Verified solution · Exam FM

SOA Exam FM Sample Question #183 Solution

This Exam FM sample reference tests Duration and Convexity. With equal payments, payment size cancels from the present-value weights; the seven-payment weighted mean time is 3.62 years, choice E.

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 Duration and Convexity question

Setup

List the seven equal year-end loan payments and discount each at 10%.

D_M=\frac{\sum_{t=1}^{7}tX(1.10)^{-t}}{\sum_{t=1}^{7}X(1.10)^{-t}}

Model

Macaulay duration is their present-value-weighted average time; the common payment amount cancels.

D_M=\frac{\sum_{t=1}^{7}t(1.10)^{-t}}{a_{\overline{7}|\,0.10}}

Compute

The time-weighted present-value sum is 17.6315 and the annuity present value factor is 4.8684.

D_M=\frac{17.6315}{4.8684}=3.6215

Answer

Their ratio is 3.62 years, which corresponds to choice E.

\boxed{D_M\approx3.62\text{ years}\quad\text{(E)}}

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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 is inconsistent with the discounted payment-time weighted mean; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the discounted payment-time weighted mean; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the discounted payment-time weighted mean; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the discounted payment-time weighted mean; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: infer a level-payment contract term from duration

A contract makes equal payments at each year-end and is valued at 6% effective annually. Its Macaulay duration is 4.613 years. How many payments does the contract make?

A
7
B
8
C
9
D
10
E
11

Variant answer in brief

A nine-payment annuity at 6% has Macaulay duration 4.6133 years, so the contract makes nine payments, choice C.

Setup

Because payments are equal, test each candidate term using only discounted time weights.

D_M(n)=\frac{\sum_{t=1}^{n}t(1.06)^{-t}}{\sum_{t=1}^{n}(1.06)^{-t}}

Model

For nine payments, form the time-weighted numerator and the ordinary-annuity denominator.

D_M(9)=\frac{\sum_{t=1}^{9}t(1.06)^{-t}}{a_{\overline{9}|\,0.06}}

Compute

The resulting duration is 4.613331 years, matching the stated measure.

D_M(9)=4.613331

Answer

The contract therefore makes nine payments, corresponding to choice C.

\boxed{n=9\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #183 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Duration and Convexity question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: infer a level-payment contract term from duration

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution