To determine the value increase of the company by selecting the better machine, we need to compute the Net Present Value (NPV) for both Machine A and Machine B using the replacement chain approach. Then, the difference between the NPVs will reflect how much the value of the company would increase. Here's the step-by-step process:

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1. Machine A NPV Calculation

Machine A operates for 4 years and then must be replaced. Since the time horizon for Machine B is 8 years, we will consider two cycles of Machine A (two replacements) to match Machine B's time horizon.

• Cost: $10 million
• Inflows: $4 million per year for 4 years
• Cost of Capital (Discount Rate): 10%
• Time Horizon: 8 years (two 4-year cycles)

NPV for 4-year cycle of Machine A: $\text{NPV} = -10 + \sum_{t=1}^{4} \frac{4}{(1+0.10)^t}$

After calculating this NPV, we repeat this cash flow for a second cycle (adjusted for the replacement cost in year 4).

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2. Machine B NPV Calculation

Machine B operates for 8 years without replacement.

• Cost: $15 million
• Inflows: $3.5 million per year for 8 years
• Cost of Capital (Discount Rate): 10%
• Time Horizon: 8 years

$\text{NPV} = -15 + \sum_{t=1}^{8} \frac{3.5}{(1+0.10)^t}$

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3. Value Increase

The value increase of the company will be: $\text{Value Increase} = \text{NPV of Better Machine} - \text{NPV of Other Machine}$

Let me compute the exact values.import numpy as np

Given values

cost_of_capital = 0.10 # Discount rate machine_a_cost = 10 # million dollars machine_a_inflows = 4 # million dollars per year machine_a_lifetime = 4 # years

machine_b_cost = 15 # million dollars machine_b_inflows = 3.5 # million dollars per year machine_b_lifetime = 8 # years

Function to calculate NPV

def calculate_npv(cost, inflows, lifetime, discount_rate): inflows_array = np.array([inflows / (1 + discount_rate)**t for t in range(1, lifetime + 1)]) return -cost + inflows_array.sum()

NPV of Machine A for 4 years

npv_a_4_years = calculate_npv(machine_a_cost, machine_a_inflows, machine_a_lifetime, cost_of_capital)

Since Machine A repeats for 8 years (2 cycles)

npv_a_8_years = npv_a_4_years + npv_a_4_years / (1 + cost_of_capital)**machine_a_lifetime

NPV of Machine B for 8 years

npv_b_8_years = calculate_npv(machine_b_cost, machine_b_inflows, machine_b_lifetime, cost_of_capital)

Value increase by choosing the better machine

value_increase = npv_b_8_years - npv_a_8_years

npv_a_8_years, npv_b_8_years, value_increase(4.509570237959954, 3.6722416926593233, -0.8373285453006307)(4.509570237959954, 3.6722416926593233, -0.8373285453006307)successThe calculated NPVs are as follows:

• NPV for Machine A (over 8 years): 4.51 million
• NPV for Machine B (over 8 years): 3.67 million

The value increase by selecting Machine A (the better machine in this scenario) is:

$\text{Value Increase} = -0.84 \, \text{million dollars}$

This means Machine A is more cost-effective by 0.84 million dollars, so the company would decrease in value by selecting Machine B.

Let me know if you'd like details on these calculations or have any questions! Here are 5 related questions to consider:

1. How does the replacement chain approach compare to the equivalent annual annuity approach?
2. What factors influence the selection of the discount rate in NPV calculations?
3. How would inflation adjustments impact the NPV results?
4. What are the risks of using only NPV to make investment decisions?
5. Can alternative methods, such as IRR, provide different insights in this scenario?

Tip: Always align project time horizons when comparing mutually exclusive investments.