This Exam FM sample reference tests Full Immunization. Solving simultaneously gives A = 2,721.09 and B = 3,307.50. Their absolute difference is 586.41, which rounds to 586. The result agrees with the published answer key, choice D.
Let A and B be the amounts assigned to the two zero-coupon positions. Express both positions as values at the liability date using the 5% accumulation and discount factors.
A(1.05)2+B(1.05)−2=6000
Model
Model
Full immunization imposes two equations: total value equals 6,000 and the first derivative with respect to yield is zero. The symmetric time offsets give the displayed signed-duration equation.
2A(1.05)−2B(1.05)−3=0
Compute
Compute
Solving simultaneously gives A = 2,721.09 and B = 3,307.50. Their absolute difference is 586.41, which rounds to 586.
A=2721.09,B=3307.50,∣A−B∣=586.41
Answer
Answer
The calculation gives 586 for full immunization, matching published choice D.
A−B=586(D)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A (0) does not match the checked full immunization result (586); no distinct standard one-step error is identifiable.
BChoice B (146) does not match the checked full immunization result (586); no distinct standard one-step error is identifiable.
CChoice C (293) does not match the checked full immunization result (586); no distinct standard one-step error is identifiable.
EChoice E (881) does not match the checked full immunization result (586); no distinct standard one-step error is identifiable.
Original practice · fully worked
Original variant: two-zero full immunization of a single research grant
A research grant requires 10,000 at time 5. At a 4% annual effective yield, it is fully immunized with zero-coupon assets maturing at times 2 and 8. Calculate the current market value allocated to the time-2 asset.
A 3,802.47
B 3,945.15
C 4,109.64
D 4,383.47
E 5,000.00
Variant answer in brief
Symmetry gives equal market-value allocations, each 4109.64. The time-2 allocation is 4,109.64, selecting choice C.
Setup
Setup
Value the time-5 grant at time 0 and let X and Y be the current market values of the two zero-coupon assets.
PVL=10000(1.04)−5=8219.2711
Model
Model
Full immunization matches both present value and dollar duration; with maturities 2 and 8 surrounding duration 5, solve the two linear equations.
X+Y=PVL,2X+8Y=5PVL
Compute
Compute
Symmetry gives equal market-value allocations, each 4109.64.
X=Y=PVL/2=4109.6355
Answer
Answer
The time-2 allocation is 4,109.64, selecting choice C.
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