Present value of the remaining payments: $ B_t = P\,a_{\overline{n-t}|} $

Loan accumulated minus payments accumulated: $ B_t = L(1+i)^t - P\,s_{\overline{t}|} $

When to use prospective

You know the remaining schedule (level payments)

When to use retrospective

Payments have varied, or only past payments are known

Premium (coupon > yield)

Coupon exceeds interest $ iB_{t-1} $; the excess writes the premium DOWN

The redemption amount C at maturity

Sum of cash flows discounted at the required rate; accept if positive

The single rate that makes NPV = 0

The fund's IRR; sensitive to size/timing of cash flows (simple-interest approx common)

Product of subperiod growth factors between cash flows; removes timing effect

How is Exam FM scored and what do I need to pass?

Your raw score is converted to a scaled score from 0 to 10, and a 6 or higher passes. That corresponds to roughly 70% of the scored questions correct. There is no penalty for guessing, so you should answer every question.

Who administers Exam FM?

Exam FM (Financial Mathematics) is jointly administered by the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS). It is one of the first preliminary exams on the path toward actuarial credentials and carries VEE-independent credit on both tracks.

What topics does Exam FM cover?

Six topics: Time Value of Money (~10–15%), Annuities and cash flows with non-contingent payments (~15–20%), Loans (~10–15%), Bonds (~10–15%), General Cash Flows, Portfolios and Asset-Liability Management including duration, convexity and immunization (~15–20%), and Interest Rate Swaps plus the Determinants of Interest Rates (~10–20%).

What calculator do I need for Exam FM?

The exam allows specific Texas Instruments models, most commonly the BA II Plus (or BA II Plus Professional) and the TI-30X series. The BA II Plus time-value-of-money and cash-flow worksheets save large amounts of time on annuity, loan, and bond questions, so practice with it until it is automatic.

Is calculus required for Exam FM?

A working knowledge of calculus is assumed. You need it mainly for the force of interest (integrals of a varying force), continuous annuities, and the derivation of duration and convexity. Most exam questions, however, reduce to algebra once the right formula is set up.

How should I use this Exam FM study guide?

Work through the six topics in order — each builds on the time-value-of-money foundation. After every topic take the checkpoint quiz to find gaps, then drill that topic with our free practice questions and flashcards. Master the calculator alongside the formulas, because speed is what separates a pass from a fail.

Is this Exam FM study guide free?

Yes — the full guide, the checkpoints, the glossary, the concept questions, the practice questions, and the flashcards are 100% free, with no account required.

What is the difference between the effective annual rate of interest and the rate of discount?

The effective annual rate of interest i applies to the balance at the start of a period, while the rate of discount d applies to the balance at the end. They are linked by d = i ÷ (1 + i) and by v = 1 − d, where v = 1 ÷ (1 + i) is the discount factor. Interest is paid at the end of the period on the amount invested; discount is taken at the start on the amount to be received. Other key links: d = iv, i = d ÷ (1 − d), and (1 − d) = v, so the same investment can be described with either rate.

What is the force of interest?

The force of interest δ is the instantaneous, continuously compounded rate of growth of an investment. For a constant force, the accumulation over t years is e raised to (δ × t), and it relates to the effective rate by δ = ln(1 + i), so e^δ = 1 + i. When the force varies with time, the accumulation factor from time a to time b is e raised to the integral of δ(t) dt over that interval. The force of interest is the limit of the nominal rate as the compounding frequency goes to infinity.

How do you convert a nominal interest rate to an effective annual rate?

A nominal annual rate i⁽ᵐ⁾ convertible m times a year means each subperiod earns i⁽ᵐ⁾ ÷ m. The effective annual rate is (1 + i⁽ᵐ⁾ ÷ m) raised to the m, minus 1. More frequent compounding gives a higher effective rate for the same nominal rate. Example: 9% convertible monthly earns 0.75% per month, so the effective annual rate is 1.0075 to the 12th minus 1, about 9.38%. A nominal discount rate d⁽ᵐ⁾ works the same way: the effective annual discount is 1 − (1 − d⁽ᵐ⁾ ÷ m) to the m.

What is the difference between an annuity-immediate and an annuity-due?

An annuity-immediate pays at the END of each period, so its present value is aₙ = (1 − vⁿ) ÷ i. An annuity-due pays at the START of each period, so its present value is äₙ = (1 − vⁿ) ÷ d. Because each due payment lands one period earlier, äₙ = aₙ × (1 + i). Accumulated values follow the same pattern: sₙ = ((1 + i)ⁿ − 1) ÷ i for immediate, s̈ₙ = sₙ × (1 + i) for due. Read the wording: 'first payment now' or 'beginning of each year' signals a due annuity.

