Math Problem Statement

An investment offers $9,600 per year for 16 years, with the first payment occurring one year from now. Assume the required return is 12 percent.

What is the value of the investment today?

Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

What would the value be if the payments occurred for 41 years?

Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

What would the value be if the payments occurred for 76 years?

Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

What would the value be if the payments occurred forever?

Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

Solution

To calculate the present value of the investment, we are dealing with an annuity. Since the payments occur annually and the first payment is one year from now, we will use the formula for the present value of an ordinary annuity:

PV=C×(11(1+r)t)÷rPV = C \times \left( 1 - \frac{1}{(1 + r)^t} \right) \div r

Where:

  • PVPV = present value of the annuity
  • CC = annual payment ($9,600)
  • rr = discount rate (12% or 0.12)
  • tt = number of periods (16, 41, 76, or forever)

1. Present value for 16 years:

PV16=9600×(11(1+0.12)16)÷0.12PV_{16} = 9600 \times \left( 1 - \frac{1}{(1 + 0.12)^{16}} \right) \div 0.12

2. Present value for 41 years:

PV41=9600×(11(1+0.12)41)÷0.12PV_{41} = 9600 \times \left( 1 - \frac{1}{(1 + 0.12)^{41}} \right) \div 0.12

3. Present value for 76 years:

PV76=9600×(11(1+0.12)76)÷0.12PV_{76} = 9600 \times \left( 1 - \frac{1}{(1 + 0.12)^{76}} \right) \div 0.12

4. Present value if payments occurred forever:

For perpetual payments (a perpetuity), the present value is calculated using the perpetuity formula:

PV=Cr=96000.12PV_{\infty} = \frac{C}{r} = \frac{9600}{0.12}

Let me calculate each value.It seems like I can’t do more advanced data analysis right now. Please try again later.

Feel free to upload any additional questions, and I’ll be happy to assist once things are working properly!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Finance
Present Value
Annuities
Perpetuities

Formulas

PV = C × [(1 - 1 / (1 + r)^t) / r]
PV_perpetuity = C / r

Theorems

Present Value of an Ordinary Annuity
Present Value of a Perpetuity

Suitable Grade Level

College-Level Finance