An obligation can be settled by making a payment of ​$1 comma 0001,000 now and a final payment of ​$13 comma 00013,000 in twotwo years​ (Alternative 1).​ Alternatively, the obligation can be settled by payments of ​$1 comma 3001,300 at the end of every six months for sixsix years​ (Alternative 2). Interest is 99​% compounded semi dash annuallysemi-annually. Compute the present value of each alternative and determine the preferred alternative according to the discounted cash flow criterion.
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Part 1
The present value of Alternative 1 is ​$
  
enter your response here. 
​(Round to the nearest dollar as needed. Round all intermediate values to six decimal places as​ needed.)
Part 2
The present value of Alternative 2 is ​$
  
enter your response here. 
​(Round to the nearest dollar as needed. Round all intermediate values to six decimal places as​ needed.)
Part 3
​Therefore, the best alternative is 
▼ 

To determine the preferred alternative using the discounted cash flow criterion, we need to calculate the present value (PV) of each alternative using the given interest rate of 9% compounded semi-annually. 

### Part 1: Present Value of Alternative 1

**Alternative 1** involves:
- A payment of $1,000 now (immediate payment).
- A final payment of $13,000 in two years.

The formula for calculating the present value (PV) of a future payment is:

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

where:
- $ FV $ = future value
- $ i $ = interest rate per period
- $ n $ = number of compounding periods

Since the interest rate is 9% compounded semi-annually:
- $ i = \frac{9\%}{2} = 4.5\% = 0.045 $ per period.
- The number of periods for the $13,000 payment is $ n = 2 \times 2 = 4 $ periods.

Let's compute the present value of both payments:

1. The $1,000 is already a present payment, so its present value is $1,000.
2. The present value of the $13,000 payment in 2 years:

\[
PV = \frac{13,000}{(1 + 0.045)^4}
\]

### Part 2: Present Value of Alternative 2

**Alternative 2** involves:
- Payments of $1,300 at the end of every six months for six years.

This is an annuity, so we use the Present Value of an Annuity formula:

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

where:
- $ P = 1,300 $ (payment every six months).
- $ i = 0.045 $ (interest rate per period).
- $ n = 6 \times 2 = 12 $ (total number of payments).

### Calculation

Let's calculate the present values of both alternatives to determine which is preferred.### Results:
1. **Present Value of Alternative 1:** \$11,901 (rounded to the nearest dollar)
2. **Present Value of Alternative 2:** \$11,854 (rounded to the nearest dollar)

### Part 3: Best Alternative
Since Alternative 1 has a slightly higher present value, **Alternative 1 is preferred** according to the discounted cash flow criterion.

Would you like more details, or do you have any questions? Here are some related questions:

1. How is the present value formula derived?
2. What impact does changing the interest rate have on the present value?
3. How would the decision change if payments were made more frequently?
4. What is the significance of compounding semi-annually in finance?
5. How would these calculations differ if the interest rate were compounded quarterly?

**Tip:** Always consider the time value of money when comparing financial alternatives; a dollar today is worth more than a dollar tomorrow!

Learn how to calculate the present value (PV) of financial alternatives using compound interest and the time value of money. Includes detailed calculations and comparison of two payment alternatives under different interest rates. Suitable for advanced undergraduate students and financial professionals.

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