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

Question 1: You invest $1,000 at an annual interest rate of 5% compounded annually. What will the investment be worth after 3 years?

$1,150
$1,215.50
$1,157.63 (Correct answer)
$1,200

Correct answer: $1,157.63

The formula for compound interest is: FV=PV⋅(1+r) t Here, PV=1,000, 𝑟=0.05 and t=3: FV=1,000⋅(1+0.05) 3=1,000⋅(1.157625)=1,157.63

Question 2: An insurance company covers losses up to $10,000 per claim. If the probability distribution of losses is: P(X≤10,000)=0.7, P(X>10,000)=0.3, and the average claim for losses exceeding $10,000 is $15,000, what is the expected claim amount?

$10,000
$10,500
$9,500 (Correct answer)
$9,000

Correct answer: $9,500

The expected claim amount is calculated as: E(X)=P(X≤10,000)⋅10,000+P(X>10,000)⋅15,000 E(X)=(0.7⋅10,000)+(0.3⋅15,000)=7,000+4,500=9,500

Question 3: In insurance pricing, the pure premium is calculated as:

Loss ratio multiplied by earned premium
Gross premium minus expenses
Expected losses divided by earned exposure units (Correct answer)
Expected losses divided by expected claim count

Correct answer: Expected losses divided by earned exposure units

The pure premium equals E[aggregate losses] / E[exposure], representing the expected loss cost per unit of exposure.

Question 4: Under PPA 2006, what is the consequence if a single-employer plan's funding shortfall triggers the 'at-risk' status?

Higher minimum required contributions using at-risk actuarial assumptions apply (Correct answer)
The plan must immediately convert to a defined contribution plan
The plan sponsor receives a tax credit for additional contributions
The PBGC takes over plan administration immediately

Correct answer: Higher minimum required contributions using at-risk actuarial assumptions apply

At-risk plans must use prescribed at-risk assumptions that increase the funding target and potentially the minimum required contribution.

Question 5: The force of mortality μ(x) is best described as:

The instantaneous rate of mortality at exact age x (Correct answer)
The number of survivors at age x
The probability of death in the next year
The expected lifetime from birth

Correct answer: The instantaneous rate of mortality at exact age x

The force of mortality μ(x) is the instantaneous hazard rate of death, analogous to an intensity in a Poisson process.

Question 6: For a select-and-ultimate life table, the select period represents:

The age at which the life was selected
The first years after underwriting when mortality differs from the ultimate table (Correct answer)
The duration of the insurance policy
The period after which all underwriting effects have worn off

Correct answer: The first years after underwriting when mortality differs from the ultimate table

The select period is the duration immediately following underwriting during which recently selected lives have different (usually lower) mortality than the ultimate rates.

Question 7: What does the curtate future lifetime K(x) represent for a life aged x?

The complete future lifetime in continuous time
The integer part of the future lifetime (Correct answer)
The expected remaining years at exact age x
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 8: The Kolmogorov forward equations in a Markov chain model describe which relationship?

How to compute backward expectations from terminal conditions
The stationary distribution of an ergodic chain
The spectral decomposition of the generator matrix
How occupation probabilities evolve forward in time as a function of transition intensities (Correct answer)

Correct answer: How occupation probabilities evolve forward in time as a function of transition intensities

The Kolmogorov forward equations express the time derivative of transition probabilities p_{ij}(t) in terms of the current state j and its outgoing intensities.

Question 9: Within a given risk (conditional on its parameter Θ), successive annual observations X₁, X₂, ..., Xₙ in the Bühlmann model are:

Independent but NOT identically distributed given Θ
Positively correlated because later years depend on earlier claims
Conditionally independent and identically distributed given Θ (Correct answer)
Negatively correlated through regression to the mean

Correct answer: Conditionally independent and identically distributed given Θ

Given Θ, observations are conditionally i.i.d.; unconditional positive correlation between same-risk observations arises only from the unknown shared Θ, not from serial dependence.

Question 10: The variance of the present value of a whole life insurance Āx can be expressed using the second moment as:

Āx(1 − Āx)
²Āx − (Āx)² (Correct answer)
(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 11: Which relationship correctly connects the whole life insurance APV and the endowment insurance APV?

Āx = Āx:n| + n|Āx
Āx = A¹x:n| + nEx · Āx+n (Correct answer)
Āx = A¹x:n| + Ax:n|1
Āx = äx · δ

Correct answer: Āx = A¹x:n| + nEx · Āx+n

A whole life can be split into an n-year term (A¹x:n|) plus an n-year deferred whole life (nEx · Āx+n), which together cover all possible death times.

Question 12: Makeham's mortality law adds which component to Gompertz's law?

An age-dependent frailty multiplier
A quadratic age term
A maximum age omega
A constant background (accident) hazard A (Correct answer)

Correct answer: A constant background (accident) hazard A

Makeham's law is μ(x) = A + Bc^x, extending Gompertz's μ(x) = Bc^x by adding a constant term A representing age-independent background mortality.

Question 13: In a select-and-ultimate mortality table, what does the 'select' period represent?

The initial period when recent underwriting affects mortality (Correct answer)
The period after age 65
The period of highest mortality rates
The ultimate steady-state mortality

Correct answer: The initial period when recent underwriting affects mortality

The select period captures the effect of recent underwriting or selection, during which mortality differs from the population average.

Question 14: Under classical credibility, if full credibility requires n₀ = 1,082 claims and a risk has n = 271 claims, the partial credibility factor Z is:

0.71
0.50 (Correct answer)
0.27
0.25

Correct answer: 0.50

Using the square-root formula: Z = √(n/n₀) = √(271/1082) = √(0.25) = 0.50.

Question 15: Which life table function gives the expected number of deaths between exact ages x and x+1 out of a cohort of l0 lives?

lx
qx
dx (Correct answer)
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 16: Which actuarial concept measures the expected loss for an insurer net of a proportional reinsurance cession of fraction α?

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

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 17: The concept of immunization in fixed-income portfolio management aims to:

Maximize coupon income regardless of rate changes
Eliminate credit risk through diversification
Maximize convexity to benefit from rate volatility
Match the duration of assets and liabilities to protect against interest rate risk (Correct answer)

Correct answer: Match the duration of assets and liabilities to protect against interest rate risk

Immunization matches asset and liability durations so that changes in interest rates affect both sides equally, protecting the surplus.

Question 18: What is a drug contraindication?

A condition that makes a particular treatment inadvisable (Correct answer)
A positive drug interaction
A dosage calculation
A drug recommendation

Correct answer: A condition that makes a particular treatment inadvisable

A contraindication is a specific situation where a drug should not be used because it may be harmful to the patient.

Question 19: Under the Uniform Distribution of Deaths (UDD) assumption, qx+t for 0 < t < 1 equals:

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

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 20: A 'catastrophe load' is added to insurance premiums primarily to:

Build margin for infrequent but severe events that distort expected loss ratios (Correct answer)
Cover routine high-frequency, low-severity claims
Satisfy state filing requirements for rate increases
Compensate agents for catastrophe response services

Correct answer: Build margin for infrequent but severe events that distort expected loss ratios

Catastrophe loads ensure premiums reflect the long-run expected cost of rare, severe events (hurricanes, earthquakes) that may not appear in recent historical data but represent real expected losses.

Question 21: The Wang transform in risk pricing applies a distortion g(u) = Φ(Φ⁻¹(u) + λ) to the survival function. What does increasing λ accomplish?

Removes the heavy tail of the original distribution
Increases the risk loading, assigning higher weight to tail events (Correct answer)
Converts the distribution to lognormal
Decreases the risk-adjusted premium toward the expected value

Correct answer: Increases the risk loading, assigning higher weight to tail events

A larger λ shifts the survival function upward via the distortion, placing more weight on adverse outcomes and thus increasing the risk-adjusted premium.

Question 22: Which of the following is a key property of the Poisson distribution that makes it useful in claim count modeling?

Its mean always exceeds its variance
It always produces overdispersed data
Its mean equals its variance (Correct answer)
It is defined on all real numbers

Correct answer: Its mean equals its variance

For a Poisson distribution, E[N] = Var[N] = λ, which is the equidispersion property central to many insurance models.

Question 23: Which goodness-of-fit test statistic compares the squared differences between observed and expected frequencies divided by expected frequencies?

Anderson-Darling statistic
Kolmogorov-Smirnov statistic
Log-likelihood ratio
Chi-square statistic (Correct answer)

Correct answer: Chi-square statistic

The chi-square goodness-of-fit statistic is Σ(Oi − Ei)²/Ei, measuring the discrepancy between observed and expected cell frequencies.

Question 24: Using k = 0.80 from the portfolio above, a risk with n = 4 years of data has X̄ = 140 and the portfolio mean μ = 110. What is the Bühlmann credibility premium?

135.0 (Correct answer)
140.0
125.0
110.0

Correct answer: 135.0

Z = 4/(4 + 0.80) = 4/4.80 ≈ 0.833; premium = 0.833(140) + 0.167(110) = 116.67 + 18.33 ≈ 135.0.

Question 25: If X follows a lognormal distribution with parameters μ and σ², what is E[X]?

μe^(σ²/2)
e^(μ + σ²/2) (Correct answer)
e^(μ + σ²)
e^μ

Correct answer: e^(μ + σ²/2)

For a lognormal distribution, E[X] = e^(μ + σ²/2).

Question 26: 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:

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

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 27: The net premium for an n-year endowment insurance is calculated by:

Multiplying the APV of benefit by the interest rate
Dividing the APV of benefit by the APV of an n-year life annuity-due (Correct answer)
Setting the benefit APV equal to the expense loading
Dividing the APV of benefit by the APV of a whole life annuity

Correct answer: Dividing the APV of benefit by the APV of an n-year life annuity-due

The net annual premium P equals the APV of the insurance benefit divided by the APV of the premium-paying annuity, here an n-year temporary life annuity-due.

Question 28: In a multiple-decrement table, the total decrement qx(τ) represents:

The probability of surviving all decrements
The force of decrement for cause 2
The probability of leaving the group from cause 1 only
The probability of leaving the group from any cause (Correct answer)

Correct answer: The probability of leaving the group from any cause

qx(τ) is the probability that a life aged x leaves the group within one year due to any of the competing causes of decrement.

Question 29: A reinsurer offers 'excess of loss' coverage with a retention of $500,000 and a limit of $1,000,000. If a claim is $1,800,000, how much does the primary insurer pay?

$800,000
$1,800,000
$1,000,000
$500,000 (Correct answer)

Correct answer: $500,000

The primary insurer retains the first $500,000; the reinsurer covers $1,000,000 (its limit); the remaining $300,000 exceeds the reinsurer's cover and reverts to the primary insurer — but typically the primary retains only the first $500,000 as the retention layer.

Question 30: In the Bühlmann-Straub model, the grand mean estimator μ̂ is computed as:

Simple average of all individual risk means X̄ᵢ
Σ X̄ᵢ / r where r is the number of risks
Σ(mᵢ × X̄ᵢ) / Σmᵢ
Σ(Zᵢ × X̄ᵢ) / ΣZᵢ (Correct answer)

Correct answer: Σ(Zᵢ × X̄ᵢ) / ΣZᵢ

The grand mean μ̂ is the credibility-weighted average Σ(Zᵢ X̄ᵢ)/ΣZᵢ, using each risk's credibility factor as its weight.

Question 31: In the collective risk model, what does the aggregate loss S = X1 + X2 + ... + XN represent?

Sum of individual claim severities over a random number of claims (Correct answer)
Average claim severity multiplied by exposure
Total number of claims in a period
Net premium income minus expenses

Correct answer: Sum of individual claim severities over a random number of claims

S is the aggregate loss formed by summing N individual claim amounts, where N is a random claim count variable.

Question 32: If claim severity follows an exponential distribution (CV_S = 1.0), the full credibility standard for pure premiums at 90% confidence with ±5% accuracy requires approximately:

1,082 expected claims
3,246 expected claims
2,164 expected claims (Correct answer)
4,328 expected claims

Correct answer: 2,164 expected claims

n₀ = 1,082 × (1 + CV²_S) = 1,082 × (1 + 1²) = 2,164 expected claims for full credibility of pure premiums.

Question 33: In the Bühlmann credibility premium P = Z × X̄ + (1−Z) × μ, the credibility factor Z equals:

k / (n + k)
n / k
n / (n + k) (Correct answer)
√(n / k)

Correct answer: n / (n + k)

Z = n/(n+k) is the Bühlmann credibility factor, where n is the number of observations and k = EVPV/VHM.

Question 34: An actuary advising a pension plan sponsor on contribution strategy uses the concept of 'funding corridor.' What does this corridor represent?

The band around the actuarial accrued liability within which no contribution is required
The spread between the discount rate and expected asset return
The range of investment returns consistent with meeting obligations
Minimum and maximum allowable contribution levels under IRS rules (Correct answer)

Correct answer: Minimum and maximum allowable contribution levels under IRS rules

IRS rules for qualified plans prescribe minimum required contributions and limits on deductible contributions, forming a regulatory funding corridor.

Question 35: A term life insurance pays a death benefit only if the insured dies within n years. Its APV for (x) is written as:

Āx:n|
n|Āx
äx: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.

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%

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