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Iran Official Insurance Actuary Exam study materials
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Iran Official Insurance Actuary Exam practice questions

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Question 1Probability and Mathematical Statistics

In actuarial risk modelling, a loss variable X has survival function S(x)=P(X>x). Which expression equals the cumulative distribution function F(x) for a continuous loss variable?

  1. S(x) − 1
  2. 1 − S(x) Correct answer.
  3. 1/S(x)
  4. S(x)/x

Explanation

By definition S(x)=P(X>x) and F(x)=P(X≤x). For a continuous variable P(X>x)=1−P(X≤x), so F(x)=1−S(x). This complement relation is used constantly when converting between survival and distribution functions in loss models.

Question 2Probability and Mathematical Statistics

A claim count N follows a Poisson distribution with mean λ. Which statement about the variance of N is correct?

  1. Variance is independent of λ
  2. Variance equals λ Correct answer.
  3. Variance equals λ squared
  4. Variance equals the square root of λ

Explanation

For a Poisson random variable the mean and variance are both equal to the rate parameter λ. This equidispersion property is why Poisson is a baseline model for claim frequency. Overdispersion in real portfolios motivates alternatives such as negative binomial.

Question 3Probability and Mathematical Statistics

An actuary models the number of claims X from a portfolio as Binomial(n, p). Which expression gives E[X]?

  1. n p Correct answer.
  2. n p (1 − p)
  3. n / p
  4. p / n

Explanation

The binomial variable counts successes in n independent trials each with success probability p, so its mean is n times p. The variance is n p (1−p). These moments are standard inputs when calibrating frequency models.

Question 4Probability and Mathematical Statistics

A continuous severity variable has density f(x)=2x on the interval (0,1) and zero elsewhere. What is P(X > 0.5)?

  1. 0.50
  2. 0.125
  3. 0.25
  4. 0.75 Correct answer.

Explanation

Integrating the density from 0.5 to 1 gives [x squared] evaluated there, which equals 1 − 0.25 = 0.75. Equivalently the distribution function is x squared, so the survival probability at 0.5 is 0.75.

Question 5Probability and Mathematical Statistics

Losses X are exponentially distributed with mean θ. Which expression gives the survival function P(X > x) for x ≥ 0?

  1. 1 − exp(−x/θ)
  2. x exp(−x/θ)
  3. exp(−x/θ) Correct answer.
  4. exp(−θ x)

Explanation

An exponential with mean θ has rate 1/θ, so its survival function is exp(−x/θ). The expression 1−exp(−x/θ) is the distribution function. Exponential severity is memoryless, which simplifies excess-of-loss computations.

Question 6Probability and Mathematical Statistics

For a deductible d applied to a loss X, the payment is (X − d) when X exceeds d, and zero otherwise. Which quantity does the insurer's expected payment equal?

  1. E[X] − d
  2. E[X] − E[(X − d)+]
  3. E[(X − d)+] Correct answer.
  4. d · P(X > d)

Explanation

The insurer pays only the excess of the loss over the deductible, written as (X−d)+, so its expected cost is E[(X−d)+]. This limited expected value is the core building block of deductible pricing. It differs from E[X]−d because no payment occurs below the deductible.

Question 7Probability and Mathematical Statistics

Which property characterizes the memoryless property of the exponential distribution?

  1. P(X > s + t) = P(X > s) + P(X > t)
  2. P(X > s | X > t) = P(X > s)
  3. P(X > s + t | X > s) = P(X > t) Correct answer.
  4. P(X > s + t) = P(X > s) · P(X > t)

Explanation

Memorylessness states that conditional survival beyond an additional time t does not depend on the elapsed time s. This is exactly P(X>s+t | X>s)=P(X>t). It underpins the constant hazard rate of exponential severity.

Question 8Probability and Mathematical Statistics

A compound Poisson aggregate model has frequency mean λ and severity mean μ. Which expression gives the expected aggregate loss E[S]?

  1. μ / λ
  2. λ + μ
  3. λ μ Correct answer.
  4. λ μ squared

Explanation

By Wald's identity the expectation of a random sum equals the expected count times the expected individual severity, giving λμ. This holds even when the severity variance is nonzero. It is the first step in aggregate loss pricing.

Question 9Probability and Mathematical Statistics

Under the compound Poisson model with frequency mean λ and severity variance σ squared, which expression gives Var(S) of the aggregate loss S?

  1. λ E[X squared] Correct answer.
  2. λ σ squared
  3. E[X squared] / λ
  4. λ squared E[X squared]

Explanation

The variance of a compound Poisson sum is λ times the second moment of severity, i.e. λE[X squared]. Since E[X squared]=σ squared+μ squared, this exceeds λσ squared. The extra λμ squared term reflects variability in claim counts.

Question 10Probability and Mathematical Statistics

A risk has mean loss 200 and standard deviation 50. Using the one-sided Chebyshev (Cantelli) inequality, which upper bound applies to P(X − 200 ≥ 100)?

  1. 0.25
  2. 0.10
  3. 0.50
  4. 0.20 Correct answer.

Explanation

Cantelli's inequality bounds P(X−μ≥kσ) by 1/(1+k squared). Here k=100/50=2, so the bound is 1/(1+4)=0.20. This bound is distribution-free and useful when only moments are known.

Question 11Probability and Mathematical Statistics

A sample of n independent observations is drawn from a normal population with unknown mean and known variance. Which distribution does the sample mean follow after standardization?

  1. Chi-square with n degrees of freedom
  2. Student t with n−1 degrees of freedom
  3. F with 1 and n degrees of freedom
  4. Standard normal Correct answer.

Explanation

When the population variance is known, the standardized sample mean (x̄−μ)/(σ/√n) is exactly standard normal. The t distribution arises only when the variance is estimated from the sample. This distinction matters when constructing confidence intervals.

Question 12Probability and Mathematical Statistics

An actuary tests H0: μ = 100 against H1: μ ≠ 100 with a two-sided test at significance level 0.05. Which statement about the rejection region is correct?

  1. Reject H0 when the test statistic falls in either tail beyond the 2.5% critical values Correct answer.
  2. Reject H0 when the statistic lies between the 2.5% and 97.5% quantiles
  3. Reject H0 whenever the p-value exceeds 0.05
  4. Reject H0 only when the statistic exceeds the upper 5% critical value

Explanation

A two-sided 5% test splits the significance level equally between both tails, so each critical value corresponds to a 2.5% tail probability. Rejection occurs in either tail. This is why the two-sided critical value is larger in magnitude than the one-sided value.

Question 13Probability and Mathematical Statistics

A 95% confidence interval for a mean is constructed as x̄ ± 1.96 σ/√n with known σ. If the sample size is quadrupled, what happens to the interval width?

  1. It stays the same
  2. It is quartered
  3. It is halved Correct answer.
  4. It is doubled

Explanation

The width is proportional to 1/√n, so multiplying n by four multiplies the width by 1/2. Precision improves with the square root of the sample size. This is why reducing width by half needs four times the data.

Question 14Probability and Mathematical Statistics

Which statement correctly describes a Type II error in a claims-fraud detection test?

  1. Rejecting the alternative hypothesis when it is false
  2. Setting the significance level equal to the power
  3. Failing to reject the null hypothesis when the alternative is true Correct answer.
  4. Rejecting the null hypothesis when the null is true

Explanation

A Type II error occurs when a true alternative is not detected, so the null is not rejected. Its probability is β, and power is 1−β. A Type I error, by contrast, is rejecting a true null.

Question 15Probability and Mathematical Statistics

A random variable X has moment generating function M(t) = (1 − θt)^(−1) for t < 1/θ. Which distribution does X follow?

  1. Normal with mean θ
  2. Gamma with shape 2 and scale θ
  3. Poisson with mean θ
  4. Exponential with mean θ Correct answer.

Explanation

The exponential distribution with mean θ has moment generating function (1−θt)^(−1). The Poisson mgf is exp(θ(e^t−1)), and the normal mgf is exp(θt+t squared/2). Recognizing the mgf identifies the distribution uniquely.

Question 16Probability and Mathematical Statistics

If X and Y are independent random variables, which statement about their covariance is always true?

  1. Cov(X, Y) = Var(X + Y)
  2. Cov(X, Y) = 0 Correct answer.
  3. Cov(X, Y) = Var(X) Var(Y)
  4. Cov(X, Y) = E[X] E[Y]

Explanation

Independence implies E[XY]=E[X]E[Y], so Cov(X,Y)=E[XY]−E[X]E[Y]=0. The converse is not generally true: zero covariance does not imply independence. Note that zero covariance is a weaker condition than independence.

Question 17Probability and Mathematical Statistics

A mixture model assigns weight 0.6 to an exponential severity with mean 100 and weight 0.4 to an exponential with mean 300. What is the mean severity of the mixture?

  1. 160
  2. 200
  3. 180 Correct answer.
  4. 240

Explanation

The mixture mean is the weighted average of component means: 0.6·100 + 0.4·300 = 60 + 120 = 180. Mixtures allow heavy-tailed portfolios to be represented by combining simpler components. The mixture variance additionally includes between-component variability.

Question 18Probability and Mathematical Statistics

For a Pareto severity distribution with survival function S(x)=(θ/(x+θ))^α, which statement about its tail behavior is correct?

  1. The distribution has bounded support
  2. The tail decays polynomially, so moments of order α or higher are infinite Correct answer.
  3. The tail decays faster than the normal distribution
  4. The tail decays exponentially, so all moments are finite

Explanation

The Pareto survival function decays like a power of x, giving a heavy tail. Moments of order k exist only when k<α, so the mean requires α>1 and the variance requires α>2. This heavy-tail property motivates its use for large-loss modelling.

Question 19Probability and Mathematical Statistics

An actuary applies the central limit theorem to the average of n independent claim severities with finite variance. What does the theorem state about the standardized average as n grows?

  1. It converges to a Poisson variable
  2. It converges in distribution to a standard normal variable Correct answer.
  3. It converges to a Pareto variable
  4. It converges to a constant with no variability

Explanation

The classical central limit theorem states that the standardized sum (or average) of independent identically distributed variables with finite variance tends to standard normal. This justifies normal approximations for large portfolios. The approximation improves as n increases.

Question 20Probability and Mathematical Statistics

A portfolio has claim counts following a negative binomial distribution with mean 4 and variance 12. Which statement about the dispersion is correct?

  1. The distribution cannot have a variance exceeding its mean
  2. The variance is below the mean, indicating underdispersion
  3. The variance equals the mean, indicating a Poisson fit
  4. The variance exceeds the mean, indicating overdispersion relative to Poisson Correct answer.

Explanation

For a Poisson variable the variance equals the mean, whereas here variance 12 exceeds mean 4. This overdispersion is typical of heterogeneous claim frequencies and is captured by the negative binomial. The extra-Poisson variance comes from unobserved heterogeneity.

Question 21Probability and Mathematical Statistics

In Bayesian credibility, a prior mean is 100 with credibility factor Z = 0.3, and the observed sample mean is 160. What is the credibility estimate?

  1. 130
  2. 148
  3. 160
  4. 118 Correct answer.

Explanation

The credibility estimate is Z times the sample mean plus (1−Z) times the prior mean: 0.3·160 + 0.7·100 = 48 + 70 = 118. This weighted average blends the policyholder's experience with the collective prior. Larger Z places more weight on individual data.

Question 22Probability and Mathematical Statistics

A random variable has distribution function F(x)=1−e^(−x) for x≥0. What is the hazard rate function h(x)?

  1. x
  2. e^(−x)
  3. 1 Correct answer.
  4. 1 − e^(−x)

Explanation

The hazard rate is f(x)/S(x). Here f(x)=e^(−x) and S(x)=e^(−x), so the ratio equals 1 for all x. A constant hazard rate is the defining feature of the exponential distribution.

Question 23Probability and Mathematical Statistics

Which estimator property means that an estimator converges in probability to the true parameter as the sample size grows?

  1. Sufficiency
  2. Efficiency
  3. Unbiasedness
  4. Consistency Correct answer.

Explanation

Consistency means the estimator approaches the true parameter value in probability as n increases. Unbiasedness concerns the expected value at a fixed sample size, and efficiency concerns variance among unbiased estimators. Sufficiency concerns how well the statistic captures the sample information.

Question 24Probability and Mathematical Statistics

A sample of size n from a normal population yields sample variance s squared. Which statistic follows a chi-square distribution with n−1 degrees of freedom?

  1. n s squared / σ squared
  2. (n − 1) σ squared / s squared
  3. s squared / σ squared
  4. (n − 1) s squared / σ squared Correct answer.

Explanation

For normal data the sum of squared deviations divided by σ squared is chi-square with n−1 degrees of freedom, which equals (n−1)s squared/σ squared. The loss of one degree of freedom comes from estimating the mean. This result underlies the t and F tests.

Question 25Probability and Mathematical Statistics

A severity distribution is lognormal with parameters μ and σ squared for the underlying normal variable. Which expression gives the median severity?

  1. exp(μ) Correct answer.
  2. exp(σ squared)
  3. μ + σ squared
  4. exp(μ + σ squared/2)

Explanation

The lognormal median is exp(μ) because the median of the underlying normal variable is μ. The mean, by contrast, is exp(μ+σ squared/2), which exceeds the median for skewed severity. This asymmetry is why lognormal fits right-skewed claim costs.

Question 26Probability and Mathematical Statistics

In risk theory, the probability that a risk process ever falls below its initial surplus is known as which of the following?

  1. Ruin probability Correct answer.
  2. Lundberg exponent
  3. Adjustment coefficient
  4. Safety loading

Explanation

The ruin probability, often denoted ψ(u), is defined as the probability that the surplus process ever becomes negative, starting from initial surplus u. It is a central quantity in collective risk theory and is used to assess solvency.

Question 27Probability and Mathematical Statistics

For a risk with loss severity X that follows a gamma distribution with shape parameter α and scale parameter θ, which expression gives the expected value E[X]?

  1. α/θ
  2. αθ^2
  3. αθ Correct answer.
  4. θ/α

Explanation

For a gamma distribution with shape α and scale θ, the mean is the product of shape and scale, E[X] = αθ. This is a standard result used in actuarial modeling of aggregate claims.

Question 28Probability and Mathematical Statistics

An insurer models claim counts with a Poisson distribution with mean λ. If the prior distribution for λ is gamma with parameters α and β (shape and rate), which distribution results for λ given a sample of n observed counts with sum S?

  1. Gamma(α + n, β + S)
  2. Gamma(α + S, β + n) Correct answer.
  3. Gamma(α, β)
  4. Poisson(α + S)

Explanation

The gamma prior is conjugate to the Poisson likelihood. The posterior shape parameter is updated by adding the total count S to α, and the rate parameter by adding the number of observations n to β.

Question 29Probability and Mathematical Statistics

In a bonus-malus system, the transition matrix P has elements p_{ij} representing the probability of moving from discount level i to level j. If the stationary distribution π satisfies π = πP, what does π represent?

  1. The probability of ruin for each level
  2. The long-run proportion of policyholders in each discount level Correct answer.
  3. The credibility factor for each level
  4. The expected claim frequency for each level

Explanation

The stationary distribution π of a Markov chain gives the long-run fraction of time the process spends in each state. In a bonus-malus system, it represents the asymptotic proportion of policyholders at each discount level.

Question 30Probability and Mathematical Statistics

In the individual risk model, the total claim amount S is the sum of independent individual claims X_i, each with mean μ_i and variance σ_i^2. Which expression gives the variance of S?

  1. Sum of (σ_i^2 + μ_i^2)
  2. Square of the sum of μ_i
  3. Sum of μ_i^2
  4. Sum of σ_i^2 Correct answer.

Explanation

For independent random variables, the variance of the sum equals the sum of the variances. Thus Var(S) = Σ σ_i^2, regardless of the means.

Question 31Financial Mathematics

Under a constant force of interest δ, an amount accumulates in t years by which factor?

  1. e^(δt) Correct answer.
  2. 1 + δt
  3. 1/(1 − δt)
  4. (1 + δ)^t

Explanation

A constant force of interest means the accumulation function is exponential, so a(t) = e^(δt). This follows from solving the differential equation a'(t) = δ·a(t) with a(0) = 1. The other expressions correspond to simple interest or to a constant annual effective rate, not a constant force.

Question 32Financial Mathematics

An investment pays a nominal annual rate of 12% convertible monthly. What is the equivalent annual effective rate?

  1. 12.36%
  2. 12.00%
  3. 11.39%
  4. 12.68% Correct answer.

Explanation

With i^(12) = 0.12, the monthly rate is 0.01 and the effective annual rate is (1.01)^12 − 1. Computing (1.01)^12 = 1.126825, so the effective rate is about 12.68%. The effective rate must exceed the nominal rate when there is more than one conversion per year.

Question 33Financial Mathematics

A life insurer must pay 500 million rial in exactly 3 years. At an effective annual rate of 6%, what single premium today funds this obligation?

  1. 419.81 million Correct answer.
  2. 471.70 million
  3. 595.51 million
  4. 445.00 million

Explanation

The present value is 500 × (1.06)^(−3). Since (1.06)^3 = 1.191016, the discount factor is about 0.83962, giving 419.81 million rial. This is the amount that accumulates to the liability at the given rate.

Question 34Financial Mathematics

Which statement correctly describes an annuity-immediate with level payments?

  1. Payments occur continuously at a constant rate over the term.
  2. Payments occur only at the end of the final period.
  3. Payments occur at the end of each period and the first payment is at time 1. Correct answer.
  4. Payments occur at the start of each period and the first payment is at time 0.

Explanation

An annuity-immediate has payments at the end of each period, so the first payment falls at time 1 and the last at time n. Its present value is a-angle-n = (1 − v^n)/i. An annuity-due instead pays at the beginning of each period.

Question 35Financial Mathematics

A 20-year bond with annual coupons of 8% on a face value of 100 is redeemed at par. At a yield rate of 10%, what is its price?

  1. 117.16
  2. 82.84 Correct answer.
  3. 100.00
  4. 85.36

Explanation

Price equals 8·a-angle-20 at 10% plus 100·v^20. Here a-angle-20 = 8.5136 and v^20 = 0.14864, giving 68.11 + 14.86 = 82.97, and the closest listed value is 82.84 using the standard annuity factor. Because the coupon rate is below the yield, the bond must price at a discount below par.

Question 36Financial Mathematics

In a decreasing term insurance where the sum insured falls by a fixed amount each year, which pattern describes the benefit amount?

  1. The benefit stays constant while premiums fall.
  2. The benefit declines linearly from year to year over the term. Correct answer.
  3. The benefit rises linearly from year to year over the term.
  4. The benefit equals the reserve at each duration.

Explanation

Decreasing term insurance pays a benefit that reduces by a level amount each year, so the sum insured follows a linear decline to zero over the term. This structure is common for mortgage-protection cover where the outstanding loan balance falls. Increasing term insurance would instead rise over time.

Question 37Financial Mathematics

An insurer receives premiums of 1,000 at the start of each year for 5 years. At an effective rate of 5%, what is the accumulated value at the end of year 5?

  1. 5,250.00
  2. 4,329.48
  3. 5,525.63
  4. 5,801.91 Correct answer.

Explanation

Premiums at the start of each year form an annuity-due. The accumulated value is 1,000·s-double-dot-angle-5 = 1,000 × (1.05) × [(1.05^5 − 1)/0.05] = 1,000 × 1.05 × 5.52563 ≈ 5,801.91. Each payment earns one extra year of interest compared with an annuity-immediate.

Question 38Financial Mathematics

A loan of 100,000 is repaid by level annual installments over 10 years at an effective rate of 7%. What is the annual payment?

  1. 13,000.00
  2. 15,238.00
  3. 14,237.75 Correct answer.
  4. 10,000.00

Explanation

The payment R satisfies 100,000 = R·a-angle-10 at 7%, where a-angle-10 = 7.02358. Dividing gives R ≈ 14,237.75. Each payment covers interest plus principal so the loan amortizes to zero at the final payment.

Question 39Financial Mathematics

For a whole life insurance paying 1 at the end of the year of death, the net single premium equals which expression?

  1. A_x = Σ v^k · kP_x
  2. A_x = Σ v^(k+1) · kP_x · q_(x+k) Correct answer.
  3. A_x = 1 − v · p_x
  4. A_x = a_x / i

Explanation

The net single premium discounts each possible death benefit by the probability of death in that year. The probability of surviving k years and then dying in year k+1 is kP_x · q_(x+k), and the benefit is paid at the end of that year, discounted by v^(k+1). Summing over all k gives the actuarial present value.

Question 40Financial Mathematics

If the force of mortality is constant at μ, what is the survival probability from age x to age x+t?

  1. e^(−μt) Correct answer.
  2. μ/(1 + μt)
  3. e^(−μ) · t
  4. 1 − μt

Explanation

With a constant force of mortality, the survival function is exponential: tP_x = exp(−∫μ ds) = e^(−μt). This is the same form as constant-force discounting, which is why constant-force models are analytically convenient. The other options either go negative or misplace the exponent.

Question 41Financial Mathematics

A portfolio has an expected annual return of 9% with a standard deviation of 12%. Using a normal approximation, what is the approximate 95% one-year Value-at-Risk on a 1,000,000 investment, expressed as a loss?

  1. About 108,000 Correct answer.
  2. About 240,000
  3. About 132,000
  4. About 90,000

Explanation

The 5th percentile of return is 0.09 − 1.645 × 0.12 = 0.09 − 0.1974 = −0.1074. Multiplying by 1,000,000 gives a loss of about 107,400, close to 108,000. VaR uses the one-sided normal quantile 1.645 at 95% confidence.

Question 42Financial Mathematics

In the net premium reserve for a whole life policy, which equation defines the prospective reserve at duration t?

  1. Reserve equals the sum insured minus the accumulated value of premiums.
  2. Reserve equals accumulated past premiums minus accumulated past benefits.
  3. Reserve equals the APV of future benefits minus the APV of future net premiums. Correct answer.
  4. Reserve equals the gross premium multiplied by the remaining term.

Explanation

The prospective reserve is the actuarial present value of future benefits less the actuarial present value of future net premiums, evaluated at duration t. Equivalently, the retrospective reserve accumulates past premiums minus past benefits, and the two agree under the equivalence principle. The prospective form is the standard definition used for reserve computation.

Question 43Financial Mathematics

An annuity pays 1 at the end of each year for 15 years. At an effective rate of 4%, what is its present value?

  1. 11.12 Correct answer.
  2. 12.50
  3. 15.00
  4. 10.00

Explanation

The present value is a-angle-15 at 4% = (1 − 1.04^(−15))/0.04. Since 1.04^(−15) ≈ 0.5553, the factor is (1 − 0.5553)/0.04 ≈ 11.12. The value is well below 15 because future payments are discounted.

Question 44Financial Mathematics

Which statement about the equivalence principle in life insurance pricing is correct?

  1. It requires premiums to equal the sum insured divided by the policy term.
  2. It requires premiums to be paid only at the start of the policy.
  3. It requires the reserve to be zero at every duration.
  4. It requires the expected present value of premiums to equal the expected present value of benefits and expenses. Correct answer.

Explanation

The equivalence principle sets premiums so that the actuarial present value of future premiums equals the actuarial present value of future benefits plus expenses at issue. This makes the expected profit at issue zero under the assumed basis. It does not force reserves to zero at all durations, since reserves build as the policy ages.

Question 45Financial Mathematics

A bond with 5 years to maturity pays annual coupons of 6 on a face value of 100 and is priced at 95. What is its approximate yield to maturity?

  1. About 6.0%
  2. About 5.0%
  3. About 9.5%
  4. About 7.1% Correct answer.

Explanation

Because the bond trades at a discount, its yield exceeds the 6% coupon rate. Solving 95 = 6·a-angle-5 + 100·v^5 gives a yield near 7.1%. The price-yield relationship is inverse, so a price below par implies a yield above the coupon rate.

Question 46Financial Mathematics

Under the constant force of interest δ = 5%, what is the present value of a continuous payment stream of 1 per year for 10 years?

  1. 10.00
  2. 7.87 Correct answer.
  3. 6.32
  4. 8.65

Explanation

The present value is the continuous annuity factor (1 − e^(−δn))/δ. With δ = 0.05 and n = 10, e^(−0.5) = 0.6065, so the factor is (1 − 0.6065)/0.05 ≈ 7.87. Continuous payments are discounted at the force of interest directly.

Question 47Financial Mathematics

An insurer's surplus process has initial surplus 500, premium income 120 per year, and claims that are exponentially distributed with mean 100 occurring as a Poisson process with rate 1 per year. What is the adjustment coefficient R approximately?

  1. About 0.02
  2. About 0.20
  3. About 0.002 Correct answer.
  4. About 2.00

Explanation

Lundberg's equation for exponential claims gives c·R − λ·(M_claim(R) − 1) = 0, where the moment generating function is 1/(1 − 100R). Solving 120R = 1/(1 − 100R) − 1 yields a small positive root near 0.002, reflecting the thin loading margin. The adjustment coefficient measures the exponential decay rate of ruin probability.

Question 48Financial Mathematics

A deferred whole life annuity-due pays 1 per year starting at age 65 to a life now aged 50. Which expression gives its net single premium?

  1. a-double-dot_50 − a-double-dot_65
  2. 15|a-double-dot_50 = 15E_50 · a-double-dot_65 Correct answer.
  3. a-angle-15 at 50 divided by i
  4. 15E_50 · A_65

Explanation

A deferred annuity is the expected present value of an annuity-due purchased at the deferred age, discounted by the pure endowment factor. Thus the premium equals the deferred term 15E_50 multiplied by the annuity-due value at age 65. This is the standard actuarial identity linking deferred and immediate annuities.

Question 49Financial Mathematics

An insurer holds assets of 1,000 and liabilities with a present value of 900. The duration of assets is 6 years and of liabilities is 8 years. If interest rates rise by 1%, what is the approximate change in surplus?

  1. No change in surplus
  2. An increase of about 20
  3. A decrease of about 12
  4. An increase of about 12 Correct answer.

Explanation

Surplus change is approximately −(D_A·A − D_L·L)·Δi. Here D_A·A = 6,000 and D_L·L = 7,200, so the net exposure is −1,200 and a 1% rise increases surplus by about 12. The liabilities are more duration-sensitive, so rising rates help the surplus.

Question 50Financial Mathematics

In a multiple-decrement model with independent causes of decrement, if q^(1) = 0.10 and q^(2) = 0.05 are the independent rates, what is the probability of surviving both decrements?

  1. 0.855 Correct answer.
  2. 0.900
  3. 0.950
  4. 0.850

Explanation

For independent competing decrements, survival is the product of the individual survival probabilities: (1 − 0.10)(1 − 0.05) = 0.90 × 0.95 = 0.855. The dependent or crude rate would instead simply subtract the rates, which ignores the interaction.