This Exam FM sample reference tests Geometrically Varying Annuities. The 20-payment geometric annuity factor is 17.7267, so the first payment is 11,282.42 and the final payment is 16,436.35, choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A is the first payment, 11,282, rather than the twentieth payment requested.
CChoice C is obtained by applying one extra 2% increase. There are 19 increases from payment 1 to payment 20.
DChoice D does not satisfy the 200,000 present-value equation. The growth and discount factors must be applied payment by payment.
EChoice E is too large because it treats the 2% payment growth as additional investment return instead of part of the cash-flow pattern.
Original practice · fully worked
Original variant: present value of declining maintenance payments
A service contract pays 1,600 one year from now. Each of the following 14 annual payments is 3% smaller than the preceding payment. At an annual effective discount rate of 5%, determine the present value of the 15 payments.
A 10,445
B 12,716
C 13,908
D 15,264
E 17,831
Variant answer in brief
Discounting the 15-payment geometric sequence gives a present value of 13,907.91, which is choice C.
Setup
Setup
Payment t is 1,600 times 0.97 to the power t minus 1, while each payment is discounted at 5%.
Ct=1600(0.97)t−1
v=(1.05)−1
Model
Model
Combine payment decline and financial discounting into one finite geometric series.
PV=1600vj=0∑14(0.97v)j
PV=1600v1−0.97v1−(0.97v)15
Compute
Compute
Substitution gives a present value a little under 13,908.
PV=13907.9070
Answer
Answer
The contract is worth approximately 13,908 today, so choice C is correct.
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