Verified solution · Exam FM

SOA Exam FM Sample Question #446 Solution

This Exam FM sample reference tests Arithmetic Annuity-Due. Pairing equal adjacent payments gives a 42-term two-month arithmetic series multiplied by one plus the monthly discount factor. Its present value is 14062.03, choice D.

Independent analysis by ActuaryProof · reviewed August 4, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Arithmetic Annuity-Due question

Setup

The monthly effective rate is 0.4%, and payments at months 2k and 2k plus 1 both equal 100 plus 5k.

j=0.048/12=0.004

C_{2k}=C_{2k+1}=100+5k

Model

Factor each adjacent pair and sum over k from zero through 41.

P=(1+v)\sum_{k=0}^{41}(100+5k)v^{2k}

v=(1.004)^{-1}

Compute

Evaluate the finite level and arithmetic components of the two-month series.

P=14{,}062.0281

Answer

The present value rounds to 14062, choice D.

\boxed{P\approx14{,}062\quad\text{(D)}}

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Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AChoice A treats each pair as one payment and omits the second discounted payment in every pair.
BChoice B increases the payment after each month rather than after each pair, changing the arithmetic schedule.
CChoice C discounts the immediate first payment for one month and shifts the entire annuity later.
EChoice E converts the 4.8% nominal rate to an annual effective rate and then divides that rate by twelve.

Original practice · fully worked

Original variant: solve the first payment of an increasing annuity

An annuity-due makes 24 quarterly payments. Payment k, numbered from zero, is P plus 10k. Interest is a nominal annual rate of 6% compounded monthly. The annuity's present value is 6,000. Determine P.

A
172.43
B
181.25
C
187.51
D
197.51
E
230.00

Variant answer in brief

The equivalent quarterly rate is 1.5075125%. Separating the level and gradient present values and solving the linear equation gives P = 187.51, choice C.

Setup

Convert three monthly compounding periods into one quarterly effective rate.

j_q=(1+0.06/12)^3-1=0.015075125

Model

Separate the unknown level payment from the known increase by quarter.

6{,}000=P\ddot a_{\overline{24}|j_q}+10\sum_{k=0}^{23}kv_q^k

Compute

The due factor is 20.314637 and the gradient sum is 219.084828.

P=\frac{6{,}000-10(219.084828)}{20.314637}=187.5077

Answer

The first payment is 187.51, choice C.

\boxed{P=187.51\quad\text{(C)}}

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The 2210-page Financial Mathematics Proof Manual reorganizes 461 verified Exam FM solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #446 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Arithmetic Annuity-Due question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: solve the first payment of an increasing annuity

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution