Worked solution · Exam FM

SOA Exam FM Sample Question #106 Solution

This Exam FM sample reference tests Cash-Flow and Risk Matching. The liabilities have present value 1,000, duration 3.5, and convexity 14.5; only portfolio A matches value and duration while having convexity 18.5 greater than the liabilities, so choice A.

Independent analysis by ActuaryProof · reviewed October 6, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Cash-Flow and Risk Matching question

Setup

Convert the two liabilities to current values before calculating their duration and convexity.

PV_L=\frac{573}{1.07^2}+\frac{701}{1.07^5}=1000,\qquad D_L=3.5

Model

A Redington candidate must first match present value and duration; only then is the convexity inequality relevant.

C_L=\frac{2^2(573)/1.07^2+5^2(701)/1.07^5}{1000}=14.5

Compute

Portfolio A has equal current values in one- and six-year zeros, yielding duration 3.5 and convexity 18.5.

D_A=\frac{500(1)+500(6)}{1000}=3.5,\qquad C_A=18.5>14.5

Answer

Its convexity exceeds the liability value 14.5, so portfolio A is the valid Redington immunization.

\boxed{\text{portfolio A}\quad\text{(A)}}

Test the method on unseen problems

Financial Mathematics Duration & Immunization Workbook

30 original general cash flows, portfolios, and asset liability management problems with complete solutions and wrong-choice analysis. Written for Exam FM, not copied from the SOA set. One-time $19 PDF.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

BPortfolio B has value 1,000 but duration 3.14, so it fails before convexity is considered.
CThe amounts in portfolio C total more than the liability present value and cannot satisfy the value condition.
DPortfolio D also has total value above 1,000, so it is not a present-value match.
EThe single 3.5-year zero matches value and duration but has convexity 12.25, below the liability convexity 14.5.

Original practice · fully worked

Original variant: selecting a Redington hedge for a civic archive

A civic archive's liability has present value 2,000, duration 4 years, and convexity 18. Three zero-coupon portfolios all cost 2,000: R invests equally in years 2 and 6; S invests equally in years 3 and 5; T invests only in year 4. Which is Redington immunized?

A
R only
B
S only
C
T only
D
R and S
E
All three

Answer and calculation

All three match duration four, but only R has convexity 20 exceeding 18; choice A.

Setup

Equal investments at symmetric maturities have duration at the midpoint, so each candidate matches four years.

D_R=D_S=D_T=4

Model

The distinguishing Redington test is whether asset convexity exceeds the liability convexity of 18.

C_R=(2^2+6^2)/2=20

Compute

R has value-weighted squared maturity 20; S and T have 17 and 16, respectively.

C_S=17,\qquad C_T=16

Answer

Only R supplies the required excess convexity, so choice A is correct.

\boxed{\text{R only}\quad\text{(A)}}

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How to solve this Cash-Flow and Risk Matching question

Setup

Model

Compute

Answer

Financial Mathematics Duration & Immunization Workbook

Why the other letters tempt

Original variant: selecting a Redington hedge for a civic archive

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

Next steps for Exam FM General Cash Flows, Portfolios, and Asset Liability Management