Jesse has been considering an investment opportunity that will pay $20,000 at the end of each of the next 7 years. Jesse must invest $20,000 at the start of the first year and an additional $20,000 at the end of the third year. What would the present value of this investment (to the nearest dollar) if the interest rate is 5%?[Fill in the blank]单项选择题
A
A. $95,728.00
B
B. $78,451.00
C
C. $77,857.00
D
D. $113,004.00
E
E. $118,451.00

Question

Jesse has been considering an investment opportunity that will pay $20,000 at the end of each of the next 7 years. Jesse must invest $20,000 at the start of the first year and an additional $20,000 at the end of the third year. What would the present value of this investment (to the nearest dollar) if the interest rate is 5%?[Fill in the blank]单项选择题

A

A. $95,728.00

B

B. $78,451.00

C

C. $77,857.00

D

D. $113,004.00

E

E. $118,451.00

BlackTom AI · Accepted Answer

The net present value is the present value (PV) of the seven end-of-year receipts minus the PV of the required investments. Use the PV formula for an ordinary annuity for the receipts and discount each investment to time 0.

1) Relevant concepts
- PV of an ordinary annuity: PV = C * [(1 - (1+r)^-n)/r].
- Discounting a single future payment: PV = FV/(1+r)^t.
- Net PV = PV(receipts) − PV(outflows).

2) Step-by-step reasoning and calculations
- Receipts: $20,000 at the end of each year for 7 years (C = 20,000, r = 0.05, n = 7).
  Compute (1.05)^7 ≈ 1.40710042265625, so (1.05)^-7 ≈ 0.710681286.
  PV(receipts) = 20,000 * [(1 − 0.710681286)/0.05] = 20,000 * 5.78637428 ≈ 115,727.49.

- Outflow 1: $20,000 at the start of the first year = time 0, so PV = 20,000.

- Outflow 2: $20,000 at the end of year 3. Compute (1.05)^3 = 1.157625 so PV = 20,000 / 1.157625 ≈ 17,276.75.

- Total PV of outflows = 20,000 + 17,276.75 = 37,276.75.

3) Net present value (all intermediate steps shown)
- Net PV = PV(receipts) − PV(outflows)
- Net PV = 115,727.49 − 37,276.75 = 78,450.74
- Rounded to the nearest dollar: $78,451.

4) Why this answer is correct
The calculation directly applies standard time-value-of-money formulas: the receipts form an ordinary annuity and both investments are single cash flows discounted to time 0. Subtracting the correctly discounted outflows from the annuity PV yields $78,450.74, which rounds to $78,451.

5) Why other choices are incorrect (brief)
- A ($95,728), C ($77,857), D ($113,004), E ($118,451) do not match the correct net PV computed using the annuity formula and correct discounting; they are either missing the subtraction of one or both outflows or use incorrect discounting/rounding.

6) Conclusion
The correct present value (to the nearest dollar) is $78,451, matching answer B.

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