This Exam FM sample reference tests Annuities and Perpetuities. The continuous-annuity ratio gives exp(−10δ) = 0.5 and δ = 0.069315; accumulating a continuous unit stream for seven years then gives 9.01, choice D.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A is inconsistent with the ratio of the two continuous-annuity integrals; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the ratio of the two continuous-annuity integrals; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the ratio of the two continuous-annuity integrals; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the ratio of the two continuous-annuity integrals; no distinct standard single-step error producing it is identifiable.
Original practice · fully worked
Original variant: infer a force from a continuous accumulation target
A fund receives contributions continuously at a rate of one per year for four years. Its value at year 4 is exactly 6. Determine the constant annual force of interest.
A 12.00%
B 15.00%
C 17.50%
D 19.07%
E 22.50%
Variant answer in brief
Solving the continuous accumulation integral for its force gives 19.07%, choice D.
Setup
Setup
A contribution arriving at time t accumulates for four minus t years.
6=∫04eδ(4−t)dt
Model
Model
Integrating those continuously arriving contributions gives the accumulation factor as a function of the force.
6=δe4δ−1
Compute
Compute
The positive numerical root of the resulting equation is 0.19067214.
δ=0.19067214
Answer
Answer
The annual force of interest is 19.07%, corresponding to choice D.
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