Society of Actuaries (SOA) Actuarial Certification Exam — Questions and Answers

Question 1: In a Bühlmann credibility model, the credibility factor Z = n/(n+k) where k is:

The ratio of expected process variance to structural variance (Correct answer)
The prior mean of the claim frequency
The square root of the number of observations
The number of years of data

Correct answer: The ratio of expected process variance to structural variance

k = v/a where v is the expected value of process variance (EVPV) and a is the variance of hypothetical means (VHM), the structural variance.

Question 2: Which actuarial concept describes the difference between the expected claims cost before and after considering policyholder behavioral responses to a rate change?

Loss development
Off-balance correction
Retaliatory loadings
Elasticity adjustment (Correct answer)

Correct answer: Elasticity adjustment

Elasticity adjustment (or demand elasticity) accounts for how policyholders' decisions to purchase, lapse, or switch coverage respond to price changes.

Question 3: In the context of the exam STAM (Short-Term Actuarial Mathematics), which topic is NOT typically covered?

Life contingencies and survival models (Correct answer)
Severity and frequency distributions
Reinsurance and policy modifications
Credibility theory

Correct answer: Life contingencies and survival models

Life contingencies and survival models are covered in Exam LTAM (Long-Term Actuarial Mathematics), not Exam STAM, which focuses on short-term loss models.

Question 4: Under the Own Risk and Solvency Assessment (ORSA) framework required by NAIC, what is an insurer primarily required to demonstrate?

That all risk models have been approved by a Big Four accounting firm
That investment returns will exceed the risk-free rate by at least 2%
That the insurer maintains a combined ratio below 100% in all stress scenarios
That the company understands its risk profile and holds sufficient capital relative to its own risk assessment (Correct answer)

Correct answer: That the company understands its risk profile and holds sufficient capital relative to its own risk assessment

ORSA requires insurers to internally assess their risk profile, quantify capital needs relative to those risks, and demonstrate prospective solvency under stressed conditions to regulators.

Question 5: A life annuity-due of 1 per year pays at the beginning of each year while a life (x) survives. Its APV is denoted:

Ax
ax
ex
äx (Correct answer)

Correct answer: äx

äx (a-umlaut-x) is the actuarial present value of a life annuity-due, with payments at the start of each year the life survives.

Question 6: What is the median of: 3, 7, 2, 9, 5?

3
5 (Correct answer)
7
9

Correct answer: 5

Arrange in order: 2, 3, 5, 7, 9. The middle value is 5.

Question 7: The recursive formula for the terminal reserve satisfies which recursion?

tVx = t+1Vx / (1+i)
tVx = Ax+t − Px · äx+t
(tVx)(1+i) = dx+t + t+1Vx
(tVx + Pt)(1+i) = qx+t · b + px+t · t+1Vx (Correct answer)

Correct answer: (tVx + Pt)(1+i) = qx+t · b + px+t · t+1Vx

The retrospective recursion states that the reserve plus premium accumulated with interest equals the expected death benefit plus the next reserve discounted back.

Question 8: A die is rolled. What is the probability of rolling a number greater than 4?

1/2
2/3
1/6
1/3 (Correct answer)

Correct answer: 1/3

A standard die has six faces (1 through 6). The numbers greater than 4 are 5 and 6, which are 2 outcomes.

Question 9: A risk has n = 5 years of observations with mean X̄ = 200. The portfolio mean is μ = 160 and Bühlmann's k = 5. What is the Bühlmann credibility premium?

170
180 (Correct answer)
200
160

Correct answer: 180

Z = 5/(5+5) = 0.50; premium = 0.50(200) + 0.50(160) = 100 + 80 = 180.

Question 10: Which interpolation method for fractional ages assumes a uniform distribution of deaths (UDD)?

Hyperbolic (Balducci) assumption
De Moivre's law
Constant force of mortality
Linear interpolation of l_x (Correct answer)

Correct answer: Linear interpolation of l_x

Under UDD, the number of survivors l_{x+t} is linear in t between integer ages, implying a uniform distribution of deaths.

Question 11: In experience rating, the credibility-weighted estimate blends the insured's own experience with the class mean. As the volume of own experience increases, the credibility factor Z:

Remains at 0.5 regardless
Decreases toward 0
Equals the prior mean
Increases toward 1 (Correct answer)

Correct answer: Increases toward 1

As data volume grows, Z approaches 1, meaning the estimate relies almost entirely on the insured's own observed experience.

Question 12: In pension valuation, the 'projected benefit obligation' (PBO) differs from the 'accumulated benefit obligation' (ABO) because the PBO:

Uses current salary levels while ABO uses projected salary at retirement
Excludes vested benefits while ABO includes them
Includes future salary increases while ABO does not (Correct answer)
Discounts at a higher rate than the ABO

Correct answer: Includes future salary increases while ABO does not

PBO incorporates projected future compensation increases in the benefit formula, whereas ABO uses only current compensation levels.

Question 13: The complete expectation of life e°x is defined as:

E[K(x)]
1/μ(x)
E[T(x)] (Correct answer)
lx / l0

Correct answer: E[T(x)]

The complete expectation of life e°x equals E[T(x)], the expected value of the complete future lifetime T(x).

Question 14: Under the Uniform Distribution of Deaths (UDD) assumption, qx+t for 0 < t < 1 equals:

(1−t) · qx
t · qx (Correct answer)
qx^t
qx / t

Correct answer: t · qx

Under UDD, deaths are uniformly distributed within each year of age, so the probability of death in the first t years of age x is t·qx.

Question 15: If X ~ Gamma(α, θ), what is E[X]?

α/θ²
α²θ
αθ (Correct answer)
α/θ

Correct answer: αθ

For a Gamma distribution parameterized with shape α and scale θ, E[X] = αθ.

Question 16: Under ERISA Section 4044, when a defined benefit plan terminates in a distress termination, which category of benefits receives priority in asset allocation?

Employer discretionary contributions made in the last 5 years
Disability benefits payable as of termination date
Benefits guaranteed by the PBGC
Participant voluntary contributions with interest (Correct answer)

Correct answer: Participant voluntary contributions with interest

ERISA Section 4044 Priority Category 1 covers participant voluntary contributions plus interest, which receive the highest priority in asset allocation.

Question 17: An insurance company must maintain solvency ratios to comply with regulatory requirements. What does a solvency ratio measure?

The efficiency of claims processing
The profitability of the company
The ability to meet long-term obligations (Correct answer)
The liquidity of the company

Correct answer: The ability to meet long-term obligations

A solvency ratio measures an insurance company's financial strength and its ability to meet its long-term financial obligations, particularly its commitments to policyholders for future claims. Regulators require these ratios to ensure that insurers have adequate capital buffers to absorb unexpected losses and remain financially sound. This protects policyholders and maintains confidence in the insurance market.

Question 18: What does the curtate future lifetime K(x) represent for a life aged x?

The expected remaining years at exact age x
The integer part of the future lifetime (Correct answer)
The complete future lifetime in continuous time
The force of mortality at age x

Correct answer: The integer part of the future lifetime

K(x) is the curtate future lifetime, defined as the greatest integer less than or equal to the complete future lifetime T(x).

Question 19: The variance of the present value of a whole life insurance Āx can be expressed using the second moment as:

²Āx − (Āx)² (Correct answer)
Āx(1 − Āx)
(Āx)² − ²Āx
²Āx − Āx

Correct answer: ²Āx − (Āx)²

Var(Z) = E[Z²] − (E[Z])² = ²Āx − (Āx)², where ²Āx is evaluated at double the force of interest.

Question 20: The net single premium for a whole life insurance of 1 payable at the moment of death for (x) is denoted:

Ax
Āx (Correct answer)
ax
äx

Correct answer: Āx

Āx (A-bar-x) is the APV of a whole life insurance of 1 payable at the moment of death, using continuous discounting.

Question 21: For the Poisson/Gamma conjugate model (Λ ~ Gamma(α, β) with E[Λ] = α/β, and X|Λ ~ Poisson(Λ)), the Bayesian posterior mean after observing S total claims in n periods equals:

(α + S) / (β + n) (Correct answer)
α / β
S / n
(α + n) / (β + S)

Correct answer: (α + S) / (β + n)

For Poisson/Gamma, the posterior is Gamma(α+S, β+n) (rate parameterization), so the posterior mean is (α+S)/(β+n), which also equals the Bühlmann credibility estimate.

Question 22: By the law of total variance, the total variance of a single observation X in the Bühlmann model equals:

EVPV × VHM
EVPV − VHM
VHM − EVPV
EVPV + VHM (Correct answer)

Correct answer: EVPV + VHM

Var(X) = E[Var(X|Θ)] + Var(E[X|Θ]) = EVPV + VHM, by direct application of the law of total variance.

Question 23: As the number of observations n from a single risk increases toward infinity, the Bühlmann credibility factor Z approaches:

1 (Correct answer)
0.5
0
k

Correct answer: 1

Z = n/(n+k) → 1 as n → ∞ because the observed mean X̄ becomes a perfect estimator of the risk's true mean, so full weight is given to observed data.

Question 24: In a compound Poisson model where S has Poisson(λ) claim counts and exponential(θ) severities, the moment generating function of S is:

λ/(λ − t)
e^(λt) / (1 − θt)
exp(λ(Mx(t) − 1)) (Correct answer)
1 / (1 − λθt)

Correct answer: exp(λ(Mx(t) − 1))

The MGF of a compound Poisson S is Ms(t) = exp(λ(Mx(t)−1)) where Mx(t) is the MGF of the severity distribution.

Question 25: What is the relationship between theory and practice in Probability Theory?

Practice is unnecessary if you know theory
Theory is unnecessary
They are completely separate
Theory provides the foundation; practice applies it to real situations (Correct answer)

Correct answer: Theory provides the foundation; practice applies it to real situations

Theory provides the conceptual foundation and principles, while practice involves applying those concepts to real-world situations. Both are essential for competence.

Question 26: A credentialed actuary signs a Statement of Actuarial Opinion (SAO) for a property-casualty insurer. What does the actuary opine on in the SAO?

Whether loss and loss adjustment expense reserves are fairly stated in accordance with accepted actuarial standards (Correct answer)
The accuracy of premium rates filed with the state
The appropriateness of the insurer's investment portfolio
The insurer's compliance with underwriting guidelines

Correct answer: Whether loss and loss adjustment expense reserves are fairly stated in accordance with accepted actuarial standards

The SAO for a P&C insurer requires the appointed actuary to opine on whether the carried reserves for unpaid claims and claim adjustment expenses make a reasonable provision for the obligations.

Question 27: Which actuarial concept measures the expected loss for an insurer net of a proportional reinsurance cession of fraction α?

α·E[S]
E[S] − α·Var[S]
E[S]/(1+α)
(1−α)·E[S] (Correct answer)

Correct answer: (1−α)·E[S]

Under quota share at cession rate α, the cedant retains fraction (1−α) of total losses, so net expected loss is (1−α)·E[S].

Question 28: The prospective formula for the net premium reserve at time t, tVx, is:

Present value of future premiums only
APV of past premiums minus APV of past benefits
Accumulated value of past premiums
APV of future benefits minus APV of future premiums (Correct answer)

Correct answer: APV of future benefits minus APV of future premiums

The prospective reserve tVx = APV of future benefits − APV of future net premiums, representing the insurer's net liability at time t.

Question 29: Which life table function gives the expected number of deaths between exact ages x and x+1 out of a cohort of l0 lives?

dx (Correct answer)
lx
qx
mx

Correct answer: dx

dx = lx − lx+1 represents the number of deaths between ages x and x+1 in the life table cohort.

Question 30: The full credibility standard for pure premiums requires more expected claims than for claim frequency alone because:

Pure premiums depend on both frequency and severity, adding severity variance to total variance (Correct answer)
Insurance regulators impose stricter actuarial standards for premium filings
Pure premiums are always larger numbers than raw frequencies
The normal approximation is less accurate for pure premiums than for counts

Correct answer: Pure premiums depend on both frequency and severity, adding severity variance to total variance

Pure premiums combine frequency and severity, so total variance includes both sources; the standard becomes n₀ = (z/r)²(1 + CV²_S) where CV_S is the severity coefficient of variation.

Question 31: What is a side effect?

An expired medication
An unintended effect of a medication beyond its intended action (Correct answer)
The intended therapeutic effect
A dosing error

Correct answer: An unintended effect of a medication beyond its intended action

Side effects are unintended responses to a medication that occur alongside the desired therapeutic effect.

Question 32: A term life insurance pays a death benefit only if the insured dies within n years. Its APV for (x) is written as:

äx:n|
n|Āx
Āx:n|
A¹x:n| (Correct answer)

Correct answer: A¹x:n|

A¹x:n| (with superscript 1 over x) denotes the n-year term insurance APV, where the superscript 1 indicates the benefit is paid only on the life's death within n years.

Question 33: The probability that a life aged 30 survives to age 40 is denoted as:

10q30
p40
q30
10p30 (Correct answer)

Correct answer: 10p30

The notation nPx represents the probability that a life aged x survives at least n more years, so 10p30 is the probability of surviving from 30 to 40.

Question 34: In the Bühlmann model, the parameter k is defined as:

VHM / Total Variance
EVPV / VHM (Correct answer)
VHM / EVPV
Total Variance / VHM

Correct answer: EVPV / VHM

k = EVPV/VHM (Expected Value of Process Variance divided by Variance of Hypothetical Means); a larger k means more observations are needed to earn credibility.

Question 35: For a portfolio of risks, estimated EVPV = 400 and estimated VHM = 500. What is Bühlmann's k?

1.25
1.60
0.45
0.80 (Correct answer)

Correct answer: 0.80

k = EVPV/VHM = 400/500 = 0.80; this low k means credibility builds quickly (fewer observations needed).

Society of Actuaries (SOA) Actuarial Certification Exam

The SOA actuarial certification exams test mathematical models for actuarial science, including life contingencies, loss models, credibility theory, and insurance mathematics. Candidates must pass a series of preliminary exams to achieve Associate (ASA) or Fellow (FSA) designation.

Questions

210 min

Time Limit

67%

To Pass

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