FM FM Annuities and Loan Amortization 2 — Questions and Answers
Question 1: Under the sinking fund method of loan repayment, the borrower each period:
- Makes increasing principal payments to the lender
- Pays interest only to the lender and deposits into a separate sinking fund (Correct answer)
- Makes blended equal payments covering both principal and interest
- Pays full principal upfront then makes interest-only payments
Correct answer: Pays interest only to the lender and deposits into a separate sinking fund
In the sinking fund method, the borrower pays interest on the full original loan to the lender and simultaneously accumulates principal in a separate fund.
Question 2: If the sinking fund interest rate equals the loan interest rate, how do total periodic payments compare between the sinking fund and amortization methods?
- The sinking fund method requires higher total payments
- The amortization method requires higher total payments
- Total periodic payments are identical under both methods (Correct answer)
- It depends on whether the loan term exceeds 10 years
Correct answer: Total periodic payments are identical under both methods
When both rates are equal, the total periodic payment (interest + sinking fund deposit) under the sinking fund method equals the level payment under amortization.
Question 3: A $50,000 loan at 8% annual effective interest is amortized over 10 years with level annual payments. What is the interest component of the very first payment?
- $3,250
- $4,000 (Correct answer)
- $5,270
- $6,980
Correct answer: $4,000
Interest in the first period = outstanding balance × i = 50,000 × 0.08 = $4,000.
Question 4: Using the retrospective method, the outstanding loan balance after k payments equals:
- The present value of remaining payments discounted at time k
- The original loan accumulated at rate i for k periods minus the accumulated value of k payments (Correct answer)
- The original loan minus the sum of all principal payments made
- The original loan times (1+i)^n divided by total payment count
Correct answer: The original loan accumulated at rate i for k periods minus the accumulated value of k payments
Retrospective balance = L(1+i)^k - Payment × s(k,i), where s(k,i) is the accumulated value of payments made.
Question 5: A $100,000 loan at 6% annual interest is amortized over 20 years with level annual payments. After 10 payments, approximately what percentage of the original balance remains outstanding?
- 72%
- 50%
- 64% (Correct answer)
- 58%
Correct answer: 64%
Payment = 100,000/a(20,6%) ≈ $8,718; balance after 10 = 8,718 × a(10,6%) ≈ $64,162, which is about 64% of $100,000.
Question 6: Which statement about the amortization method for a level-payment loan is CORRECT?
- Total interest paid equals loan amount times the interest rate times the term
- Early payments consist mostly of principal repayment
- Late payments consist mostly of interest
- Early payments consist mostly of interest (Correct answer)
Correct answer: Early payments consist mostly of interest
In early periods the outstanding balance is large, so interest charges are high and only a small portion of the fixed payment reduces principal.
Question 7: The principal repaid in the k-th payment of an amortizing loan (original loan L, n-period level payments, rate i) is given by:
- L × i × v^(n-k+1)
- (L / a(n,i)) × v^(n-k+1) (Correct answer)
- L × i / (1 - (1+i)^-n)
- L × i × (1+i)^k
Correct answer: (L / a(n,i)) × v^(n-k+1)
The k-th principal payment equals the level payment divided by the annuity factor times v^(n-k+1), i.e., (L/a(n,i)) × (1+i)^(-(n-k+1)).
FM FM Annuities and Loan Amortization 2
Under the sinking fund method of loan repayment, the borrower each period: