Worked solution · Exam FM

SOA Exam FM Sample Question #162 Solution

This Exam FM sample reference tests Loan Amortization. Using quarters as the time unit, the six 800 advances occur at odd quarter indices, the 4,000 balance is at quarter 12, and 36 midmonth payments occur at n−0.5 thirds of a quarter; only choice E places every term correctly.

Independent analysis by ActuaryProof · reviewed October 6, 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 Loan Amortization question

Setup

Fix one time unit before assigning exponents; quarters are natural because the nominal rate is convertible quarterly.

j_q=0.264/4

Model

The additional borrowing dates occur at months 3, 9, and so on, while payments occur at each month's midpoint.

PV_{\rm advances}=4000+\sum_{n=1}^{6}\frac{800}{(1+0.264/12)^{6n-3}}

Compute

Translate those month positions into fractional quarter exponents and include both the initial and terminal 4,000 balances.

PV_{\rm repayments}=\frac{4000}{(1+0.264/4)^{12}}+\sum_{n=1}^{36}\frac{X}{(1+0.264/4)^{n/3-1/6}}

Answer

Only the equation in choice E preserves all those dates and rate conversions.

\boxed{\text{choice E}\quad\text{(E)}}

Test the method on unseen problems

Financial Mathematics Loans Practice Workbook

30 original loans problems with complete solutions and wrong-choice analysis. Written for Exam FM, not copied from the SOA set. One-time $19 PDF.

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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.

AThis option mixes quarterly and monthly rate divisors while leaving payment exponents on incompatible scales.
BIts advance dates use the wrong month-to-quarter conversion.
CIt places the midmonth payments at integer quarter times rather than fractional positions.
DIt discounts at least one 4,000 balance to the wrong endpoint.

Original practice · fully worked

Original variant: monthly midperiod payments with later equipment draws

A studio owes 2,000 at month 0, borrows another 500 at months 3, 9, and 15, and pays X at the middle of each month for 18 months. The balance is zero at month 18. At 1% effective per month, determine X.

A
180.00
B
195.00
C
204.68
D
215.00
E
230.00

Answer and calculation

Present-valuing the four advances and 18 half-month-shifted payments gives X = 204.68, choice C.

Setup

Month zero is the value date; later equipment draws are discounted by their exact month counts.

PV_D=2000+500(v^3+v^9+v^{15})=3373.1397

Model

A payment in month k occurs at k minus one-half, so the repayment annuity has a half-month timing shift.

PV_P=X\sum_{k=1}^{18}v^{k-1/2}=16.480056X

Compute

The advance value is 3,373.1397 and the payment factor is 16.480056.

X=3373.1397/16.480056=204.6801

Answer

Their quotient is a monthly payment of 204.68, corresponding to choice C.

\boxed{X=204.68\quad\text{(C)}}

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How to solve this Loan Amortization question

Setup

Model

Compute

Answer

Financial Mathematics Loans Practice Workbook

Why the other letters tempt

Original variant: monthly midperiod payments with later equipment draws

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

Next steps for Exam FM Loans