# Loans

Study Loans for SOA Exam FM (weight 10–15%): worked examples, exam traps, a vendor chapter map, and a 5–7 day plan.

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[Actuarial](https://exclam.ai/actuarial/index.md)›[Exam FM](https://exclam.ai/actuarial/exam-fm/index.md)›Loans

Exam FM topic · 10–15% of exam

Loan repayment, amortization schedules, sinking funds, and varying loan payments.

[Back to Exam FM](https://exclam.ai/actuarial/exam-fm/index.md)

## Per-objective worked-example outlines

For each learning objective on Loans, here is the approach an exam item would test — the setup, the ordering of your reasoning, and the formula or identity you need to bring to the page. Approaches, not full solutions, by design. Verify against the current [soa.org](https://www.soa.org/) syllabus before your sitting.

### Construct and interpret amortization schedules

Setup

A loan is amortized with level or varying payments and you must find the principal paid, interest paid, or outstanding balance at a specific payment date.

Approach

Build the schedule row by row: each row's interest = balance × i, principal = payment - interest, new balance = old balance - principal. For a target payment, jump to that row using the prospective method (PV of remaining payments) rather than iterating all rows. Recognize that for a level-payment loan, principal portions grow geometrically.

Key identity

Principal in payment t = P · v^{n - t + 1}; outstanding balance after t = P · a-angle-{n - t}.

### Compute outstanding balance by the prospective and retrospective methods

Setup

After a partial repayment history, you must compute the outstanding balance at time t.

Approach

Prospective: present value of all remaining scheduled payments at the loan rate. Retrospective: accumulated original loan minus accumulated payments to date. Both should match for any well-defined amortization, so use both as a check.

Key identity

OB\_t (prospective) = P · a-angle-{n - t}; OB\_t (retrospective) = L · (1 + i)^t - P · s-angle-t.

### Work with sinking fund loan arrangements and compare to amortized loans

Setup

A loan pays only interest periodically while a separate sinking fund accumulates the principal repayment at a different rate.

Approach

Compute the periodic interest payment (loan principal × loan rate) plus the periodic sinking fund deposit (principal / s-angle-n at the sinking fund rate). Compare the total annual outlay to the level payment on an amortized loan at the same loan rate; the difference depends on the spread between loan and sinking fund rates.

Key identity

Total periodic outlay = L · i + L / s-angle-n^{j}, where j is the sinking fund rate.

## Common exam traps on Loans

Recurring patterns where candidates lose points on Loans-style items. Each entry pairs the trap with the fix.

Trap

Using the sinking fund rate instead of the loan rate when computing periodic interest.

Fix

Interest on the loan is always charged at the loan rate; only the deposit side uses the fund rate.

Trap

Off-by-one errors in the amortization schedule (e.g., interest portion in payment 1).

Fix

In payment t, interest = previous balance × i; the very first payment uses the original loan balance.

Trap

Computing principal portion as payment minus loan rate × original balance.

Fix

Principal portion uses the current outstanding balance, not the original loan amount.

Trap

Ignoring that varying-payment loans break the geometric pattern.

Fix

For non-level payments, run the schedule row by row or use the prospective method explicitly.

## Where to find Loans in popular manuals

Pointers to where each major vendor covers this topic, so you can grab the right chapter without combing the full manual. We do not reproduce vendor content — just the location. Chapter and lesson numbers shift between editions; use these as a guide, not as a citation.

ASM

Loan amortization and sinking fund chapters

ACTEX

Loan repayment chapter including sinking funds

Coaching Actuaries

Learn modules on Loans; Adapt category "Loans"

The Infinite Actuary

Loan video block covering amortization and sinking funds

## 6-day Loans micro plan

A focused 6-day sub-schedule for Loans specifically, at roughly 1.5–2.5 hours per day. Drop it inside your full Exam FM plan as a single coverage module.

Day 1

Read the amortization chapter; build a flashcard set linking outstanding balance, interest, and principal portions.

Day 2

Drill 15 amortization problems on the BA II Plus using AMORT mode to verify hand-calculated rows.

Day 3

Prospective vs retrospective outstanding balance problems — 10 problems where you compute both ways.

Day 4

Sinking fund loans — 10 problems with mismatched loan and fund rates.

Day 5

Mixed varying-payment loans and refinancing problems; 12 problems with a 60-minute timer.

Day 6

Re-do flagged problems and create a one-page "loan flow" cheat sheet covering both amortized and sinking fund cases.

## How exclam.ai helps you master Loans

### Flashcards from your manual

Upload your ACTEX Exam FM digital edition, scanned ASM pages, TIA handouts, or your own notes. exclam.ai extracts the Loans sections and generates flashcards automatically, tuned to the exam traps above.

### Worked-example drilling

Each per-objective approach above maps to a quiz template. exclam.ai re-surfaces missed items until you can recall both the setup and the key identity from cold.

### FSRS spaced repetition

Because Loans is 10–15% of your exam, losing it during review costs you. FSRS brings it back at the optimal moment.

## Loans in the Exam FM context

SOA Exam FM has 8 topic areas. Loans is weighted at approximately 10–15% of the exam, here is where it sits relative to the other topics.

| Topic area | Weight |
| --- | --- |
| Time Value of Money | 10–15% |
| Annuities | 15–20% |
| → Loans | 10–15% |
| Bonds | 10–15% |
| General Cash Flows and Portfolios | 15–20% |
| Immunization | 10–15% |
| Interest Rate Swaps | 0–5% |
| Determinants of Interest Rates | 0–10% |

## Other Exam FM topics

[Time Value of Money 10–15% Interest rates, discount rates, present and future value, and the relationships between different interest measurement conventions. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/time-value-of-money/index.md)

[Annuities 15–20% Level and varying annuities paid annually or more frequently, annuities due and immediate, and deferred annuities. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/annuities/index.md)

[Bonds 10–15% Bond pricing, yield to maturity, premium and discount, and book value schedules. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/bonds/index.md)

[General Cash Flows and Portfolios 15–20% Valuing arbitrary cash flow streams, net present value, internal rate of return, dollar-weighted and time-weighted returns. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/general-cash-flows-and-portfolios/index.md)

[Immunization 10–15% Portfolio immunization strategies, Redington immunization, full immunization, and asset-liability matching. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/immunization/index.md)

[Interest Rate Swaps 0–5% Basic mechanics of interest rate swaps, including fixed and floating legs and swap rate determination. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/interest-rate-swaps/index.md)

[Determinants of Interest Rates 0–10% Economic and market factors that shape interest rates, including inflation, risk premiums, and the yield curve. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/determinants-of-interest-rates/index.md)