CT 6 Exam Questions Flashcards
Simplifications usually made in the basic model for short term insurance contracts.
- Model assumes that the mean and standard deviation of the claim amounts are known with certainty.
- Model assumes that claims are settled as soon as the incident occurs, with no delays.
- No allowance for expenses is made.
- No allowance for interest.
2 Examples of forms of insurance regarded as short term insurance contracts.
- