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?

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: $PV = \frac{C}{(1 + r)^n}$, where $C$ is the cash flow, $r$ is the interest rate, and $n$ is the number of years until the cash flow occurs.

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.

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)

The present value $PV$ of each cash flow is calculated using the formula: $$PV = \sum_{i=1}^{n} \frac{C}{(1 + r)^{n_i}}$$ where $r = 0.05$ and $n_i$ is the number of years until each payment.

Calculating each term:

• For $n = 3$: $\frac{8000}{(1 + 0.05)^3}$
• For $n = 4$: $\frac{8000}{(1 + 0.05)^4}$
• For $n = 5$: $\frac{8000}{(1 + 0.05)^5}$
• For $n = 6$: $\frac{8000}{(1 + 0.05)^6}$

Calculating the total present value: $$PV = \frac{8000}{(1.05)^3} + \frac{8000}{(1.05)^4} + \frac{8000}{(1.05)^5} + \frac{8000}{(1.05)^6}$$ This results in: $$PV \approx 25730.2531$$

Final Answer