How do you value a perpetuity?

A perpetuity pays forever. A level perpetuity-immediate paying 1 at the end of each period is worth 1 ÷ i, and a perpetuity-due is worth 1 ÷ d. A perpetuity whose payments grow at rate g each period (with g < i) is worth the first payment ÷ (i − g). The perpetuity formula is just the limit of the annuity formula as the number of payments goes to infinity, since vⁿ → 0. The growing-perpetuity result, P ÷ (i − g), is the basis of the dividend-discount valuation of an income stream.

How do you value an annuity whose payments increase or decrease arithmetically?

For an annuity-immediate that pays 1, 2, …, n, the present value (Ia)ₙ equals (äₙ − n·vⁿ) ÷ i. For one that pays n, n−1, …, 1, the present value (Da)ₙ equals (n − aₙ) ÷ i. A unit-increasing perpetuity-immediate is worth (1 + i) ÷ i². These come from decomposing the varying stream into a sum of level deferred annuities, then simplifying. For geometrically changing payments, treat the constant growth factor by adjusting the rate to (1 + i) ÷ (1 + g) − 1.

What is the amortization method of repaying a loan?

Under amortization, each level payment first covers the interest on the outstanding balance, and the remainder repays principal. Interest in payment t is i × Bₜ₋₁, where Bₜ₋₁ is the prior balance; the principal repaid is the payment minus that interest, and it grows by (1 + i) each period. Early payments are mostly interest; later payments are mostly principal because the balance shrinks. The principal repaid in payment t equals the level payment × v raised to (n − t + 1).

What is the difference between the prospective and retrospective methods of finding a loan balance?

The prospective method finds the outstanding balance as the present value of the REMAINING payments. The retrospective method finds it as the original loan accumulated forward, minus the accumulated value of the payments already made. For a level-payment loan the two give the same answer. Prospective is easier when you know the remaining payment schedule; retrospective is easier when payments have varied. Both rely on the equivalence principle: at any time, the balance is what still must be paid, valued at the loan rate.

How does the sinking-fund method of loan repayment work?

Under the sinking-fund method the borrower pays the lender interest on the full loan each period, and separately deposits a level amount into a sinking fund earning its own rate, so the fund accumulates to the loan principal by the end. The total outlay is the periodic interest plus the sinking-fund deposit. The deposit equals the loan ÷ sₙ at the fund's rate, where sₙ is the accumulated-value annuity factor. If the fund rate is below the loan rate, the sinking-fund method costs more than straight amortization.

How is the price of a bond determined?

A bond's price is the present value of its future coupons plus the present value of its redemption amount, all discounted at the investor's yield rate. With level coupons Fr and redemption C in n periods at yield i, the price P = Fr × aₙ + C × vⁿ. Equivalent forms include the premium/discount formula P = C + (Fr − Ci) × aₙ and the Makeham formula. If the coupon rate exceeds the yield the bond sells at a premium; if it is below the yield it sells at a discount.

What is the difference between a bond's clean price and dirty price?

The dirty (full) price is the total amount a buyer pays between coupon dates; it includes accrued interest — the seller's earned share of the upcoming coupon. The clean (market) price is the quoted price with accrued interest removed: clean price = dirty price − accrued interest. Accrued interest is the fraction of the coupon period elapsed times the coupon, typically using simple interest for the fractional period. Bond quotes are clean; the cash that changes hands is the dirty price.

How do you price a callable bond to guarantee a minimum yield?

Price a callable bond at the call date that is WORST for the investor. For a premium bond (coupon above yield) the worst case is the EARLIEST call; for a discount bond (coupon below yield) it is the LATEST possible redemption. Pricing to that date guarantees at least the desired yield whenever the issuer actually redeems. The intuition: a premium bond's value falls as redemption nears, so early call hurts the holder most. A discount bond's value rises toward redemption, so the latest date gives the lowest price and protects the yield.

What is the difference between net present value and internal rate of return?

Net present value (NPV) discounts every cash flow at a chosen required rate and sums them; a positive NPV means the project beats that rate. The internal rate of return (IRR, or yield rate) is the single rate that makes the NPV exactly zero — the rate at which discounted inflows equal outflows. Accept a project when its NPV is positive, or equivalently when its IRR exceeds the required rate of return. A cash-flow stream can have more than one IRR when its sign changes more than once.

What is the difference between the dollar-weighted and time-weighted rates of return?

The dollar-weighted return is the fund's internal rate of return, so it is sensitive to the size and timing of deposits and withdrawals. The time-weighted return links the growth factors of the subperiods between cash flows, removing the effect of contribution timing to isolate investment performance. Use the time-weighted return to judge a manager; use the dollar-weighted return to judge the investor's actual experience. A simple-interest approximation gives the dollar-weighted rate as interest ÷ (exposure-weighted average balance).

