Flaherty is considering an investment that, if paid for immediately, is expected to return $163,000 ten years from now. If Flaherty demands a 12% return, how much is she willing to pay for this investment? (PV of $1, FV of $1, PVA of $1, and FVA of $1) (Use appropriate factor(s) from the tables provided. Round your “PV of a single amount” to 4 decimal places and final answer to the nearest whole dollar.)
Future Value × p (PV of a Single Amount) = Present Value × =
The Correct Answer and Explanation is:
Problem Summary:
- Future Value (FV) = $163,000 (to be received 10 years from now)
- Required rate of return (discount rate) = 12% per year
- Time (n) = 10 years
- Find Present Value (PV), i.e., how much Flaherty should pay today for this future amount.
Step 1: Identify the formula
The present value of a single future amount is given by: PV=FV×PV factorPV = FV \times \text{PV factor}
Where PV factor is the present value of $1 at 12% for 10 years.
Step 2: Find the PV factor
Using the Present Value of $1 table for 12% and 10 years: PV factor=1(1+r)n=1(1+0.12)10=?PV\text{ factor} = \frac{1}{(1 + r)^n} = \frac{1}{(1 + 0.12)^{10}} = ?
From the table or calculation: PV factor=0.32197(rounded to 4 decimal places, as requested)PV\text{ factor} = 0.32197 \quad (\text{rounded to 4 decimal places, as requested})
Step 3: Calculate Present Value
PV=163,000×0.32197=52,710.11PV = 163,000 \times 0.32197 = 52,710.11
Final Answer:
52,710\boxed{52,710}
Flaherty should be willing to pay $52,710 today for the investment to achieve her required 12% return.
Explanation
The problem involves calculating the present value (PV) of a future amount of money, which is a fundamental concept in finance and investment decision-making. The future value (FV) of $163,000 is expected 10 years from now. Flaherty wants to know how much she should pay today for that future sum, assuming she demands a 12% annual return on her investment.
The principle behind present value is the time value of money, which states that a dollar today is worth more than a dollar received in the future due to its potential earning capacity. To account for this, future cash flows are discounted back to the present using a discount rate—in this case, 12%, reflecting Flaherty’s desired rate of return.
The formula to find the present value of a single lump sum is: PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}
where rr is the annual discount rate and nn is the number of years until the payment is received. Alternatively, using financial tables or a calculator, we can multiply the future amount by the present value factor (PV factor) corresponding to the given rate and time period.
For 12% over 10 years, the PV factor is approximately 0.32197. This means that each dollar expected 10 years from now is worth about 32 cents today at this discount rate.
Multiplying the future $163,000 by 0.32197 gives a present value of about $52,710. This is the maximum amount Flaherty should be willing to pay for the investment today if she wants to earn a 12% return. Paying more than this would result in a return lower than 12%, making it a less attractive investment.
This method is widely used in finance for valuing bonds, stocks, projects, and any future cash flows to make informed investment decisions.
Let’s break down the problem step-by-step:
Problem Summary:
- Future Value (FV) = $163,000 (to be received 10 years from now)
- Required rate of return (discount rate) = 12% per year
- Time (n) = 10 years
- Find Present Value (PV), i.e., how much Flaherty should pay today for this future amount.
Step 1: Identify the formula
The present value of a single future amount is given by: PV=FV×PV factorPV = FV \times \text{PV factor}
Where PV factor is the present value of $1 at 12% for 10 years.
Step 2: Find the PV factor
Using the Present Value of $1 table for 12% and 10 years: PV factor=1(1+r)n=1(1+0.12)10=?PV\text{ factor} = \frac{1}{(1 + r)^n} = \frac{1}{(1 + 0.12)^{10}} = ?
From the table or calculation: PV factor=0.32197(rounded to 4 decimal places, as requested)PV\text{ factor} = 0.32197 \quad (\text{rounded to 4 decimal places, as requested})
Step 3: Calculate Present Value
PV=163,000×0.32197=52,710.11PV = 163,000 \times 0.32197 = 52,710.11
Final Answer:
52,710\boxed{52,710}
Flaherty should be willing to pay $52,710 today for the investment to achieve her required 12% return.
Explanation
The problem involves calculating the present value (PV) of a future amount of money, which is a fundamental concept in finance and investment decision-making. The future value (FV) of $163,000 is expected 10 years from now. Flaherty wants to know how much she should pay today for that future sum, assuming she demands a 12% annual return on her investment.
The principle behind present value is the time value of money, which states that a dollar today is worth more than a dollar received in the future due to its potential earning capacity. To account for this, future cash flows are discounted back to the present using a discount rate—in this case, 12%, reflecting Flaherty’s desired rate of return.
The formula to find the present value of a single lump sum is: PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}
where rr is the annual discount rate and nn is the number of years until the payment is received. Alternatively, using financial tables or a calculator, we can multiply the future amount by the present value factor (PV factor) corresponding to the given rate and time period.
For 12% over 10 years, the PV factor is approximately 0.32197. This means that each dollar expected 10 years from now is worth about 32 cents today at this discount rate.
Multiplying the future $163,000 by 0.32197 gives a present value of about $52,710. This is the maximum amount Flaherty should be willing to pay for the investment today if she wants to earn a 12% return. Paying more than this would result in a return lower than 12%, making it a less attractive investment.
This method is widely used in finance for valuing bonds, stocks, projects, and any future cash flows to make informed investment decisions.
