To determine the discount rate that makes you indifferent between two cash flow streams—$3,094.00 per year forever and $5,927.00 per year for 27.00 years—you can use the present value formulas for each cash flow stream and set them equal to each other.
Cash Flow Stream 1: Perpetuity
The present value (PV) of the perpetuity that pays $3,094.00 per year is calculated using the formula:
\[ PV_{\text{perpetuity}} = \frac{C}{r} \]
Where:
- \(C = 3,094\) (annual cash flow)
- \(r\) = discount rate
Cash Flow Stream 2: Annuity
The present value of the annuity that pays $5,927.00 per year for 27 years is calculated using the formula:
\[ PV_{\text{annuity}} = C \times \left(1 - (1 + r)^{-n}\right) / r \]
Where:
- \(C = 5,927\) (annual cash flow)
- \(n = 27\) (number of years)
- \(r\) = discount rate
Setting Both Present Values Equal
To find the discount rate \(r\) that makes you indifferent between the two cash flows, set the present values equal to each other:
\[ \frac{3,094}{r} = 5,927 \times \left(1 - (1 + r)^{-27}\right) / r \]
Simplifying the Equation
Multiply both sides by \(r\) (assuming \(r \neq 0\)):
\[ 3,094 = 5,927 \times \left(1 - (1 + r)^{-27}\right) \]
Now, isolate \(1 - (1 + r)^{-27}\):
\[ 1 - (1 + r)^{-27} = \frac{3,094}{5,927} \]
Calculating the right-hand side:
\[ \frac{3,094}{5,927} \approx 0.522 \]
Now, we have:
\[ 1 - (1 + r)^{-27} \approx 0.522 \]
Thus:
\[ (1 + r)^{-27} \approx 0.478 \]
Taking the reciprocal provides:
\[ 1 + r \approx 0.478^{-1} \approx 2.09 \]
Solving for \(r\):
\[ r \approx 2.09 - 1 \approx 1.09 \]
Expressed as a percentage, the discount rate is approximately:
\[ r \approx 1.09 \times 100 \approx 109% \]
This is clearly an unusually high rate, suggesting the calculations or assumptions could be revisited based on context. To further refine or verify accuracy of this result, numerical techniques such as iteration or using software may be more suitable for precise financial modeling.
However, the concept remains: equating the present values and solving for \(r\) gives you the discount rate that establishes indifference between the two cash flows.
1 answer