Original practice · fully worked
Original variant: allocate present value between two zeroes
A pension plan owes 900 at time 1 and 1,400 at time 5. At a 4% annual effective yield it will immunize the obligation with zero-coupon assets maturing at times 2 and 6. What percentage of asset present value must be invested in the time-6 asset to match Macaulay duration?
- A
21.4% - B
28.3% - C
32.1% - D
40.0% - E
53.3%
Answer and calculation
The liability duration is 3.28304. A weighted average of asset maturities 2 and 6 equals that duration when 32.0759% of present value is in the later asset.
Setup
Setup
Compute the liability's present-value-weighted payment time at 4%.
v=(1.04)−1 DL=900v+1,400v51(900v)+5(1,400v5) Model
Model
Let w be the fraction of asset present value in the time-6 zero. The remaining fraction is invested at time 2.
DA=2(1−w)+6w Compute
Compute
Set the durations equal and isolate w.
DL=3.283037 w=4DL−2=0.320759 Answer
Answer
The later zero receives about 32.1% of present value, choice C.
100w≈32.1%(C) 