This Exam FM sample reference tests Redington Immunization. Duration matching places equal present value on the six-month and one-year assets. Discounting the six-month allocation back by one quarter year gives X = 4975.31, choice B.
Measure times in years. The liability time is 0.75, midway between asset times 0.50 and 1.00.
tX=0.50,tL=0.75,tY=1.00
Model
Model
Equal present value and duration require the asset present-value weights to average to 0.75. Hence each asset carries half the liability present value.
0.50wX+1.00wY=0.75(wX+wY)
wX=wY=5,000v0.75
Compute
Compute
Convert the time-zero present-value allocation for X into its nominal payment at time 0.50.
Xv0.50=5,000v0.75
X=5,000v0.25=4,975.308
Answer
Answer
The six-month cash flow rounds to 4975, choice B.
X≈4,975(B)
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AChoice A discounts the 5000 allocation for a full half-year instead of the quarter-year gap between asset and liability dates.
CChoice C reports the half-liability amount without adjusting from the liability date to the earlier asset date.
DChoice D accumulates 5000 for a quarter year, which moves value in the wrong direction for the earlier payment.
EChoice E applies a full-year accumulation to a cash flow only three months earlier than the liability.
Original practice · fully worked
Original variant: long-zero share for duration matching
An insurer has a liability portfolio with present value 12,000 and Macaulay duration 4 years. It will match present value and duration using zero-coupon assets maturing in 1 year and 7 years. What fraction of the asset portfolio's present value must be allocated to the 7-year zero?
A 33.3%
B 40.0%
C 50.0%
D 57.1%
E 66.7%
Variant answer in brief
The asset duration is the present-value-weighted average of 1 and 7. Setting it equal to 4 assigns exactly 50% to the 7-year zero.
Setup
Setup
Let w denote the present-value fraction invested in the 7-year zero.
0≤w≤1
Model
Model
The short zero receives fraction one minus w, so the asset Macaulay duration is a weighted average.
DA=1(1−w)+7w
Compute
Compute
Match the stated liability duration and solve.
1+6w=4
w=21
Answer
Answer
Half of asset present value belongs in the long zero, choice C.
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