Final answer:
Calculating the present value of $500,000 to be paid in five annual instalments of $100,000 each at a 10% discount rate confirms that the parish officials' determination of the present value being $416,987 is true.
Step-by-step explanation:
To establish whether the statement "The parish uses a discount rate of 10% to evaluate all long term projects, and officials determined the present value of the eventual payments to be $416,987" is true or false, we would need to calculate the present value of the $500K settled to be paid in five annual installments of $100K each, starting immediately, using your given discount rate of 10%.
Present Value (PV) is calculated using the formula:
PV = P / (1 + r)^n
Where:
- P = payment amount
- r = discount rate
- n = period number
In this case, because the payments start immediately, the first payment does not need to be discounted. Hence, the PV for the first installment is simply $100K. For the remaining four payments, we discount them back to their present value.
Calculating this:
- PV1 = $100,000 / (1 + 0.10)^0 = $100,000
- PV2 = $100,000 / (1 + 0.10)^1 = $90,909.09
- PV3 = $100,000 / (1 + 0.10)^2 = $82,644.63
- PV4 = $100,000 / (1 + 0.10)^3 = $75,131.48
- PV5 = $100,000 / (1 + 0.10)^4 = $68,301.35
Summing these amounts gives us the total present value:
Total PV = PV1 + PV2 + PV3 + PV4 + PV5 = $100,000 + $90,909.09 + $82,644.63 + $75,131.48 + $68,301.35
Total PV = $416,986.55 (rounded to $416,987)
Therefore, the statement concerning the officials determining the "present value" of the eventual payments to be $416,987 is true.