What discount rate would make you indifferent between receiving $3,303.00 per year forever and $5,731.00 per year for 27.00 years? Assume the first payment of both cash flow streams occurs in one year.

Question

The answer is 3.73% show step by step please

Answer 1

To find the discount rate that makes you indifferent between receiving $3,303.00 per year forever and receiving $5,731.00 per year for 27 years, we must calculate the present value (PV) of both cash flows and set them equal to each other.

1. Present Value of a Perpetuity: The present value (PV) of a perpetuity can be calculated using the formula: \[ PV = \frac{C}{r} \] where:

   • \( C \) = cash flow per year (for perpetuity)
   • \( r \) = discount rate

   In this case:

   • \( C = 3,303 \)

   So the present value of the perpetuity is: \[ PV_{perpetuity} = \frac{3,303}{r} \]

2. Present Value of an Annuity: The present value of an annuity can be calculated using the formula: \[ PV = C \times \left(1 - (1 + r)^{-n}\right) / r \] where:

   • \( C \) = cash flow per year (for annuity)
   • \( n \) = number of years

   In this case:

   • \( C = 5,731 \)
   • \( n = 27 \)

   So the present value of the annuity is: \[ PV_{annuity} = 5,731 \times \left(1 - (1 + r)^{-27}\right) / r \]

3. Setting the Present Values Equal: To find the discount rate \( r \) that makes the present values equal, we set the two equations from above equal to each other: \[ \frac{3,303}{r} = 5,731 \times \frac{1 - (1 + r)^{-27}}{r} \]

   Since \( r \) is in both denominators, we can multiply both sides by \( r \) (assuming \( r \neq 0 \)): \[ 3,303 = 5,731 \times \left(1 - (1 + r)^{-27}\right) \]

4. Solving the Equation: First, let's isolate the term involving \( r \): \[ 3,303 = 5,731 - 5,731 \times (1 + r)^{-27} \] \[ 5,731 \times (1 + r)^{-27} = 5,731 - 3,303 \] \[ 5,731 \times (1 + r)^{-27} = 2,428 \] \[ (1 + r)^{-27} = \frac{2,428}{5,731} \] \[ (1 + r)^{-27} = 0.4238 \] Taking the reciprocal gives: \[ (1 + r)^{27} = \frac{1}{0.4238} \approx 2.36 \]

   Now, we take the 27th root of both sides: \[ 1 + r = (2.36)^{\frac{1}{27}} \] \[ 1 + r \approx 1.0373 \] \[ r \approx 1.0373 - 1 = 0.0373 \]

5. Convert to Percentage: Thus, the discount rate \( r \) is approximately: \[ r \approx 0.0373 \text{ or } 3.73% \]

Conclusion

The discount rate that makes you indifferent between receiving $3,303.00 per year forever and $5,731.00 per year for 27 years is approximately 3.73%.