A deposit of 4,000 earns simple interest at 6% a year. What is the balance after 30 months?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome b (simple interest). SOA, "Notation and terminology used for Exam FM".
A nominal annual interest rate of 9% convertible monthly equals what annual effective rate?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome c (nominal rate convertible m-thly to effective rate). Wikipedia, "Effective interest rate".
An account earns 7% in a year when prices rise by 3%. What is the real rate of interest for the year?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome a (inflation and real rate of interest). Wikipedia, "Real interest rate": Fisher equation.
The annual effective interest rate is 8%. What is the equivalent annual effective discount rate?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome c (effective discount rate from the effective interest rate). SOA, "Notation and terminology used for Exam FM".
Show all 64 questions (60 more)
Time value of money, continued
The force of interest at time t is δ(t) = 0.02 + 0.01t. What is the accumulated value at time 4 of 1,000 invested at time 0?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome b (variable force of interest). Wikipedia, "Compound interest": force of interest.
How much must be invested today at an annual effective rate of 5% to have 10,000 in 12 years?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome b (present value at compound interest). Wikipedia, "Time value of money".
Two debts, 500 due now and 800 due in 3 years, are replaced by one payment at the end of year 2. At an annual effective rate of 6%, what single payment is equivalent?SOA Exam FM syllabus (December 2026), topic 1, Time Value of Money, outcome d (equation of value). SOA, "Notation and terminology used for Exam FM".
Annuities and cash flows 16
A perpetuity pays 60 one year from now, and each later annual payment is 2% larger than the one before. At an annual effective rate of 9%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (geometric progression, perpetuity). Wikipedia, "Perpetuity": growing perpetuity.
A perpetuity-due pays 400 at the start of each year, forever, and has a present value of 10,400. What is the annual effective interest rate?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (level perpetuity; interest rate as the remaining item). Wikipedia, "Perpetuity".
A perpetuity-immediate pays 50 at the end of year 1, 100 at the end of year 2, 150 at the end of year 3, and keeps rising by 50 a year forever. At an annual effective rate of 8%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (arithmetic progression, perpetuity).
A perpetuity-immediate pays 750 at the end of each year, forever. At an annual effective rate of 6%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (level perpetuity). Wikipedia, "Perpetuity".
An account holds 200,000 and earns an annual effective rate of 5%. A withdrawal of 20,000 is made at the end of each year. How many full withdrawals of 20,000 can be made?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (term of annuity as the remaining item).
An annuity pays 500 at the end of each quarter for 6 years. The interest rate is a nominal 8% a year convertible quarterly. What is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome a and b (annuity payable m-thly).
An annuity pays 1,000 at the end of each year for 10 years. At an annual effective rate of 5%, what is its current value just after the 4th payment?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (current value). SOA, "Notation and terminology used for Exam FM".
An annuity pays 2,000 at the end of each year for 10 years, with the first payment 6 years from now. At an annual effective rate of 5%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (level annuity, finite term; present value).
An annuity-immediate makes 20 annual payments. The first is 1,000 and each later payment is 3% larger than the one before. At an annual effective rate of 7%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (geometric progression, finite term). Wikipedia, "Time value of money": growing annuity.
An annuity-immediate pays 100 at the end of year 1, 200 at the end of year 2, and so on up to 1,000 at the end of year 10. At an annual effective rate of 5%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (arithmetic progression, finite term).
An annuity-immediate pays 500 a year for 5 years, then 800 a year for the next 5 years. At an annual effective rate of 6%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (other non-level annuities/cash flows).
An annuity-immediate pays 1,200 at the end of each year for 15 years. At an annual effective rate of 4%, what is its present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (level annuity, finite term). Wikipedia, "Annuity": annuity-immediate.
Deposits of 300 are made at the start of each month for 5 years. The account earns a nominal 6% a year convertible monthly. What is the balance at the end of the 5 years?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (level annuity-due, future value). Wikipedia, "Annuity": annuity-due.
Level deposits at the end of each year for 10 years must build a fund of 50,000 just after the last deposit. At an annual effective rate of 5%, what is each deposit?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome b (payment amount from the future value). Wikipedia, "Time value of money".
Payments are made continuously at a rate of 1,000 a year for 10 years. The force of interest is 5% throughout. What is the present value?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome a and b (annuity payable continuously).
When are the payments of an annuity-due made?SOA Exam FM syllabus (December 2026), topic 2, Annuities/cash flows with non-contingent payments, outcome a (annuity-immediate, annuity due). SOA, "Notation and terminology used for Exam FM".
Loans 13
A loan is repaid by 15 level annual payments of 2,000 at an annual effective rate of 4%. How much principal does the 10th payment repay?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (amount of interest and principal repayment in a given payment).
A loan is repaid by 20 annual payments of 3,000 at an annual effective rate of 6%. What is the outstanding balance just after the 12th payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (outstanding balance at any point in time). Wikipedia, "Annuity": amortization calculations.
A loan is repaid by level payments. How does the split of each payment between interest and principal change over the term?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome a (principal, interest, amortization). Wikipedia, "Amortization schedule".
A loan of 8,000 at an annual effective rate of 5% is repaid by payments of 2,000 at the end of each year for as long as a full payment is owed, then a smaller final (drop) payment one year later. What is the drop payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome a and b (final payment: drop payment).
A loan of 9,000 charges interest of 1.5% a month. The borrower pays 300 at the end of each month, then a smaller final payment. How many full payments of 300 are made?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (term of loan as the missing item).
A loan of 10,000 at an annual effective rate of 8% is repaid by payments of 1,500 at the end of each year. What is the outstanding balance just after the 3rd payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (outstanding balance at any point in time). Wikipedia, "Annuity": retrospective method.
A loan of 12,000 is repaid by level payments at the end of each year for 5 years, at an annual effective rate of 6%. What is the total interest paid?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (amount of interest in the payments). Wikipedia, "Amortization schedule".
A loan of 25,000 is repaid by level payments at the end of each year for 8 years, at an annual effective rate of 7%. What is each payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (payment amount as the missing item). Wikipedia, "Annuity": amortization calculations.
A loan of 30,000 at an annual effective rate of 6% is repaid by 5 payments of 3,000 at the end of each year, plus a balloon payment made with the 5th payment that clears the loan. What is the balloon payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome a and b (final payment: balloon payment). Wikipedia, "Balloon payment mortgage".
A loan of 50,000 is repaid by 10 level annual payments at an annual effective rate of 5%. How much of the 4th payment is interest?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (amount of interest and principal repayment in a given payment).
A loan of 100,000 at an annual effective rate of 8% has 20 level annual payments. Just after the 5th payment the balance is refinanced at 6% over the remaining 15 years. What is the new annual payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (calculations when refinancing is involved). SOA, "Notation and terminology used for Exam FM".
A loan of 240,000 is repaid by level monthly payments over 30 years, at a nominal 6% a year convertible monthly. What is the monthly payment?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (payment amount, monthly payment period). Wikipedia, "Compound interest": monthly amortized loan payments.
Payments of 650 at the end of each month for 4 years repay a loan at a nominal 9% a year convertible monthly. How much was borrowed?SOA Exam FM syllabus (December 2026), topic 3, Loans, outcome b (principal as the missing item).
Bonds 13
A 4-year bond with a par value of 1,000 pays annual coupons at 8% and is bought to yield 6%. What is its book value just after the 2nd coupon?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (book value).
A 5-year bond with a par value of 1,000 pays annual coupons at 7%. What price gives an annual effective yield of 5%?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (price). Wikipedia, "Bond valuation": present value approach.
A 5-year bond with a par value of 1,000 pays annual coupons at 8% and is bought at par. Each coupon is reinvested at an annual effective rate of 5%. What annual effective yield does the buyer earn over the 5 years?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (yield rate; accumulated value with reinvestment of coupons).
A 10-year bond with a par value of 1,000 pays annual coupons at 6% and is bought at par. Each coupon is reinvested at an annual effective rate of 4%. What is the accumulated value at maturity?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (accumulated value with reinvestment of coupons).
A 10-year bond with a par value of 1,000 pays coupons at 6% a year, payable semiannually. It is bought to yield a nominal 8% a year convertible semiannually. What is its price?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (price). Wikipedia, "Bond valuation": present value approach.
A 12-year bond with a par value of 1,000 and annual coupons is priced at 1,187.70 to yield an annual effective 4%. What is its annual coupon rate?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (coupon, coupon rate).
A bond redeems at par and its coupon rate is higher than its yield rate. How does its price compare with its par value?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome a (price, par value, yield rate, coupon rate). Wikipedia, "Bond valuation".
A bond with a par value of 1,000 pays annual coupons at 3% and matures in 12 years. The issuer may call it at par at the end of any year from 8 to 12. What price guarantees the buyer an annual effective yield of at least 5%?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome c (price of a callable bond to achieve a specified minimum yield). Wikipedia, "Callable bond".
A bond with a par value of 1,000 pays annual coupons at 4%, has 3 years to maturity and is bought to yield 6%. By how much does its book value rise in the first year?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (accumulation of discount).
A bond with a par value of 1,000 pays annual coupons at 5% and redeems for 1,050 in 6 years. What price gives an annual effective yield of 6%?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (redemption value, face value). SOA, "Notation and terminology used for Exam FM".
A bond with a par value of 1,000 pays annual coupons at 8% and matures in 10 years. The issuer may call it at par at the end of any year from 5 to 10. What price guarantees the buyer an annual effective yield of at least 6%?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome c (price of a callable bond to achieve a specified minimum yield).
A bond with a par value of 1,000 pays annual coupons at 9%, has 6 years to maturity and is bought to yield 7%. How much premium is amortized in the first coupon?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (amortization of premium).
A zero-coupon bond that redeems for 1,000 in 8 years sells for 627.41. What is its annual effective yield?SOA Exam FM syllabus (December 2026), topic 4, Bonds, outcome b (yield rate). Wikipedia, "Yield to maturity": zero-coupon bonds.
Duration, yield curves and immunization 15
The 1-year spot rate is 3%. The 1-year forward rates deferred 1 year and deferred 2 years are 5% and 6%. What is the present value of 10,000 due in 3 years?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (present value using a yield curve developed from forward and spot rates). Wikipedia, "Forward rate".
The 1-year, 2-year and 3-year spot rates are 3%, 4% and 5%. What is the present value of 1,000 paid at the end of each of the next 3 years?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (present value using a yield curve developed from forward and spot rates).
The 2-year spot rate is 4% and the 3-year spot rate is 5%. What is the 1-year forward rate, deferred 2 years?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome a (spot rate, forward rate). SOA, "Notation and terminology used for Exam FM".
A perpetuity-immediate pays a level amount at the end of each year. At an annual effective rate of 5%, what is its Macaulay duration?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (duration of a set of cash flows). Wikipedia, "Bond duration": consol or perpetuity.
A portfolio holds a zero-coupon bond due in 4 years with a present value of 60,000 and a zero-coupon bond due in 9 years with a present value of 40,000. What is the Macaulay duration of the portfolio?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome a and b (portfolio; duration of a set of cash flows). SOA study note FM-24-17.
A set of cash flows has a Macaulay duration of 7.35 at an annual effective rate of 5%. What is its modified duration?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (either Macaulay or modified duration given the other). SOA study note FM-24-17.
A set of cash flows has a present value of 250,000 and a modified duration of 6.4. By the first-order modified approximation, what is the present value if the interest rate rises by 50 basis points?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (1st-order linear approximation based on modified duration). SOA study note FM-24-17.
A single liability is due in 6 years. It is to be immunized with zero-coupon bonds that mature in 3 years and in 10 years. To match duration, what share of the assets' present value belongs in the 3-year bond?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome c (protect an asset-liability portfolio using Redington or full immunization).
A single payment is due in 4 years. At an annual effective rate of 5%, what is its modified convexity?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (convexity of a set of cash flows). SOA, "Notation and terminology used for Exam FM".
An investment of 1,000 today returns 600 at the end of year 1 and 550 at the end of year 2. What is its annual effective yield rate?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome a (yield rate/rate of return). Wikipedia, "Internal rate of return".
At an annual effective rate of 6%, a set of cash flows has a present value of 80,000 and a Macaulay duration of 5. By the first-order Macaulay approximation, what is its present value at 7%?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (1st-order approximation based on Macaulay duration). SOA study note FM-24-17.
At an annual effective yield of 4%, what is the Macaulay duration of a zero-coupon bond that matures in 9 years?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (duration of a set of cash flows). Wikipedia, "Zero-coupon bond".
Liabilities of 5,000 and 10,500 are due in 1 and 2 years. They are to be matched exactly with a 1-year zero-coupon bond and a 2-year bond that pays annual coupons at 5%. What maturity value of the 1-year bond is needed?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome c (exactly match a set of liability cash flows). Wikipedia, "Immunization (finance)": cash flow matching.
Payments of 1,000 are due at the end of years 1, 2 and 3. At an annual effective rate of 10%, what is the Macaulay duration of the payments?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome b (duration of a set of cash flows). SOA study note FM-24-17.
Which set of conditions, at the current interest rate, gives Redington immunization?SOA Exam FM syllabus (December 2026), topic 5, General Cash Flows, Portfolios, and Asset Liability Management, outcome a and c (Redington immunization). Wikipedia, "Immunization (finance)".
How this list was compiled: Our own problems, written to the Society of Actuaries' published syllabus of the Financial Mathematics Exam for December 2026 (https://www.soa.org/globalassets/assets/files/edu/2026/fall/syllabi/2026-12-exam-fm-syllabus.pdf, read 2026-10-10): Time Value of Money 5-15%, Annuities/cash flows with non-contingent payments 20-30%, Loans 15-25%, Bonds 15-25%, General Cash Flows, Portfolios, and Asset Liability Management 20-30%. The bank holds 7, 16, 13, 13 and 15 problems. Every numerical answer is worked from the numbers the problem prints and was worked a second way, cash flow by cash flow, on 2026-10-10. The rule each answer uses was read on 2026-10-10 in the public page named under the problem: the syllabus itself, the SOA's note "Notation and terminology used for Exam FM", the SOA study note FM-24-17 "Using Duration and Convexity to Approximate Change in Present Value", or Wikipedia. Licence: Questions, choices and explanations are original work of this site.
How it works
These free SOA FM test practice problems are our own, sorted by the five topics of the Exam FM syllabus, each with a worked solution. They are unofficial and not affiliated with or endorsed by the Society of Actuaries (SOA).
Read the full guide
Five topics, weighted
The SOA's syllabus for December 2026 weights Time Value of Money at 5–15%, Annuities at 20–30%, Loans at 15–25%, Bonds at 15–25%, and General Cash Flows, Portfolios and Asset Liability Management at 20–30%. This bank holds 7, 16, 13, 13 and 15 problems.
Full length
Exam mode draws 30 of our problems on a 2.5-hour clock, the real exam's length. Ours have four choices, not the real five.
What the syllabus assumes
A basic knowledge of calculus, one of the calculators the SOA allows, and the SOA's notation: rates are annual unless a question says otherwise, and "duration" means Macaulay duration.
The SOA scores a blank as a wrong answer, so leave nothing unanswered.
Check the timing first: payments at the end of each period make an annuity-immediate, payments at the start an annuity-due.
For a bond callable at par, price at the earliest call date when the coupon rate is above the yield, at the latest when below.
Questions and answers
How many questions are on the SOA FM exam?
30 multiple-choice questions in 2.5 hours on computer, by the SOA's syllabus for December 2026. A few of them are pilot questions that are not scored.
What topics are on Exam FM?
Five, by the SOA's syllabus: interest and discount rates and the force of interest; level, increasing and geometric annuities and perpetuities; loan payments, balances and refinancing; bond prices, book values and callable bonds; and yield curves, duration, convexity, cash flow matching and immunization.
What score do you need to pass Exam FM?
The syllabus gives no percent. Test forms are built from a pool of questions and scaled so that passing criteria are comparable from form to form. An unofficial pass or fail result reaches your email within one hour.
Where are the SOA's own FM sample questions?
On soa.org. The syllabus links the SOA's sample questions and solutions, its online sample exam and its released exam papers since 2000. None of them is copied or reworded here; every problem on this page is our own.