To solve this question, we need to determine the discount rate that makes someone indifferent between two cash flow streams: an infinite cash flow of $3,476 per year and a finite cash flow of $5,212 per year for a period of 25 years.
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Stream 1: Perpetuity of $3,476 per year
The present value (PV) of a perpetuity is given by the formula:
\[ PV = \frac{C}{r} \] where \(C\) is the cash flow per period and \(r\) is the discount rate. -
Stream 2: Annuity of $5,212 per year for 25 years
The present value (PV) of an annuity is given by the formula:
\[ PV = C \times \left( \frac{1 - (1 + r)^{-N}}{r} \right) \] where \(C\) is the cash flow per period, \(r\) is the discount rate, and \(N\) is the number of periods.
We want to find the discount rate (\(r\)) that makes the present value of both cash flows equal:
\[ \frac{3,476}{r} = 5,212 \times \left( \frac{1 - (1 + r)^{-25}}{r} \right) \]
Setting up the Finance Calculator
Assuming you want to solve for \(r\) directly without simplifying the equation, here's how to set it up on a financial calculator:
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Stream 1:
- PMT = 3,476 (cash flow)
- FV = 0 (there's no future value for a perpetuity)
- N = ∞ (but most calculators do not accept infinity; you can ignore or approximate via another means.)
- PV = ? (This is what you are solving for implicitly through the equation.)
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Stream 2:
- PMT = 5,212 (cash flow)
- FV = 0 (the annuity does not have a future value at the end of 25 years)
- N = 25 (time duration)
- PV = ? (This is also what you are solving for, using the formula for annuities.)
Note
While some financial calculators allow direct solving for \(r\), others may require you to use an iterative method or an Excel function (like IRR or Goal Seek) to approximate the discount rate that equalizes the present values of both cash flows.
Given the nature of this problem, it’s often easier to set up a spreadsheet and use numerical methods (like Goal Seek in Excel) to find \(r\) that satisfies the equation, as financial calculators may not have direct functions to solve for \(r\) in this specific configuration.
Conclusion
For a straightforward finance calculator entry, you typically set PMT, N, and FV based on the above categorizations for each cash flow stream, and then compute or iteratively adjust \(r\) until both PVs are equal.
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