Stage 2: Choose a Sinking Fund
Using the future value established in #3, compare the following sinking fund options.
Option A: A sinking fund at 2.7% interest compounded monthly.
Option B: A sinking fund at 2.8% interest compounded semiannually.
Option Payment Amount
Show formula and answer
Total Number
of Payments
Total Investment Interest Earned
4.
A
5.
B
6. Based on the total amount invested and the interest earned, which of the two options would you
recommend that Porchlight Coffee move forward with? Explain your reasoning.

To solve this problem, we'll need to use the future value (FV) formula for sinking funds, which accounts for regular payments, the interest rate, and the compounding frequency. Since we're comparing two different options, we'll calculate the payment amount, total investment, and interest earned for each.

### Future Value of a Sinking Fund

The formula to calculate the future value $ FV $ of a sinking fund is:

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

Where:
- $ P $ = payment amount per period
- $ r $ = annual interest rate (as a decimal)
- $ n $ = number of compounding periods per year
- $ t $ = total number of years

We need to rearrange this formula to find the payment amount $ P $:

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

### Information Given:
- **Future Value ($ FV $)**: As established in #3 (we'll assume this value is known; let's denote it as $ FV $).
- **Interest Rates**:
  - Option A: 2.7% compounded monthly
  - Option B: 2.8% compounded semiannually

Let's calculate each option step by step.

### Option A: 2.7% Interest Compounded Monthly

#### Details:
- Annual interest rate, $ r $: 2.7% or 0.027
- Compounding periods per year, $ n $: 12 (monthly)
- Future value, $ FV $: Assume this value is $100,000 (for example)
- Total investment duration, $ t $: Assume 5 years

#### Step 1: Find Payment Amount $ P_A $
\[
P_A = \frac{100,000 \times (0.027/12)}{(1 + 0.027/12)^{12 \times 5} - 1}
\]

Let's calculate this.

### Option B: 2.8% Interest Compounded Semiannually

#### Details:
- Annual interest rate, $ r $: 2.8% or 0.028
- Compounding periods per year, $ n $: 2 (semiannually)
- Future value, $ FV $: $100,000 (example value)
- Total investment duration, $ t $: 5 years

#### Step 2: Find Payment Amount $ P_B $
\[
P_B = \frac{100,000 \times (0.028/2)}{(1 + 0.028/2)^{2 \times 5} - 1}
\]

Now we will proceed with calculations to find the payment amounts, total number of payments, total investment, and interest earned.### Results:

#### Option A (2.7% Interest Compounded Monthly):
- **Payment Amount**: \$1,558.57 (approximately)
- **Total Number of Payments**: $ 12 \times 5 = 60 $
- **Total Investment**: \$93,514.12
- **Interest Earned**: \$6,485.88

#### Option B (2.8% Interest Compounded Semiannually):
- **Payment Amount**: \$9,386.05 (approximately)
- **Total Number of Payments**: $ 2 \times 5 = 10 $
- **Total Investment**: \$93,860.53
- **Interest Earned**: \$6,139.47

### Recommendation:
Based on the calculations, **Option A** is the more favorable choice. Even though it requires more frequent payments (monthly), the total investment is slightly lower (\$93,514.12 vs. \$93,860.53) and it earns more interest (\$6,485.88 vs. \$6,139.47). Therefore, Porchlight Coffee should consider Option A for maximizing returns and minimizing overall costs.

Do you want more details or have any questions?

### Related Questions:
1. How would changing the investment duration affect the total interest earned for each option?
2. What would the results be if the interest rates were swapped between the two options?
3. Can the calculation be adjusted for an increasing payment plan?
4. What effect does increasing the compounding frequency have on the total interest earned?
5. How can we calculate the future value if the payments are made at the beginning of each period instead of the end?

### Tip:
When comparing different compounding frequencies, remember that more frequent compounding generally results in slightly higher interest earned due to more frequent reinvestment.

Stage 2: Choose a Sinking Fund
Using the future value established in #3, compare the following sinking fund options.
Option A: A sinking fund at 2.7% interest compounded monthly.
Option B: A sinking fund at 2.8% interest compounded semiannually.
Option Payment Amount
Show formula and answer
Total Number of Payments
Total Investment Interest Earned
4. A
5. B
6. Based on the total amount invested and the interest earned, which of the two options would you recommend that Porchlight Coffee move forward with? Explain your reasoning.

Learn how to compare two sinking fund options with different compounding frequencies: 2.7% interest compounded monthly versus 2.8% interest compounded semiannually. This problem uses the future value formula for sinking funds to calculate payment amounts, total investment, and interest earned. Ideal for college-level finance or math students.

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