What is Macaulay duration and how does it relate to modified duration?

Macaulay duration is the present-value-weighted average time until a stream's cash flows are received — a measure of when, on average, you get your money. Modified duration measures price sensitivity to yield and equals Macaulay duration ÷ (1 + i). A higher duration means greater interest-rate risk. The first-order price change is approximately minus modified duration times the change in yield. Convexity is a second-order correction that captures the curvature the straight-line duration estimate misses.

What are the conditions for Redington immunization?

Redington immunization protects surplus against small interest-rate changes when three conditions hold at the valuation rate: the present value of assets equals the present value of liabilities, the asset duration equals the liability duration, and the asset convexity exceeds the liability convexity. The first two conditions set the surplus and its first derivative to zero; the convexity condition makes the surplus a local minimum. Full immunization and cash-flow matching are stronger strategies that protect against larger or arbitrary rate moves.

What is a spot rate and how does it differ from a forward rate?

A spot rate is the annual yield today on a single cash flow paid at one specific future date, so a t-year cash flow is discounted by (1 + spot rate)^t. A forward rate is the interest rate, agreed today, that will apply over a future period — derived from spot rates by no-arbitrage. No-arbitrage links them: investing two years at the 2-year spot must equal one year at the 1-year spot then one year at the forward rate. The plot of spot rates against maturity is the yield curve; an upward slope is a 'normal' term structure.

How does a plain vanilla interest rate swap work?

In a plain vanilla interest rate swap, two counterparties exchange a stream of fixed interest payments for a stream of floating payments on a notional amount; the notional itself is never exchanged. The level swap rate is set so the swap has zero value at inception, letting a party convert floating exposure to fixed or vice versa. The swap rate equals (1 − the final discount factor) ÷ (the sum of the discount factors for every settlement date). Swaps are valued off the term structure, so they tie directly to the spot- and forward-rate material.

Effective rate of interest

The rate i earned over one period, applied to the balance at the START of the period. The accumulated value of 1 after one period is $ 1 + i $.

Effective rate of discount

The rate d applied to the balance at the END of the period. It relates to interest by $ d = \dfrac{i}{1+i} $ and to the discount factor by $ d = 1 - v $.

The present value of 1 due one period from now: $ v = \dfrac{1}{1+i} = 1 - d $. Multiplying a future payment by $ v^n $ discounts it n periods.

Nominal rate of interest

A stated annual rate $ i^{(m)} $ convertible m times a year; each subperiod earns $ i^{(m)}/m $. The effective annual rate is $ (1 + i^{(m)}/m)^m - 1 $.

The instantaneous, continuously compounded rate $ \delta $. For a constant force the accumulation over t years is $ e^{\delta t} $, and $ \delta = \ln(1+i) $.

Accumulation function

a(t), the accumulated value at time t of 1 invested at time 0. Under compound interest $ a(t) = (1+i)^t $.

A series of level payments made at the END of each period. Its present value is $ a_{\overline{n}|} = \dfrac{1 - v^n}{i} $.

A series of level payments made at the START of each period. Its present value is $ \ddot{a}_{\overline{n}|} = \dfrac{1 - v^n}{d} = a_{\overline{n}|}\,(1+i) $.

An annuity that pays forever. A level perpetuity-immediate is worth $ 1/i $; a perpetuity-due is worth $ 1/d $.

An annuity whose first payment is delayed. Value it one period before the first payment, then discount over the deferral.

An annuity whose payments rise by a fixed amount each period. The unit-increasing immediate value is $ (Ia)_{\overline{n}|} = \dfrac{\ddot{a}_{\overline{n}|} - n v^n}{i} $.

An annuity paid continuously at rate 1 per period; its present value is $ \bar{a}_{\overline{n}|} = \dfrac{1 - v^n}{\delta} $.

The value of a stream at a future date. For an annuity-immediate, $ s_{\overline{n}|} = \dfrac{(1+i)^n - 1}{i} $.

FREE Exam FM Study Guide 2026 + Quizzes & Readiness Score

This free Exam FM study guide teaches to the SOA/CAS Financial Mathematics exam — every topic on the official syllabus, organized the way the exam is built.^[2] Exam FM is one of the first preliminary actuarial exams, and it is fundamentally a single skill applied six ways: move money along a timeline using the time value of money.

Everything here is interactive, not a wall of text: every topic has a built-in checkpoint quiz, hover-able glossary terms, worked cash-flow examples, and typeset formulas, so you learn by doing. The notation is the real actuarial notation — annuity symbols like $a_{\overline{n}|}$ and $\ddot{a}_{\overline{n}|}$ , the discount factor $v = \dfrac{1}{1+i}$ , and the force of interest $\delta$ — so it matches what you will see on exam day.

Work through the guide topic by topic, test yourself at each checkpoint, then round out your free Exam FM prep with our practice questions and flashcards. And practice every problem on an approved calculator (the BA II Plus) — on Exam FM, speed is as important as the math.

Exam FM Snapshot

SOA/CAS Exam FM at a glance (2026)
Detail	Exam FM (Financial Mathematics)
Questions	35 multiple-choice (five choices each); some unscored pilot items
Format	Computer-based test (CBT) at Prometric, offered year-round
Total time	3.5 hours (210 minutes) — about 6 minutes per question
Scoring	Scaled 0–10; a 6 or higher passes (~70% of scored questions)
Guessing penalty	None — answer every question
Calculator	Approved TI models (BA II Plus, BA II Plus Professional, TI-30X series)
Administered by	Society of Actuaries (SOA) and Casualty Actuarial Society (CAS)
Prerequisite math	Calculus assumed (force of interest, continuous annuities, duration)

How SOA/CAS Exam FM is built

Exam FM (Financial Mathematics) is jointly administered by the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS) and is one of the first exams on the path to associateship.

Computer-based test (CBT)Delivered year-round at Prometric centers. You get a 3.5-hour (210-minute) appointment for the test itself.
35 multiple-choice questionsEach question has five answer choices (A–E). There is no penalty for guessing, so answer every question.
Pilot (unscored) questions may appearA handful of questions are unscored pilot items the SOA is testing. You cannot tell which, so treat every question as scored.
Scored 0–10; a 6 passesYour raw score is scaled to a 0–10 reported score. A scaled score of 6 or higher passes — roughly 70% of the scored questions correct.

35 questions · 3.5 hours · calculator required (BA II Plus or an approved model). Every question is worth the same; pace yourself at about 6 minutes each.

The six syllabus topics carry roughly the weights below. Notice that Annuities and General Cash Flows / ALM are the two heaviest, and that the newer Swaps & Determinants of Interest Rates topic can be a meaningful share — so do not skip it:^[2]

Exam FM topics by approximate share of the exam (2026)

Annuities & Cash Flows18% · ~15–20%

General Cash Flows, Portfolios & ALM18% · ~15–20%

Swaps & Determinants of Interest15% · ~10–20%

Time Value of Money13% · ~10–15%

Loans13% · ~10–15%

Bonds13% · ~10–15%

The SOA publishes each topic’s weight as an approximate range, so the exact mix shifts slightly each sitting.^[2] This guide teaches all six topics in the order they build on one another — start with the time value of money, because every later topic is an application of it.

1 · Time Value of Money

About 10–15% of the exam, and the foundation for everything else. Money has a time value: a dollar today is worth more than a dollar later, because it can earn interest. This topic is the grammar of the whole exam.^[2]

The cash-flow timeline — discount back, accumulate forward

Discount (←)
multiply a future payment by v = 1 ÷ (1 + i) for each period to get its value today

Accumulate (→)
multiply a deposit by (1 + i) for each period to get its future value

Every Exam FM problem is “move money along this line.” Pick one comparison date, discount or accumulate each cash flow to it, then set inflows equal to outflows.

Interest, Discount & the Factor v

The $i$ grows 1 to $1 + i$ over a period. The $v = \dfrac{1}{1+i}$ does the reverse, pulling a future dollar back to today. The $d$ ties them together:

The core interest/discount relationships
Quantity	Formula
Discount factor	$v = \dfrac{1}{1+i} = 1 - d$
Discount from interest	$d = \dfrac{i}{1+i} = iv$
Interest from discount	$i = \dfrac{d}{1-d}$
Present value of 1 in n periods	$v^n = (1+i)^{-n}$
Accumulated value of 1 in n periods	$(1+i)^n$

Nominal Rates & Compounding

A $i^{(m)}$ convertible $m$ times a year means each subperiod earns $i^{(m)}/m$ . Convert to an effective annual rate before comparing rates with different compounding:

$1 + i = \left(1 + \dfrac{i^{(m)}}{m}\right)^{m} = \left(1 - \dfrac{d^{(m)}}{m}\right)^{-m}$

Force of Interest

The $\delta$ is the limit of the nominal rate as compounding becomes continuous. For a constant force, the is $a(t) = e^{\delta t}$ , and $\delta = \ln(1+i)$ . When the force varies with time, accumulate with the integral:

$a(t) = \exp\!\left(\int_0^{t} \delta(s)\,ds\right)$

Checkpoint · Topic 1 · Time Value of Money

Question 1 of 10

A single deposit of $5,000 grows to $6,000 in an account over 3 years. Assuming a constant rate of compounding, what is the effective annual interest rate earned, to the nearest tenth of a percent?

2 · Annuities & Cash Flows

About 15–20% of the exam — one of the two heaviest topics. An annuity is a series of payments. Master the level annuity factors and the rest of the exam falls into place, because loans and bonds are just annuities in disguise.^[2]

Annuity-immediate vs annuity-due — when does each payment land?

Because every due payment arrives one period earlier, äₙ = aₙ × (1 + i). Read the timing carefully — “end of year” means immediate, “beginning of year” means due.

Level Annuities (Immediate & Due)

An pays at the end of each period; an pays at the start. The single most important fact is that the due value is the immediate value moved one period earlier:

Level annuity factors (n payments at rate i)
Factor	Formula
Present value, immediate	$a_{\overline{n}\|} = \dfrac{1 - v^n}{i}$
Present value, due	$\ddot{a}_{\overline{n}\|} = \dfrac{1 - v^n}{d} = a_{\overline{n}\|}(1+i)$
Accumulated, immediate	$s_{\overline{n}\|} = \dfrac{(1+i)^n - 1}{i}$
Accumulated, due	$\ddot{s}_{\overline{n}\|} = s_{\overline{n}\|}(1+i)$

Perpetuities & Deferred Annuities

A pays forever. As $n \to \infty$ , $v^n \to 0$ , so the perpetuity-immediate is worth $1/i$ and the perpetuity-due $1/d$ . A is valued one period before its first payment, then discounted over the deferral.

Increasing, Decreasing & Continuous

For payments that change by a constant amount, use the and decreasing factors; for geometric growth, adjust the rate; for payments made continuously, use the factor with the force of interest:

Varying-annuity present values
Pattern	Present value
Increasing 1, 2, …, n (immediate)	$(Ia)_{\overline{n}\|} = \dfrac{\ddot{a}_{\overline{n}\|} - n v^n}{i}$
Decreasing n, n−1, …, 1 (immediate)	$(Da)_{\overline{n}\|} = \dfrac{n - a_{\overline{n}\|}}{i}$
Increasing perpetuity (immediate)	$\dfrac{1+i}{i^2}$
Geometric growth g per period	$P\cdot\dfrac{1 - \left(\frac{1+g}{1+i}\right)^n}{i - g}$
Continuous annuity	$\bar{a}_{\overline{n}\|} = \dfrac{1 - v^n}{\delta}$

Checkpoint · Topic 2 · Annuities & Cash Flows

Question 1 of 10

Which feature distinguishes an annuity-immediate from an otherwise identical annuity-due?

3 · Loans

About 10–15% of the exam.A loan is an annuity viewed from the lender’s side: the loan amount is the present value of the repayments. The two big skills are splitting each payment into interest and principal, and finding the outstanding balance at any time.^[2]

Amortization & the Payment Split

Under the , each level payment first pays the interest on the current balance, and the remainder repays principal:

Amortization — each level payment splits into interest + principal

The payment stays level, but early payments are mostly interest and later ones are mostly principal. Interest in period t equals i × (outstanding balance), and the rest repays principal.

Payment 1

Interest

Payment 2

Interest

Principal

Payment 3

Interest

Principal

Payment 4

Interest

Principal

Payment 5

Principal

Interest (i × balance) Principal repaid

Principal repaid in payment t grows geometrically by (1 + i); the final payment is almost entirely principal because little balance remains.

Amortization of a level-payment loan (payment P, rate i, n payments)
Quantity in payment t	Formula
Interest paid	$I_t = i \cdot B_{t-1} = P\left(1 - v^{\,n-t+1}\right)$
Principal repaid	$P_t = P \cdot v^{\,n-t+1}$
Outstanding balance after t	$B_t = P \cdot a_{\overline{n-t}\|}$
Level payment	$P = \dfrac{L}{a_{\overline{n}\|}}$

Outstanding Balance (Two Methods)

The can be found two equivalent ways:

Prospective vs retrospective loan balance
Method	What it computes
Prospective	Present value of the remaining payments: $B_t = P\,a_{\overline{n-t}\|}$
Retrospective	Loan accumulated minus payments accumulated: $B_t = L(1+i)^t - P\,s_{\overline{t}\|}$
When to use prospective	You know the remaining schedule (level payments)
When to use retrospective	Payments have varied, or only past payments are known

Sinking-Fund Method

Under the method the borrower pays the lender interest on the full loan each period, and separately deposits a level amount into a fund (at possibly a different rate) that accumulates to the principal by maturity. The total annual outlay is the interest plus the deposit:

Checkpoint · Topic 3 · Loans

Question 1 of 10

A home loan of $200,000 is repaid by level monthly payments over 30 years at a monthly effective interest rate of 0.5%. What is the amount of each monthly payment?

4 · Bonds

About 10–15% of the exam.A bond is a stream of coupons plus a redemption — so it is, once again, an annuity plus a lump sum. The price is the present value of both, discounted at the investor’s yield rate.^[2]

Bond Pricing & Premium/Discount

With face amount $F$ , coupon rate $r$ (so coupon $Fr$ ), redemption $C$ , $n$ periods, and $i$ , the price is:

$P = Fr\,a_{\overline{n}|} + C v^n = C + (Fr - Ci)\,a_{\overline{n}|}$

Bond pricing — coupon rate vs yield rate decides premium or discount

Coupon > YieldPremiumPrice > redemption value; premium is written down each period.

Coupon = YieldParPrice equals the redemption value exactly.

Coupon < YieldDiscountPrice < redemption value; discount is accumulated (written up) each period.

Price = (present value of coupons) + (present value of redemption), both at the yield rate. The book value glides toward the redemption value as the bond ages.

Book Value & Amortization of Premium

Between purchase and redemption the is the present value of the remaining cash flows at the original yield. Each period, the difference between the coupon and the yield-rate interest adjusts the book value toward redemption value:

Premium write-down vs discount write-up (book value Bₜ)
Case	What happens each period
Premium (coupon > yield)	Coupon exceeds interest $iB_{t-1}$ ; the excess writes the premium DOWN
Discount (coupon < yield)	Interest $iB_{t-1}$ exceeds the coupon; the shortfall writes the discount UP
Interest earned in period t	$i \cdot B_{t-1}$
Book value glides to	The redemption amount C at maturity

Callable Bonds & Clean/Dirty Price

For a callable bond, price to the redemption date that is worst for the investor: the earliest call for a premium bond, the latest date for a discount bond. Between coupon dates, distinguish the from the :

Checkpoint · Topic 4 · Bonds

Question 1 of 10

On the basic price formula for a bond, what does the price equal at the time of issue?

5 · General Cash Flows, Portfolios & ALM

About 15–20% of the exam — the other heaviest topic. Here the cash flows are arbitrary, and the tools are project-valuation measures (NPV, IRR), return measures, and the interest-rate-risk tools that drive asset-liability management: duration, convexity, and immunization.^[2]

NPV, IRR & Rates of Return

discounts every cash flow at a required rate; accept the project if NPV is positive. The is the rate that makes NPV zero. For a fund with deposits and withdrawals, distinguish the and returns:

Return measures for a fund with cash flows
Measure	What it is / how to find it
NPV	Sum of cash flows discounted at the required rate; accept if positive
IRR (yield rate)	The single rate that makes NPV = 0
Dollar-weighted	The fund's IRR; sensitive to size/timing of cash flows (simple-interest approx common)
Time-weighted	Product of subperiod growth factors between cash flows; removes timing effect

Duration & Convexity

is the present-value-weighted average time to receive a stream’s cash flows; measures price sensitivity to yield. is the second-order correction:

$D_{\text{Mac}} = \dfrac{\sum_t t\,v^t\,CF_t}{\sum_t v^t\,CF_t}, \qquad D_{\text{mod}} = \dfrac{D_{\text{Mac}}}{1+i}$

Immunization & Cash-Flow Matching

protects surplus against small rate changes; is the stronger, rebalancing-free alternative. The three Redington conditions:

Redington immunization — three conditions that protect against small rate changes

1Present values match. PV of assets = PV of liabilities at the valuation rate, so surplus starts at zero.
2Durations match. Asset (modified/Macaulay) duration = liability duration, so the first derivative of surplus with respect to i is zero.
3Asset convexity is greater. Asset convexity > liability convexity, so surplus sits at a local minimum and rises for any small rate move.

All three together immunize surplus against small rate moves. Cash-flow matching (exact dates and amounts) is the stronger, rebalancing-free alternative that handles any move.

Checkpoint · Topic 5 · General Cash Flows, Portfolios & ALM

Question 1 of 10

A project requires an outflow of $5,000 today and returns $2,000 at the end of year 1, $2,500 at the end of year 2, and $3,000 at the end of year 3. Using an annual interest rate of 10%, what is the net present value of the project?

