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Math  /  Algebra

Question11 Given $250,000\$ 250,000 today, determine the equivalent series of 10 annual payments which could be generated beginning in 1 year. Assume interest is 12 percent compounded annually.

Studdy Solution

STEP 1

1. The present value of the investment is \$250,000.
2. The interest rate is 12% compounded annually.
3. The series of payments is an annuity with 10 annual payments.
4. Payments begin in 1 year.

STEP 2

1. Identify the formula for the present value of an annuity.
2. Define the variables in the formula.
3. Substitute the known values into the formula.
4. Solve for the annual payment.

STEP 3

Identify the formula for the present value of an annuity. The formula is:
PV=P×1−(1+r)−nr PV = P \times \frac{1 - (1 + r)^{-n}}{r}
where PV PV is the present value, P P is the annual payment, r r is the annual interest rate, and n n is the number of payments.

STEP 4

Define the variables in the formula.
- PV=250,000 PV = 250,000 - r=0.12 r = 0.12 (12% interest rate) - n=10 n = 10 (10 annual payments)

STEP 5

Substitute the known values into the formula.
250,000=P×1−(1+0.12)−100.12 250,000 = P \times \frac{1 - (1 + 0.12)^{-10}}{0.12}

STEP 6

Solve for the annual payment P P .
First, calculate (1+0.12)−10 (1 + 0.12)^{-10} :
(1.12)−10≈0.3220 (1.12)^{-10} \approx 0.3220
Next, calculate the annuity factor:
1−0.32200.12≈0.67800.12≈5.6500 \frac{1 - 0.3220}{0.12} \approx \frac{0.6780}{0.12} \approx 5.6500
Now, solve for P P :
250,000=P×5.6500 250,000 = P \times 5.6500
P=250,0005.6500 P = \frac{250,000}{5.6500}
P≈44,247.79 P \approx 44,247.79
The equivalent series of 10 annual payments is approximately:
44,247.79 \boxed{44,247.79}

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