Worked solution · Exam FM

SOA Exam FM Sample Question #432 Solution

This Exam FM sample reference tests Redington Immunization Conditions. Redington immunization protects surplus locally against small parallel yield shifts when present value and duration match and asset convexity exceeds liability convexity. Therefore E is true.

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 Redington Immunization Conditions question

Setup

State the local immunization objective in terms of surplus, the value of assets minus liabilities.

S(i)=A(i)-L(i)

Model

At the immunization yield, Redington conditions set surplus to zero or positive, match first derivatives through duration, and require favorable second-order curvature.

A(i_0)=L(i_0)

D_A(i_0)=D_L(i_0)

C_A(i_0)>C_L(i_0)

Compute

These conditions make the matched point a local minimum of surplus as a function of a parallel interest-rate shift.

S'(i_0)=0,\qquad S''(i_0)>0

Answer

That is precisely the small-parallel-change protection described in choice E.

\boxed{\text{choice E}}

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.

AChoice A removes duration matching, even though equality of first-order interest sensitivity is a core Redington condition.
BChoice B reverses the practical comparison: cash-flow matching needs no interest-rate rebalancing, while immunization generally does.
CChoice C reverses the convexity inequality; asset convexity must exceed liability convexity for local surplus protection.
DChoice D assigns zero duration to the entire portfolio and negative convexity, neither of which states the Redington conditions.

Original practice · fully worked

Original variant: diagnose a failed immunization

At a chosen yield, an asset portfolio and its liability have equal present value and equal Macaulay duration. The asset convexity is 84 while the liability convexity is 91. Which conclusion is correct under the Redington criterion?

A
The portfolio is immunized because present values match.
B
The portfolio is immunized because durations match.
C
The portfolio fails because asset convexity is too low.
D
The portfolio fails because liability duration must be zero.
E
No conclusion is possible without coupon rates.

Answer and calculation

Present-value and duration equalities remove the zero- and first-order gaps, but the asset convexity must exceed liability convexity. Since 84 is below 91, the portfolio fails, choice C.

Setup

The first two Redington requirements are already supplied.

A=L,\qquad D_A=D_L

Model

Inspect the remaining curvature requirement for local surplus protection.

C_A>C_L

Compute

The stated figures have the opposite ordering.

84<91

Answer

The asset portfolio is not Redington immunized, so choice C is correct.

\boxed{\text{fails because }C_A<C_L\quad\text{(C)}}

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How to solve this Redington Immunization Conditions question

Setup

Model

Compute

Answer

Financial Mathematics Duration & Immunization Workbook

Why the other letters tempt

Original variant: diagnose a failed immunization

Answer and calculation

Setup

Model

Compute

Answer

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