Purple Industrial just bought supplies from Mountain Sugar. There are 3 payment options. Option A involves paying $99,000 in 4 years. Option B involves paying $144,127 in 8 years. Option C involves paying $173,672 in 10 years. The discount rate is 9.76 percent. What is the present value of the cash flow associated with the option most preferred by Purple Industrial? Be careful with the sign of your answer.

We can calculate the present value (PV) of each payment option using the following formula for present value:

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

Where:
- $FV$ is the future value of the payment.
- $r$ is the discount rate (in decimal form).
- $t$ is the number of years until the payment is due.

### Given Data:
- Discount rate: $r = 9.76\% = 0.0976$
- Option A: Payment of $99,000 in 4 years.
- Option B: Payment of $144,127 in 8 years.
- Option C: Payment of $173,672 in 10 years.

### Step 1: Calculate the PV for Option A
For Option A, we have:
\[
PV_A = \frac{99,000}{(1 + 0.0976)^4}
\]

### Step 2: Calculate the PV for Option B
For Option B, we have:
\[
PV_B = \frac{144,127}{(1 + 0.0976)^8}
\]

### Step 3: Calculate the PV for Option C
For Option C, we have:
\[
PV_C = \frac{173,672}{(1 + 0.0976)^10}
\]

I'll now calculate these present values.The present values (PVs) for each payment option are as follows:

- **Option A**: $68,211.69
- **Option B**: $68,421.50
- **Option C**: $68,436.66

The highest present value corresponds to **Option C**. Therefore, **Option C** is the most preferred by Purple Industrial, with a present value of **$68,436.66**.

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How does changing the discount rate affect the present value of the options?
2. What if the payments were due at different intervals (e.g., semi-annually)?
3. Can you calculate the future value based on a known present value and interest rate?
4. How would inflation influence the decision for selecting the best option?
5. How can net present value (NPV) analysis be applied in comparing these options?

**Tip**: The higher the discount rate, the lower the present value of future cash flows.

Discover how to calculate the present value of three different payment options with a 9.76% discount rate. This example problem is perfect for students studying finance or economics, as it covers essential concepts like the time value of money and discount rate calculations. Learn which option provides the highest present value and is most preferred by Purple Industrial.

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