# Time Value of Money

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

Canonical URL: https://exclam.ai/actuarial/exam-fm/topics/time-value-of-money/

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

Exam FM topic · 10–15% of exam

Interest rates, discount rates, present and future value, and the relationships between different interest measurement conventions.

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

## Per-objective worked-example outlines

For each learning objective on Time Value of Money, 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.

### Convert between effective and nominal interest rates at different compounding frequencies

Setup

A rate is quoted at one compounding frequency (e.g., nominal 6% convertible monthly) and you need the effective annual rate or another nominal frequency to value a cash flow.

Approach

Convert the quoted rate to a per-period effective rate, then accumulate across the conversion period to get the target effective rate. Distinguish "nominal interest convertible m-thly" from "nominal discount convertible m-thly" — they accumulate differently. Verify the conversion direction by checking that more frequent compounding gives a higher effective rate.

Key identity

(1 + i)^t = (1 + i^{(m)} / m)^{mt} = (1 - d^{(m)} / m)^{-mt} = e^{δt}.

### Compute present value and future value using simple, compound, and continuous compounding

Setup

A lump sum is moved through time using simple interest, compound interest, or a continuous force of interest, possibly with mixed conventions over different sub-periods.

Approach

For each sub-period, identify the convention and apply the matching accumulation factor: (1 + it) for simple, (1 + i)^t for compound, or e^{∫δ(s) ds} for continuous force. Multiply accumulation factors across sub-periods. Always anchor your time line to a consistent reference date before discounting back.

Key identity

a(t) = e^{∫₀^t δ(s) ds}; PV = FV · a(t)^{-1}.

### Work with force of interest and varying interest rates

Setup

A force of interest δ(t) is given as a function of time and you must accumulate or discount a payment.

Approach

Integrate δ(s) over the relevant interval to get a(t) = e^{∫δ(s) ds}. For PV, divide by a(t); for FV from time s to t, use a(t) / a(s). Keep the integration limits straight — common error is integrating from the payment date when you should integrate from the valuation date.

Key identity

δ(t) = a'(t) / a(t); a(t) = exp(∫₀^t δ(s) ds).

## Common exam traps on Time Value of Money

Recurring patterns where candidates lose points on Time Value of Money-style items. Each entry pairs the trap with the fix.

Trap

Confusing nominal rate of interest with nominal rate of discount.

Fix

Translate each to the effective rate i first; then convert again to the target convention.

Trap

Integrating δ(t) on the wrong interval or sign.

Fix

Set up a(t) = e^{∫₀^t δ(s) ds}; for PV factor v(t) = 1 / a(t), which is e^{-∫δ}.

Trap

Treating simple interest like compound interest over multi-year horizons.

Fix

Simple interest accumulates linearly, compound geometrically; check which the problem specifies.

Trap

Skipping the m-thly conversion when payments and rate use different periods.

Fix

Match the period of the rate to the period of the cash flow before applying any annuity formula.

## Where to find Time Value of Money 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

Introductory chapters on interest measurement and accumulation functions

ACTEX

TVM and interest measurement chapter

Coaching Actuaries

Learn modules on Interest Theory Basics; Adapt category "Time Value of Money"

The Infinite Actuary

Opening video lessons on interest, discount, and force of interest

## 6-day Time Value of Money micro plan

A focused 6-day sub-schedule for Time Value of Money 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 interest measurement chapter; build flashcards mapping i, i^{(m)}, d, d^{(m)}, and δ to one another.

Day 2

Drill 20 rate-conversion problems including both interest and discount conventions.

Day 3

Force-of-interest problems — 10 problems with δ(t) given as a function of t requiring integration.

Day 4

Mixed accumulation and present value drills across simple, compound, and continuous conventions.

Day 5

Calculator practice: 30 minutes converting between conventions on the BA II Plus using both ICONV and manual entry.

Day 6

Re-do flagged problems and write a one-page cheat sheet linking every rate to the effective annual rate.

## How exclam.ai helps you master Time Value of Money

### Flashcards from your manual

Upload your ACTEX Exam FM digital edition, scanned ASM pages, TIA handouts, or your own notes. exclam.ai extracts the Time Value of Money 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 Time Value of Money is 10–15% of your exam, losing it during review costs you. FSRS brings it back at the optimal moment.

## Time Value of Money in the Exam FM context

SOA Exam FM has 8 topic areas. Time Value of Money 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

[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)

[Loans 10–15% Loan repayment, amortization schedules, sinking funds, and varying loan payments. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/loans/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)