To solve this problem, we will calculate the Profitability Index (PI) and Net Present Value (NPV) for each project.

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a-1. Profitability Index Formula:

$PI = \frac{\text{Present Value of Future Cash Flows}}{\text{Initial Investment}}$

The present value (PV) of cash flows is calculated as: $PV = \sum \frac{\text{Cash Flow in Year } t}{(1 + r)^t}$ Where $r = 16\% = 0.16$.

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a-2. Decision Rule for PI:

• If $PI > 1$, the project is acceptable.
• The company should select the project with the higher PI.

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b-1. Net Present Value Formula:

$NPV = \text{PV of Future Cash Flows} - \text{Initial Investment}$

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b-2. Decision Rule for NPV:

• If $NPV > 0$, the project is acceptable.
• The company should select the project with the higher NPV.

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Let's compute these values step by step for both projects. Define the cash flows and required return required_return = 0.16

Cash flows for project I

cash_flows_I = [-83000, 32900, 43000, 49000]

Cash flows for project II

cash_flows_II = [-41000, 12900, 30500, 24500]

Function to calculate Present Value

def calculate_pv(cash_flows, rate): return sum(cf / ((1 + rate) ** t) for t, cf in enumerate(cash_flows) if t > 0)

Function to calculate Profitability Index

def calculate_pi(pv, initial_investment): return pv / abs(initial_investment)

Function to calculate NPV

def calculate_npv(pv, initial_investment): return pv - abs(initial_investment)

Calculations for Project I

pv_I = calculate_pv(cash_flows_I, required_return) pi_I = calculate_pi(pv_I, cash_flows_I[0]) npv_I = calculate_npv(pv_I, cash_flows_I[0])

Calculations for Project II

pv_II = calculate_pv(cash_flows_II, required_return) pi_II = calculate_pi(pv_II, cash_flows_II[0]) npv_II = calculate_npv(pv_II, cash_flows_II[0])

Results

results = { "Project I": {"PI": round(pi_I, 3), "NPV": round(npv_I, 2)}, "Project II": {"PI": round(pi_II, 3), "NPV": round(npv_II, 2)}, }

import pandas as pd import ace_tools as tools

results_df = pd.DataFrame(results).T tools.display_dataframe_to_user(name="Profitability Index and NPV Results for Projects", dataframe=results_df)I have calculated the Profitability Index (PI) and Net Present Value (NPV) for both projects. You can review the results in the displayed table.

Summary:

• a-1: Profitability Index (PI) indicates that both projects are acceptable ($PI > 1$).
• a-2: Project II has a higher PI ($1.207 > 1.105$), so it should be selected based on the PI decision rule.
• b-1: Both projects have positive NPVs, indicating they are profitable.
• b-2: Project I has a slightly higher NPV ($8710.30 > 8483.27$), so it should be selected based on the NPV decision rule.

Let me know if you have questions or would like further details!

Related Questions:

1. What factors could lead to discrepancies between PI and NPV decision rules?
2. How does the required return affect the profitability of these projects?
3. Can we calculate the Internal Rate of Return (IRR) for these projects for comparison?
4. Would the decision change if the initial investments were significantly different?
5. What are the risks of relying solely on the NPV decision rule?

Tip:

Always consider the context of the project and resource constraints when making decisions using PI and NPV, as they might favor different outcomes in specific scenarios.