Verified solution · Exam FM

SOA Exam FM Sample Question #252 Solution

This Exam FM sample reference tests Interest Rate Measures. Integrating the force 2/(1 + 2t) from 0 to 20 gives log 41, so Fund X grows to 41. Equality at year 20 makes Fund Y's five-year value the fourth root of 41, or 2.53, choice B.

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 Interest Rate Measures question

Setup

A time-varying force accumulates one unit by the exponential of its integral.

A_X(20)=\exp\left(\int_0^{20}\frac{2}{1+2t}\,dt\right)

Model

The integral is the logarithm of 1 + 2t evaluated from 0 to 20.

A_X(20)=e^{\ln41}=41=(1+i)^{20}

Compute

Five years is one quarter of the common 20-year horizon, so Fund Y's value is 2.530440.

A_Y(5)=41^{1/4}=2.53043953

Answer

Fund Y is worth approximately 2.53 after five years, selecting choice B.

\boxed{A_Y(5)\approx2.53\quad\text{(B)}}

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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 (2.46) does not satisfy the integrated force through year 20 and the one-quarter horizon relation; no distinct standard one-step error is identifiable.
CChoice C (2.60) does not satisfy the integrated force through year 20 and the one-quarter horizon relation; no distinct standard one-step error is identifiable.
DChoice D (2.67) does not satisfy the integrated force through year 20 and the one-quarter horizon relation; no distinct standard one-step error is identifiable.
EChoice E (2.74) does not satisfy the integrated force through year 20 and the one-quarter horizon relation; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: force coefficient calibrated to a constant annual return

A fund has force of interest a divided by 1 + 2t at time t. Over ten years it must produce the same accumulation as a fund earning 5% annually. Calculate a.

A
0.27
B
0.29
C
0.30
D
0.32
E
0.34

Variant answer in brief

The integrated force is one-half a times log 21. Equating it to ten times log 1.05 gives a = 0.3205, choice D.

Setup

Integrate the parameterized force through ten years.

\int_0^{10}\frac{a}{1+2t}\,dt=\frac{a}{2}\ln21

Model

Match that integrated force to the logarithm of ten years of 5% annual accumulation.

\frac{a}{2}\ln21=10\ln(1.05)

Compute

Solving gives coefficient a = 0.32051112.

a=\frac{20\ln(1.05)}{\ln21}=0.3205111157

Answer

The force coefficient is 0.3205, selecting choice D.

\boxed{a\approx0.3205\quad\text{(D)}}

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ActuaryProof. “SOA Exam FM Sample Question #252 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Interest Rate Measures question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: force coefficient calibrated to a constant annual return

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution