To find the discount rate that makes you indifferent between the two cash flow streams, we need to set the present value (PV) of both cash flows equal.
- Cash Flow Stream 1: An annuity of $3,282 per year forever. The present value of a perpetuity can be calculated using the formula:
\[ PV = \frac{C}{r} \]
where \(C\) is the cash flow per period, and \(r\) is the discount rate.
So, for the perpetuity:
\[ PV_1 = \frac{3,282}{r} \]
- Cash Flow Stream 2: An annuity of $5,840 per year for 24 years. The present value of a finite annuity can be computed using the formula:
\[ PV = C \times \left(1 - (1 + r)^{-n}\right) \div r \]
where \(C\) is the cash flow per period, \(r\) is the discount rate, and \(n\) is the number of periods.
So, for the annuity:
\[ PV_2 = 5,840 \times \left(1 - (1 + r)^{-24}\right) \div r \]
- Setting the Present Values Equal: We need to set these two present values equal to find \(r\):
\[ \frac{3,282}{r} = 5,840 \times \left(1 - (1 + r)^{-24}\right) \div r \]
Since \(r\) is in the denominator on both sides, we can multiply through by \(r\) (assuming \(r \neq 0\)) to simplify:
\[ 3,282 = 5,840 \times \left(1 - (1 + r)^{-24}\right) \]
- Solving for r: Now, we can simplify this equation:
\[ 3,282 = 5,840 - 5,840 \times (1 + r)^{-24} \]
Rearranging gives us:
\[ 5,840 \times (1 + r)^{-24} = 5,840 - 3,282 \]
\[ 5,840 \times (1 + r)^{-24} = 2,558 \]
We divide both sides by 5,840:
\[ (1 + r)^{-24} = \frac{2,558}{5,840} \]
Calculating the right-hand side:
\[ \frac{2,558}{5,840} \approx 0.437 \]
Now, we flip both sides:
\[ 1 + r = \left(0.437\right)^{-1/24} \]
Taking the inverse gives:
\[ 1 + r \approx \left(2.287\right)^{1/24} \]
Calculating this using a calculator or estimation methods:
\[ 1 + r \approx 1.1009 \]
Thus,
\[ r \approx 1.1009 - 1 \approx 0.1009 \text{ or } 10.09% \]
So, the discount rate that would make you indifferent between receiving $3,282 per year forever and $5,840 per year for 24 years is approximately 10.09%.
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