To find the discount rate (r) that makes you indifferent between receiving $3,094 per year forever (a perpetuity) and $5,927 per year for 27 years, we'll first calculate the present value (PV) of both cash flow streams and set them equal to each other.
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Present Value of a Perpetuity: The present value of a perpetuity is calculated using the formula: \[ PV = \frac{C}{r} \] where \( C \) is the annual cash flow and \( r \) is the discount rate.
For the perpetuity with payments of $3,094: \[ PV_{perpetuity} = \frac{3094}{r} \]
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Present Value of an Annuity: The present value of an annuity is calculated using the formula: \[ PV = C \times \left(1 - (1 + r)^{-n}\right) / r \] where \( C \) is the annual cash flow, \( r \) is the discount rate, and \( n \) is the number of periods.
For the annuity of $5,927 for 27 years: \[ PV_{annuity} = 5927 \times \left(1 - (1 + r)^{-27}\right) / r \]
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Set the Present Values Equal: We want to find the discount rate \( r \) such that: \[ \frac{3094}{r} = 5927 \times \left(1 - (1 + r)^{-27}\right) / r \]
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Simplify the Equation: Multiply both sides by \( r \) to eliminate it from the denominator. \[ 3094 = 5927 \times \left(1 - (1 + r)^{-27}\right) \]
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Rearranging the Equation: Divide both sides by 5927: \[ \frac{3094}{5927} = 1 - (1 + r)^{-27} \] Rearranging: \[ (1 + r)^{-27} = 1 - \frac{3094}{5927} \] Simplifying the right side: \[ (1 + r)^{-27} = \frac{5927 - 3094}{5927} = \frac{2833}{5927} \]
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Take the Inverse: Taking the inverse, we have: \[ 1 + r = \left(\frac{5927}{2833}\right)^{\frac{1}{27}} \] Then, solve for \( r \): \[ r = \left(\frac{5927}{2833}\right)^{\frac{1}{27}} - 1 \]
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Calculate Using a Finance Calculator:
- First, calculate \( \frac{5927}{2833} \).
- Then, raise this result to the power of \( \frac{1}{27} \).
- Subtract 1 from the result to find \( r \).
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Final Calculation:
- Calculating \( \frac{5927}{2833} \approx 2.094 \).
- Raising \( 2.094 \) to the power of \( \frac{1}{27} \) gives you a result close to \( 1.07404 \).
- Therefore, \( r \approx 1.07404 - 1 = 0.07404 \) or \( 7.40% \).
Thus, the discount rate that makes you indifferent between the two cash flow streams is approximately 7.40%.
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