A company has two investment opportunities. Alternative 1 (Alt. 1) pays  $12,000 (inflow) two years from now, and $18,000 (inflow) four years from now.
Alternative 2 (Alt. 2) pays  $7,000 (inflow) at the end of every year for five years.
Interest is 7.88% compounded annually. Which is the preferable alternative? 

To determine which investment alternative is preferable, we need to compare the present value (PV) of each alternative. The present value is the current worth of a future sum of money given a specific rate of return (interest rate). 

The formula for the present value of a future amount is:

\[ PV = \frac{FV}{(1 + r)^n} \]

where:
- $ PV $ = Present Value
- $ FV $ = Future Value
- $ r $ = interest rate (expressed as a decimal)
- $ n $ = number of periods

**Alternative 1:**

For Alternative 1, we have two cash inflows:
1. $12,000 received 2 years from now
2. $18,000 received 4 years from now

First, calculate the present value of each cash inflow:

\[ PV_1 = \frac{12,000}{(1 + 0.0788)^2} \]
\[ PV_2 = \frac{18,000}{(1 + 0.0788)^4} \]

Calculate the values:

\[ PV_1 = \frac{12,000}{(1.0788)^2} = \frac{12,000}{1.16343904} \approx 10,312.38 \]
\[ PV_2 = \frac{18,000}{(1.0788)^4} = \frac{18,000}{1.35808858} \approx 13,256.90 \]

Sum the present values for Alternative 1:

\[ PV_{\text{Alt1}} = 10,312.38 + 13,256.90 = 23,569.28 \]

**Alternative 2:**

For Alternative 2, we have an annuity of $7,000 received at the end of each year for 5 years. The present value of an annuity can be calculated using the formula:

\[ PV = P \times \frac{1 - (1 + r)^{-n}}{r} \]

where:
- $ P $ = annuity payment
- $ r $ = interest rate (expressed as a decimal)
- $ n $ = number of periods

Substitute the given values:

\[ PV_{\text{Alt2}} = 7,000 \times \frac{1 - (1 + 0.0788)^{-5}}{0.0788} \]

Calculate the value:

\[ PV_{\text{Alt2}} = 7,000 \times \frac{1 - (1.0788)^{-5}}{0.0788} \approx 7,000 \times 4.016 \approx 28,112 \]

**Comparison:**

- Present value of Alternative 1: $ 23,569.28 $
- Present value of Alternative 2: $ 28,112 $

Since $ 28,112 $ is greater than $ 23,569.28 $, Alternative 2 is the preferable investment.

Do you have any questions or need further details on the calculations?

Here are 5 numbered questions you might want to ask:
1. How is the present value of a series of cash flows calculated?
2. What is the formula for the present value of an annuity?
3. Can you explain the significance of the interest rate in present value calculations?
4. How does compounding frequency affect the present value of future cash flows?
5. What are some common applications of present value calculations in finance?

**Tip:** Always ensure you are using the correct formula and values for the specific type of cash flow (single lump sum vs. annuity) when calculating present value.

Learn how to compare investment alternatives using present value calculations. This problem involves determining which investment option is preferable based on future cash flows and compound interest rates. Detailed step-by-step solution provided.

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