6 · Term Structure & Interest Rate Swaps

About 10–20% of the exam. This topic moves from a single interest rate to a whole term structure: spot rates, forward rates, the yield curve, and the interest rate swaps priced off them. Do not skip it — it can be a large share of a given sitting.^[2]

Spot & Forward Rates, the Yield Curve

A $s_t$ is today’s yield on a single cash flow paid at time $t$ , so it discounts by $(1+s_t)^{-t}$ . A is set today for a future period, derived by no-arbitrage:

$(1+s_2)^2 = (1+s_1)\,(1+f_{1,2})$

The plots spot rates against maturity. When long rates exceed short rates the curve slopes up — a “normal” term structure; when short rates are higher it is inverted.

Interest Rate Swaps

In a plain vanilla , two counterparties exchange fixed interest payments for floating ones on a notional amount — the notional itself is never exchanged. The is set so the swap is worth zero at inception:

$R = \dfrac{1 - P_n}{\sum_{t=1}^{n} P_t}$

where $P_t$ is the price today of 1 paid at time $t$ (the discount factor from the spot curve).

Determinants of Interest Rates

Finally, the syllabus covers why interest rates are what they are: the components of a rate (a real risk-free rate plus premiums for inflation, default, liquidity, and maturity/term), and how supply, demand, and central-bank policy shape the term structure. These are qualitative questions — know the building blocks of a quoted rate and what shifts each one.

Checkpoint · Topic 6 · Term Structure & Swaps

Question 1 of 7

The one-year spot rate is 4% and the two-year spot rate is 5%, both expressed as annual effective rates. What is the one-year forward rate covering the period from the end of year 1 to the end of year 2?

How to Use This Study Guide

Exam FM rewards fluency: you must recognize what kind of cash-flow problem you are looking at and reach for the right formula and calculator keystrokes without hesitation. A study guide is a map, not the whole territory — use it alongside the official SOA sample questions^[5] and lots of timed practice. Because every later topic rests on the time value of money, study in order, and do not move on until the earlier factors are automatic.

A study loop that actually works

1
Read a topic here
Work through one of the six topics at a time, in order — each builds on the last.
2
Take the checkpoint
The quick check at the end of each topic exposes what didn't stick.
3
Drill on the calculator
Re-solve every problem on the BA II Plus until the TVM, cash-flow, and bond worksheets are second nature.
4
Take full, timed practice
Sit full 35-question practice sets under the 3.5-hour clock, then review every miss and the official solutions.

Exam FM Concept Questions

Core Financial Mathematics ideas the exam actually measures — at least one per official topic. Tap any card for a short, exam-ready answer backed by the official source (Society of Actuaries), then test yourself on them as flashcards.

Exam FM Glossary

Quick definitions for the interest-theory terms you’ll see most across Exam FM:

