This Exam FM sample reference tests Present Value and Duration Matching. Matching dollar duration makes the time-3 asset cash flow 1650.06. The value rounds to 1650, so choice C is correct.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A copies the first liability amount and does not match the time-weighted present values.
BChoice B is below the payment required by the dollar-duration equation 3Y times v cubed equals N sub L.
DChoice D is the unadjusted average of liability amounts scaled to time 3 and misses discount-weighted timing.
EChoice E places both liability amounts at their arithmetic-average time rather than using present-value weights.
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%
Variant answer in brief
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.
The 2210-page Financial Mathematics Proof Manual reorganizes 461 verified Exam FM solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
