Let's solve the problem **step by step**, following the yellow-highlighted blanks. ## **a) Effective Annual Rate (EAR)** ### **Given:** - **Notional purchase = $1,000** - **Discount (%) = 0.25%** - **Days difference = 8 days** (20 days - 12 days) --- ### **Step 1: Calculate Discount ($)** \[ \text{Discount (\$)} = \text{Notional purchase} \times \text{Discount (\%)} \] \[ \text{Discount (\$)} = \$1,000 \times 0.25\% = \$1,000 \times 0.0025 = \$2.50 \] --- ### **Step 2: Calculate Rate (%)** The implied rate is the discount *not* taken over the cash paid for the 8-day period: \[ \text{Rate (\%)} = \frac{\text{Discount (\$)}}{\text{Amount Paid}} \] \[ \text{Amount Paid} = \$1,000 - \$2.50 = \$997.50 \] \[ \text{Rate} = \frac{\$2.50}{\$997.50} \approx 0.25\% \] But, the correct way is: the discount is for paying 8 days early, so the periodic rate is: \[ \text{Periodic Rate} = \frac{\text{Discount \%}}{1 - \text{Discount \%}} \] \[ = \frac{0.25\%}{1 - 0.25\%} = \frac{0.0025}{0.9975} \approx 0.251\% \] --- ### **Step 3: Days difference in 1 year** \[ \text{Days difference in 1 year} = \frac{365}{8} = 45.625 \] --- ### **Step 4: Calculate EAR** \[ \text{EAR} = \left(1 + \text{Periodic Rate}\right)^{n} - 1 \] Where \( n = \frac{365}{8} \approx 45.625 \) \[ \text{EAR} = (1 + 0.00251)^{45.625} - 1 \] \[ \text{EAR} \approx (1.00251)^{45.625} - 1 \approx 1.122 - 1 = 0.122 = **12.2\%** \] --- ### **Summary Table (a):** | | Value | |------------|--------------| | Discount ($)| $2.50 | | Rate (%) | 0.251% | | Days diff in 1 year | 45.625 | | **EAR** | **12.2%** | --- ## **b) Average Collection Period** ### **Given:** - Gross revenue = $16,000,000 - % paying early = 30% - Cost of capital = 7% - Gross margin = 30% --- ### **Step 1: Avg. receivables before new policy** Assume average collection is 20 days (since no discount). \[ \text{Avg Receivables (before)} = \text{Gross Revenue} \times \frac{20}{365} \] \[ = 16,000,000 \times \frac{20}{365} = 16,000,000 \times 0.0548 = \$876,712 \] --- ### **Step 2: Avg. receivables after new policy** Now, 30% pay in 12 days and 70% pay in 20 days: \[ \text{Avg Collection Period} = (30\% \times 12) + (70\% \times 20) = (0.3 \times 12) + (0.7 \times 20) = 3.6 + 14 = 17.6 \text{ days} \] \[ \text{Avg Receivables (after)} = 16,000,000 \times \frac{17.6}{365} = 16,000,000 \times 0.04822 = \$771,452 \] --- ### **Step 3: Change in receivables** \[ \text{Change} = \$771,452 - \$876,712 = -\$105,260 \] --- ### **Step 4: Projected savings in capital costs** \[ \text{Savings} = |\text{Change}| \times \text{Cost of capital} \] \[ = \$105,260 \times 7\% = \$7,368 \] --- ### **Step 5: Less Discounts given** 30% take the discount: \[ \text{Annual Discount} = \text{Gross Revenue} \times 30\% \times 0.25\% \] \[ = \$16,000,000 \times 0.3 \times 0.0025 = \$16,000,000 \times 0.00075 = \$12,000 \] --- ### **Step 6: Projected savings net of discounts** \[ \text{Net Savings} = \$7,368 - \$12,000 = -\$4,632 \] --- ### **Step 7: Gross revenues must rise by** To offset the loss, gross margin is 30%: \[ \text{Required Increase in Revenue} = \frac{|\text{Net Savings}|}{\text{Gross Margin}} \] \[ = \frac{4,632}{0.3} = \$15,440 \] \[ \text{In percent} = \frac{15,440}{16,000,000} \times 100 = 0.0965\% \] --- ## **Final Answers (filled in yellow):** | a) Effective Annual Rate (EAR) | b) Average Collection Period | |---------------------------------------|-------------------------------------| | Notional purchase: $1,000.00 | Gross revenue: $16,000,000 | | Discount ($): $2.50 | Avg. receivables before: $876,712 | | Rate (%): 0.251% | % paying early: 30% | | Days difference in 1 year: 45.625 | Avg. receivables after: $771,452 | | EAR: **12.2%** | Change in receivables: -$105,260 | | | Cost of capital: 7% | | | Projected savings in capital costs: $7,368 | | | minus: discounts: $12,000 | | | Projected savings net of discounts: -$4,632| | | Gross margin: 30% | | | Gross revenues must rise by: | | | - in dollars: $15,440 | | | - in percent: 0.0965% | --- ## **Summary Table for Submission** | a) Effective Annual Rate (EAR) | b) Average Collection Period | |-------------------------------|-----------------------------------| | Discount ($): $2.50 | Avg. receivables before: $876,712 | | Rate (%): 0.251% | Avg. receivables after: $771,452 | | Days diff in 1 year: 45.625 | Change: -$105,260 | | EAR: 12.2% | Savings: $7,368 | | | Discounts: $12,000 | | | Net savings: -$4,632 | | | Gross margin: 30% | | | Gross revenue must rise by: | | | - $15,440 | | | - 0.0965% | --- ### **If you need a spreadsheet file or markdown table for submission, let me know!**