This Exam FM sample reference tests Cash-Flow and Risk Matching. Cash-flow set X has zero surplus value and zero first derivative at 25% with positive second derivative; Y fails the derivative condition and Z fails value equality, so choice A.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BSet Y has nonzero first derivative at 25%, so its first-order rate exposure is not neutralized.
CSet Z does not have zero present-value surplus at the immunization rate.
DY and Z each fail at least one required condition, so all three cannot qualify.
EChoice A identifies the single row satisfying value, slope, and curvature conditions.
Original practice · fully worked
Original variant: measure Redington surplus curvature
A net cash-flow stream pays 2,000 at time 0, pays out 3,300 at time 1, and receives 1,331 at time 3. At 10% effective annually its present-value surplus and first yield derivative are both zero. Determine the second yield derivative.
A 3,967
B 4,463
C 4,959
D 5,455
E 5,950
Variant answer in brief
Differentiating the time-one and time-three present values twice gives positive curvature 4,958.68, choice C.
Setup
Setup
Write the present-value surplus with signs determined by receipts and outflows.
h(i)=2000−1+i3300+(1+i)31331
Model
Model
Differentiate the discounted time-one and time-three terms twice with respect to yield.
h′′(i)=−(1+i)32(3300)+(1+i)512(1331)
Compute
Compute
At 10%, the two curvature contributions combine to 4,958.6777.
h′′(0.10)=4958.6777
Answer
Answer
The second yield derivative is approximately 4,959, corresponding to choice C.
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