This Exam FM sample reference tests Immunization and Asset-Liability Management. Redington immunization is a local result: matching present value and duration with greater asset convexity protects surplus against sufficiently small yield movements in either direction. Therefore choice C is correct.
How to solve this Immunization and Asset-Liability Management question
Setup
Setup
At the immunization yield, asset and liability present values and first derivatives are matched.
A(i0)=L(i0)
A′(i0)=L′(i0)
Model
Model
The convexity condition makes the asset value curve locally more curved than the liability value curve.
A′′(i0)>L′′(i0)
Compute
Compute
A second-order expansion then makes surplus nonnegative for sufficiently small positive or negative changes, but it does not guarantee protection for large changes.
A(i0+Δi)−L(i0+Δi)≈21[A′′(i0)−L′′(i0)](Δi)2
Answer
Answer
The guaranteed statement is protection against a small change in either direction, which is choice C.
small change in yield(C)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A does not satisfy the local second-order Redington conditions rather than a claim about arbitrary yield movements; no distinct standard one-step error is identifiable.
BChoice B does not satisfy the local second-order Redington conditions rather than a claim about arbitrary yield movements; no distinct standard one-step error is identifiable.
DChoice D (Any decrease in the yield rate) does not satisfy the local second-order Redington conditions rather than a claim about arbitrary yield movements; no distinct standard one-step error is identifiable.
EChoice E (Any change in the yield rate) does not satisfy the local second-order Redington conditions rather than a claim about arbitrary yield movements; no distinct standard one-step error is identifiable.
Original practice · fully worked
Original variant: direction of surplus change under matched duration and greater asset convexity
At a reference yield, an asset portfolio and a liability have equal present value and equal modified duration. Asset convexity is strictly greater. For sufficiently small parallel yield shifts, which statement describes the asset-minus-liability surplus?
A It rises only when yields rise.
B It rises only when yields fall.
C It rises for either direction of shift.
D It remains exactly zero for either shift.
E Its direction cannot be determined locally.
Variant answer in brief
Equal value and duration eliminate the constant and first-order surplus changes. Greater asset convexity leaves a positive second-order term for either sign of a small yield shift, so choice C is correct.
Setup
Setup
Let surplus be asset value minus liability value and expand it around the reference yield.
S(i0)=0
S′(i0)=0
Model
Model
The convexity inequality makes the second derivative of surplus positive at the reference yield.
S′′(i0)=A′′(i0)−L′′(i0)>0
Compute
Compute
The leading local change is proportional to the square of the yield shift and is therefore positive for either direction.
S(i0+Δi)≈21S′′(i0)(Δi)2>0
Answer
Answer
The surplus rises for either sufficiently small upward or downward shift, selecting choice C.
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