To determine which option is more valuable, we need to calculate the present value (PV) of the annuity option (Option 1) and compare it to the lump-sum payment (Option 2).
Option 1: Annuity of $1,000,000 per year for 30 years
The present value of an annuity can be calculated using the formula:
\[ PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \(C\) = Cash flow per period ($1,000,000)
- \(r\) = Interest rate (6% or 0.06)
- \(n\) = Number of periods (30 years)
Plugging in the values:
\[ PV = 1,000,000 \times \left( \frac{1 - (1 + 0.06)^{-30}}{0.06} \right) \]
Calculating \( (1 + 0.06)^{-30} \):
\[ (1 + 0.06)^{-30} = (1.06)^{-30} \approx 0.1741 \]
Now, substitute that value back into the formula:
\[ PV = 1,000,000 \times \left( \frac{1 - 0.1741}{0.06} \right) \] \[ = 1,000,000 \times \left( \frac{0.8259}{0.06} \right) \] \[ = 1,000,000 \times 13.765 \] \[ = 13,765,000 \]
Option 2: Lump-sum payment of $15,000,000 today
This is straightforward since it's a one-time payment with no discounting necessary. It's simply $15,000,000.
Conclusion
- Present Value of Option 1 (Annuity) = $13,765,000
- Present Value of Option 2 (Lump Sum) = $15,000,000
Since $15,000,000 > $13,765,000, the one-time payment of $15,000,000 today is the more valuable option.
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