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:
Let’s solve this step-by-step.
Problem Recap:
Flaherty wants to invest in something that will pay $163,000 in 10 years. She demands a 12% annual return. The question is: How much should she pay now?
Step 1: Understand the Problem
The problem asks for the Present Value (PV) of a future amount.
- Future Value (FV) = $163,000
- Time (n) = 10 years
- Interest rate (i) = 12% per year
We want to find the Present Value (PV) — the amount to pay now to get $163,000 in 10 years, assuming a 12% return.
Step 2: Use the Present Value Formula
The formula for PV of a single future amount is: PV=FV×(1(1+i)n)PV = FV \times \left(\frac{1}{(1 + i)^n}\right)
Here, 1(1+i)n\frac{1}{(1 + i)^n} is the Present Value factor for a single amount, often found in tables or calculated.
Step 3: Find the PV Factor from Table or Calculate
The PV factor for 12% and 10 years is: p=1(1+0.12)10=1(1.12)10p = \frac{1}{(1 + 0.12)^{10}} = \frac{1}{(1.12)^{10}}
Calculating: (1.12)10=3.10585(approx.)(1.12)^{10} = 3.10585 \quad (\text{approx.})
Therefore, p=13.10585=0.32197(rounded to 4 decimal places)p = \frac{1}{3.10585} = 0.32197 \quad (\text{rounded to 4 decimal places})
Step 4: Calculate Present Value
PV=163,000×0.32197=52,609.11PV = 163,000 \times 0.32197 = 52,609.11
Final Answer:
Flaherty should be willing to pay approximately $52,609 today for this investment.
Explanation (300 words):
The concept of present value (PV) is fundamental in finance and investment decision-making. It represents the current worth of a sum of money to be received in the future, discounted back at a rate that reflects the required return or interest rate.
In this scenario, Flaherty expects to receive $163,000 ten years from now, but because money today is worth more than the same amount in the future (due to inflation, opportunity cost, and risk), we must calculate how much that future amount is worth today. This is done using the present value formula.
The required return rate of 12% per year is the discount rate. It accounts for the risk and opportunity cost of the investment. By discounting the $163,000 back ten years at 12%, we find the amount Flaherty should pay now to earn that expected return.
We used the formula: PV=FV(1+i)nPV = \frac{FV}{(1 + i)^n}
where ii is the interest rate (12%) and nn is the number of years (10). Calculating (1.12)10(1.12)^{10} gave approximately 3.10585, meaning that $1 invested today grows to about $3.11 in 10 years at 12%. To reverse this, the present value factor is the reciprocal, roughly 0.32197.
Multiplying the future value by this factor discounts it to today’s dollars, resulting in $52,609. This is the maximum amount Flaherty should be willing to invest now if she demands a 12% return, since investing more than this would yield less than her required return.
This calculation ensures rational investment decisions by comparing the present cost to the expected future benefits, adjusting for time and risk.
If you want, I can also explain how this calculation would change if the investment involved multiple payments or if the return rate varied!