Finance Quiz: Loan Amortization
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Loan AmortizationQuestion 1 of 20

A loan has a fixed monthly payment of $1,000. For the current month's payment, 70% is allocated to interest and 30% is allocated to principal. Assuming the monthly interest rate is 0.5%, what is the outstanding principal balance on the loan before the current payment is made?

$60,000
$140,000
$200,000
$30,000
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Finance Quiz

Finance Quiz: Loan Amortization

Practice Loan Amortization in 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 Loan Amortization, giving you a quick way to practice the rules, question types, and explanations that matter most for Finance.

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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.

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Question 1

A loan has a fixed monthly payment of $1,000. For the current month's payment, 70% is allocated to interest and 30% is allocated to principal. Assuming the monthly interest rate is 0.5%, what is the outstanding principal balance on the loan before the current payment is made?

  1. $60,000
  2. $140,000 (correct answer)
  3. $200,000
  4. $30,000
Explanation: This problem requires working backward from the payment composition.\n
  1. Calculate the dollar amount of interest paid: The total payment is $1,000, and 70% of it is interest. Interest paid = $1,000 × 0.70 = $700.
  2. Calculate the principal portion: Principal paid = $1,000 × 0.30 = $300.
  3. Calculate the beginning balance: The interest paid for any period is calculated on the beginning balance for that period. Interest Paid = Beginning Balance × Periodic Rate. We can rearrange this to find the balance.\n Beginning Balance = Interest Paid / Periodic Rate.\n Beginning Balance = $700 / 0.005 = $140,000.\n\nDistractor Analysis:\n* A: 60,000isthebalanceifyouincorrectlyusetheprincipalportion(60,000 is the balance if you incorrectly use the principal portion (300) in the numerator: $300 / 0.005.\n* C: $200,000 is the total loan if the payment were interest only (1000/0.005). This ignores the principal component.\n* D: $30,000 is not directly calculable but represents a misunderstanding of the relationship between payment components and the loan balance.

Question 2

A loan's amortization schedule shows that for the 24th monthly payment, the interest portion is $800 and the principal portion is $400. For the 25th monthly payment, the principal portion will be $402. What is the annual interest rate of the loan?

  1. 4.8%
  2. 5.4%
  3. 6.0% (correct answer)
  4. 7.2%
Explanation: The key insight is that the increase in the principal portion from one payment to the next is equal to the interest saved on the principal that was paid in the earlier payment. \n\nIncrease in principal = (P_{25} - P_{24} = 402−402 - 400 = 2\). \nThis $2 increase is the interest that is *not* being charged in month 25 because the principal was reduced by \(P_{24} = 400) in month 24. \nTherefore, Interest saved=P24×monthly rate\text{Interest saved} = P_{24} \times \text{monthly rate}. \n(2=2 = 400 \times \text{monthly rate}). \n(\text{Monthly rate} = 2/2 / 400 = 0.005). \n\nTo find the annual rate, multiply the monthly rate by 12: \nAnnual rate=0.005×12=0.06=6.0\text{Annual rate} = 0.005 \times 12 = 0.06 = 6.0%. \n\nDistractor Analysis:\n* A: 4.8% (0.4% monthly) would mean an increase in principal of $400 * 0.004 = $1.60.\n* B: 5.4% (0.45% monthly) would mean an increase in principal of $400 * 0.0045 = $1.80.\n* D: 7.2% (0.6% monthly) would mean an increase in principal of $400 * 0.006 = $2.40.

Question 3

If a borrower makes bi-weekly payments on a standard monthly-amortizing loan, and each bi-weekly payment is exactly half of the calculated monthly payment, which of the following is the primary reason this strategy reduces the total interest paid?

  1. The interest rate is effectively lowered because payments are made more frequently.
  2. The bank charges fewer administrative fees for bi-weekly payment plans.
  3. Making 26 bi-weekly payments per year results in the equivalent of one extra monthly payment annually. (correct answer)
  4. The principal balance is reduced more rapidly within each month, leading to lower interest accrual.
Explanation: The primary advantage of a typical bi-weekly payment plan comes from making an extra payment each year, which accelerates principal reduction.\nThere are 52 weeks in a year, which means 26 bi-weekly periods. If each payment is half of a monthly payment, the borrower pays 26 * (0.5 * Monthly PMT) = 13 * Monthly PMT per year, instead of the standard 12. This one extra monthly payment is applied directly to the principal, which significantly shortens the loan term and reduces the total interest paid over the life of the loan.\n\nDistractor Analysis:\n* A: The nominal interest rate on the loan does not change. While more frequent compounding can change the effective rate, the loan's contractual rate remains the same.\n* B: This is unrelated to the mechanics of loan amortization and is not the mathematical reason for the interest savings.\n* D: While there is a very minor benefit from reducing principal mid-month, this effect is negligible compared to the impact of the extra annual payment. Most standard mortgages still calculate interest monthly, so this benefit may not even be realized.

Question 4

A corporation takes out a 10-year, $2 million loan at an 8% annual interest rate, with monthly payments. After exactly 4 years, prevailing interest rates drop. The corporation refinances the remaining balance for the remaining 6 years at a new annual rate of 6%. What will be the new monthly payment after refinancing?

  1. $20,285.45 (correct answer)
  2. $21,795.53
  3. $22,452.87
  4. $24,242.60
Explanation: This requires a three-step calculation.\n\n1. Calculate the original monthly payment: N = 120 (10 * 12), I/Y = 8 / 12 = 0.6667%, PV = 2,000,000, FV = 0. Solving for PMT gives -$24,265.45. (Note: using a high-precision calculator is key). Using the formula gives PMT=2M×0.08/121−(1+0.08/12)−120=24,265.45PMT = 2M \times \frac{0.08/12}{1 - (1+0.08/12)^{-120}} = 24,265.45
  1. Find the remaining balance after 4 years (48 payments): There are 6 years (72 payments) remaining on the original loan. N = 72, I/Y = 8 / 12, PMT = -24,265.45, FV = 0. Solving for PV gives the remaining balance of $1,378,633.24.
  2. Calculate the new payment on the refinanced balance: The new loan terms are: PV = 1,378,633.24, N = 72 (remaining 6 years), I/Y = 6 / 12 = 0.5%. Solving for PMT gives $20,285.45.\n\nDistractor Analysis:\n* B: $21,795.53 is the result of incorrectly calculating the remaining balance after 4 years by simply taking 60% of the original principal (2M * 0.6 = 1.2M) and then amortizing that over 72 months at 6%.\n* C: 22,452.87istheresultifyouamortizethecorrectremainingbalance(22,452.87 is the result if you amortize the correct remaining balance (1,378,633.24) over the original remaining term (72 months) but forget to change the interest rate from 8% to 6%.\n* D: $24,242.60 is the original monthly payment, which is incorrect as the loan has been refinanced at a lower rate.

Question 5

A fully amortizing loan of $50,000 has a term of 5 years and a fixed annual interest rate of 7.2%, compounded monthly. For the first payment, the principal portion is $P_1andtheinterestportionisand the interest portion isI_1.Forthesecondpayment,theprincipalportionis. For the second payment, the principal portion is P_2.Whatistherelationshipbetween. What is the relationship between P_1andandP_2$?

  1. P2=P1+(I1×0.006)P_2 = P_1 + (I_1 \times 0.006)
  2. P2=P1×(1+0.006)P_2 = P_1 \times (1 + 0.006) (correct answer)
  3. P2=P1+I1P_2 = P_1 + I_1
  4. P2=P1P_2 = P_1
Explanation: The principal portion of each subsequent payment increases by a factor of (1 + periodic interest rate). The interest saved from the principal reduction in the first payment is applied to the principal portion of the second payment.\n\nLet ii be the monthly interest rate (0.072 / 12 = 0.006). Let PMT be the monthly payment. \nI1=PV×iI_1 = \text{PV} \times i \nP1=PMT−I1P_1 = \text{PMT} - I_1 \n\nThe balance after the first payment is PV−P1\text{PV} - P_1. \nI2=(PV−P1)×i=(PV×i)−(P1×i)=I1−(P1×i)I_2 = (\text{PV} - P_1) \times i = (\text{PV} \times i) - (P_1 \times i) = I_1 - (P_1 \times i) \nP2=PMT−I2=PMT−(I1−(P1×i))=(PMT−I1)+(P1×i)=P1+(P1×i)=P1(1+i)P_2 = \text{PMT} - I_2 = \text{PMT} - (I_1 - (P_1 \times i)) = (\text{PMT} - I_1) + (P_1 \times i) = P_1 + (P_1 \times i) = P_1(1+i). \nTherefore, P2=P1×(1+0.006)P_2 = P_1 \times (1 + 0.006).\n\nDistractor Analysis:\n* A: This incorrectly suggests the increase in principal is the interest on the first interest payment, not the interest on the first principal payment. \n* C: This incorrectly implies the entire first interest payment is added to the second principal payment, which would dramatically alter the loan balance. \n* D: This is incorrect as the principal portion of the payment must increase over time for a standard amortizing loan.

Question 6

A company is considering two loan options for a $100,000 equipment purchase. Loan A is a 5-year loan at 6% APR. Loan B is a 7-year loan at 6% APR. How much more in total interest will the company pay if it chooses Loan B over Loan A?

  1. $6,412 (correct answer)
  2. $6,887
  3. $7,194
  4. $16,396
Explanation: This question requires calculating the total interest paid for both loans and finding the difference.\n\nLoan A (5-year):
  1. Calculate monthly payment: N = 60, I/Y = 6 / 12 = 0.5, PV = 100,000, FV = 0. PMT = -$1,933.28.
  2. Total payments: 60 * $1,933.28 = $115,996.80.
  3. Total interest: $115,996.80 - $100,000 = 15,996.80.\n\n∗∗LoanB(7−year):∗∗\n1.Calculatemonthlypayment:N=84,I/Y=6/12=0.5,PV=100,000,FV=0.PMT=−15,996.80.\n\n**Loan B (7-year):**\n1. Calculate monthly payment: N = 84, I/Y = 6 / 12 = 0.5, PV = 100,000, FV = 0. PMT = -1,460.94.
  4. Total payments: 84 * $1,460.94 = $122,718.96.
  5. Total interest: $122,718.96 - $100,000 = 22,408.96.\n\n∗∗DifferenceinInterest:∗∗\n22,408.96.\n\n**Difference in Interest:**\n22,408.96 (Loan B) - $15,996.80 (Loan A) = $6,412.16.\n\nDistractor Analysis:\n* B: This might be the result of a small calculation error or rounding difference.\n* C: This might result from miscalculating one of the payment amounts.\n* D: $16,396 is close to the total interest on Loan A, not the difference between the two loans.

Question 7

A borrower secures a $450,000 mortgage for 30 years at a fixed annual interest rate of 5.4%, compounded monthly. What is the total amount of principal paid down during the second year of the loan (i.e., from payment 13 through payment 24)?

  1. $6,109.44
  2. $6,389.21 (correct answer)
  3. $24,195.48
  4. $30,364.56
Explanation: This is a multi-step problem. First, calculate the monthly payment (PMT). Second, find the remaining balance at the end of year 1 (payment 12). Third, find the remaining balance at the end of year 2 (payment 24). The difference between these two balances is the principal paid during the second year.\n
  1. Calculate PMT: N = 360 (30 * 12), I/Y = 5.4 / 12 = 0.45, PV = 450,000, FV = 0. Solving for PMT gives $2,530.38.
  2. Find balance after 12 payments: N = 348 (360 - 12), I/Y = 0.45, PMT = -2,530.38, FV = 0. Solving for PV gives the remaining balance of $443,610.79.
  3. Find balance after 24 payments: N = 336 (360 - 24), I/Y = 0.45, PMT = -2,530.38, FV = 0. Solving for PV gives the remaining balance of $437,221.58.
  4. Calculate principal paid: $443,610.79 - $437,221.58 = $6,389.21.\n\nDistractor Analysis:\n* A: $6,109.44 is the principal paid during the first year (450,000 - 443,610.79).\n* C: $24,195.48 is the total interest paid during the second year (12 * 2,530.38 - 6,389.21).\n* D: $30,364.56 is the total payments made during the second year (12 * $2,530.38).

Question 8

Two individuals, Alex and Blair, each take out a $300,000 loan. Alex's loan is for 15 years at a 4.0% annual rate. Blair's loan is for 30 years at a 4.0% annual rate. What is the approximate difference in the total interest paid over the life of the two loans?

  1. $82,000
  2. $115,000 (correct answer)
  3. $146,000
  4. $292,000
Explanation: This requires calculating the total interest for each loan and then finding the difference. For Alex (15-year loan at 4.0%):
  1. Calculate monthly payment: N = 180, I/Y = 4.0 / 12 = 0.3333, PV = 300,000, FV = 0. PMT = -$2,219.95.
  2. Calculate total payments: 180 * $2,219.95 = $399,591.
  3. Calculate total interest: $399,591 - $300,000 = $99,591.
For Blair (30-year loan at 4.0%):
  1. Calculate monthly payment: N = 360, I/Y = 4.0 / 12 = 0.3333, PV = 300,000, FV = 0. PMT = -$1,432.25.
  2. Calculate total payments: 360 * $1,432.25 = $515,610.
  3. Calculate total interest: $515,610 - $300,000 = $215,610.
Difference: $215,610 (Blair's Interest) - $99,591 (Alex's Interest) = $116,019, which is approximately $115,000. Distractor Analysis:
  • A: $82,000 is too low, might result from a calculation error or using a much lower interest rate.
  • C: $146,000 would be the difference for a higher interest rate (approx 5.5%).
  • D: $292,000 is approximately double the correct difference, perhaps from doubling one of the interest totals instead of taking the difference.

Question 9

A borrower has a 30-year mortgage for $500,000 at a 4.8% annual interest rate. The monthly payment is $2,623.33. After making payments for 10 years, what is the total amount of equity attributed to principal reduction, assuming the property value has not changed?

  1. $74,799.60
  2. $115,268.41 (correct answer)
  3. $166,666.67
  4. $314,799.60
Explanation: Equity from principal reduction is simply the original loan amount minus the remaining balance. We need to find the loan balance after 10 years (120 payments).\n
  1. Identify remaining term: The original term was 30 years (360 months). After 10 years (120 months), there are 20 years (240 months) remaining.
  2. Calculate remaining balance: We find the present value of the remaining 240 payments.\n N = 240, I/Y = 4.8 / 12 = 0.4, PMT = -2,623.33, FV = 0. \n Solving for PV gives the remaining balance of $384,731.59.
  3. Calculate principal reduction: This is the original principal minus the remaining balance.\n Principal Reduction = $500,000 - $384,731.59 = $115,268.41.\n\nDistractor Analysis:\n* A: $74,799.60 is the total principal paid in just the first 5 years, underestimating the amount.\n* C: $166,666.67 is one-third of the original loan amount, assuming a simple linear payoff, which is incorrect.\n* D: $314,799.60 is the total amount of payments made over 10 years (120 * $2,623.33). This value includes both principal and a large amount of interest.

Question 10

A $400,000 loan is being amortized over 25 years with monthly payments at an annual interest rate of 6.6%. After the 60th payment, the borrower makes an additional lump-sum principal payment of $20,000. If the borrower continues to make the same original monthly payment, approximately how many months sooner will the loan be paid off?

  1. 18 months
  2. 24 months
  3. 30 months (correct answer)
  4. 36 months
Explanation: This is a multi-step problem involving calculating the effect of a prepayment.\n
  1. Calculate the original monthly payment: N = 300 (25 * 12), I/Y = 6.6 / 12 = 0.55, PV = 400,000, FV = 0. Solving for PMT gives -$2,709.61.
  2. Find the remaining balance after 60 payments: There are 240 payments remaining. N = 240, I/Y = 0.55, PMT = -2,709.61, FV = 0. Solving for PV gives the remaining balance of $358,074.08.
  3. Calculate the new balance after the lump-sum payment: $358,074.08 - $20,000 = $338,074.08.
  4. Calculate the number of remaining payments for this new balance: PV = 338,074.08, I/Y = 0.55, PMT = -2,709.61, FV = 0. Solving for N gives approximately 209.6, so 210 payments.
  5. Find the number of months saved: Originally, there were 240 payments remaining. Now there are 210. The loan will be paid off 240 - 210 = 30 months sooner.\n\nDistractor Analysis:\n* A: Too small of an effect, underestimating the power of the prepayment.\n* B: A common guess, but the calculation shows the effect is larger.\n* D: Overstates the effect. This might result from miscalculating the remaining balance or the original payment.

Question 11

A loan has a negative amortization feature. A borrower takes a $300,000 loan at a 6% annual interest rate, but the required monthly payments for the first year are fixed at $1,200. What will be the outstanding loan balance after exactly one year?

