%/year compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 5%/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.)
Well, Yumi sure lucked out with those generous grandparents! Let's crunch some numbers, shall we?
First, we need to calculate how much Yumi's initial investment grew to by the time she turned 17. We use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal (initial investment)
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, Yumi's initial investment (P) is $22,000. The annual interest rate (r) is 4.5%, compounded monthly (n = 12). And she has invested for 8 years (t).
Using the formula, we have:
A = 22000(1 + 0.045/12)^(12*8)
After doing some calculations, we find that Yumi's investment grew to approximately $31,984.65.
Okay, now let's move on to Yumi's withdrawals. She plans on withdrawing equal annual installments over 4 years, starting at age 18. Yumi wants to determine the size of each installment.
We can use the formula for the future value of an ordinary annuity:
A = P * ((1 + r)^n - 1) / r
Where:
A = the total future value of the annuity
P = the size of each annual installment
r = the annual interest rate
n = the number of years
In this case, the total future value (A) is $31,984.65. The annual interest rate (r) is 5%. Yumi plans to make withdrawals over 4 years (n).
We rearrange the formula to solve for P:
P = A * r / ((1 + r)^n - 1)
Plugging in the values, we get:
P = 31984.65 * 0.05 / ((1 + 0.05)^4 - 1)
After some calculations, we find that Yumi should withdraw approximately $8,879.15 each year.
So, Yumi can expect to receive an installment of around $8,879.15. It's not a bad deal for a college fund, huh? Time for Yumi to hit the books!
To find the size of each installment, we need to calculate the future value of the college fund.
Step 1: Calculate the future value of the initial investment and interest earned from age 9 to 17.
Principal (P1) = $22,000
Interest rate (r1) = 4.5% per year
Time (t1) = 8 years
Interest is compounded monthly, so the monthly interest rate (n1) = 4.5%/12 = 0.375% per month
Number of compounding periods (n1) = t1 x 12 = 8 x 12 = 96
Future value (F1) = P1 * (1 + r1/n1)^(n1*t1)
F1 = $22,000 * (1 + 0.045/12)^(12*8)
Step 2: Calculate the future value of the college fund when Yumi reaches age 18.
Interest rate (r2) = 5% per year
Time (t2) = 4 years
Future value (F2) = F1 * (1 + r2)^t2
Step 3: Calculate the size of each installment.
Number of installments (n2) = 4
Size of each installment (S) = F2 / n2
Let's now calculate these values step by step:
Step 1: Calculate F1
F1 = $22,000 * (1 + 0.045/12)^(12*8)
F1 ≈ $22,000 * (1 + 0.00375)^(96)
F1 ≈ $22,000 * (1.00375)^(96)
F1 ≈ $22,000 * (1.4420897)
F1 ≈ $31,725.97
Step 2: Calculate F2
F2 = F1 * (1 + 0.05)^4
F2 ≈ $31,725.97 * (1 + 0.05)^4
F2 ≈ $31,725.97 * (1.05)^4
F2 ≈ $31,725.97 * 1.21550625
F2 ≈ $38,591.69
Step 3: Calculate S
S = F2 / n2
S ≈ $38,591.69 / 4
S ≈ $9,647.92
Therefore, the size of each installment will be approximately $9,647.92.
To find the size of each installment, we first need to calculate the accumulated value of the initial $22,000 after 8 years.
We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = accumulated value
P = principal (initial amount)
r = annual interest rate (as a decimal)
n = number of compounding periods per year
t = number of years
Given:
P = $22,000
r = 4.5% = 0.045 (compounded monthly)
n = 12 (monthly compounding)
t = 8 years
Using this information, we can calculate the accumulated value after 8 years:
A = 22000 * (1 + 0.045/12)^(12*8)
A ≈ $30,672.89
Now, we need to find the equal annual installment over 4 years, starting at age 18. Since the college fund is expected to earn interest at the rate of 5% compounded annually, we can determine the present value of these future installments.
We can use the present value of an annuity formula:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
Where:
PV = present value
PMT = equal annual installment
r = annual interest rate (as a decimal)
n = number of years
Given:
PV = $30,672.89 (accumulated value after 8 years)
r = 5% = 0.05 (compounded annually)
n = 4 years
Substituting the values into the formula:
30672.89 = PMT * ((1 - (1 + 0.05)^(-4)) / 0.05)
Now we can solve for PMT:
PMT = 30672.89 / ((1 - (1 + 0.05)^(-4)) / 0.05)
Calculating this expression will give us the size of each installment.
first find the amount in the fund after the initial 8 years:
Amount = 22000(1 + .045/12)^96
= ....
This will become the "present value" of 4 payments at 5% per annum,
but the payment is made at the beginning of the year. Let that payment be p
p + p(1 - 1.05^-3)/.05 = amount
factor out the p and solve for p
I wrote my answer down, let me know what you got