Exam FM Duration and Convexity Practice Problems

Try this original problem before opening the solution. Then compare the method with SOA Exam FM sample question 100.

Original practice · fully worked

Original variant: duration of community-theatre royalty stream

A community-theatre royalty stream pays 90 at year 0, 60 at year 2, 140 at year 5. The valuation rate is 5.5% effective annually. Determine the Macaulay duration of these promised cash flows.

A
2.113
B
2.363
C
2.563
D
2.813
E
3.113

Show the answer and worked solution

Answer and calculation

Discounted time weights give a Macaulay duration of 2.563 years, choice C.

Setup

Place each promised amount at its payment year and use the same annual discount factor for every term.

v=(1+0.0550)^{-1}

Model

Macaulay duration is the present-value-weighted average payment time.

D_M=\frac{\sum_t tC_tv^t}{\sum_t C_tv^t}

Compute

The denominator is the cash-flow present value; dividing the time-weighted numerator by it gives 2.563.

D_M=\frac{0(90)v^{0}+2(60)v^{2}+5(140)v^{5}}{90v^{0}+60v^{2}+140v^{5}}=2.563

Answer

Thus the center of present value occurs 2.563 years after valuation, corresponding to choice C.

\boxed{D_M=2.563\text{ years}\quad\text{(C)}}

Technique orientation

What duration and convexity questions are testing

Duration summarizes the first-order sensitivity of present value to yield, while convexity captures the curvature left after the linear approximation. Both are computed from the same discounted cash flows and must use one yield convention, one payment period, and one valuation date throughout.

Exam FM connects duration to price changes, portfolio weights, and immunization. A reliable setup first matches present value, then matches the required duration measure, and finally checks the convexity condition when Redington immunization is involved. Converting Macaulay to modified duration must reflect the yield per payment period.

Recognition audit

Signals that this is the right method

The question asks for a weighted-average payment time or for sensitivity to a small yield change.
A bond’s price at nearby yields is used to infer duration or convexity numerically.
Portfolio weights must be chosen to match a liability’s present value and duration.
An immunization condition asks whether an asset surplus has a local minimum when yields shift.

Before calculation

A disciplined first pass

Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.

Formula core

Duration and Convexity formulas to know

These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.

Macaulay duration

D_{\mathrm{Mac}}=\frac{\sum_t t\,C_t v^t}{\sum_t C_t v^t}

Times and the yield period must use compatible units.

Modified duration

D_{\mathrm{mod}}=\frac{D_{\mathrm{Mac}}}{1+i},\qquad \frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\,\Delta i

The approximation is first order and the yield change is entered as a decimal.

Second-order price approximation

\frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\,\Delta i+\frac{1}{2}\mathcal{C}(\Delta i)^2

Use the convexity definition consistent with the yield variable in the problem.

Three failure modes to catch

Averaging component durations by face amount instead of by present-value weights.
Using Macaulay duration directly in a price-change approximation that requires modified duration.
Claiming immunization from duration matching alone without matching present value and checking the required convexity inequality.

Verified practice ledger

Exam FM duration and convexity problems and solutions

The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.

General Cash Flows, Portfolios, and Asset Liability Management

31 verified references

Continue the study loop

Keep every worked reference in one searchable manual

The complete Exam FM manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.

View contents and sample pagesGet the free formula sheet

Related drills

Other Exam FM technique collections

Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.

28 verified problems

Duration and Convexity practice problems