Salvatore has the opportunity to invest in a scheme which will pay $5000 at the end of each of the next 5 years. He must invest $10,000 at the start of the first year and an additional $10,000 at the end of the first year. What is the present value of this investment if the interest rate is 3%?

Question

BlackTom AI · Accepted Answer

Net present value at time 0 using a 3% discount rate is the key concept. Compute the present value (PV) of the five $5,000 year-end receipts, subtract the two investments (one at time 0 and one at time 1 discounted to time 0).

1) Relevant knowledge points and concepts
- Present value (PV) discounts future payments by (1 + i)^t.
- PV of an n-period level annuity (payments at year-ends) is PV = R * [(1 − (1+i)^(−n)) / i].
- A payment at the end of year 1 is discounted by (1+i)^1 to get its PV at time 0.

2) Complete reasoning process step by step
- Compute PV of the five $5,000 receipts (years 1–5) at 3%:
  (1.03)^5 = 1.1592740743
  (1.03)^(−5) = 1 / 1.1592740743 = 0.862608784
  Annuity factor = (1 − 0.862608784) / 0.03 = 0.137391216 / 0.03 = 4.5797072
  PV_annuity = 5,000 * 4.5797072 = 22,898.536

- PV of the second investment of $10,000 paid at end of year 1:
  PV_payment_t1 = 10,000 / 1.03 = 9,708.737864

- The first $10,000 is invested at the start of the first year (time 0), so its PV is simply −10,000.

3) All intermediate calculation steps shown above.

4) Why the correct answer is correct
Net PV = PV_receipts − PV_investments = 22,898.536 − 9,708.737864 − 10,000
Net PV = 22,898.536 − 19,708.737864 = 3,189.798136 ≈ $3,189.80
This positive value shows the inflows, discounted to time 0, exceed the two investments; the arithmetic yields $3,189.80.

5) Why other options are incorrect
- $5,907.57 (or −$5,907.57) would result from incorrect discounting (for example, failing to discount the year-1 investment to time 0 or mis-evaluating the annuity factor).
- −$3,189.80 is simply the wrong sign; it would imply net outflow rather than net positive PV.

6) Conclusion
The calculated net present value is $3,189.80, which matches the provided correct answer $3189.80.

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