1. An investment offers $4,000 every quarter for 20 years, with the first payment occurring 1 quarter from now. If the required return is 8 percent, what is the value of the invest... 1. An investment offers $4,000 every quarter for 20 years, with the first payment occurring 1 quarter from now. If the required return is 8 percent, what is the value of the investment? 2. An investment offers $4,000 every quarter for 20 years, with the first payment just occurred. If the required return is 8 percent, what is the value of the investment? 3. An investment offers $4,000 every quarter forever. If the required return is 8 percent, what is the value of the investment?
Understand the Problem
The user is asking three related present value questions. All three questions involve calculating the present value of an investment that pays $4,000 quarterly but differ in payment timing and duration. The first question asks for the present value of an ordinary annuity. The second question asks for the present value of an annuity due. The third question asks for the present value of a perpetuity.
Answer
1. Ordinary annuity: \$65,405.80 2. Annuity due: \$66,713.92 3. Perpetuity: \$200,000
Answer for screen readers
- Present value of the ordinary annuity: $65,405.80
- Present value of the annuity due: $66,713.92
- Present value of the perpetuity: $200,000
Steps to Solve
- Present Value of an Ordinary Annuity
The formula for the present value of an ordinary annuity is:
$$ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} $$
Where:
- $PV$ is the present value of the annuity
- $PMT$ is the payment amount per period ($4,000)
- $r$ is the interest rate per period (8% per year compounded quarterly, so $r = 0.08 / 4 = 0.02$)
- $n$ is the number of periods (5 years compounded quarterly, so $n = 5 \times 4 = 20$)
Plug in the values:
$$ PV = 4000 \times \frac{1 - (1 + 0.02)^{-20}}{0.02} $$
Calculate $(1 + 0.02)^{-20}$:
$$ (1.02)^{-20} \approx 0.672971 $$
Subtract this value from 1:
$$ 1 - 0.672971 = 0.327029 $$
Divide by $0.02$:
$$ \frac{0.327029}{0.02} = 16.35145 $$
Multiply by $4000$:
$$ PV = 4000 \times 16.35145 = 65405.80 $$
- Present Value of an Annuity Due
The formula for the present value of an annuity due is:
$$ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r) $$
Using the values from the previous step:
- $PMT = 4000$
- $r = 0.02$
- $n = 20$
We already calculated the present value of the ordinary annuity, so we can just multiply that by $(1 + r)$:
$$ PV = 65405.80 \times (1 + 0.02) $$
$$ PV = 65405.80 \times 1.02 = 66713.92 $$
- Present Value of a Perpetuity
The formula for the present value of a perpetuity is:
$$ PV = \frac{PMT}{r} $$
Where:
- $PV$ is the present value of the perpetuity
- $PMT$ is the payment amount per period ($4,000)
- $r$ is the interest rate per period (8% per year compounded quarterly, so $r = 0.08 / 4 = 0.02$)
Plug in the values:
$$ PV = \frac{4000}{0.02} = 200000 $$
- Present value of the ordinary annuity: $65,405.80
- Present value of the annuity due: $66,713.92
- Present value of the perpetuity: $200,000
More Information
An annuity due is always worth more than an ordinary annuity because the payment is received sooner. A perpetuity represents an infinite series of payments and is greatly impacted by the discount rate $r$.
Tips
A common mistake is using the annual interest rate without converting it to the per-period rate. Also, the number of years should be multiplied by the number of compounding periods per year to get the correct number of periods, $n$. Forgetting to multiply by $(1 + r)$ in Annuity Due is also a common mistake
AI-generated content may contain errors. Please verify critical information
