Fundamentals of Actuarial Mathematics - Long Term - Solutions to Sample Multiple Choice Questions - July 1, 2022

Q: How does higher-than-expected mortality affect actuarial present value for a single premium immediate annuity?

Higher mortality reduces the number of benefit payments, lowering the actuarial present value, so the company benefits under the stated reasoning.

Q: Why can combining savings and insurance products make underwriting assumptions different in Question 1.2?

The solution states that with certain combined products (e.g., limited underwriting), insurers may have increased interest in the combination, making the option that claims otherwise false.

Q: What approach is used to compute survival-related quantities like forces of mortality and survival probabilities across Questions 2 and 3?

The solutions apply relationships among survival functions (S), hazard/force of mortality (µ), and present values, often using exponentials of µ and calculus or conditional expectation formulas; some items also provide an alternative derivation.

Sep 27, 2026

Uploaded by Emma Wilson

This document provides worked solutions to sample multiple-choice questions for “Fundamentals of Actuarial Mathematics - Long Term,” dated July 1, 2022. It explains why each selected option is correct or false using actuarial concepts such as mortality assumptions, underwriting, life annuities, and pure endowments. Additional problems address survival probabilities, present values, force of mortality, conditional expectations, mixed distributions, and alternative solution methods for computed quantities.

LanguageEnglish

CategoryExam

FormatPDF

Pages76

Reading Time3 h 12 min

AccessFree to download

Question 1.1

Question 1.2

Question 2.1

Question 2.2

Question 2.3

Question 2.4

Question 2.5

Question 2.6

Question 2.7

Question 2.8

Question 3.1

Question 3.2

Fundamentals of Actuarial Mathematics – Long Term Solutions to Sample Multiple Choice Questions July 1, 2022 Versions: July 1, 2022 Original version for FAM-L. These solutions are written to provide evidence that the stated answer is correct and to offer at least one approach to solving the problem. There may be additional approaches that can be employed. Copyright © 2022 Society of Actuaries Question 1.1 Answer C Answer C is false. If the purchaser of a single premium immediate annuity has higher mortality than expected, this reduces the number of payments that will be paid. Therefore, the Actuarial Present Value will be less and the insurance company will benefit. Therefore, single premium life annuities do not need to be underwritten. The other items are true. A: Life insurance is typically underwritten to prevent adverse selection as higher mortality than expected will result in the Actuarial Present Value of the benefits being higher than expected. B: In some cases, such as direct marketed products for low face amounts, there may be very limited underwriting. The actuary would assume that mortality will be higher than normal, but the expenses related to selling the business will be low and partially offset the extra mortality. D: If the insured’s occupation or hobby is hazardous, then the insured life may be rated. E: If the purchaser of the pure endowment has higher mortality than expected, this reduces the number of endowments that will be