To determine the best option between receiving a one-time payment of $1,000,000 or receiving $200 daily indefinitely, we need to consider the present value (PV) of these cash flows. PV calculations will help us understand how much future cash flows are worth today, given a specific interest rate (or discount rate). Let's assume an annual interest rate of 5% for this calculation, which is a reasonable average rate for long-term investments. This interest rate needs to be converted into a daily rate for our calculations, as the $200 is a daily payment.
Step 1: Convert Annual Interest Rate to Daily Rate
If we assume the annual interest rate is compounded daily, the daily interest rate \( r \) is calculated from the annual rate \( R \) as follows:
\[ r = (1 + R)^{\frac{1}{365}} - 1 \]
\[ r = (1 + 0.05)^{\frac{1}{365}} - 1 \approx 0.000136 \text{ per day} \]
Step 2: Calculate the Present Value of Each Option
Option 1: $1,000,000 One-Time Payment
The present value of receiving $1,000,000 today is straightforward:
\[ PV_1 = \$1,000,000 \]
Option 2: $200 Daily Payment Indefinitely (Perpetuity)
The present value of a perpetuity is calculated using the formula:
\[ PV = \frac{C}{r} \]
Where \( C \) is the daily payment, and \( r \) is the daily interest rate. Substituting the values:
\[ PV_2 = \frac{\$200}{0.000136} \approx \$1,470,588 \]
Step 3: Comparison of Present Values
- Option 1 PV: $1,000,000
- Option 2 PV: Approximately $1,470,588
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