An annuity pays $100 per year for 20 years, with the first payment starting 2 years from now. What is the present value (PV) of this annuity (rounded to the closest dollar), given that the discount rate is 5%?[Fill in the blank]单项选择题
A
A. $1,169.0
B
B. $1,130.00
C
C. $3,307.00
D
D. $1,187.00
E
E. $1,246.00

Question

An annuity pays $100 per year for 20 years, with the first payment starting 2 years from now. What is the present value (PV) of this annuity (rounded to the closest dollar), given that the discount rate is 5%?[Fill in the blank]单项选择题

A

A. $1,169.0

B

B. $1,130.00

C

C. $3,307.00

D

D. $1,187.00

E

E. $1,246.00

BlackTom AI · Accepted Answer

The value of a 20-year level annuity deferred one year (first payment at t = 2) is obtained by taking the standard present-value factor for 20 payments and discounting that one extra year to today.

1) Relevant formulas and concepts
- Present value of an ordinary annuity (payments of PMT at t = 1,2,...,n) is PV = PMT * [1 − (1 + r)^(−n)] / r.
- If the first payment is deferred and occurs at t = k+1, you multiply the ordinary-annuity PV by v^k where v = 1/(1 + r) (i.e., discount by k extra years). Here the first payment is at t = 2, so k = 1 and v = 1/1.05.

2) Step-by-step reasoning
- PMT = $100, n = 20, r = 0.05.
- Compute the annuity-immediate factor for 20 years at 5%:
  a_{\overline{20}|} = [1 − (1.05)^(−20)] / 0.05.
- Evaluate (1.05)^(−20). Using the standard power value gives (1.05)^20 ≈ 2.653297705, so (1.05)^(−20) ≈ 0.376889483.
- Then 1 − (1.05)^(−20) = 0.623110517.
- Divide by r = 0.05: a_{\overline{20}|} = 0.623110517 / 0.05 = 12.46221034.
- Multiply by PMT to get the PV one period before the first payment (i.e., at t = 1): PV_{t=1} = 100 * 12.46221034 = $1,246.221034.

3) Discount to time 0 (today)
- Because the first payment is at t = 2, discount PV_{t=1} back one more year to t = 0:
  PV_{t=0} = PV_{t=1} / (1 + r) = 1,246.221034 / 1.05.
- Compute the division: 1,246.221034 / 1.05 ≈ $1,169.00 (rounded to the nearest dollar).

4) Why this is correct
- The method uses the standard annuity-immediate PV factor for the full 20 payments (which gives the value one period before the first payment) and then discounts that single-period difference to today. All intermediate algebra follows directly from the annuity PV formula and the time-value-of-money discounting principle.

5) Why the other choices are incorrect
- B ($1,130.00) would correspond to discounting the annuity two full years instead of one or using a significantly higher discounting factor.
- C ($3,307.00) is far too large; it resembles the PV of a much larger payment stream or a perpetuity.
- D ($1,187.00) is close to the undiscounted annuity-immediate PV (or results from slightly different intermediate rounding), but it does not reflect the required one-year additional discount back to t = 0.
- E ($1,246.00) is the PV one period before the first payment (i.e., at t = 1), not the PV today.

Conclusion: Following the annuity-immediate formula and discounting the one extra year gives a present value of approximately $1,169, so the correct choice is A.

An annuity pays $100 per year for 20 years, with the first payment starting 2 years from now. What is the present value (PV) of this annuity (rounded to the closest dollar), given that the discount rate is 5%?[Fill in the blank]单项选择题

类似问题

The Good Life Insurance Co. wants to sell you an annuity which will pay you $770 per quarter for 25 years. You want to earn a minimum rate of return of 6.2 percent. What is the most you are willing to pay as a lumpsum today to buy this annuity?

You are scheduled to receive annual payments of $10,100 for each of the next 23 years. Your discount rate is10 percent. What is the difference in the present value if you receive these payments at the beginning ofeach year rather than at the end of each year?

What is the value today of $26,839 payments per year, at a discount rate of 5% p.a. if the first payment is received eight years from now and the final payment is received thirty years from now? Please write the answer in two decimal places.

Peter paid annual instalments of $1,000 on 15 March 2010 for 5 years. The effective annual interest rate is 6%. Calculate the value of the loan on 15 March 2005.

George will receive a study excellence award of $10,000 per year at the end of the next 3 years. Calculate the present value today of George's cash flows if the effective annual interest rate is 10%.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单