Questions: On January 1, 2024, you are considering making an investment that will pay 4 annual payments of 8,000. The first payment is not expected until December 31, 2026. You are eager to earn 5%. What is the present value of the investment on January 1, 2024?
Transcript text: Question 10
2 pts
On January 1, 2024, you are considering making an investment that will pay 4 annual payments of $\$ 8,000$. The first payment is not expected until December 31, 2026. You are eager to earn $5 \%$. What is the present value of the investment on January 1, 2024?
Solution
Solution Steps
To find the present value of the investment, we need to discount each of the future cash flows back to the present value using the given interest rate. The cash flows occur at the end of each year starting from December 31, 2026, so we will discount each payment back to January 1, 2024, using the formula for present value: PV=(1+r)nC, where C is the cash flow, r is the interest rate, and n is the number of years until the cash flow occurs.
Step 1: Identify Cash Flows and Timing
The investment consists of 4 annual payments of C=8000 starting from December 31, 2026. The present value needs to be calculated as of January 1, 2024.
Step 2: Determine Discounting Periods
The cash flows occur at the following times:
Payment 1: n=3 years (December 31, 2026)
Payment 2: n=4 years (December 31, 2027)
Payment 3: n=5 years (December 31, 2028)
Payment 4: n=6 years (December 31, 2029)
Step 3: Calculate Present Value of Each Cash Flow
The present value PV of each cash flow is calculated using the formula:
PV=i=1∑n(1+r)niC
where r=0.05 and ni is the number of years until each payment.
Calculating each term:
For n=3: (1+0.05)38000
For n=4: (1+0.05)48000
For n=5: (1+0.05)58000
For n=6: (1+0.05)68000
Step 4: Sum the Present Values
Calculating the total present value:
PV=(1.05)38000+(1.05)48000+(1.05)58000+(1.05)68000
This results in:
PV≈25730.2531
Final Answer
The present value of the investment on January 1, 2024, is approximately \\(\boxed{25730.25}\\).