Your rich godfather has offered you a choice of one of the three following alternatives: $10,000 now; $2,000 a year for eight years; or $24,000 at the end of eight years. Assuming you could earn 11 percent annually, which alternative should you choose? If you could earn 12 percent annually, would you still choose the same alternative?
20
To determine which alternative to choose, we need to calculate the value of each option. We can use the concept of Present Value (PV) and Future Value (FV) to compare the alternatives.
1. $10,000 now: This option has a PV of $10,000 since it is the amount received immediately.
2. $2,000 a year for eight years: This option represents a series of equal cash flows, which we can calculate using the Present Value of Annuity (PVA) formula. The formula is as follows:
PV = CF × [(1 - (1 + r)^(-n)) / r]
Where PV is the present value, CF is the annual cash flow, r is the interest rate, and n is the number of years.
Plugging in the values: CF = $2,000, r = 11%, and n = 8, we can calculate the PV of this option.
3. $24,000 at the end of eight years: This option is a single cash flow at the end of eight years. To calculate its PV, we can use the Present Value (PV) formula:
PV = FV / (1 + r)^n
Where PV is the present value, FV is the future value, r is the interest rate, and n is the number of years.
Plugging in the values: FV = $24,000, r = 11%, and n = 8, we can calculate the PV of this option.
By comparing the PVs of each option, we can determine which one is the most beneficial.
Now, let's calculate the PVs for each option using an interest rate of 11%:
1. PV of $10,000 now: $10,000
2. PV of $2,000 a year for eight years:
PV = $2,000 × [(1 - (1 + 0.11)^(-8)) / 0.11]
PV ≈ $12,403.23
3. PV of $24,000 at the end of eight years:
PV = $24,000 / (1 + 0.11)^8
PV ≈ $8,611.59
Based on these calculations, the option with the highest PV is receiving $2,000 a year for eight years, which results in approximately $12,403.23 in today's value. Therefore, if you could earn 11% annually, you should choose this option.
Now let's evaluate the same options assuming an interest rate of 12%:
2. PV of $2,000 a year for eight years:
PV = $2,000 × [(1 - (1 + 0.12)^(-8)) / 0.12]
PV ≈ $11,084.45
3. PV of $24,000 at the end of eight years:
PV = $24,000 / (1 + 0.12)^8
PV ≈ $6,174.68
Based on these calculations, with an interest rate of 12%, the option with the highest PV is still receiving $2,000 a year for eight years, with an approximate value of $11,084.45. Therefore, you should choose the same option even if the interest rate is 12%.
In summary, if you could earn 11% annually, you should choose the option of receiving $2,000 a year for eight years. Even with an interest rate of 12%, the same option is still the most beneficial.