To determine which option has the greatest present value (PV), we will calculate the present value of each option using the formula:
\[ PV = \frac{FV}{(1 + r)^n} \]
where:
- \(FV\) is the future value,
- \(r\) is the interest rate (6% or 0.06),
- \(n\) is the number of years until the payment is received.
We will evaluate each option:
a. $300 paid in two years
\[ PV_a = \frac{300}{(1 + 0.06)^2} = \frac{300}{1.1236} \approx 266.22 \]
b. $285 today
\[ PV_b = 285 \text{ (since it is already in present value)} \]
c. $150 paid in one year plus $140 paid in two years
Calculate the PV of each amount separately and then sum them: \[ PV_c = \frac{150}{(1 + 0.06)^1} + \frac{140}{(1 + 0.06)^2} \] \[ PV_c = \frac{150}{1.06} + \frac{140}{1.1236} \] \[ PV_c \approx 141.51 + 124.97 \approx 266.48 \]
d. $100 paid today plus $100 paid in one year plus $100 paid in two years
Calculate the PV of each amount: \[ PV_d = 100 + \frac{100}{(1 + 0.06)^1} + \frac{100}{(1 + 0.06)^2} \] \[ PV_d = 100 + \frac{100}{1.06} + \frac{100}{1.1236} \] \[ PV_d = 100 + 94.34 + 89.00 \approx 283.34 \]
Summary of Present Values
- \(PV_a \approx 266.22\)
- \(PV_b = 285\)
- \(PV_c \approx 266.48\)
- \(PV_d \approx 283.34\)
Conclusion
The option with the greatest present value is option b. $285 today.
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