This Exam FM sample reference tests Cash-Flow and Risk Matching. Writing the two liability constraints in terms of the coupon-bond amount makes portfolio cost decrease as that bond increases; the second-year constraint binds, producing the investments in choice E.
How to solve this Cash-Flow and Risk Matching question
Setup
Setup
Let H and I be redemption amounts of the zero-coupon positions and J the par amount of the coupon bond.
H+0.12J=11000,I+1.12J=12100
Model
Model
Exact matching supplies one constraint at each liability date; substituting those constraints into purchase cost leaves a linear function of J.
C=1.10H+1.112I+J
Compute
Compute
Because the coefficient of J is negative, cost is minimized at the greatest feasible coupon-bond holding, where its final cash flow exactly covers year two.
C=19820−0.0181J,Jmax=1.1212100=10803.57
Answer
Answer
The corresponding one-year investment is about 8,821 and the coupon-bond amount is about 10,804, matching choice E.
H/1.10=8821.43,J=10803.57(E)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis feasible-looking mix leaves unnecessary holdings after the liabilities and therefore costs more than the binding solution.
BThe two zero positions do not exploit the cheaper liability coverage available through the higher-coupon bond.
CThis allocation is not the minimum of the substituted cost function because it sets the coupon-bond holding to zero.
DThe stated amounts do not arise at the endpoint where the year-two residual first reaches zero.
Original practice · fully worked
Original variant: least-cost funding for an arts venue
An arts venue must pay 600 in one year and 900 in two years. It can buy one-year zeros yielding 5%, two-year zeros yielding 6%, or units of a note priced at 500 that pay 40 in year one and 540 in year two. Fractional units are allowed. Find the least current cost of an exact match.
A 1,300.00
B 1,325.50
C 1,341.27
D 1,360.00
E 1,390.00
Variant answer in brief
Using 5/3 notes exhausts the second-year liability; the residual one-year zero makes total cost 1,341.27, choice C.
Setup
Setup
Represent residual liability coverage by zero-coupon redemption amounts after selecting J note units.
H+40J=600,I+540J=900
Model
Model
The exact-match equations express both zero holdings as linear functions of J, making current cost linear as well.
C=1.05H+1.062I+500J
Compute
Compute
The negative cost slope favors the largest feasible note position; the year-two constraint limits it to five-thirds units.
dJdC=−1.0540−1.062540+500<0,J=900/540
Answer
Answer
Those notes plus a one-year zero cost 1,341.27 in total, which is choice C.
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