ACAS exams & assessments

FM / Exam 2

Financial Mathematics · Shared with SOA · One discussion board for SOA and CAS

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Study strategy

FM: Use duration as an approximation, then check its limits

Modified duration approximates percentage price sensitivity to a small yield change. It is not a promised exact price change, and Macaulay duration cannot be inserted without the appropriate conversion. Convexity and changing cash flows can matter.

Worked example or practice scenario

For a two-year zero-coupon bond at effective yield 5%, Macaulay duration is 2 and modified duration is 2/1.05. A +.01 yield change gives an approximate −1.9048% price change. The exact ratio is (1.05/1.06)², about −1.8778%.

Try this next

Compare the approximation with exact repricing for a larger move. Explain the growing discrepancy rather than adjusting duration to force a match. For cash flows with options, identify what could change as yields move.

Reading sources

Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.

0 repliesOct 1

Explanation

FM: A bond price check begins with the cash-flow equation

A bond’s coupon rate determines coupon cash flows, while the yield discounts them. Confusing these two rates hides whether the bond trades above or below redemption value. Include the redemption payment at the same time as the final coupon.

Worked example or practice scenario

A two-year bond pays annual coupons of 40 and redeems for 1,000. At yield 5%, price is 40/1.05 + 1,040/(1.05)² ≈ 981.41. Coupon rate 4% is below yield 5%, so a below-par price is a useful reasonableness check.

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Reprice at 3% yield before calculating the exact number. Predict the direction first. Then separate present value of coupons and redemption to locate a missing final payment or a period mismatch.

Reading sources

Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.

0 repliesOct 1

Explanation

FM: Nominal, effective and force-of-interest rates are not interchangeable

The compounding convention belongs to the rate. Convert the rate to the period of your cash flows before using an annuity or accumulation formula. Using an annual nominal percentage as an annual effective percentage silently changes the result.

Worked example or practice scenario

A nominal annual rate of 12% convertible monthly has monthly rate 1% and annual effective rate (1.01)^12−1 ≈ 12.6825%. A constant force of .12 gives annual effective rate e^.12−1 ≈ 12.7497%, a different contract.

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Find the force equivalent to 12% nominal convertible monthly: 12 ln(1.01). Verify that accumulating 1 for a year under either equivalent description gives the same amount. Keep rates and durations in consistent units.

Reading sources

Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.

0 repliesOct 1

Explanation

FM: Prospective and retrospective loan balances must agree

Specify whether the balance is immediately before or after a payment. The prospective method values remaining payments at that time; the retrospective method accumulates earlier cash flows. Their agreement is a useful timing check.

Worked example or practice scenario

A loan of 1,000 is repaid by two equal annual payments at 10%. The payment is 576.19. Immediately after the first payment, balance is 1,000×1.10−576.19 = 523.81. Prospectively it is the one remaining payment discounted one year: 576.19/1.10.

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Calculate the balance immediately before that first payment: 1,100. State why the difference is exactly the payment. For longer loans, label payment number and timing on every balance formula.

Reading sources

Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.

0 repliesOct 1

Explanation

FM: Place a deferred annuity at the correct valuation date

An annuity-immediate factor values payments one period before the first payment. Deferral problems become much easier when you mark that date before applying discounting. A correct annuity factor at the wrong date still gives a wrong answer.

Worked example or practice scenario

Payments of 100 occur at times 4, 5 and 6, with annual effective interest 5%. The value at time 3 is 100 a-angle-3. The value at time 0 is v³ × 100 a-angle-3 ≈ 235.25. It is not v⁴ times that factor.

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Calculate the value directly as 100(v⁴+v⁵+v⁶). Then move the first payment to time 3 and explain which discount exponent changes. Draw the timeline before using your calculator.

Reading sources

Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.

0 repliesOct 1