Let's break this down step by as requested. --- ## **Step 1: Calculate Annual Cash Flows** ### **Given Data:** - **Selling Price per Set:** $900 - **Variable Cost per Set:** $435 - **Annual Sales (units):** - Year 1: 8,100 - Year 2: 8,550 - Year 3: 9,000 - Year 4: 9,000 - Year 5: 8,100 - **Fixed Costs:** $520,000 per year - **Marketing Cost (Year ):** $210,000 - **High-Price Clubs Research Cost (Year ):** $130,000 - **Plant and Equipment Cost (Year ):** $4,800,000 (depreciated straight-line to $ over 5 years) - **Working Capital (Year ):** $367,500 (recovered at the end) - **Tax Rate:** 21% - **Cost of Capital:** 12% ### **A. Calculate Revenues, Variable Costs, and Contribution Margin** For each year, calculate: - **Revenue = Units Sold × Price** - **Variable Costs = Units Sold × Variable Cost per Set** - **Contribution Margin = Revenue – Variable Costs** | Year | Units Sold | Revenue | Variable Costs | Contribution Margin | |------|------------|----------------|-----------------|--------------------| | 1 | 8,100 | $7,290,000 | $3,523,500 | $3,766,500 | | 2 | 8,550 | $7,695,000 | $3,719,250 | $3,975,750 | | 3 | 9,000 | $8,100,000 | $3,915,000 | $4,185,000 | | 4 | 9,000 | $8,100,000 | $3,915,000 | $4,185,000 | | 5 | 8,100 | $7,290,000 | $3,523,500 | $3,766,500 | **Explanation:** Revenue and variable costs are calculated for each year. Contribution margin is the remainder after subtracting variable costs from revenue. --- ### **B. Subtract Fixed Costs and Depreciation** - **Fixed Costs:** $520,000 per year - **Depreciation:** $4,800,000 / 5 = $960,000 per year | Year | Contribution Margin | Fixed Costs | Depreciation | EBIT | |------|--------------------|-------------|--------------|--------------| | 1 | $3,766,500 | $520,000 | $960,000 | $2,286,500 | | 2 | $3,975,750 | $520,000 | $960,000 | $2,495,750 | | 3 | $4,185,000 | $520,000 | $960,000 | $2,705,000 | | 4 | $4,185,000 | $520,000 | $960,000 | $2,705,000 | | 5 | $3,766,500 | $520,000 | $960,000 | $2,286,500 | **Explanation:** EBIT (Earnings Before Interest and Taxes) is calculated by subtracting fixed costs and depreciation from the contribution margin. --- ### **C. Calculate Taxes and Net Income** - **Tax = EBIT × 21%** - **Net Income = EBIT – Tax** | Year | EBIT | Tax (21%) | Net Income | |------|-------------|---------------|---------------| | 1 | $2,286,500 | $480,165 | $1,806,335 | | 2 | $2,495,750 | $524,108 | $1,971,643 | | 3 | $2,705,000 | $568,050 | $2,136,950 | | 4 | $2,705,000 | $568,050 | $2,136,950 | | 5 | $2,286,500 | $480,165 | $1,806,335 | **Explanation:** Tax is 21% of EBIT, net income is EBIT minus tax. --- ### **D. Add Back Depreciation (Non-Cash Charge) to Get Annual Cash Flows** | Year | Net Income | Depreciation | Annual Cash Flow | |------|-------------|---------------|------------------| | 1 | $1,806,335 | $960,000 | $2,766,335 | | 2 | $1,971,643 | $960,000 | $2,931,643 | | 3 | $2,136,950 | $960,000 | $3,096,950 | | 4 | $2,136,950 | $960,000 | $3,096,950 | | 5 | $1,806,335 | $960,000 | $2,766,335 | **Explanation:** Depreciation is a non-cash charge, so it is added back to net income to get the annual cash flow. --- ### **E. Year Cash Outflows** - **Marketing:** $210,000 - **Research:** $130,000 - **Plant/Equipment:** $4,800,000 - **Working Capital:** $367,500 - **Total Initial Outflow:** $5,507,500 **Year 5: Add back recovery of working capital: +$367,500** --- ## **Step 2: Calculate Payback Period, NPV, and IRR** --- ### **A. List All Cash Flows** | Year | Cash Flow | |------|----------------| | | -$5,507,500 | | 1 | $2,766,335 | | 2 | $2,931,643 | | 3 | $3,096,950 | | 4 | $3,096,950 | | 5 | $3,133,835 | *(includes $2,766,335 + $367,500)* --- ### **B. Payback Period** Add annual cash flows until initial investment is recovered: 1. End of Year 1: $2,766,335 2. End of Year 2: $2,766,335 + $2,931,643 = $5,697,978 Initial investment ($5,507,500) is recovered between years 2 and 3. Payback in Year 2 + (Amount remaining / Year 3 cash flow): $5,507,500 - $2,766,335 - $2,931,643 = ($5,507,500 - $5,697,978) = -$190,478 (i.e., fully recovered in Year 2) But actually, after Year 2, cumulative = $5,697,978. So at some point in Year 2, the investment is recovered. Between Year 1 and 2: Cumulative after Year 1: $2,766,335 Left to recover: $5,507,500 - $2,766,335 = $2,741,165 Year 2 cash flow: $2,931,643 Fraction of year: $2,741,165 / $2,931,643 = .935 Payback period = 1 + .935 = **1.94 years** --- ### **C. Net Present Value (NPV) at 12%** \[ NPV = \sum \frac{CF_t}{(1 + r)^t} \] Where \( CF_t \) is the cash flow at year \( t \), and \( r = .12 \). \[ NPV = -5,507,500 + \frac{2,766,335}{1.12^1} + \frac{2,931,643}{1.12^2} + \frac{3,096,950}{1.12^3} + \frac{3,096,950}{1.12^4} + \frac{3,133,835}{1.12^5} \] Calculate each term: - Year 1: $2,766,335 / 1.12 = $2,470,841 - Year 2: $2,931,643 / (1.12^2) = $2,335,996 - Year 3: $3,096,950 / (1.12^3) = $2,210,259 - Year 4: $3,096,950 / (1.12^4) = $1,973,437 - Year 5: $3,133,835 / (1.12^5) = $1,779,429 Sum: $2,470,841 + $2,335,996 + $2,210,259 + $1,973,437 + $1,779,429 = $10,770,962 NPV = $10,770,962 - $5,507,500 = **$5,263,462** --- ### **D. Internal Rate of Return (IRR)** Solve for IRR where NPV = : \[ = -5,507,500 + \frac{2,766,335}{(1+IRR)^1} + \frac{2,931,643}{(1+IRR)^2} + \frac{3,096,950}{(1+IRR)^3} + \frac{3,096,950}{(1+IRR)^4} + \frac{3,133,835}{(1+IRR)^5} \] Using a financial calculator or Excel's IRR function: \[ \text{IRR} \approx 41.1\% \] --- ## **Final Answers** | | Value | |--------------------|-------------| | Payback period | **1.94 years** | | Net present value | **$5,263,462** | | Internal rate of return | **41.10%** | --- ## **Summary** - **Step 1:** Calculated annual cash flows by accounting for revenues, costs, depreciation, and taxes. - **Step 2:** Used these cash flows to calculate payback period, NPV, and IRR. **Final Answers:** - Payback period: **1.94 years** - Net present value: **$5,263,462** - Internal rate of return: **41.10%**