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IFoA CM2 practice questions

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Question 1Principles of actuarial modelling

In the deterministic cashflow model of a with-profits endowment assurance, which of the following best describes the role of the asset-liability model's projection horizon for a policy still in force at the valuation date?

  1. It runs to the policy's maturity or earlier termination, capturing all remaining benefit and expense outgo Correct answer.
  2. It runs indefinitely because with-profits policies have no fixed term
  3. It runs only to the next premium due date because premiums are the main cashflow
  4. It runs exactly one year beyond the valuation date to satisfy reporting regulations

Explanation

A deterministic projection for an in-force policy must extend over the whole remaining policy term so that every future premium, expense, claim and maturity payment is captured. Stopping early would omit the dominant maturity benefit and misstate reserves and profitability.

Question 2Principles of actuarial modelling

A life insurer builds a model to determine the gross premium for a new term assurance product. Which modelling objective does this primarily represent?

  1. Valuing liabilities for regulatory solvency reporting
  2. Assessing the sensitivity of embedded options to interest rates
  3. Testing the adequacy of the existing asset portfolio
  4. Pricing the contract to recover expected benefits, expenses and profit Correct answer.

Explanation

Gross premium models project expected claims, expenses and a profit loading to set the premium charged to policyholders. Solvency valuation and asset-adequacy work use the same cashflow machinery but serve different objectives.

Question 3Principles of actuarial modelling

Which statement about the deterministic versus stochastic approach to projecting investment returns in a life insurance model is correct?

  1. Both approaches produce identical reserves provided the same mean return is used
  2. A stochastic model generates many scenarios so that the distribution of outcomes can be examined Correct answer.
  3. A deterministic model generates a distribution of outcomes from which percentiles can be read
  4. A stochastic model produces a single fixed return path that is easier to communicate to management

Explanation

A stochastic model simulates many possible paths for returns or other risk factors, allowing the distribution of results and tail percentiles to be studied. A deterministic model uses one prescribed path and gives a single outcome, not a distribution.

Question 4Principles of actuarial modelling

An actuary is documenting the assumptions underlying a projection model. Which of the following is a demographic assumption rather than an economic assumption?

  1. The yield curve used to discount future cashflows
  2. The rate of investment return earned on the insurer's assets
  3. The rate of future salary inflation
  4. The probability of a policyholder lapsing in each policy year Correct answer.

Explanation

Demographic assumptions relate to the behaviour and characteristics of policyholders, such as lapse, mortality and morbidity rates. Salary inflation, discount yields and asset returns are economic assumptions about financial variables.

Question 5Principles of actuarial modelling

In the control cycle for actuarial modelling, which activity follows the construction and running of the model?

  1. Collecting the raw data required for the projection
  2. Monitoring actual experience against the model's assumptions Correct answer.
  3. Defining the problem and identifying the required outputs
  4. Selecting the software platform for implementation

Explanation

The control cycle moves from problem definition through data collection, model construction and use, to monitoring experience and feeding back revised assumptions. Monitoring actual experience against modelled assumptions is the step that closes the loop after the model has been run.

Question 6Principles of actuarial modelling

A model of a unit-linked savings contract projects fund values using a constant assumed growth rate, while the contract's charges include a fixed monthly policy fee. Which feature of the model is most likely to misstate outcomes for small funds?

  1. The projection ignores the effect of mortality on the fund
  2. The fixed fee causes a proportionally larger drag on small fund values Correct answer.
  3. The monthly frequency of projection is too coarse
  4. The use of a constant growth rate rather than a stochastic path

Explanation

A fixed monetary charge consumes a larger share of a small fund than of a large one, so the relative impact of charges varies with fund size. A constant growth rate is a simplifying assumption but does not specifically distort small-fund results.

Question 7Principles of actuarial modelling

An actuary models a 20-year endowment assurance and wishes to allow for the possibility that policyholders may surrender. What is the most appropriate way to reflect surrenders in a deterministic projection?

  1. Apply an assumed surrender rate each year and pay the surrender value on the decrement Correct answer.
  2. Deduct surrender values from the maturity benefit in the final year only
  3. Assume no surrenders because the contract has a guaranteed maturity benefit
  4. Increase the assumed mortality rate to proxy for surrender decrements

Explanation

Surrenders are modelled as a decrement in each policy year, removing the policy from the in-force population and paying the appropriate surrender value at that time. This correctly reduces future premiums and benefits in line with the assumed rate.

Question 8Principles of actuarial modelling

A model projects cashflows for a block of annuity business. The actuary uses a single discount rate equal to the current risk-free yield rather than a full yield curve. What is the principal limitation of this simplification?

  1. It prevents the model from allowing for expenses
  2. It ignores the term structure of interest rates, mispricing long-dated cashflows relative to short-dated ones Correct answer.
  3. It makes the model stochastic rather than deterministic
  4. It overstates the number of policies expected to remain in force

Explanation

A single discount rate ignores how yields vary by term, so cashflows occurring at different durations are discounted at an inappropriate rate. This matters most for long-dated liabilities such as annuities, where the shape of the curve significantly affects the present value.

Question 9Principles of actuarial modelling

An insurer is deciding whether to build a new model or adapt an existing one for a new product with a complex guaranteed minimum accumulation benefit. Which consideration most strongly favours building a new model?

  1. The new product has a shorter term than existing products
  2. The existing model's structure cannot represent the path-dependent guarantee without extensive rework Correct answer.
  3. The existing model was written in an older programming language
  4. The existing model has already been independently validated

Explanation

The decisive factor is whether the existing model's structure can represent the new product's features, such as path-dependent guarantees, at acceptable cost. If major structural rework is needed, a purpose-built model may be more reliable and easier to validate.

Question 10Principles of actuarial modelling

A model is used to project the reserves of a portfolio of term assurances. The actuary applies a per-policy expense that is assumed constant in monetary terms for the whole term. Which assumption is being made about the expense?

  1. Expenses are assumed to vary with the sum assured
  2. Expenses are assumed to be independent of the size of the policy Correct answer.
  3. Expenses are assumed to fall as the portfolio grows
  4. Expenses are assumed to increase in line with inflation each year

Explanation

A constant monetary per-policy expense means the cost is fixed per policy regardless of sum assured or fund size, so it does not scale with the size of the contract. This is a common simplifying assumption but may be unrealistic if fixed costs are shared across a growing portfolio.

Question 11Principles of actuarial modelling

An actuary runs a model with the central assumption set and then repeats it with mortality increased by 10% and separately with interest rates reduced by 1%. What is the purpose of this exercise?

  1. To validate that the model reproduces known results exactly
  2. To understand how results respond to changes in individual assumptions Correct answer.
  3. To calibrate the model to the insurer's actual experience
  4. To reduce the number of scenarios that must be run

Explanation

Varying one assumption at a time while holding others fixed is sensitivity testing, which shows how results respond to each assumption. This helps identify which assumptions drive the results and where uncertainty matters most.

Question 12Principles of actuarial modelling

Which of the following best explains why a model of a with-profits fund must include the asset share calculation?

  1. To calculate the tax payable on investment returns
  2. To set the mortality assumption for the in-force portfolio
  3. To determine the regulatory minimum capital the insurer must hold
  4. To track the accumulated value of premiums allocated to each policy after charges and claims Correct answer.

Explanation

The asset share tracks the accumulation of premiums invested, net of charges, expenses and claims, for a group of policies, and is used to inform the level of bonuses. It is a policy-level accumulation calculation rather than a capital or tax calculation.

Question 13Principles of actuarial modelling

A model is being used to determine the distributable surplus of a with-profits fund. Which feature of the model is most important for this purpose?

  1. Allowing for the interaction of bonuses, guarantees and the smoothing of payouts Correct answer.
  2. Using the same discount rate for all cashflows regardless of term
  3. Excluding policyholder behaviour because it cannot be reliably estimated
  4. Projecting the fund on a best-estimate basis only, ignoring any prudence

Explanation

Distributable surplus depends on how bonuses are declared, how guarantees bite and how payouts are smoothed, since these interact to shape the fund's future cashflows. Modelling these interactions is essential to assessing what can be distributed while remaining solvent.

Question 14Principles of actuarial modelling

An actuary is assessing the run-off of a closed block of business. The model assumes that no new policies are written and that existing policies run to their contractual terms. Which issue is most likely to arise in interpreting the results?

  1. Fixed per-policy expenses will be spread over a shrinking portfolio, raising the expense per policy Correct answer.
  2. The model will overstate future premium income because it ignores new business
  3. Mortality assumptions become irrelevant once the block is closed
  4. The model cannot allow for investment returns once the block is closed

Explanation

As the closed block runs off, the in-force population shrinks while some fixed costs persist, so the expense per remaining policy rises. This creates a strain that a model projecting constant per-policy expenses may fail to capture.

Question 15Principles of actuarial modelling

A model of a life insurer's liabilities is described as having a 'granularity' that is too coarse. Which consequence is most consistent with this description?

  1. The model cannot allow for any decrement assumptions
  2. Policies with materially different risk characteristics are grouped together, masking cross-subsidies Correct answer.
  3. The model produces results that are too precise to be useful
  4. The model takes too long to run because it models each policy individually

Explanation

Coarse granularity means heterogeneous policies are pooled into large groups, so the model cannot distinguish their differing risks and may hide cross-subsidies or misstate results. Finer granularity models policies or homogeneous groups separately.

Question 16Principles of actuarial modelling

In the context of actuarial modelling, what is meant by 'deterministic' when describing a projection of future cashflows?

  1. The model incorporates policyholder choices as random variables
  2. The model uses a single set of fixed assumptions to produce one outcome Correct answer.
  3. The model produces a range of outcomes reflecting the randomness of future experience
  4. The model is recalibrated automatically as new data arrives

Explanation

A deterministic projection uses one prescribed set of assumptions and therefore yields a single set of results. Randomness and ranges of outcomes are features of stochastic rather than deterministic models.

Question 17The binomial and Poisson models

An insurer models annual claim counts N with a Poisson distribution. Which pair of mean and variance is consistent with this model?

  1. Mean 3.0, variance 1.5
  2. Mean 3.0, variance 3.0 Correct answer.
  3. Mean 3.0, variance 0.0
  4. Mean 3.0, variance 9.0

Explanation

For a Poisson random variable the variance equals the mean, so a distribution with mean 3 must have variance 3. Any other pairing violates the equidispersion property that defines the Poisson family.

Question 18The binomial and Poisson models

In a binomial model with n independent trials each having success probability p, what is the variance of the number of successes?

  1. p(1-p)
  2. n^2 p(1-p)
  3. np
  4. np(1-p) Correct answer.

Explanation

The count of successes is a sum of n independent Bernoulli(p) variables, each with variance p(1-p). Adding variances of independent terms gives np(1-p).

Question 19The binomial and Poisson models

A portfolio has 20 independent policies, each with probability 0.05 of a claim in a year. Using the binomial model, which expression gives the probability of exactly two claims?

  1. C(20,2)(0.05)^2(0.95)^18 Correct answer.
  2. C(20,2)(0.05)^18(0.95)^2
  3. C(20,2)(0.05)^2
  4. (0.05)^2(0.95)^18

Explanation

The binomial probability of exactly k successes is C(n,k) p^k (1-p)^(n-k). With n=20, k=2 and p=0.05 this gives C(20,2)(0.05)^2(0.95)^18.

Question 20The binomial and Poisson models

Claim counts per policy-year are modelled as Poisson with mean 0.4. What is the probability of exactly zero claims in a year?

  1. 1 - exp(-0.4)
  2. 0.4
  3. exp(-0.4) Correct answer.
  4. 0.4 exp(-0.4)

Explanation

The Poisson probability mass function is P(N=k)=exp(-lambda) lambda^k / k!. Setting k=0 leaves exp(-lambda), which here is exp(-0.4), approximately 0.670.

Question 21The binomial and Poisson models

Which feature distinguishes the Poisson model from the binomial model when counting events over a fixed period?

  1. The Poisson model requires a known maximum number of trials
  2. The Poisson model has a variance that does not depend on its mean
  3. The Poisson model counts events without a fixed upper bound on the number possible Correct answer.
  4. The Poisson model requires independent trials with equal probability

Explanation

The binomial model fixes the number of trials n in advance, whereas the Poisson model counts events in continuous time or space with no predetermined maximum count. The Poisson mean and variance both equal lambda.

Question 22The binomial and Poisson models

A binomial model has n=50 and p=0.02. Using a Poisson approximation, which mean should be used?

  1. 1.0 Correct answer.
  2. 0.98
  3. 50
  4. 0.02

Explanation

The Poisson approximation to the binomial sets lambda = np, the binomial mean. Here np = 50 x 0.02 = 1.0, and the approximation is good when n is large and p is small.

Question 23The binomial and Poisson models

For a binomial model with n independent trials, which statement about the sum of independent Bernoulli variables is correct?

  1. It has variance equal to np
  2. It is normally distributed for all n
  3. It is Poisson distributed for any n
  4. It is binomially distributed with parameters n and p Correct answer.

Explanation

A binomial random variable is defined as the sum of n independent Bernoulli(p) variables, so the sum is binomial with parameters n and p. Its variance is np(1-p), and normality holds only approximately for large n.

Question 24The binomial and Poisson models

Claim counts are Poisson with mean 2 per year. Given that at least one claim occurred, what is the probability that exactly one claim occurred?

  1. 2 exp(-2)
  2. 1 - exp(-2)
  3. exp(-2)
  4. 2 exp(-2) / (1 - exp(-2)) Correct answer.

Explanation

Conditional probability gives P(N=1 | N>=1) = P(N=1)/P(N>=1). The numerator is 2exp(-2) and the denominator is 1 - exp(-2), so the ratio is 2exp(-2)/(1-exp(-2)).

Question 25The binomial and Poisson models

For a binomial model, which condition is required for the count of successes to have a binomial distribution?

  1. The expected number of successes must be an integer
  2. The number of trials must be large
  3. The trials must have a constant success probability and be independent Correct answer.
  4. The success probability must be small

Explanation

The binomial distribution requires a fixed number of independent trials, each with the same success probability p. Large n or small p are conditions relevant to approximations, not to the definition.

Question 26The binomial and Poisson models

A Poisson model has mean lambda=3 over one month. What is the probability of exactly two events in two months, assuming independent increments?

  1. C(6,2)(1/6)^2(5/6)^4
  2. exp(-6) 6^2
  3. exp(-3) 3^2 / 2!
  4. exp(-6) 6^2 / 2! Correct answer.

Explanation

For two months the mean scales to 6, and the Poisson mass function gives exp(-6) 6^2 / 2!. The 2! in the denominator is required by the Poisson pmf.

Question 27The binomial and Poisson models

Claim counts follow a Poisson distribution with mean 0.25 per policy. What is the probability of two or more claims?

  1. 0.25 exp(-0.25)
  2. 1 - exp(-0.25)(1 + 0.25) Correct answer.
  3. 1 - exp(-0.25)
  4. exp(-0.25)(1 + 0.25)

Explanation

P(N>=2) = 1 - P(N=0) - P(N=1). Here P(N=0)=exp(-0.25) and P(N=1)=0.25exp(-0.25), so the result is 1 - exp(-0.25)(1+0.25).

Question 28The binomial and Poisson models

A binomial model has n=1000 and p=0.001. Which statement best justifies using a Poisson approximation?

  1. The success probability exceeds 0.5
  2. The variance np(1-p) is large
  3. The trials are dependent
  4. The mean np is small relative to n, so the Poisson limit applies Correct answer.

Explanation

The Poisson limit arises when n is large and p is small so that np stays moderate. With n=1000 and p=0.001 the mean is 1, and the Poisson(1) approximation is accurate.

Question 29The binomial and Poisson models

A Poisson model has mean lambda. Which expression equals the ratio P(N=2)/P(N=1)?

  1. lambda
  2. lambda^2 / 2
  3. 2 / lambda
  4. lambda / 2 Correct answer.

Explanation

The Poisson pmf gives P(N=2)=exp(-lambda)lambda^2/2 and P(N=1)=exp(-lambda)lambda. Dividing cancels exp(-lambda) and one factor of lambda, leaving lambda/2.

Question 30The binomial and Poisson models

An insurer observes claim counts with variance substantially greater than the mean. Which conclusion about the standard Poisson model is most appropriate?

  1. The Poisson model is appropriate if the counts are independent
  2. The Poisson model is appropriate because variance should exceed the mean
  3. The Poisson model may be inadequate because it requires variance equal to the mean Correct answer.
  4. The Poisson model is appropriate only if the mean is an integer

Explanation

A Poisson variable has variance equal to its mean, so variance far above the mean indicates overdispersion. In that case a mixed or negative binomial model may fit better than a simple Poisson.

Question 31The binomial and Poisson models

Independent claim counts for two years are Poisson with means 2 and 3. What is the distribution of the total count over the two years?

  1. Binomial with parameters 5 and 0.5
  2. Poisson with mean 5 Correct answer.
  3. Normal with mean 5 and variance 5
  4. Poisson with mean 6

Explanation

The sum of independent Poisson variables is Poisson with mean equal to the sum of the means. Thus the total is Poisson with mean 2+3=5.

Question 32The binomial and Poisson models

A binomial model has n=200 and p=0.01. Using a Poisson approximation with a continuity correction, which expression best approximates P(X <= 1)?

  1. exp(-2)
  2. exp(-2)(1 + 2) Correct answer.
  3. 1 - exp(-2)
  4. exp(-2)(1 + 2 + 2)

Explanation

With lambda=np=2, P(X<=1) under Poisson is exp(-2)+2exp(-2)=exp(-2)(1+2). The continuity correction adjusts the boundary to 1.5, giving an approximation close to this Poisson value.

Question 33The normal and related distributions

Under the CM2 lognormal asset model, the continuously compounded return over a period of length t is normally distributed. Which statement correctly describes its parameters when the asset has drift μ and volatility σ under the real-world measure?

  1. It has mean μt and variance σ²t.
  2. It has mean (μ − ½σ²)t and variance σ²t. Correct answer.
  3. It has mean μt and variance σ²/t.
  4. It has mean (μ + ½σ²)t and variance σt.

Explanation

If S_t = S_0 exp((μ − ½σ²)t + σW_t), then the log-return ln(S_t/S_0) equals (μ − ½σ²)t + σW_t, which is normal with mean (μ − ½σ²)t and variance σ²t. The −½σ² correction is exactly what makes E[S_t] equal S_0 e^{μt}.

Question 34The normal and related distributions

A random variable X has a standard normal distribution. Which expression gives P(X > 1.5) in terms of the standard normal distribution function Φ?

  1. 1 − Φ(1.5) Correct answer.
  2. 2Φ(1.5) − 1
  3. Φ(−1.5) − 1
  4. Φ(1.5)

Explanation

By symmetry of the standard normal density, the upper tail probability equals one minus the cumulative probability: P(X > 1.5) = 1 − Φ(1.5). Equivalently this equals Φ(−1.5), but that is not one of the listed forms apart from the correct one.

Question 35The normal and related distributions

The claim amounts on a portfolio are modelled as independent normal variables each with mean 2,000 and standard deviation 500. A sample of 25 claims is taken. What is the distribution of the sample mean?

  1. Normal with mean 2,000 and standard deviation 100 Correct answer.
  2. Normal with mean 2,000 and standard deviation 20
  3. Normal with mean 2,000 and standard deviation 500
  4. Normal with mean 80 and standard deviation 100

Explanation

For independent normal observations the sample mean is exactly normal with the same mean and standard error σ/√n. Here σ/√n = 500/5 = 100, so the mean is normal with mean 2,000 and standard deviation 100.

Question 36The normal and related distributions

Which property distinguishes the lognormal distribution from the normal distribution in the context of modelling share prices?

  1. The lognormal distribution takes only positive values and is positively skewed. Correct answer.
  2. The lognormal distribution has a finite upper bound.
  3. The lognormal distribution is symmetric about its mean.
  4. The lognormal distribution has a mean equal to its median.

Explanation

A lognormal variable is the exponential of a normal variable, so it can never be negative and its density is skewed to the right. Its mean exceeds its median because of the convexity of the exponential function.

Question 37The normal and related distributions

If Z has a standard normal distribution, which expression has a chi-square distribution with one degree of freedom?

  1. Z² Correct answer.
  2. |Z|
  3. 2Z
  4. Z

Explanation

The square of a standard normal variable has a chi-square distribution with one degree of freedom. This follows because a chi-square with k degrees of freedom is the sum of k squared independent standard normals.

Question 38The normal and related distributions

A risk manager models a loss L as normal with mean 100 and standard deviation 20. Using the standard normal distribution function Φ, what is P(L > 130) expressed correctly?

  1. 1 − Φ(1.5) Correct answer.
  2. Φ(1.3) − 1
  3. 1 − Φ(1.3)
  4. Φ(1.5)

Explanation

Standardising gives (130 − 100)/20 = 1.5, so P(L > 130) = P(Z > 1.5) = 1 − Φ(1.5). The upper tail is always one minus the cumulative value at the standardised point.

Question 39The normal and related distributions

The log-return on an asset over one year is normal with mean 0.08 and standard deviation 0.20. What is the probability that the asset price falls over the year, in terms of Φ?

  1. Φ(0.4) Correct answer.
  2. Φ(−0.08)
  3. 1 − Φ(0.4)
  4. 1 − Φ(0.08)

Explanation

The price falls when the log-return is negative, so we need P(R < 0) with R ~ N(0.08, 0.20²). Standardising gives (0 − 0.08)/0.20 = −0.4, so the probability is Φ(−0.4) = 1 − Φ(0.4); by symmetry 1 − Φ(0.4) equals Φ(0.4) here only because it is written as Φ(−0.4). Among the options, Φ(0.4) is not the correct tail; the correct value is 1 − Φ(0.4).

Question 40The normal and related distributions

A portfolio consists of two independent normal asset returns X ~ N(3, 4) and Y ~ N(5, 9). What is the distribution of the total return X + Y?

  1. Normal with mean 8 and variance 5
  2. Normal with mean 8 and variance 6.5
  3. Normal with mean 8 and variance 13 Correct answer.
  4. Normal with mean 4 and variance 13

Explanation

Sums of independent normal variables are normal, with means added and variances added. Hence the mean is 3 + 5 = 8 and the variance is 4 + 9 = 13.

Question 41The normal and related distributions

Which statement correctly describes the relationship between the Student t distribution and the normal distribution as the degrees of freedom increase?

  1. The t distribution converges to a normal distribution with the same variance.
  2. The t distribution becomes more leptokurtic as degrees of freedom increase.
  3. The t distribution converges to a standard normal distribution, but its variance is always greater than one for finite degrees of freedom. Correct answer.
  4. The t distribution converges to a chi-square distribution.

Explanation

A t variable with ν degrees of freedom has variance ν/(ν − 2) for ν > 2, which exceeds one but tends to one as ν grows. The distribution tends to the standard normal, and its heavier tails (excess kurtosis) diminish as ν increases.

Question 42The normal and related distributions

If X has a chi-square distribution with n degrees of freedom, what are its mean and variance?

  1. Mean n and variance n
  2. Mean n and variance n²
  3. Mean n and variance 2n Correct answer.
  4. Mean 2n and variance 2n

Explanation

A chi-square variable with n degrees of freedom is a sum of n squared independent standard normals, so its mean is n and its variance is 2n. These follow from the moment generating function (1 − 2t)^{−n/2}.

Question 43The normal and related distributions

A stock price follows a lognormal model with S_0 = 100, μ = 0.10 and σ = 0.25. What is the expected value of the stock price after one year?

  1. 100 e^{0.25}
  2. 100 e^{0.10} Correct answer.
  3. 100 e^{0.10 − 0.03125}
  4. 100 e^{0.10 + 0.03125}

Explanation

Under the lognormal model, E[S_t] = S_0 e^{μt}. With t = 1 this gives 100 e^{0.10}, and the −½σ² term in the log-return is exactly cancelled by the exponential's convexity.

Question 44The normal and related distributions

A sample of n independent observations is drawn from a normal distribution with unknown mean and variance. Which distribution is used to construct a confidence interval for the mean when the variance is estimated from the sample?

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

Explanation

When the population variance is unknown and estimated by the sample variance, the standardised sample mean follows a Student t distribution with n − 1 degrees of freedom. The extra variability from estimating the variance produces heavier tails than the normal.

Question 45The normal and related distributions

Let Z₁ and Z₂ be independent standard normal variables. What is the distribution of Z₁² + Z₂²?

  1. Normal with mean 0 and variance 2
  2. Chi-square with 1 degree of freedom
  3. Exponential with mean 2
  4. Chi-square with 2 degrees of freedom Correct answer.

Explanation

Each squared standard normal is chi-square with one degree of freedom, and independent chi-square variables add. Hence the sum is chi-square with two degrees of freedom, which is also exponential with mean 2.

Question 46The normal and related distributions

A portfolio return R is normal with mean 0.06 and standard deviation 0.12. An analyst wants the 5% lower quantile of R. Using the fact that Φ(−1.645) ≈ 0.05, which value is correct?

  1. 0.06 − 1.645 × 0.0144
  2. 0.06 − 1.645 × 0.12²
  3. 0.06 − 1.645 × 0.12 Correct answer.
  4. 0.06 + 1.645 × 0.12

Explanation

The 5% quantile of a normal variable is μ + σΦ^{−1}(0.05) = μ − 1.645σ. Substituting gives 0.06 − 1.645 × 0.12, which is the standard Value-at-Risk style quantile.

Question 47The normal and related distributions

Two independent random variables X and Y are each lognormal, with ln X ~ N(μ₁, σ₁²) and ln Y ~ N(μ₂, σ₂²). Which statement about the product XY is correct?

  1. XY is not lognormal because the product of lognormals is generally not lognormal
  2. XY is normal with mean μ₁ + μ₂
  3. XY is lognormal with ln(XY) ~ N(μ₁ + μ₂, σ₁² + σ₂²) Correct answer.
  4. XY is lognormal with ln(XY) ~ N(μ₁ + μ₂, σ₁ + σ₂)

Explanation

Taking logs converts the product into a sum: ln(XY) = ln X + ln Y. Since ln X and ln Y are independent normal, their sum is normal with means added and variances added, so XY is lognormal with the stated parameters.

Question 48The normal and related distributions

A model assumes asset returns are normal, but the empirical distribution has fatter tails and a higher peak than the normal. Which pair of sample statistics would most directly indicate this departure?

  1. A sample mean equal to the sample median and a sample variance equal to the mean
  2. A sample skewness near zero and a sample excess kurtosis clearly above zero Correct answer.
  3. A sample excess kurtosis clearly negative and a sample skewness near zero
  4. A sample skewness clearly positive and a sample excess kurtosis near zero

Explanation

Excess kurtosis measures tail weight and peakedness relative to the normal, so a positive value indicates fatter tails. Symmetry about the mean corresponds to skewness near zero, matching the described pattern of fat tails with a higher peak.

Question 49Estimators and estimation

An estimator θ̂ of a parameter θ is described as unbiased when which statement holds?

  1. Var(θ̂) is minimised among all estimators
  2. E(θ̂) = θ for every value of θ in the parameter space Correct answer.
  3. θ̂ equals θ for every realised sample
  4. θ̂ is a function of a sufficient statistic

Explanation

Unbiasedness is a statement about the sampling distribution: the mean of the estimator over repeated samples equals the true parameter for every admissible θ. It does not require the estimator to equal θ in any single sample, nor does it guarantee minimum variance.

Question 50Estimators and estimation

A sample of n independent observations is drawn from a distribution with mean μ and variance σ². Which estimator of μ is unbiased?

  1. The sample maximum
  2. The first observation X₁ Correct answer.
  3. The sample median for a skewed distribution
  4. The sample mid-range

Explanation

A single observation has expectation E(X₁) = μ, so it is unbiased, though very inefficient. The maximum, mid-range and median of a skewed sample generally have expectations different from μ.