Amortization Schedules: Principal and Interest Breakdown
Master loan amortization schedules and payment breakdowns for Exam FM.
Level Payment Amortization
When a loan of L is repaid with n level payments of R at interest rate i, R = L / a-angle-n. Each payment is split into interest and principal: the interest portion of payment t is I_t = i * B_{t-1}, where B_{t-1} is the outstanding balance after payment t-1. The principal portion is P_t = R - I_t. The outstanding balance after payment t is B_t = B_{t-1} - P_t.
Prospective formula: B_t = R * a-angle-(n-t). Retrospective formula: B_t = L*(1+i)^t - R*s-angle-t. Both give the same answer and either may be more convenient depending on the problem. The prospective method is usually simpler when you know R and the remaining term. The retrospective method works well when you know the original loan terms and need to find the balance partway through.
Key Relationships in Level-Payment Loans
In a level-payment amortization, P_t = R * v^(n-t+1), so the principal portions form a geometric sequence increasing by factor (1+i). The interest portions decrease correspondingly. The sum of all principal portions equals L, and the sum of all interest portions equals n*R - L (total interest paid).
These formulas mean you can find the principal and interest in any specific payment without building the entire schedule. For example, P_1 = R * v^n and P_n = R * v. If you know the principal in one payment, you can find the principal in any other payment: P_t = P_1 * (1+i)^(t-1). This geometric progression property is one of the most powerful tools for amortization problems on Exam FM.
Example: A $50,000 loan is repaid over 20 years with annual payments at i = 6%. The payment is R = 50,000 / a-angle-20 at 6% = 50,000 / 11.4699 = $4,359.29. The principal in the first payment is P_1 = 4,359.29 * (1.06)^(-20) = $1,359.29. The principal in the 10th payment is P_10 = 1,359.29 * (1.06)^9 = $2,296.37.
Building an Amortization Schedule
An amortization schedule shows the breakdown of each payment into interest and principal, along with the outstanding balance. For the example above, the first few rows would be: Payment 1 has interest I_1 = 0.06 * 50,000 = $3,000, principal P_1 = $1,359.29, and balance B_1 = $48,640.71. Payment 2 has interest I_2 = 0.06 * 48,640.71 = $2,918.44, principal P_2 = $1,440.85, and balance B_2 = $47,199.86.
Notice how the interest portion decreases and the principal portion increases with each payment. Early in the loan, most of the payment goes toward interest. By the end, most goes toward principal. The crossover point, where interest and principal are equal, occurs when I_t = R/2, which can be solved for t.
Total Interest Over a Range of Payments
For total principal paid from payment j+1 to payment k, the sum equals B_j - B_k = R * (a-angle-(n-j) - a-angle-(n-k)). For total interest over the same range, the sum of interest portions equals (k-j)*R - (B_j - B_k). These formulas avoid the need to compute each individual payment.
Example: For the $50,000 loan above, the total interest paid in payments 6 through 10 is: 5*R - (B_5 - B_10). Using the prospective formula, B_5 = R * a-angle-15 and B_10 = R * a-angle-10. The total interest is 5*4,359.29 - 4,359.29*(a-angle-15 - a-angle-10). Compute a-angle-15 = 9.7122 and a-angle-10 = 7.3601, so total interest = 21,796.45 - 4,359.29*2.3521 = 21,796.45 - 10,254.07 = $11,542.38.
Adjustments and Special Cases
When the interest rate changes mid-loan, compute the outstanding balance at the changeover point using the original rate, then treat it as a new loan at the new rate. The outstanding balance at the changeover is the same regardless of whether you use the prospective or retrospective formula.
When extra payments are made, they reduce the outstanding balance immediately. The subsequent interest portion decreases, and the borrower either shortens the loan term or reduces future payments. On Exam FM, you may be told that the borrower makes an extra lump-sum payment at a specific time and then asked for the new remaining term or payment amount.
For loans with a balloon payment (a larger final payment), the first n-1 payments are level at R, and the final payment is the outstanding balance B_{n-1} plus interest: Balloon = B_{n-1} * (1+i). Set up the equation L = R * a-angle-(n-1) + Balloon * v^n and solve for the unknown.
Study Tips and Common Pitfalls
The most frequent mistake is using the wrong time index. Remember that I_t uses B_{t-1} (the balance before payment t, not after). Another common error is confusing the prospective and retrospective formulas. The prospective formula counts the remaining payments; the retrospective formula accumulates from the start.
Practice finding the principal and interest in a specific payment directly using P_t = R * v^(n-t+1) rather than building the entire schedule. This saves significant time on Exam FM. Also practice problems where the payment or rate changes partway through, as these are among the most commonly tested variations.