What would you pay today for an investment that makes ten annual payments of $2,500 and an additional lump sum payment of $10,000 at the end of the tenth year if the interest rate is 8% compounded semi-annually.

Question

BlackTom AI · Accepted Answer

Nominal 8% compounded semi-annually converts to an effective annual rate of (1 + 0.08/2)^2 − 1 = (1.04)^2 − 1 = 0.0816 = 8.16%. Use this effective annual rate because payments are annual.

1) Relevant knowledge points and formulas
- Convert nominal rate compounded m times per year to effective annual rate: (1 + r/m)^m − 1.
- Present value of an n-year level annuity immediate: PVannuity = PMT * [1 − (1 + i)^−n] / i.
- Present value of a single future lump sum: PVlump = FV * (1 + i)^−n.
- Combine PVs because of linearity.

2) Step-by-step reasoning and calculations
- Periodic semiannual rate: 0.08/2 = 0.04. Effective annual rate i = (1.04)^2 − 1 = 0.0816.
- Compute (1 + i)^10 = (1.0816)^10.
  Calculate (1.0816)^5 = 1.480244285776 (intermediate multiplication shown by successive powers).
  Then (1.0816)^10 = (1.480244285776)^2 = 2.191123085576.
  So discount factor for 10 years v^10 = (1 + i)^−10 = 1 / 2.191123085576 = 0.4563874989.
- Annuity factor: [1 − v^10] / i = (1 − 0.4563874989) / 0.0816 = 0.5436125011 / 0.0816 = 6.661919.
- Present value of the ten annual $2,500 payments: 2,500 * 6.661919 = 16,654.7975.
- Present value of the $10,000 lump sum at year 10: 10,000 * 0.4563874989 = 4,563.874989.
- Total present value = 16,654.7975 + 4,563.874989 = 21,218.672489 ≈ $21,218.68.

3) Why the correct answer is correct
Because the nominal 8% compounded semi-annually was correctly converted to an effective annual rate (8.16%), each annual cash flow and the year-10 lump sum were discounted by (1.0816)^t. The algebra and numeric evaluation above yield the present value $21,218.68.

4) Why the other choices are incorrect
- $19,778.17 and $20,854.82 are too low — they likely used 8% as an effective annual rate (ignoring semiannual compounding) or miscomputed the annuity factor.
- $22,309.39 and $22,979.25 are too high — they likely used a lower discount rate or mis-applied the lump-sum discounting.

Conclusion: The calculated present value rounded to cents is $21,218.68, matching the provided correct answer.

What would you pay today for an investment that makes ten annual payments of $2,500 and an additional lump sum payment of $10,000 at the end of the tenth year if the interest rate is 8% compounded semi-annually.单项选择题

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