Hello! Welcome back to our journey through Present Worth analysis.
In our last lesson, we established a clear rule for choosing between mutually exclusive alternatives: select the one with the highest Present Worth (PW), assuming they have equal lives. This works perfectly when you're comparing two software licenses that both last for 5 years.
But what if you're comparing a 3-year cloud server lease against purchasing hardware that will last 7 years? A direct PW comparison would be fundamentally flawed—it's like comparing the cost of a three-course meal to a seven-course meal. Today, we'll tackle this exact problem.
Your learning outcome is to be able to handle alternatives with unequal useful lives using the least common multiple or a defined study period. By mastering this, you'll be equipped to make fair and accurate comparisons in almost any real-world technology investment scenario.
The Fundamental Rule: A Common Time Horizon
When you compare investment alternatives, you are trying to determine which one provides a required service at the lowest equivalent cost or generates the highest equivalent value. To be a fair comparison, this evaluation must cover the same period of service for all alternatives.
Directly comparing the PW of a 3-year project with the PW of a 6-year project is misleading. The 3-year analysis ignores the costs that will be incurred to provide the service for years 4, 5, and 6.
As the MIT course notes you've seen before state succinctly, the guiding principle is:
All alternatives must be compared over the same time span.
To achieve this, we define a study period (or "analysis period") and use it for every alternative. There are two primary methods to determine this study period.
Method 1: The Defined Study Period
In many business situations, you have a specific time frame in mind. Your company might need a customer support platform for a 5-year strategic plan, or you might be developing a product with a projected 4-year market window. This pre-determined time frame is your study period.
When project lives don't align with this study period, we make adjustments:
- If Project Life > Study Period: The asset has remaining useful life at the end of the study period. We account for this by including a salvage value as a positive cash flow (an inflow) in the final year of the study period.
- If Project Life < Study Period: We assume the project will be repeated with identical costs until the study period is covered. If the last repetition is cut short by the end of the study period, we calculate a salvage value for it at that point.
Example: Choosing a Development Server
Your team needs a dedicated server for a 3-year project. This is our study period. The MARR is 10%.
- Alternative A (Robust Server):
- Initial Cost: $9,000
- Annual Maintenance: $500
- Useful Life: 5 years
- Estimated Salvage Value after 3 years: $4,000
- Alternative B (Basic Server):
- Initial Cost: $5,000
- Annual Maintenance: $800
- Useful Life: 2 years
- (We assume you can buy another identical server after 2 years)
Let's analyze both over the 3-year study period. The (P/A, 10%, n) factors are 1.7355 for n=2 and 2.4869 for n=3. The (P/F, 10%, n) factors are 0.8264 for n=2 and 0.7513 for n=3.
PW of Costs for Alternative A (Life > Study Period):
We truncate the project at year 3 and include its salvage value.
PW of Costs for Alternative B (Life < Study Period):
This is trickier. We need the service for 3 years.
- Year 0: Buy the first server for $5,000.
- Years 1, 2: Pay $800 maintenance.
- Year 2: The first server's life ends. We buy a second identical server for $5,000.
- Year 3: Pay $800 maintenance for the second server. Its life is cut short. We'll assume a salvage value based on linear depreciation: .
The cash flow stream is:
- Year 0: -$5,000
- Year 1: -$800
- Year 2: -$800 (maintenance) - $5,000 (new server) = -$5,800
- Year 3: -$800 (maintenance) + $2,500 (salvage) = +$1,700
Decision:
Comparing the two present worth values of cost:
Since is numerically larger (less negative), Alternative A is the more economical choice.
Method 2: The Least Common Multiple (LCM) Approach
What if there's no obvious study period? If you're choosing equipment for an ongoing operation, you might use the Least Common Multiple (LCM) of the alternatives' lives as the study period.
The core assumption is that the service is needed over a long horizon, and each alternative's cash flow cycle will be repeated with identical costs to cover this period. The LCM represents the first point in time where the replacement cycles of all alternatives align.
A fantastic, detailed guide on this method is available in the resource below. It walks through a clear example comparing three options with 4, 5, and 10-year lives.
5.3 Analysis Period and Project Lives
The following resource provides an excellent, detailed walkthrough of the LCM method. It compares three alternatives with lives of 5, 4, and 10 years, setting the study period to their LCM of 20 years.
Please read the subsection 'B3a: Analysis Period = Least Common Multiple of Project Lives'. Follow the example of the three garbage truck models. Pay close attention to how the cash flows for each model are repeated over the 20-year LCM period and how their total Present Worth is calculated.
As you read, notice how the calculation for each alternative works:
- Truckasaurus (5-year life): The initial cost of $5M is incurred at years 0, 5, 10, and 15. The annual cost of $375k runs for all 20 years.
- Trashgobbler (4-year life): The initial cost of $4M is incurred at years 0, 4, 8, 12, and 16. The annual cost of $475k runs for all 20 years.
- Garbotron 3000 (10-year life): The initial cost of $10M is incurred at years 0 and 10. The annual cost of $325k runs for all 20 years.
The PW of each of these extended cash flow streams is then calculated and compared to find the most economical choice. The resource shows that despite the different approaches, the final decision remains consistent, which underscores the robustness of these methods.
Test your understanding!
You're choosing between two open-source frameworks for a long-term project. The MARR is 8%. You decide to use the LCM method.
- Framework X: Requires a major integration effort every 2 years at a cost of $20,000. Annual support costs are $5,000.
- Framework Y: Requires a major integration effort every 3 years at a cost of $25,000. Annual support costs are $4,000.
What is the appropriate study period for this comparison, and which framework has a higher cost over that period? (You don't need to do the full calculation, just set up the PW equations).
Show answer
The appropriate study period is the LCM of 2 and 3, which is 6 years.
The setup for the Present Worth of Costs for each framework would be:
Framework X (2-year cycle):
The major integration cost of $20k happens at year 0, year 2, and year 4. The annual support cost of $5k runs for all 6 years.
Framework Y (3-year cycle):
The major integration cost of $25k happens at year 0 and year 3. The annual support cost of $4k runs for all 6 years.
By calculating these two values, you could determine which framework has the lower total PW of costs (the numerically larger PW).
Conclusion
You now have the tools to make "apples-to-apples" comparisons of projects even when their lifespans differ. This is a critical skill for making sound long-term investment decisions.
- Key Takeaways:
- To compare alternatives with unequal lives, you must use a common study period.
- If a study period is defined by the business need, you adjust cash flows by assuming project repetition or calculating salvage values.
- If no study period is defined, the Least Common Multiple (LCM) of the project lives is a common and logical choice, assuming that projects can be repeated with identical cash flows.
- The goal is always to compare the Present Worth of each alternative over the exact same time horizon.
Preview of the Next Lesson
We've repeatedly touched on the idea of an asset's remaining value. In the next lesson, we will formalize this by learning how to properly incorporate salvage values and disposal costs in present worth calculations. This will add another layer of precision to your project evaluations.