  1. $285,600
  2. $296,354
  3. $300,000
  4. $303,646 (correct answer)
Explanation: Negative amortization occurs when the payment made is less than the interest accrued, causing the loan balance to increase.\n
  1. Calculate monthly interest accrued: The monthly interest rate is 6% / 12 = 0.5%. The interest accrued in the first month is $300,000 * 0.005 = $1,500.
  2. Compare payment to interest: The payment is $1,200, which is 300lessthantheinterestaccrued(300 less than the interest accrued (1,500 - $1,200 = $300).\n3. Calculate balance increase: This $300 shortfall is added to the principal each month. This is a growing annuity problem. The future value of the 300monthlyshortfallover12months,plustheoriginalprincipal,willbethenewbalance.\nWecancalculatethisiteratively,orusetheTVMsolver:N=12,I/Y=0.5,PV=300,000,PMT=1,200(enteredasapositivecashflowsinceitreducestheloan).SolvingforFVgives−300 monthly shortfall over 12 months, plus the original principal, will be the new balance.\n We can calculate this iteratively, or use the TVM solver: N = 12, I/Y = 0.5, PV = 300,000, PMT = 1,200 (entered as a positive cash flow since it reduces the loan). Solving for FV gives -303,646. The negative sign indicates it is still a debt.\n\nDistractor Analysis:\n* A: 285,600istheresultofsubtractingtotalpaymentsfromtheprincipal(285,600 is the result of subtracting total payments from the principal (300,000 - 12*$1,200), ignoring interest completely.\n* B: This reflects a misunderstanding of how the unpaid interest compounds.\n* C: 300,000wouldbethebalanceifthepaymentswereinterest−only(300,000 would be the balance if the payments were interest-only (1,500/month), which they are not.

Question 12

A client takes out a $25,000 auto loan for 60 months with a monthly payment of $500. After exactly two years (24 payments), the client receives a bonus and wants to pay off the remaining balance. Assuming no prepayment penalties, what is the lump-sum amount required to pay off the loan?

  1. $13,000.00
  2. $16,047.53 (correct answer)
  3. $16,408.81
  4. $18,000.00
Explanation: To find the remaining balance, we must calculate the present value of the remaining payments. First, we need to find the interest rate implied by the loan terms. Then, use that rate to find the balance after 24 payments.\n
  1. Calculate the monthly interest rate (I/Y): N = 60, PV = 25,000, PMT = -500, FV = 0. Solving for I/Y gives a monthly rate of 0.77014%.
  2. Calculate the remaining balance: The remaining term is 60 - 24 = 36 months. We calculate the present value of these 36 remaining payments using the rate we just found. N = 36, I/Y = 0.77014, PMT = -500, FV = 0. Solving for PV gives $16,047.53.\n\nDistractor Analysis:\n* A: $13,000 is calculated by (Total Payments - Payments Made) = (50060) - (50024) = 30,000 - 12,000 = 18,000. No, that doesn't work. It's calculated by (Original Principal - Principal Paid in a simple sense) = 25,000 - (24 * 500) = 25,000 - 12,000 = 13,000. This ignores interest completely.\n* C: $16,408.81 is the future value of the original loan after 24 months, which is an incorrect application of the TVM formula. (25000 * (1.0077014)^24) = 30,047.53, minus payments made... this is a messy but plausible calculation error.\n* D: $18,000 is the total amount of remaining payments (36 * $500). This value is not discounted and thus overstates the payoff amount.

Question 13

A fully amortizing loan of $50,000 has a term of 5 years and a fixed annual interest rate of 7.2%, compounded monthly. For the first payment, the principal portion is $P_1andtheinterestportionisand the interest portion isI_1.Forthesecondpayment,theprincipalportionis. For the second payment, the principal portion is P_2.Whatistherelationshipbetween. What is the relationship between P_1andandP_2$?

  1. P2=P1+(I1×0.006)P_2 = P_1 + (I_1 \times 0.006)
  2. P2=P1×(1+0.006)P_2 = P_1 \times (1 + 0.006) (correct answer)
  3. P2=P1+I1P_2 = P_1 + I_1
  4. P2=P1P_2 = P_1
Explanation: The principal portion of each subsequent payment increases by a factor of (1 + periodic interest rate). The interest saved from the principal reduction in the first payment is applied to the principal portion of the second payment.\n\nLet ii be the monthly interest rate (0.072 / 12 = 0.006). Let PMT be the monthly payment. \nI1=PV×iI_1 = \text{PV} \times i \nP1=PMT−I1P_1 = \text{PMT} - I_1 \n\nThe balance after the first payment is PV−P1\text{PV} - P_1. \nI2=(PV−P1)×i=(PV×i)−(P1×i)=I1−(P1×i)I_2 = (\text{PV} - P_1) \times i = (\text{PV} \times i) - (P_1 \times i) = I_1 - (P_1 \times i) \nP2=PMT−I2=PMT−(I1−(P1×i))=(PMT−I1)+(P1×i)=P1+(P1×i)=P1(1+i)P_2 = \text{PMT} - I_2 = \text{PMT} - (I_1 - (P_1 \times i)) = (\text{PMT} - I_1) + (P_1 \times i) = P_1 + (P_1 \times i) = P_1(1+i). \nTherefore, P2=P1×(1+0.006)P_2 = P_1 \times (1 + 0.006).\n\nDistractor Analysis:\n* A: This incorrectly suggests the increase in principal is the interest on the first interest payment, not the interest on the first principal payment. \n* C: This incorrectly implies the entire first interest payment is added to the second principal payment, which would dramatically alter the loan balance. \n* D: This is incorrect as the principal portion of the payment must increase over time for a standard amortizing loan.

Question 14

Two individuals, Alex and Blair, each take out a $300,000 loan. Alex's loan is for 15 years at a 4.0% annual rate. Blair's loan is for 30 years at a 4.0% annual rate. What is the approximate difference in the total interest paid over the life of the two loans?

  1. $82,000
  2. $115,000 (correct answer)
  3. $146,000
  4. $292,000
Explanation: This requires calculating the total interest for each loan and then finding the difference. For Alex (15-year loan at 4.0%):
  1. Calculate monthly payment: N = 180, I/Y = 4.0 / 12 = 0.3333, PV = 300,000, FV = 0. PMT = -$2,219.95.
  2. Calculate total payments: 180 * $2,219.95 = $399,591.
  3. Calculate total interest: $399,591 - $300,000 = $99,591.
For Blair (30-year loan at 4.0%):
  1. Calculate monthly payment: N = 360, I/Y = 4.0 / 12 = 0.3333, PV = 300,000, FV = 0. PMT = -$1,432.25.
  2. Calculate total payments: 360 * $1,432.25 = $515,610.
  3. Calculate total interest: $515,610 - $300,000 = $215,610.
Difference: $215,610 (Blair's Interest) - $99,591 (Alex's Interest) = $116,019, which is approximately $115,000. Distractor Analysis:
  • A: $82,000 is too low, might result from a calculation error or using a much lower interest rate.
  • C: $146,000 would be the difference for a higher interest rate (approx 5.5%).
  • D: $292,000 is approximately double the correct difference, perhaps from doubling one of the interest totals instead of taking the difference.

Question 15

A $400,000 loan is being amortized over 25 years with monthly payments at an annual interest rate of 6.6%. After the 60th payment, the borrower makes an additional lump-sum principal payment of $20,000. If the borrower continues to make the same original monthly payment, approximately how many months sooner will the loan be paid off?

  1. 18 months
  2. 24 months
  3. 30 months (correct answer)
  4. 36 months
Explanation: This is a multi-step problem involving calculating the effect of a prepayment.\n
  1. Calculate the original monthly payment: N = 300 (25 * 12), I/Y = 6.6 / 12 = 0.55, PV = 400,000, FV = 0. Solving for PMT gives -$2,709.61.
  2. Find the remaining balance after 60 payments: There are 240 payments remaining. N = 240, I/Y = 0.55, PMT = -2,709.61, FV = 0. Solving for PV gives the remaining balance of $358,074.08.
  3. Calculate the new balance after the lump-sum payment: $358,074.08 - $20,000 = $338,074.08.
  4. Calculate the number of remaining payments for this new balance: PV = 338,074.08, I/Y = 0.55, PMT = -2,709.61, FV = 0. Solving for N gives approximately 209.6, so 210 payments.
  5. Find the number of months saved: Originally, there were 240 payments remaining. Now there are 210. The loan will be paid off 240 - 210 = 30 months sooner.\n\nDistractor Analysis:\n* A: Too small of an effect, underestimating the power of the prepayment.\n* B: A common guess, but the calculation shows the effect is larger.\n* D: Overstates the effect. This might result from miscalculating the remaining balance or the original payment.

Question 16

A loan's amortization schedule shows that for the 24th monthly payment, the interest portion is $800 and the principal portion is $400. For the 25th monthly payment, the principal portion will be $402. What is the annual interest rate of the loan?

  1. 4.8%
  2. 5.4%
  3. 6.0% (correct answer)
  4. 7.2%
Explanation: The key insight is that the increase in the principal portion from one payment to the next is equal to the interest saved on the principal that was paid in the earlier payment. \n\nIncrease in principal = (P_{25} - P_{24} = 402−402 - 400 = 2\). \nThis $2 increase is the interest that is *not* being charged in month 25 because the principal was reduced by \(P_{24} = 400) in month 24. \nTherefore, Interest saved=P24×monthly rate\text{Interest saved} = P_{24} \times \text{monthly rate}. \n(2=2 = 400 \times \text{monthly rate}). \n(\text{Monthly rate} = 2/2 / 400 = 0.005). \n\nTo find the annual rate, multiply the monthly rate by 12: \nAnnual rate=0.005×12=0.06=6.0\text{Annual rate} = 0.005 \times 12 = 0.06 = 6.0%. \n\nDistractor Analysis:\n* A: 4.8% (0.4% monthly) would mean an increase in principal of $400 * 0.004 = $1.60.\n* B: 5.4% (0.45% monthly) would mean an increase in principal of $400 * 0.0045 = $1.80.\n* D: 7.2% (0.6% monthly) would mean an increase in principal of $400 * 0.006 = $2.40.

Question 17

A loan has a fixed monthly payment of $1,000. For the current month's payment, 70% is allocated to interest and 30% is allocated to principal. Assuming the monthly interest rate is 0.5%, what is the outstanding principal balance on the loan before the current payment is made?

  1. $60,000
  2. $140,000 (correct answer)
  3. $200,000
  4. $30,000
Explanation: This problem requires working backward from the payment composition.\n
  1. Calculate the dollar amount of interest paid: The total payment is $1,000, and 70% of it is interest. Interest paid = $1,000 × 0.70 = $700.
  2. Calculate the principal portion: Principal paid = $1,000 × 0.30 = $300.
  3. Calculate the beginning balance: The interest paid for any period is calculated on the beginning balance for that period. Interest Paid = Beginning Balance × Periodic Rate. We can rearrange this to find the balance.\n Beginning Balance = Interest Paid / Periodic Rate.\n Beginning Balance = $700 / 0.005 = $140,000.\n\nDistractor Analysis:\n* A: 60,000isthebalanceifyouincorrectlyusetheprincipalportion(60,000 is the balance if you incorrectly use the principal portion (300) in the numerator: $300 / 0.005.\n* C: $200,000 is the total loan if the payment were interest only (1000/0.005). This ignores the principal component.\n* D: $30,000 is not directly calculable but represents a misunderstanding of the relationship between payment components and the loan balance.

Question 18

Consider a standard 30-year, fixed-rate mortgage. At which point in the loan's life does the total cumulative principal paid first exceed the total cumulative interest paid?

  1. At approximately 1/3 of the way through the loan term (Year 10).
  2. Exactly halfway through the loan term (Year 15).
  3. At approximately 2/3 of the way through the loan term (Year 20).
  4. Significantly more than 2/3 of the way through the loan term (e.g., Year 25). (correct answer)
Explanation: In a long-term loan like a 30-year mortgage, the initial payments are heavily weighted towards interest. The point where total principal paid equals total interest paid happens much later in the loan's life. The exact point depends on the interest rate, but it is always significantly past the halfway mark. For typical mortgage rates (e.g., 4-7%), this crossover point often occurs around years 22-26. Therefore, 'significantly more than 2/3 of the way' is the best description.\n\nLet's test with an example: $300,000 loan, 30 years, 6% rate. Total payments = 360 * $1798.65 = $647,514. Total interest = $347,514. We need to find when cumulative principal paid > cumulative interest paid. Cumulative principal paid = Original Loan - Remaining Balance. Cumulative interest paid = (N * PMT) - Cumulative principal paid. This crossover happens when the remaining balance is less than (N * PMT) - Original Loan. Solving this shows the crossover is very late in the loan term.\n\nDistractor Analysis:\n* A: In year 10, the loan balance is still very high, and the cumulative interest paid far exceeds the principal paid.\n* B: At the halfway point (15 years), the borrower has paid far more in interest than principal. The remaining balance is typically still well over 60-70% of the original loan amount.\n* C: This is getting closer, but for most standard rates, the crossover has not yet occurred by year 20.

Question 19

A client takes out a $25,000 auto loan for 60 months with a monthly payment of $500. After exactly two years (24 payments), the client receives a bonus and wants to pay off the remaining balance. Assuming no prepayment penalties, what is the lump-sum amount required to pay off the loan?

  1. $13,000.00
  2. $16,047.53 (correct answer)
  3. $16,408.81
  4. $18,000.00
Explanation: To find the remaining balance, we must calculate the present value of the remaining payments. First, we need to find the interest rate implied by the loan terms. Then, use that rate to find the balance after 24 payments.\n
  1. Calculate the monthly interest rate (I/Y): N = 60, PV = 25,000, PMT = -500, FV = 0. Solving for I/Y gives a monthly rate of 0.77014%.
  2. Calculate the remaining balance: The remaining term is 60 - 24 = 36 months. We calculate the present value of these 36 remaining payments using the rate we just found. N = 36, I/Y = 0.77014, PMT = -500, FV = 0. Solving for PV gives $16,047.53.\n\nDistractor Analysis:\n* A: $13,000 is calculated by (Total Payments - Payments Made) = (50060) - (50024) = 30,000 - 12,000 = 18,000. No, that doesn't work. It's calculated by (Original Principal - Principal Paid in a simple sense) = 25,000 - (24 * 500) = 25,000 - 12,000 = 13,000. This ignores interest completely.\n* C: $16,408.81 is the future value of the original loan after 24 months, which is an incorrect application of the TVM formula. (25000 * (1.0077014)^24) = 30,047.53, minus payments made... this is a messy but plausible calculation error.\n* D: $18,000 is the total amount of remaining payments (36 * $500). This value is not discounted and thus overstates the payoff amount.

Question 20

A loan has a negative amortization feature. A borrower takes a $300,000 loan at a 6% annual interest rate, but the required monthly payments for the first year are fixed at $1,200. What will be the outstanding loan balance after exactly one year?

  1. $285,600
  2. $296,354
  3. $300,000
  4. $303,646 (correct answer)
Explanation: Negative amortization occurs when the payment made is less than the interest accrued, causing the loan balance to increase.\n
  1. Calculate monthly interest accrued: The monthly interest rate is 6% / 12 = 0.5%. The interest accrued in the first month is $300,000 * 0.005 = $1,500.
  2. Compare payment to interest: The payment is $1,200, which is 300lessthantheinterestaccrued(300 less than the interest accrued (1,500 - $1,200 = $300).\n3. Calculate balance increase: This $300 shortfall is added to the principal each month. This is a growing annuity problem. The future value of the 300monthlyshortfallover12months,plustheoriginalprincipal,willbethenewbalance.\nWecancalculatethisiteratively,orusetheTVMsolver:N=12,I/Y=0.5,PV=300,000,PMT=1,200(enteredasapositivecashflowsinceitreducestheloan).SolvingforFVgives−300 monthly shortfall over 12 months, plus the original principal, will be the new balance.\n We can calculate this iteratively, or use the TVM solver: N = 12, I/Y = 0.5, PV = 300,000, PMT = 1,200 (entered as a positive cash flow since it reduces the loan). Solving for FV gives -303,646. The negative sign indicates it is still a debt.\n\nDistractor Analysis:\n* A: 285,600istheresultofsubtractingtotalpaymentsfromtheprincipal(285,600 is the result of subtracting total payments from the principal (300,000 - 12*$1,200), ignoring interest completely.\n* B: This reflects a misunderstanding of how the unpaid interest compounds.\n* C: 300,000wouldbethebalanceifthepaymentswereinterest−only(300,000 would be the balance if the payments were interest-only (1,500/month), which they are not.