SOA Exam FM is a 2.5-hour, 30-question multiple-choice computer-based test covering time value of money, annuities, loans, bonds, and general cash flows with immunization. This guide gives you the verified format and logistics, the official topic weights, and a study approach built on one discipline: convert every quoted rate to a single effective rate per payment period and place every valuation factor at the correct date before touching a formula. Four fully worked exercises show the method in action.
Format, length, and how it is scored
Exam FM is a 2.5-hour computer-based test with 30 multiple-choice questions. Answer every question, because blanks are scored incorrect.
The Financial Mathematics exam is a 2.5-hour computer-based test (CBT) consisting of 30 multiple-choice questions. Each question offers five answer choices, A through E, with exactly one correct, and some answers are rounded. Unanswered questions are scored incorrect, so the SOA advises entering an answer for every question even under time pressure.
A few pilot questions are randomly mixed into each form and do not count toward your score; everything else does. Because the exam runs over a window of several days, candidates receive different question sets drawn from a pool, and statistical scaling keeps the forms comparable within a testing period. Unofficial pass or fail results are sent to your scheduling email within about an hour.
The syllabus assumes a basic knowledge of calculus, and the learning outcomes are written on the understanding that you will use a specified calculator. That assumption shapes the exam: approximation methods that a calculator makes unnecessary are not examined.
Syllabus topics and weights
Five topics are tested, with annuities and general cash flows carrying the heaviest weights. An individual exam can occasionally fall outside the published ranges.
The December 2026 syllabus organizes the exam into five topics. The SOA publishes weight ranges to indicate relative emphasis and cautions that a particular sitting may fall outside them, and that some questions cover more than one learning outcome. Use the ranges to budget study time, not to predict an exact question count per topic.
| Topic | Published weight range |
|---|---|
| Time Value of Money | 5–15% |
| Annuities and cash flows with non-contingent payments | 20–30% |
| Loans | 15–25% |
| Bonds | 15–25% |
| General Cash Flows, Portfolios, and Asset Liability Management | 20–30% |
Fees, dates, and how to register
The 2026 fee is $275 USD. Exam FM runs in several windows each year; the next is December 3–14, 2026, with registration closing November 4, 2026.
The Financial Mathematics exam fee is $275 in U.S. dollars for 2026, and the preliminary-exam fees include electronic access to the required study notes. Reduced fees, reimbursement, and scholarship opportunities exist for eligible candidates, and all outstanding fees must be paid before you can enroll in future exams.
Most SOA exams, including FM, are administered by CBT through Prometric test centers; candidates without access to a Prometric center can take select exams in paper-and-pencil format on published dates. The next FM window is December 3–14, 2026 (CBT), with a registration deadline of November 4, 2026, 10:00 AM. Further sittings follow roughly every two months through December 2027 — for example February 4–15, April 1–12, June 10–21, and October 1–12, 2027 — each with its own deadline, so confirm dates on the SOA exam schedule when you register.
Sources used: SOA Exam Schedule — ASA exam listings, 2026–2027 FM windows; SOA Exam and e-Learning Module Fees (updated December 16, 2025) — ASA Exams and Modules 2026 table
Fix the rate basis before any formula
Start every FM question by converting the quoted rate to one effective rate per payment period. Most silent errors are rate-basis errors.
A nominal rate i(m) convertible m times per year is a label, not a usable rate: each period earns i(m)/m, so the effective annual rate is (1 + i(m)/m)^m minus 1. Take 12% convertible monthly: 1% per month compounds to about 12.68% effective annually. Entering 12% as an annual effective rate in a time-value-of-money calculation understates growth every period, and the distortion compounds across an annuity factor, a loan balance, or a bond price.
The force of interest δ is the continuously compounded limit: a fund grows as e raised to the integral of δ, and constant δ is equivalent to an effective annual rate of e^δ minus 1. The effective discount rate d is different again — interest paid at the start of the period on an amount of 1 due at the end, with d = i/(1+i). Name the measure, convert it once, and write the converted rate at the top of your work so later steps inherit the right basis.
Worked exercise
A fund credits interest at a nominal discount rate of 7.2% convertible monthly. You deposit 1,000 today. What amount has accumulated after exactly three years, and what is the equivalent effective annual rate?
Show answer
The accumulated value is about 1,241.91, and the equivalent effective annual rate is about 7.49%.
Assume the nominal discount rate applies for the full three years with no other cash flows. A nominal discount rate of 7.2% convertible monthly means a discount of 0.072/12 = 0.006 per month, so each month a balance of 1 grows by the factor 1/(1 − 0.006) = 1/0.994. Thirty-six months gives 1,000 × 0.994^(−36) = 1,241.91 to two decimals. Twelve months gives 0.994^(−12) = 1.074889, so the equivalent effective annual rate is about 7.49%. Raising the unrounded annual factor 1.074889 to the third power reproduces the three-year accumulation, which is a good consistency check. The trap is treating 7.2% as an effective discount rate applied once per year, or converting the discount rate as if it were an interest rate.
| Quoted measure | What it means | Convert to effective annual | Common misreading |
|---|---|---|---|
| i(m), nominal convertible m-thly | Rate i(m)/m charged each of m periods | (1 + i(m)/m)^m − 1 | Used directly as an effective annual rate |
| i, effective annual | Growth of 1 over one full year | i itself | Divided by m when payments are monthly |
| d, effective discount | Interest on 1 due at period end, paid upfront | d/(1−d) | Swapped with i when discounting |
| δ, force of interest | Continuously compounded growth rate | e^δ − 1 | Typed into TVM keys as if it were annual effective |
Annuities: pin the valuation date
An annuity factor is meaningless until you know where time zero sits relative to the payments. Decide immediate versus due first, then shift with (1+i).
The annuity-immediate factor a(n) values payments at the end of each period, and it places the value one period before the first payment. The annuity-due factor values payments at the start of each period, at the moment the first payment occurs, and equals the immediate factor times (1+i). The recurring error is using a(n) when payments begin immediately: the answer lands one period early, and failing to adjust moves every payment's value by a full period of interest.
Non-standard intervals need the same discipline at a larger scale. If payments of 100 occur every two years for twenty years at 12% effective annual, the two-year rate is (1.12)^2 − 1 = 25.44%, and the annuity runs for ten two-year periods. Using 12% per period with ten payments values a completely different cash flow stream. Increasing annuities follow the same rule as level ones: the standard increasing factor values the stream one period before the first payment of 1, so state that date explicitly before you add or multiply anything.
Loans: split interest from principal
Amortization splits each payment into interest on the outstanding balance plus principal reduction. The December 2026 syllabus asks candidates to calculate payments, balances, principal, and interest using that structure.
Every amortization row starts from the same identity: interest for the period is the rate times the outstanding balance, and the rest of the payment reduces principal. The rate you use must be the effective rate per payment period, whatever the question quoted. The exercise below shows how a quoted nominal rate changes the entire schedule.
Worked exercise
A loan of 10,000 is repaid with five equal annual payments, the first due one year from now. Interest is quoted as a nominal 8% convertible quarterly. Calculate the annual payment, and state what payment results if the 8% is mistakenly treated as an annual effective rate.
Show answer
The correct annual payment is about 2,520.64. Treating 8% as an effective annual rate gives about 2,504.56, understating every payment.
Assume payments are annual in arrears and interest converts quarterly. Each quarter earns 0.08/4 = 2%, so the effective annual rate is (1.02)^4 − 1 = 8.243216%. The payment is 10,000 / a(5) at that rate, where a(5) = (1 − 1.08243216^(−5)) / 0.08243216 ≈ 3.967246, giving about 2,520.64. At an 8% effective annual rate, a(5) ≈ 3.992710 and the payment drops to about 2,504.56. The error is not cosmetic: every row of the amortization schedule inherits the wrong rate, so the interest and principal split in each period and the implied balance at any date all drift from the true values.
Bonds: what the syllabus actually tests
The syllabus covers price, book value, amortization of premium, accumulation of discount, and callable bonds — but explicitly excludes valuation between coupon payment dates.
The bond outcomes ask you to define and work with price, book value, market value, redemption and par value, yield rate, coupon and coupon rate, term, and call features. The core calculation prices a bond as the present value of its coupons plus the redemption value, all at the yield rate, then tracks book value between dates: when the coupon rate exceeds the yield, the bond sells at a premium and the book value falls toward par, with the excess of coupon over yield interest recorded as amortization of premium. When the yield exceeds the coupon rate, the bond sells at a discount and the book value accumulates upward. Recording the whole coupon as return on a premium bond overstates income.
Two scope notes save you wasted effort. First, the syllabus states that valuation of bonds between coupon payment dates will not be covered, so practice should focus on pricing on coupon dates and building amortization-of-premium or accumulation-of-discount schedules. Second, for a callable bond the syllabus asks for the price that achieves a specified minimum yield: compute the price at each possible redemption date and take the smallest price, which is the worst case for the buyer and therefore guarantees the stated minimum yield. Rehearse that computation for both a premium and a discount bond so the direction of the adjustment is automatic.
Duration and immunization
Keep Macaulay duration — the cash-flow-weighted time — distinct from modified duration, the price sensitivity multiplier. Redington immunization requires three conditions, not one.
Topic 5 asks you to calculate duration and convexity (Macaulay and modified), convert one duration to the other, and approximate the change in present value for a change in interest rates. Macaulay duration is the weighted average time of a cash flow stream, with weights equal to each flow's present value. Modified duration converts it into price sensitivity: the relative price change is approximately minus the modified duration times the yield change, and modified duration equals Macaulay duration divided by (1+i) under annual effective compounding. Using the Macaulay figure directly in the sensitivity formula misstates the price impact by the factor 1/(1+i), which is not negligible at typical rates.
The syllabus also asks you to construct portfolios that use Redington or full immunization, or that exactly match a set of liability cash flows. Redington immunization requires matching present values, matching duration, and greater asset convexity — in the classical setup, asset cash-flow times spread around the liability time. The exercise below shows why matching present value alone protects nothing.
Worked exercise
At a flat 6% effective annual yield, a liability has a present value of 100,000 and is paid as a single amount in 10 years. An investor buys 60,000 present value of 5-year zero-coupon bonds and 40,000 present value of 15-year zero-coupon bonds. Show that present value is matched but Redington immunization fails, then find the present-value split of the two zeros that satisfies all three conditions, assuming small parallel shifts in the flat yield.
Show answer
The portfolio's Macaulay duration is 9, not 10, so immunization fails. A 50/50 present-value split — 50,000 in each zero — sets the asset duration to exactly 10, and the asset times 5 and 15 straddle the liability time, satisfying the spread condition.
Assumptions: the Macaulay duration of a zero-coupon bond equals its maturity, portfolio duration is the present-value-weighted average of the asset durations, and the liability's duration equals its 10-year payment time. Present value matches: 60,000 + 40,000 = 100,000. But the asset duration is 0.6(5) + 0.4(15) = 9, one year short of the liability, so a small parallel yield shift produces a surplus or deficit the portfolio was supposed to neutralize. With present-value weight w in the 5-year zero, w(5) + (1−w)(15) = 10 gives w = 0.5. With equal weights the durations match, and because the asset times 5 and 15 lie on both sides of the liability time, the portfolio's convexity exceeds the liability's, so the protection holds for both upward and downward shifts.
Official study materials
Work from the syllabus for your sitting, the required duration-and-convexity study note, and the free online sample exam; one suggested textbook of your choice covers the rest.
The SOA publishes a syllabus with learning objectives, outcomes, and readings for each sitting — August, October, and December 2026 are listed on the FM study page — so confirm you are reading the edition matching your exam date. There is no single required textbook; the syllabus lists representative texts such as Broverman's Mathematics of Investment and Credit, Vaaler-Harper-Daniel's Mathematical Interest Theory, Brown and Kopp's Financial Mathematics: Theory and Practice, Francis and Ruckman's Interest Theory, and Chan and Tse's Financial Mathematics for Actuaries, each with specific included and excluded sections. Whichever you choose, the SOA notes that understanding is significantly enhanced by working the textbook exercises for the listed readings.
Three official resources deserve special attention. The study note FM-24-17, Using Duration and Convexity to Approximate Change in Present Value, is required reading (Sections 1–4). The free Online Sample Exam FM draws coded questions from the released pool in a CBT-like experience, and all released exam papers since 2000 are available on the SOA site. Note that the texts, study notes, and notation materials will not be available during the exam itself.
On calculators, the SOA strongly recommends the BA II Plus or BA II Plus Professional because they can solve directly for interest rates, and states that some exam problems require one of those models to find the answer. Check the SOA's calculator policy for the current approved list before exam day.
Sources used: SOA Financial Mathematics Exam December 2026 syllabus, format, learning outcomes, weights, and readings — Syllabus front matter and learning objectives; Studying for Financial Mathematics (FM) Exam (SOA) — Syllabus editions and supplemental study materials
A practice sequence to adapt
Rotate through topic blocks, then run mixed timed sets, scoring your setups rather than only your final answers. This is editorial advice, not an official requirement.
The sequence below is a study suggestion you can compress or stretch around the official syllabus and your own schedule; the SOA does not prescribe a study method. What it protects is the habit this article has emphasized: naming the rate basis and the valuation date before computing.
An error log makes the sequence concrete. For each practice problem, record one line stating the effective rate per payment period you used, the valuation date of every factor, and the formula family. Score the setup before the answer: rate basis correct, valuation date correct, units consistent. Reviewing the log by category, rather than by raw score, shows you which habit to repair next.
- Weeks 1–2: rate conversion drills plus annuities of every timing pattern — immediate, due, perpetuities, m-thly, and increasing or decreasing streams.
- Week 3: loan amortization alongside bond pricing, premium and discount schedules, and callable bonds at a minimum yield.
- Week 4: general cash flows, spot and forward rates, yield curves, and present-value calculations using the curve.
- Week 5: duration, convexity, the required study note's approximation methods, and immunization conditions, then begin mixed sets.
- Week 6: full mixed practice under exam timing, using the online sample exam and released papers, with the error log reviewed after every set.
Sources used
Key facts verified against SOA sources: