To solve this problem, we need to figure out how much Mr. and Mrs. Yates must invest today, when Mr. Yates is 45, so they can receive 10 payments of $10,000 every six months beginning six months after his 60th birthday.
The key is to work backward using the time value of money. We will first determine how much money they need at age 60, then calculate how much they must invest at age 45 to reach that amount.
Step 1: Identify the information given.
- Current age = 45 years
- Retirement age = 60 years
- Time until retirement = 60 - 45 = 15 years
- Payment amount = $10,000
- Number of payments = 10
- Annual interest rate = 10%
- Compounding frequency = Semi-annually (twice per year)
Since interest is compounded every six months, we must use a six-month interest rate.
Semi-annual interest rate = 10% / 2 = 5%
Therefore, the interest rate per period is 0.05.
Step 2: Calculate how much money is needed at age 60.
The couple wants to receive $10,000 every six months for 10 payments.
Since the first payment occurs six months after Mr. Yates turns 60, this is an ordinary annuity.
An ordinary annuity means payments occur at the end of each period rather than at the beginning.
The present value of an ordinary annuity is calculated using:
PV = PMT × [1 - (1 + r)^(-n)] / r
Where:
- PV = Present value of the payments
- PMT = $10,000
- r = 0.05
- n = 10 payments
Substitute the values:
PV = 10,000 × [1 - (1.05)^(-10)] / 0.05
First, calculate:
(1.05)^(-10) ≈ 0.613913
Next:
1 - 0.613913 = 0.386087
Divide by 0.05:
0.386087 / 0.05 ≈ 7.721735
Multiply by $10,000:
PV ≈ 10,000 × 7.721735
PV at age 60 = $77,217.35
This means the couple needs approximately $77,217.35 in their investment account on Mr. Yates' 60th birthday to fund all 10 payments.
Step 3: Calculate how many compounding periods occur between ages 45 and 60.
The couple has 15 years before Mr. Yates turns 60.
Because interest is compounded twice per year:
Number of periods = 15 × 2
Number of periods = 30
Their investment will earn interest for 30 six-month periods.
Step 4: Calculate how much they must invest today.
We know they need $77,217.35 at age 60.
Now we must determine how much money invested today will grow to that amount over 30 periods at 5% per period.
The present value formula is:
PV = FV / (1 + r)^n
Where:
- FV = $77,217.35
- r = 0.05
- n = 30
Substitute:
PV = 77,217.35 / (1.05)^30
Calculate the denominator:
(1.05)^30 ≈ 4.321942
Now divide:
PV ≈ 77,217.35 / 4.321942
PV = $17,866.35
This means they need to invest approximately $17,866.35 today.
Step 5: Explain why this answer makes sense.
Although the couple wants to receive a total of $100,000 in payments, they do not need to invest $100,000 today.
Why?
Because their investment earns compound interest over time.
First, the initial investment grows for 15 years before any withdrawals begin.
Then, even after they start receiving payments, the remaining money continues earning interest during the five-year payment period.
This is why a much smaller initial investment can eventually fund all 10 payments.
Final Answer:
Mr. and Mrs. Yates should invest approximately $17,866.35 today.
At an annual interest rate of 10% compounded semi-annually, this amount will grow to approximately $77,217.35 by Mr. Yates' 60th birthday.
That balance will then be sufficient to provide 10 payments of $10,000 each, beginning six months after his 60th birthday.
The $250,000 lottery prize is more than enough to fund this investment. They only need to set aside approximately $17,866.35 of their winnings, assuming the stated interest rate remains constant and ignoring taxes and fees.