Accrued interest: The seller's earned share of the next coupon when a bond is sold between coupon dates, usually prorated by the fraction of the period elapsed.
Accumulated value: The value of a stream at a future date. For an annuity-immediate, $s_{\overline{n}|} = \dfrac{(1+i)^n - 1}{i}$ .
Accumulation function: a(t), the accumulated value at time t of 1 invested at time 0. Under compound interest $a(t) = (1+i)^t$ .
Amortization method: Repaying a loan with level payments, each covering interest on the balance first and repaying the rest of the principal. Interest in payment t is $i \cdot B_{t-1}$ .
Annuity-due: A series of level payments made at the START of each period. Its present value is $\ddot{a}_{\overline{n}|} = \dfrac{1 - v^n}{d} = a_{\overline{n}|}\,(1+i)$ .
Annuity-immediate: A series of level payments made at the END of each period. Its present value is $a_{\overline{n}|} = \dfrac{1 - v^n}{i}$ .
Bond: A security paying periodic coupons plus a redemption amount. Its price is the present value of the coupons plus the redemption, at the yield rate.
Book value: The value of a bond between purchase and redemption, equal to the present value of its remaining cash flows at the original yield rate.
Cash-flow matching: Funding each liability with an asset cash flow of identical date and amount, eliminating reinvestment and interest-rate risk without rebalancing.
Clean price: The quoted (market) price of a bond, equal to the dirty price minus accrued interest.
Continuous annuity: An annuity paid continuously at rate 1 per period; its present value is $\bar{a}_{\overline{n}|} = \dfrac{1 - v^n}{\delta}$ .
Convexity: The second-order measure of how a price-yield curve bends. Adding it to duration improves the estimated price change for large rate moves.
Coupon rate: The bond's stated rate r applied to its face amount F, giving each coupon $Fr$ . Compared with the yield, it determines premium vs discount.
Deferred annuity: An annuity whose first payment is delayed. Value it one period before the first payment, then discount over the deferral.
Dirty price: The full price actually paid for a bond, equal to the clean price plus accrued interest.
Discount (bond): When a bond's price is below its redemption value, because the coupon rate is below the yield. The discount is accumulated (written up) each period.
Discount factor (v): The present value of 1 due one period from now: $v = \dfrac{1}{1+i} = 1 - d$ . Multiplying a future payment by $v^n$ discounts it n periods.
Dollar-weighted return: A fund's internal rate of return; it is sensitive to the size and timing of deposits and withdrawals.
Effective rate of discount: The rate d applied to the balance at the END of the period. It relates to interest by $d = \dfrac{i}{1+i}$ and to the discount factor by $d = 1 - v$ .
Effective rate of interest: The rate i earned over one period, applied to the balance at the START of the period. The accumulated value of 1 after one period is $1 + i$ .
Force of interest: The instantaneous, continuously compounded rate $\delta$ . For a constant force the accumulation over t years is $e^{\delta t}$ , and $\delta = \ln(1+i)$ .
Forward rate: An interest rate, agreed today, that will apply over a future period, derived from spot rates by no-arbitrage.
Increasing annuity: An annuity whose payments rise by a fixed amount each period. The unit-increasing immediate value is $(Ia)_{\overline{n}|} = \dfrac{\ddot{a}_{\overline{n}|} - n v^n}{i}$ .
Interest rate swap: A contract to exchange fixed interest payments for floating ones on a notional amount; the notional itself is never exchanged.
Internal rate of return: The single rate that makes a project's net present value zero — the rate at which discounted inflows equal outflows.
Macaulay duration: The present-value-weighted average time to receive a stream's cash flows: $D = \dfrac{\sum t\, v^t \, CF_t}{\sum v^t \, CF_t}$ .
Modified duration: A measure of price sensitivity to yield, equal to Macaulay duration divided by $(1+i)$ . Price change $\approx -D_{mod}\,\Delta i$ .
Net present value: The sum of all project cash flows discounted at a required rate. A positive NPV means the project beats that rate.
Nominal rate of interest: A stated annual rate $i^{(m)}$ convertible m times a year; each subperiod earns $i^{(m)}/m$ . The effective annual rate is $(1 + i^{(m)}/m)^m - 1$ .
Outstanding balance: The principal still owed. Prospectively it is the present value of remaining payments; retrospectively it is the loan accumulated minus payments accumulated.
Perpetuity: An annuity that pays forever. A level perpetuity-immediate is worth $1/i$ ; a perpetuity-due is worth $1/d$ .
Premium: When a bond's price exceeds its redemption value, because the coupon rate is above the yield. The premium is written down each period.
Redington immunization: Structuring assets and liabilities so surplus is protected against small rate changes: equal present values, equal durations, and asset convexity greater than liability convexity.
Sinking fund: A separate fund the borrower deposits into so it accumulates to the loan principal, while paying the lender interest each period on the full loan.
Spot rate: The annual yield today on a single cash flow paid at one future date; a t-year cash flow is discounted by $(1 + s_t)^t$ .
Swap rate: The level fixed rate that gives a swap zero value at inception: $(1 - \text{final discount factor}) \div (\text{sum of discount factors})$ .
Time-weighted return: The product of the growth factors of the subperiods between cash flows, which removes the effect of contribution timing.
Yield curve: A plot of spot rates against time to maturity at a single point in time. An upward slope is a normal term structure.
Yield rate: The investor's required return i used to discount a bond's cash flows. The price solves $P = Fr\,a_{\overline{n}|} + C v^n$ .

Free Exam FM Study Materials & Resources

Everything you need to prepare for Exam FM is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free Exam FM study materials for active recall, timed practice, and last-minute review:

Exam FM Practice Test — exam-style questions across all six topics, with full worked solutions.
Exam FM Flashcards — active-recall decks for the formulas, annuity symbols, and interest-theory definitions.

Exam FM Study Guide FAQ

Exam FM is a 3.5-hour (210-minute) computer-based test with 35 multiple-choice questions, each with five answer choices. It is administered year-round at Prometric centers. A small number of unscored pilot questions may be embedded, so treat every question as if it counts.

References

1.Society of Actuaries. “Exam FM: Financial Mathematics.” Society of Actuaries. ↑
2.Society of Actuaries. “Exam FM (Financial Mathematics) Syllabus.” Society of Actuaries. ↑
3.Society of Actuaries. “Financial Mathematics (Exam FM) — Learning Objectives & Outcomes.” Society of Actuaries. ↑
4.Casualty Actuarial Society. “Exam 2 / Financial Mathematics (jointly administered with the SOA).” Casualty Actuarial Society. ↑
5.Society of Actuaries. “FM Sample Questions and Solutions.” Society of Actuaries. ↑

Sources for the concept answers

Every answer in the Exam FM concept questions above is drawn from the official primary source:

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