1. **Problem Statement:** You want to evaluate two franchise projects, Franchise A and Franchise F, each with cash flows based on the last five digits of your student ID (let's denote this number as $C$). Franchise A has a constant annuity cash flow of $C$ for 4 years. Franchise F starts with $C$ in year 1 and decreases by 5000 each subsequent year for 4 years. 2. **Capital Budgeting Technique Based on Time Period:** The appropriate technique here is the **Payback Period** because it measures how quickly the initial investment is recovered, focusing on the time period without considering the time value of money. 3. **Calculating Payback Period for Franchise A:** - Initial outlay = $2 \times C$ - Annual cash flow = $C$ - Payback period = $\frac{2C}{C} = 2$ years 4. **Calculating Payback Period for Franchise F:** - Cash flows: Year 1 = $C$, Year 2 = $C - 5000$, Year 3 = $C - 10000$, Year 4 = $C - 15000$ - Cumulative cash flows: - After Year 1: $C$ - After Year 2: $C + (C - 5000) = 2C - 5000$ - After Year 3: $2C - 5000 + (C - 10000) = 3C - 15000$ - After Year 4: $3C - 15000 + (C - 15000) = 4C - 30000$ - To find payback period, find when cumulative cash flow equals initial outlay $2C$: - After Year 1: $C < 2C$ - After Year 2: $2C - 5000$; if $2C - 5000 \geq 2C$, payback is 2 years, else between 2 and 3 years. - Solve for exact payback time $t$ between years 2 and 3: $$C + (C - 5000) + (t - 2)(C - 10000) = 2C$$ Simplify: $$2C - 5000 + (t - 2)(C - 10000) = 2C$$ $$ (t - 2)(C - 10000) = 5000$$ $$ t = 2 + \frac{5000}{C - 10000}$$ 5. **Decision Based on Payback Period (Independent Projects):** - Franchise A payback = 2 years - Franchise F payback = $2 + \frac{5000}{C - 10000}$ years - If $C > 10000$, payback for F is slightly more than 2 years. - Both projects recover investment within 4 years, so both can be accepted if payback period is the only criterion. 6. **Capital Budgeting Techniques Incorporating Time Value of Money (TVM):** - Techniques like **Net Present Value (NPV)** and **Internal Rate of Return (IRR)** consider TVM. - Payback period does not consider TVM. 7. **Applying NPV for Both Projects:** - Discount rate $r = 10\%$ - Initial outlay = $2C$ For Franchise A (annuity): $$NPV_A = -2C + C \times \frac{1 - (1 + r)^{-4}}{r}$$ For Franchise F (decreasing cash flows): Cash flows: $C$, $C-5000$, $C-10000$, $C-15000$ $$NPV_F = -2C + \frac{C}{(1+r)^1} + \frac{C-5000}{(1+r)^2} + \frac{C-10000}{(1+r)^3} + \frac{C-15000}{(1+r)^4}$$ 8. **Decision Based on NPV (Independent Projects):** - Calculate $NPV_A$ and $NPV_F$ using your $C$ value. - Accept projects with positive NPV. - Choose the project with the higher NPV if you must select one. **Summary:** - Use Payback Period to evaluate time to recover investment. - Use NPV to incorporate time value of money and make financially sound decisions. - Both projects can be accepted if independent and NPVs are positive. - Choose the project with the higher NPV if mutually exclusive.