Free SOA Exam FM (Financial Mathematics) Annuities and Non-Contingent Cash Flows Practice Questions

Annuities on SOA Exam FM cover annuities-due, deferred annuities, increasing and decreasing annuities, and perpetuities. Questions test both formula application and the ability to set up complex payment streams using annuity notation.

214 Questions

79 Easy

79 Medium

56 Hard

2026 Syllabus

Sample Questions

Question 1 Easy

What is the term of an annuity?

Solution

E is correct.

The **term** of an annuity is the duration of time from when payments begin to when they end, typically measured by the number of payment periods.

Question 2 Medium

An investor receives $250 at the end of each quarter for 6 years. The effective annual interest rate is 8%. Calculate the present value.

Solution

C is correct.

Convert effective annual rate to quarterly rate:

j = (1.08)^{1/4} - 1 = 1.01943 - 1 = 0.01943

Number of payments:

n = 6 \times 4 = 24

PV = 250 \cdot a_{\overline{24}|j} = 250 \cdot \frac{1 - (1+j)^{-24}}{j}

(1+j)^{24} = (1.08)^{6} = 1.58687

v^{24} = 0.63017

a_{\overline{24}|j} = \frac{1 - 0.63017}{0.01943} = \frac{0.36983}{0.01943} = 19.034

a_{\overline{24}|0.01943} = \frac{1 - 0.63017}{0.01943} = \frac{0.36983}{0.01943} = 19.033

PV = 250 \times 19.033 = 4{,}758

(1.08)^{1/4}

: We need the fourth root of 1.08.

1.08^{0.25} = e^{0.25 \ln 1.08} = e^{0.25 \times 0.07696} = e^{0.01924} = 1.01943

. So

j = 0.01943

(1.01943)^{24} = (1.08)^6 = 1.586874

v^{24} = 0.630170

a_{\overline{24}|} = (1 - 0.630170)/0.01943 = 0.369830/0.01943 = 19.033

PV = 250 \times 19.033 = 4{,}758

Annual payment equivalent: $1,000 per year. Using the annual annuity-immediate and the m-thly adjustment:

a_{\overline{6}|0.08}^{(4)} = \frac{1 - v^6}{i^{(4)}}

where

i^{(4)} = 4[(1.08)^{1/4} - 1] = 4(0.01943) = 0.07771

PV = 250 \times 19.033 = 4{,}758

. The closest choice is 4,622.

(1.08)^2 = 1.1664

(1.08)^3 = 1.259712

(1.08)^6 = (1.259712)^2 = 1.586874

. Yes.

So

PV = 4{,}758

. But the answer choice (A) is 4,622. My calculation may be off.

PV = 1{,}000 \cdot a_{\overline{6}|0.08}^{(4)}

a_{\overline{6}|0.08} = \frac{1 - (1.08)^{-6}}{0.08} = \frac{1 - 0.63017}{0.08} = \frac{0.36983}{0.08} = 4.62288

a_{\overline{6}|0.08}^{(4)} = \frac{i}{i^{(4)}} \cdot a_{\overline{6}|0.08} = \frac{0.08}{0.07771} \times 4.62288 = 1.02948 \times 4.62288 = 4.7592

PV = 1{,}000 \times 4.7592 = 4{,}759

. Still 4,759.

OK, both methods agree at about 4,758-4,759. Choice (A) at 4,622 =

1{,}000 \times a_{\overline{6}|0.08} = 4{,}623

. That would be the PV if payments were annual ($1,000/year), not quarterly.

Question 3 Hard

An annuity-immediate pays $600 per year for 20 years. The effective annual interest rate for the first 10 years is 5%, and 7% for the last 10 years. Calculate the present value at time 0.

Solution

D is correct.

Split into two parts:

Part 1: Payments at times 1-10, valued at 5%.

PV_1 = 600 \cdot a_{\overline{10}|0.05}

v^{10}_{5\%} = (1.05)^{-10} = 0.61391

a_{\overline{10}|0.05} = \frac{1 - 0.61391}{0.05} = 7.72173

PV_1 = 600 \times 7.72173 = 4{,}633

Part 2: Payments at times 11-20. First find their value at time 10:

PV_{10} = 600 \cdot a_{\overline{10}|0.07}

v^{10}_{7\%} = (1.07)^{-10} = 0.50835

a_{\overline{10}|0.07} = \frac{1 - 0.50835}{0.07} = 7.02358

PV_{10} = 600 \times 7.02358 = 4{,}214

Discount to time 0 at 5%:

PV_2 = 4{,}214 \times (1.05)^{-10} = 4{,}214 \times 0.61391 = 2{,}587

Total:

PV = 4{,}633 + 2{,}587 = 7{,}220

The closest answer is 7,211.

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Free SOA Exam FM (Financial Mathematics) Annuities and Non-Contingent Cash Flows Practice Questions

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