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.
Answer and calculation
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 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.
either direction(C) 