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.Single choice

Similar Questions

Jacqueline deposited $20,000 at 9% effective annual interest rate. She will make equal withdrawals at the end of each year over a 6-year term. What would be the amount of equal withdrawal?

An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?

You want to receive 5,000 per month in retirement. If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at the beginning of retirement?

Which answer choice explains why furanoses and pyranoses can be reduced by sodium borohydride despite having no carbonyl functional groups.

Which of the following statements describes a glycoside?

Which of the following structures represents chitin?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler