Verified solution · Exam FM

SOA Exam FM Sample Question #265 Solution

This Exam FM sample reference tests Interest Rate Measures. The first loan's observed monthly interest is P1 − P0 + m, so its nominal annual rate is 12 times that amount divided by P0. Loan B credits one 365th of the nominal rate daily for 365/12 days per month; this is the expression in 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.

Read the solution Open official source

Independent solution

How to solve this Interest Rate Measures question

Setup

Use the first loan's one-month balance equation to isolate the month's interest.

P_0-m+I=P_1

I=P_1-P_0+m

Model

Convert the resulting monthly effective rate to the shared nominal annual rate, then to Loan B's daily periodic rate.

i^{(12)}=12\frac{P_1-P_0+m}{P_0}

r_d=\frac{12}{365}\frac{P_1-P_0+m}{P_0}

Compute

Accumulate the daily rate through 365/12 days in one equal month.

j=\left[1+\frac{12}{365}\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{365/12}-1

Answer

The resulting expression is listed in choice E.

\boxed{\text{choice E}}

Continue without hunting through PDFs

All 461 Exam FM sample solutions in syllabus order

The searchable 2210-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam FM study manual PDF

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 (1 +   −1) does not satisfy the balance-implied nominal rate followed by daily compounding over 365/12 days; no distinct standard one-step error is identifiable.
BChoice B (1 + 12   −1) does not satisfy the balance-implied nominal rate followed by daily compounding over 365/12 days; no distinct standard one-step error is identifiable.
CChoice C (1 +   −1) does not satisfy the balance-implied nominal rate followed by daily compounding over 365/12 days; no distinct standard one-step error is identifiable.
DChoice D (1 +   −1) does not satisfy the balance-implied nominal rate followed by daily compounding over 365/12 days; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: daily-compounded monthly rate from one observed loan balance

A loan starts a month at 10,000, receives a month-end payment of 900, and then has balance 9,150. Its nominal annual rate is shared by another loan that compounds daily using 365 days and twelve equal months. Calculate the second loan's effective monthly rate.

A
0.43%
B
0.45%
C
0.48%
D
0.50%
E
0.53%

Variant answer in brief

The observed interest is 50, implying 0.5% monthly and a 6% nominal annual rate. Compounding one 365th of 6% for 365/12 days gives effective monthly rate 0.5012%, choice D.

Setup

Recover the first loan's monthly interest from its balance change.

I=9150-10000+900=50

j_A=50/10000=0.005

Model

Annualize nominally and apply the daily rate for one equal month.

i^{(12)}=12(0.005)=0.06

j_B=(1+0.06/365)^{365/12}-1

Compute

The daily-compounded monthly rate is 0.501211%.

j_B=0.0050121079

Answer

The second loan's effective monthly rate is 0.5012%, selecting choice D.

\boxed{j_B\approx0.5012\%\quad\text{(D)}}

Keep practicing

Turn this solution into a complete study sequence

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.

See the Exam FM study manual PDF — $49 More Exam FM Time Value of Money practice problems Free Exam FM formula sheet

Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #265 — 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: daily-compounded monthly rate from one observed loan balance

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution