Corporate Finance Quiz: Annuities And Perpetuities
19 questions · exam conditions
0:00
Annuities And PerpetuitiesQuestion 1 of 19

An employee contributes $5,000 at the end of each year to a retirement account for 10 years. The account earns a stated annual rate of 6%, but interest is compounded semi-annually. What is the future value of the account after 10 years?

$67,176
$66,035
$65,904
$57,319
← Back to quizzes

Corporate Finance Quiz

Corporate Finance Quiz: Annuities And Perpetuities

Practice Annuities And Perpetuities in Corporate Finance with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Annuities And Perpetuities, giving you a quick way to practice the rules, question types, and explanations that matter most for Corporate Finance.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

An employee contributes $5,000 at the end of each year to a retirement account for 10 years. The account earns a stated annual rate of 6%, but interest is compounded semi-annually. What is the future value of the account after 10 years?

  1. $67,176
  2. $66,035 (correct answer)
  3. $65,904
  4. $57,319
Explanation: When the compounding frequency does not match the payment frequency, the Effective Annual Rate (EAR) must be used. The payments are annual, but compounding is semi-annual.\n\nStep 1: Calculate the EAR. EAR = (1 + (APR/m))^m - 1 = (1 + (0.06/2))^2 - 1 = (1.03)^2 - 1 = 1.0609 - 1 = 6.09%.\n\nStep 2: Use the EAR as the interest rate to calculate the future value (FV) of the annual annuity. FV = PMT × [ ((1 + EAR)^N - 1) / EAR ] = $5,000 × [ ((1.0609)^10 - 1) / 0.0609 ] = $5,000 × [ (1.80430 - 1) / 0.0609 ] = $5,000 × 13.2069 = $66,034.50.

Question 2

A company is leasing a piece of equipment for 5 years. The lease agreement requires payments of $20,000 to be made at the beginning of each year. Using a discount rate of 8% APR, compounded annually, what is the present value of these lease payments?

  1. $86,243 (correct answer)
  2. $79,854
  3. $66,242
  4. $100,000
Explanation: When you encounter lease payments or any series of cash flows made at the beginning of each period, you're dealing with an annuity due. The key distinction is timing: these payments occur at the start of each year, not the end. To find the present value of an annuity due, you can use the formula: PV=PMT×1−(1+r)−nr×(1+r)PV = PMT \times \frac{1-(1+r)^{-n}}{r} \times (1+r), where the final (1+r)(1+r) factor accounts for payments occurring at the beginning of each period. With $20,000 annual payments, 8% discount rate, and 5 years: First, calculate the ordinary annuity factor: $1−(1.08)−50.08=1−0.68060.08=3.9927\frac{1-(1.08)^{-5}}{0.08} = \frac{1-0.6806}{0.08} = 3.9927 $ Then multiply by (1+r)(1+r): 3.9927×1.08=4.31213.9927 \times 1.08 = 4.3121 Finally: PV = \20,000 \times 4.3121 = $86,242$ This confirms answer A is correct at $86,243 (slight rounding difference). Answer B (79,854)representsthepresentvalueifthesewereordinaryannuitypaymentsmadeatyear−endinsteadofthebeginning.AnswerC(79,854) represents the present value if these were ordinary annuity payments made at year-end instead of the beginning. Answer C (66,242) appears to use an incorrect discount rate or formula application. Answer D ($100,000) is simply the sum of all payments without any discounting—a common trap that ignores the time value of money entirely. Study tip: Always identify whether cash flows occur at the beginning (annuity due) or end (ordinary annuity) of each period. Lease payments, insurance premiums, and rent typically occur at the beginning, requiring that extra $(1+r)$ multiplier in your calculation.

Question 3

A mature company's dividend per share is expected to be $5.00 next year. Due to declining market share, analysts project that the dividend will decrease by 2% per year indefinitely. If the required rate of return for the company's stock is 10%, what is the estimated value of one share?

  1. $62.50
  2. $50.00
  3. $41.67 (correct answer)
  4. $37.88
Explanation: This scenario describes a growing perpetuity with a negative growth rate. The formula for the present value (PV) of a growing perpetuity is PV = D₁ / (r - g), where D₁ is the dividend next year, r is the required rate of return, and g is the constant growth rate.\n\nGiven: D₁ = $5.00, r = 10% or 0.10, and g = -2% or -0.02.\n\nPV = $5.00 / (0.10 - (-0.02)) = $5.00 / (0.10 + 0.02) = $5.00 / 0.12 = $41.67.

Question 4

An individual will receive a 10-year annuity of $50,000 per year, with the first payment to be received exactly 6 years from today. Assuming a discount rate of 7% per annum, what is the present value of this annuity today?

  1. $351,179
  2. $267,914
  3. $250,387 (correct answer)
  4. $233,995
Explanation: This is a deferred annuity. The calculation requires two steps.\n\nStep 1: Calculate the present value of the ordinary annuity one period before its start. The first payment is at year 6, so we calculate the value at year 5. PV₅ = PMT × [ (1 - (1 + r)^-N) / r ] = $50,000 × [ (1 - (1.07)^-10) / 0.07 ] = $50,000 × 7.02358 = $351,179.13.\n\nStep 2: Discount this lump-sum value from year 5 back to today (year 0). PV₀ = PV₅ / (1 + r)^5 = $351,179.13 / (1.07)^5 = $351,179.13 / 1.40255 = $250,387.11.

Question 5

A company wants to accumulate $500,000 to purchase a new machine. It plans to deposit $40,000 at the end of each year into an account that earns 7% annually. What is the minimum number of full years the company must make these deposits to reach its goal?

  1. 9 years
  2. 30 years
  3. 13 years
  4. 10 years (correct answer)
Explanation: This question tests your understanding of ordinary annuity calculations—specifically, finding the time period needed to reach a future value goal through regular deposits. When you see a problem involving equal periodic payments to accumulate a target amount, you're working with the future value of an ordinary annuity formula. To solve this, you need to find how many years of $40,000 annual deposits at 7% interest will accumulate to at least $500,000. Using the future value of ordinary annuity formula: $FV=PMT×(1+r)n−1rFV = PMT \times \frac{(1+r)^n - 1}{r} ,where$500,000=, where $500,000 = 40,000 × \frac{(1.07)^n - 1}{0.07}$. Solving algebraically: 12.5=(1.07)n−10.0712.5 = \frac{(1.07)^n - 1}{0.07}, which gives us (1.07)n=1.875(1.07)^n = 1.875. Taking the natural log: n=ln⁡(1.875)ln⁡(1.07)≈9.64n = \frac{\ln(1.875)}{\ln(1.07)} ≈ 9.64 years. Since the company can only make deposits for full years and needs at least $500,000, they must deposit for 10 full years to reach their goal. Answer D) 10 years is correct because after 9 years, they'll have accumulated less than $500,000, but after 10 years, they'll exceed their target. Answer A) 9 years falls short—the calculation shows 9.64 years needed, so 9 full years isn't sufficient. Answer C) 13 years significantly overshoots the requirement, representing unnecessary over-saving. Answer B) 30 years is dramatically excessive and likely represents a calculation error or confusion with a different type of problem. Study tip: Always round up when finding the minimum number of periods needed to reach a financial goal, since partial periods typically aren't practical in real business scenarios.

Question 6

A company takes out a loan of $1,000,000 at 8% annual interest. The loan agreement calls for 10 annual end-of-year payments. The first 9 payments are of an equal amount, PMT. The 10th and final payment is PMT + $200,000. What is the value of PMT?

  1. $119,224
  2. $135,223 (correct answer)
  3. $149,029
  4. $160,080
Explanation: The present value of the loan must equal the sum of the present values of all its payments. This loan can be viewed as a standard 10-year annuity of PMT plus an additional single payment (lump sum) of $200,000 at year 10.\n\nLoan PV = (PV of 10-year annuity of PMT) + (PV of 200,000lumpsumatyear10)\n200,000 lump sum at year 10)\n1,000,000 = PMT × [ (1 - (1.08)^-10) / 0.08 ] + 200,000/(1.08)10\n200,000 / (1.08)^10\n1,000,000 = PMT × 6.71008 + 200,000/2.158925\n200,000 / 2.158925\n1,000,000 = PMT × 6.71008 + 92,639.78\n92,639.78\n1,000,000 - 92,639.78=PMT×6.71008\n92,639.78 = PMT × 6.71008\n907,360.22 = PMT × 6.71008\nPMT = $907,360.22 / 6.71008 = $135,223.18.

Question 7

The stock of a company is trading at $50 per share. The company just paid a dividend of $2.00 per share (D₀). It maintains a constant dividend growth rate. If the required rate of return is 12%, what is the implied annual growth rate (g) of the dividends?

  1. 4.00%
  2. 7.69% (correct answer)
  3. 8.00%
  4. 8.33%
Explanation: The Gordon Growth Model (constant growth model) for stock valuation is P₀ = D₁ / (r - g). The dividend D₁ is the dividend expected next year, which is D₀(1+g). We are given D₀, the dividend just paid.\n\nP₀ = D₀(1 + g) / (r - g)\n$50 = 2.00(1+g)/(0.12−g)\n2.00(1 + g) / (0.12 - g)\n50(0.12 - g) = 2.00(1+g)\n2.00(1 + g)\n6.00 - 50g = 2.00+2g\n2.00 + 2g\n4.00 = 52g\ng = $4.00 / 52 = 0.07692, or 7.69%.

Question 8

A rental property generates $2,000 in cash flow at the exact middle of each month, for 12 months. Assuming a nominal annual interest rate of 12% compounded monthly, what is the present value of this stream of cash flows?

  1. $22,735
  2. $22,622 (correct answer)
  3. $22,510
  4. $22,400
Explanation: This is a non-standard annuity where payments occur in the middle of each period. The monthly interest rate is 12% / 12 = 1%. We can solve this by first calculating the present value of a standard ordinary annuity and then adjusting for the timing.\n\nStep 1: Calculate the PV of a 12-month ordinary annuity of $2,000 at 1% per month. This formula values the payments as if they occurred at the end of each month (t=1, 2, ..., 12). PV_ord = $2,000 × [ (1 - (1.01)^-12) / 0.01 ] = $2,000 × 11.2551 = $22,510.16.\n\nStep 2: The actual cash flows occur at t=0.5, 1.5, ..., 11.5. This is half a period earlier than the standard annuity calculated in Step 1. To find the present value, we must compound the PV_ord forward by half a period. PV = PV_ord × (1.01)^0.5 = $22,510.16 × 1.00498756 = $22,622.35.

Question 9

A company's next dividend (at year 1) is expected to be $2.00. Dividends are expected to grow at 15% per year for the following 2 years (i.e., for years 2 and 3). After year 3, the dividend will grow at a constant rate of 4% per year in perpetuity. If the required rate of return is 10%, what is the current value of a share of stock?

  1. $37.02
  2. $38.82
  3. $40.15 (correct answer)
  4. $43.81
Explanation: This is a two-stage dividend discount model problem. \nStep 1: Calculate the dividends for years 1, 2, and 3. D₁ = $2.00; D₂ = $2.00(1.15) = $2.30; D₃ = $2.30(1.15) = $2.645. \nStep 2: Calculate the present value of these three dividends. PV(Divs) = $2.00/1.10 + $2.30/1.10² + $2.645/1.10³ = $1.818 + $1.901 + $1.987 = $5.706. \nStep 3: Calculate the terminal value (the value of the growing perpetuity) at year 3. First, find D₄ = D₃(1+g_stable) = $2.645(1.04) = $2.7508. The terminal value at year 3 is P₃ = D₄ / (r - g_stable) = $2.7508 / (0.10 - 0.04) = $45.847. \nStep 4: Discount the terminal value back to year 0. PV(P₃) = $45.847 / (1.10)³ = $34.445. \nStep 5: Sum the present values. Total Value = $5.706 + $34.445 = $40.151.

Question 10

An investor deposits $10,000 at the end of each year for 10 years. For the first 4 years, the account earns 5% annual interest. For the remaining 6 years, the account earns 8% annual interest. What is the total value of the account at the end of 10 years?

  1. $116,460
  2. $136,900
  3. $141,757 (correct answer)
  4. $144,866
Explanation: This problem involves an annuity with a change in interest rate. We must calculate the future value (FV) in two separate parts. \nPart 1: The first 4 deposits at 5%. \nStep 1: Find the FV of the first 4 deposits at the end of year 4. FV₄ = $10,000 × [((1.05)⁴ - 1) / 0.05] = $10,000 × 4.310125 = $43,101.25. \nStep 2: Compound this amount forward for the remaining 6 years at the new 8% rate. FV₄_at_₁₀ = $43,101.25 × (1.08)⁶ = $68,397.74. \nPart 2: The last 6 deposits at 8%. \nStep 3: Find the FV of the annuity of the last 6 deposits at the 8% rate. FV_last₆ = $10,000 × [((1.08)⁶ - 1) / 0.08] = $10,000 × 7.335929 = $73,359.29. \nStep 4: Add the results. Total FV = $68,397.74 + $73,359.29 = $141,757.03.

Question 11

A person wishes to have $1,000,000 in a retirement fund on their 65th birthday. They will begin making equal annual contributions on their 35th birthday and will make the last contribution on their 65th birthday. The fund is expected to earn 9% annually. What is the required annual contribution?

  1. $5,959 (correct answer)
  2. $6,495
  3. $6,731
  4. $7,336
Explanation: The contributions represent an annuity due, as the first payment is made at the beginning of the investment period (35th birthday) and the last is on the target date (65th birthday). The number of payments is 65 - 35 + 1 = 31 payments. \nWe use the future value of an annuity due formula: FV = PMT × [ ((1+r)^N - 1) / r ] × (1+r).\n1,000,000=PMT×[((1.09)31−1)/0.09]×(1.09).\n1,000,000 = PMT × [ ((1.09)^31 - 1) / 0.09 ] × (1.09).\n1,000,000 = PMT × [ (14.8566 - 1) / 0.09 ] × 1.09.\n1,000,000=PMT×[153.9622]×1.09.\n1,000,000 = PMT × [ 153.9622 ] × 1.09.\n1,000,000 = PMT × 167.8188.\nPMT = $1,000,000 / 167.8188 = $5,958.80.

Question 12

A foundation plans to fund an annual scholarship of $30,000 in perpetuity. The first scholarship payment will be awarded 4 years from today. If the foundation's endowment earns a return of 6% annually, how much money must be in the endowment today to fund this scholarship?

  1. $500,000
  2. $445,000
  3. $419,810 (correct answer)
  4. $396,047
Explanation: This is a deferred perpetuity. The calculation requires two steps.\n\nStep 1: Calculate the present value of the perpetuity one period before its start. The first payment is at year 4, so we calculate the value at year 3. The formula for the PV of a perpetuity is P / r. PV₃ = $30,000 / 0.06 = $500,000.\n\nStep 2: Discount this lump-sum value from year 3 back to today (year 0). PV₀ = PV₃ / (1 + r)³ = $500,000 / (1.06)³ = $500,000 / 1.191016 = $419,809.92.

Question 13

A project is expected to generate cash flows of $100,000 at the end of the first year. The cash flows are projected to grow at a rate of 4% per year for a total of 15 years. If the appropriate discount rate is 9%, what is the present value of these cash flows?

  1. $2,000,000
  2. $1,450,920
  3. $1,000,878 (correct answer)
  4. $806,070
Explanation: This is a growing annuity problem. The formula for the present value (PV) of a growing annuity is: PV = C₁/(r - g) × [1 - ((1 + g)/(1 + r))^N].\n\nGiven: C₁ = 100,000,r=9100,000, r = 9%, g = 4%, N = 15.\n\nPV = (100,000 / (0.09 - 0.04)) × [1 - ((1.04 / 1.09)^15)]\nPV = ($100,000 / 0.05) × [1 - (0.954128)^15]\nPV = $2,000,000 × [1 - 0.499561]\nPV = $2,000,000 × 0.500439 = $1,000,878.

Question 14

A philanthropist has an investment that will pay her $100,000 at the end of each year for the next 20 years. She plans to use the future value of this 20-year annuity to endow a perpetual scholarship fund. The first scholarship will be awarded at the end of year 21. If both the annuity and the perpetuity endowment earn an interest rate of 5% per year, what annual scholarship amount can be provided forever?

  1. $100,000
  2. $50,000
  3. $157,457
  4. $165,330 (correct answer)
Explanation: This problem tests your understanding of time value of money concepts, specifically how annuities and perpetuities interact across different time periods. When you see questions involving both accumulation and distribution phases, you need to carefully track the timing and apply the appropriate formulas sequentially. First, calculate the future value of the 20-year annuity at the end of year 20. Using the future value of annuity formula: FV=PMT×(1+r)n−1rFV = PMT \times \frac{(1+r)^n - 1}{r}, where PMT = $100,000, r = 5%, and n = 20 years. This gives us: $FV = 100,000 \times \frac{(1.05)^{20} - 1}{0.05} = 100,000 \times 33.066 = \3,306,600 This accumulated amount becomes the principal for the perpetuity starting in year 21. For a perpetuity, the annual payment equals the principal times the interest rate: Annual scholarship = $3,306,600 × 0.05 = $165,330. Answer A (100,000)incorrectlyassumestheannualscholarshipequalstheoriginalannuitypayment,ignoringcompoundgrowth.AnswerB(100,000) incorrectly assumes the annual scholarship equals the original annuity payment, ignoring compound growth. Answer B (50,000) appears to arbitrarily halve the original payment without any mathematical basis. Answer C ($157,457) likely results from a calculation error, possibly in computing the future value factor or applying an incorrect perpetuity formula. The key insight is recognizing this as a two-stage problem: first accumulate the annuity payments with compound interest, then determine what perpetual payment that accumulated amount can support. Always identify whether you're in the accumulation phase (building wealth) or distribution phase (spending wealth), as the formulas and timing differ significantly.

Question 15

Consider a growing annuity with a positive initial cash flow, a fixed term, a growth rate (g) that is positive but less than the discount rate (r). If the growth rate g were to increase, but still remain less than r, what would be the effect on the annuity's present value (PV)?

  1. The PV will decrease because the denominator (r-g) increases.
  2. The PV will remain unchanged as the changes in the numerator and denominator of the formula cancel each other out.
  3. The effect on PV is ambiguous because the denominator (r-g) decreases while the discount factor ((1+g)/(1+r)) also decreases.
  4. The PV will increase because later cash flows are larger and the (r-g) denominator is smaller. (correct answer)
Explanation: When you encounter growing annuity questions, focus on how changes in parameters affect both the cash flows themselves and the present value formula components. A growing annuity's present value formula is: PV=C1r−g[1−(1+g1+r)n]PV = \frac{C_1}{r-g} \left[1 - \left(\frac{1+g}{1+r}\right)^n\right], where C1C_1 is the first cash flow, rr is the discount rate, gg is the growth rate, and nn is the number of periods. When the growth rate increases (while staying below rr), two key effects occur. First, all future cash flows become larger since each payment grows at a higher rate. Second, the denominator (r−g)(r-g) becomes smaller, which increases the overall multiplier effect. Both changes work in the same direction to increase present value. Let's examine why the other answers miss the mark. Choice A incorrectly states that (r−g)(r-g) increases when gg rises - this is backwards since a larger gg makes (r−g)(r-g) smaller. Choice B suggests the effects cancel out, but this ignores that both the larger cash flows and smaller (r−g)(r-g) denominator push present value in the same positive direction. Choice C claims the effect is ambiguous, but while the discount factor (1+g1+r)n\left(\frac{1+g}{1+r}\right)^n does increase (making the bracketed term smaller), this effect is dominated by the combination of larger cash flows and the smaller (r−g)(r-g) denominator. The correct answer is D because both mechanisms - larger future cash flows and a smaller (r−g)(r-g) denominator - work together to increase present value. Study tip: For growing annuity questions, always trace through how parameter changes affect both the cash flows and the mathematical components separately, then consider their combined impact.

Question 16

An investor is choosing between two options. Option A offers a guaranteed 20-year annuity of $10,000 per year, with the first payment in one year. Option B offers a single lump-sum payment of $100,000 today. The investor can earn 8% per year on other investments of similar risk. Which option should the investor choose, and what is the approximate difference in their present values?

  1. Choose Option A; its PV is $6,036 greater.
  2. Choose Option A; its undiscounted cash flows are $100,000 greater.
  3. Choose Option B; its PV is $3,964 greater.
  4. Choose Option B; its PV is $1,819 greater. (correct answer)
Explanation: When comparing investment alternatives with different payment timing, you need to calculate the present value (PV) of each option using the appropriate discount rate to make an apples-to-apples comparison. Option B is straightforward: $100,000 today has a present value of exactly $100,000. For Option A, you need to find the present value of a 20-year ordinary annuity (payments at year-end) of $10,000 using the 8% discount rate. Using the annuity formula: $PV=PMT×1−(1+r)−nrPV = PMT \times \frac{1-(1+r)^{-n}}{r} $ Where PMT = 10,000, r = 0.08, and n = 20: $$PV = 10,000 \times \frac{1-(1.08)^{-20}}{0.08} = 10,000 \times 9.8181 = 98,181$$ Since Option B (100,000)exceedsOptionA(100,000) exceeds Option A (98,181) by $1,819, you should choose Option B. Answer choice A incorrectly suggests Option A has the higher present value and uses a completely wrong difference amount. Answer choice B makes the classic mistake of comparing undiscounted total cash flows ($200,000 for Option A vs. $100,000 for Option B) rather than present values—this ignores the time value of money entirely. Answer choice C correctly identifies Option B as superior but uses an incorrect difference of $3,964, likely from a calculation error. Remember: Never compare investment alternatives by looking at undiscounted cash flows. Always convert everything to present value using the appropriate discount rate. The time value of money means that $10,000 received years from now is worth less than $10,000 today.

Question 17

An ordinary annuity and an annuity due have the same number of identical payments and the same positive interest rate. Which of the following statements correctly describes their relationship?

  1. The future value of the annuity due is higher because each payment earns interest for one additional period. (correct answer)
  2. Their future values are identical because the total principal paid is the same for both.
  3. The present value of the annuity due is always less than the present value of the ordinary annuity.
  4. The difference in their present values is equal to the last payment discounted to the present.
Explanation: When comparing annuities in corporate finance, timing of payments fundamentally affects their values. An ordinary annuity makes payments at the end of each period, while an annuity due makes payments at the beginning of each period. For future values, this timing difference is crucial. In an annuity due, each payment gets deposited one period earlier than in an ordinary annuity, giving every payment exactly one additional period to earn interest. If you have a 5-year annuity due versus a 5-year ordinary annuity, the first payment in the annuity due earns interest for all 5 years, while the first payment in the ordinary annuity only earns interest for 4 years. This pattern holds for every payment, making the annuity due's future value consistently higher. Choice A correctly identifies this relationship—the annuity due's future value is higher because each payment compounds for one additional period. Choice B is wrong because identical principal amounts don't guarantee identical future values when timing differs. The extra compounding period matters significantly. Choice C incorrectly states that annuity due present values are always lower. Actually, since annuity due payments occur sooner, they require less discounting, making their present value higher than ordinary annuities. Choice D mischaracterizes the present value difference. The difference isn't simply the last payment discounted—it's more complex, involving the timing shift of all payments. Remember this pattern: annuity due values (both present and future) are always higher than ordinary annuity values when all other factors are equal, due to the favorable timing of earlier payments.

Question 18

An investment is expected to pay $1,000 one year from today. After that, it will pay $500 per year in perpetuity. If the discount rate is 8%, what is the present value of this investment?

  1. $12,500
  2. $6,713 (correct answer)
  3. $6,284
  4. $6,250
Explanation: This stream of cash flows can be broken down into two components: an extra payment of $500 at year 1 and a standard perpetuity of $500 starting at year 1.\n\nComponent 1: A perpetuity of $500 starting in one year. The present value (PV) is P / r = $500 / 0.08 = $6,250.\n\nComponent 2: An additional one-time payment of $500 at year 1 (since the first payment is $1,000, which is $500 more than the perpetual payment). The PV of this single payment is $500 / (1.08)¹ = $462.96.\n\nTotal PV = PV of perpetuity + PV of extra payment = $6,250 + $462.96 = $6,712.96.

Question 19

A company is offered a loan of $100,000. The terms require the company to make 5 equal annual payments of $25,046 at the end of each year. What is the implied annual interest rate (r) of this loan?

  1. 5.0%
  2. 7.0%
  3. 8.0% (correct answer)
  4. 9.0%
Explanation: To find the implied interest rate of the loan, we need to solve for 'r' in the present value of an ordinary annuity formula: PV = PMT × [ (1 - (1 + r)^-N) / r ].\n\nHere, PV = $100,000, PMT = 25,046,andN=5.\n\n25,046, and N = 5.\n\n100,000 = $25,046 × PVIFA(r, 5). Therefore, the Present Value Interest Factor of an Annuity (PVIFA) is $100,000 / $25,046 = 3.99265.\n\nUsing a financial calculator or a PVIFA table, we can find the interest rate that corresponds to this factor for N=5. Testing r = 8%: PVIFA(8%, 5) = (1 - (1.08)^-5) / 0.08 = 3.9927. This matches the calculated factor, so the implied annual interest rate is 8.0%.