paid. Therefore, the Actuarial Present Value will be less and the insurance company will benefit. Therefore, pure endowments do not need to be underwritten. Question 1.2 Answer E Insurers have an increased interest in combining savings and insurance products so Item E is false. The other items are all true. Question 2.1 Answer: B 1 􀂧 t 􀂷 4 1 􀂪 ω − t 􀂺 Since S0 (t) = 1− F0 (t) = 􀂩􀂨 1 − ~~ ~~ω~~ ~~ 􀂹􀂸 , we have ln[S0 (t)] = 4 ln 􀂫􀂬~~ ~~ω~~ ~~􀂼􀂻 . d 1 1 1 1 1 Then µ = − ~~ ~~ logS0 (t) = ~~ ~~ ~~ ~~ , and µ65 = ~~ ~~ = ~~ ~~ ~~ ~~ ⇒ ω = 110. t dt 4 ω − t 180 4 ω − 65 e106 = ∑t1 t p106 , since 4 p106 = 0 S0 (106 + t) 􀂨􀂧􀂩1 − ~~ ~~10161+0~~ ~~t~~ ~~ 􀂷􀂹􀂸 1/ 4 􀂧 4 − t 􀂷 1/ 4 t p106 = S0 (106) = 􀂧 1 − 106 􀂷 1/ 4 = 􀂩􀂨 4 􀂹􀂸 􀂩􀂨 110 􀂹􀂸 e 106 = ∑ii41 t p 106 = ~~ ~~41025~~ ~~ ( 10.25 + 2 0.25 + 30.25 ) = 2.4786 Question 2.2 Answer: D This is a mixed distribution for the population, since the vaccine will apply to all once available. S = \# of survivors Available? (A) Pr( A) 2 p | A E (S A) Var (S A) E (S2 A) Yes 0.2 0.9702 97,020 2,891 9,412,883,291 No 0.8 0.9604 96,040 3,803 9,223,685,403 E (S ) E (S2 ) 96,236 9,261,524,981 Var (S) 157,285 SD (S ) 397 As an example, the formulas for the “No” row are Pr(No) = 1 – 0.2 = 0.8 2 p given No = (0.98 during year 1)(0.98 during year 2) = 0.9604. E (S | No), Var(S | No) and E(S2 | No) are just binomial, n = 100, 000; p(success) = 0.9604 E (S), E(S2 ) are weighted averages, Var (S) = E (S2 ) − E(S)2 Or, by the conditional variance formula: Var (S) = Var [E(S | A)] + E[Var(S | A)] = 0.2(0 . 8)(97, 020 − 96, 040)2 + 0 .2(2,891) + 0 . 8(3,803) = 153, 664 + 3, 621 = 157, 285 StdDev (S) = 397 Question 2.3 Answer: A fx ( t ) = − ~~ ~~ddt~~ ~~ Sx ( t ) = − ~~ ~~ddt~~ ~~ 􀂨􀂧􀂩e − ~~ ~~Blnc~~ ~~ (cx)(ct −1) 􀂷􀂹􀂸 = −e − ~~ ~~Blnc~~ ~~ (cx)(ct −1) ⋅ 􀂩􀂧􀂨 − ~~ ~~lBnc ⋅ cx 􀂷􀂹􀂸 ⋅ ct ⋅ln c = e − ~~ ~~Blnc~~ ~~ (cx)(ct −1) ⋅ Bcx+t = 0.00027 × 1.1x+t ⋅ e − ~~ ~~0l.0n021(1.1x)(1.1t −1) f50 (10) = 0.00027 × 1.150+10 ⋅ e − ~~ ~~0l.0n021(1.150 )(1.110 −1) = 0.04839 Alternative Solution: fx (t) = t px ⋅ µx+t Then we can use the formulas given for Makeham with A = 0, B = 0.00027 and c = 1.1 fx (t) = 􀂧􀂩􀂨 e − ~~ ~~0l.0n021(1.150 )(1.110 −1) 􀂷􀂹􀂸 ( 0.00027 × 1.150+10 ) = 0.04839 Question 2.4 Answer: E 􀁄 e75:10 = ∫t010 t p75dt where t +x p0 1 − ~~ ~~(t~~ ~~+~~ ~~x10002~~ ~~ 10000 − (t + x )2 p = = = t x p x2 10000 − x2 x 0 1 − 10000 for 0 < t

FAQs

How does higher-than-expected mortality affect actuarial present value for a single premium immediate annuity?

Higher mortality reduces the number of benefit payments, lowering the actuarial present value, so the company benefits under the stated reasoning.

Why can combining savings and insurance products make underwriting assumptions different in Question 1.2?

The solution states that with certain combined products (e.g., limited underwriting), insurers may have increased interest in the combination, making the option that claims otherwise false.

What approach is used to compute survival-related quantities like forces of mortality and survival probabilities across Questions 2 and 3?

The solutions apply relationships among survival functions (S), hazard/force of mortality (µ), and present values, often using exponentials of µ and calculus or conditional expectation formulas; some items also provide an alternative derivation.

Sep 27, 2026

Uploaded by Emma Wilson