Exam FM Loan Amortization and Balance Problems

Technique orientation

What loan amortization and balances questions are testing

A loan balance at time k is the value at that same time of what remains to be paid. The prospective method values future payments; the retrospective method accumulates the original advance and subtracts accumulated past payments. They are two views of one equation of value and provide a strong built-in cross-check.

Exam FM asks for balances immediately before or after a payment, principal and interest portions, revised payments, and refinancing values. Draw the time-k boundary explicitly. Then count remaining payments from that boundary and use the contractual loan rate unless a separate fund or refinancing rate is stated.

Recognition audit

Signals that this is the right method

A principal is repaid through a stated sequence of payments at a contractual effective rate.
The question asks for an outstanding balance after a specified number of payments.
A payment must be split into interest on the prior balance and principal reduction.
Terms change through refinancing, extra payments, a balloon payment, or a new repayment schedule.

Before calculation

A disciplined first pass

Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.

Formula core

Loan Amortization and Balances formulas to know

These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.

Original loan equation

L=R\,a_{\overline{n}|i}

R is the level end-of-period payment when the loan is an annuity-immediate.

Balance after the kth payment

B_k=R\,a_{\overline{n-k}|i}=L(1+i)^k-R\,s_{\overline{k}|i}

The prospective and retrospective expressions must agree at the same point in time.

Interest and principal portions

I_{k+1}=iB_k,\qquad P_{k+1}=R-I_{k+1},\qquad B_{k+1}=B_k-P_{k+1}

Interest for the next payment period is calculated from the balance immediately after the preceding payment.

Three failure modes to catch

Using n rather than n − k as the remaining term in a prospective balance.
Computing a payment’s interest portion from the balance after that payment.
Valuing a refinancing or sinking-fund cash flow at the original loan rate when a different rate is specified.

Verified practice ledger

Exam FM loan amortization and balances problems and solutions

The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.

Loans

49 verified references

Continue the study loop

Keep every worked reference in one searchable manual

The complete Exam FM manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.

View contents and sample pagesGet the free formula sheet

Related drills

Other Exam FM technique collections

Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.

54 verified problems

Loan Amortization and Balances practice problems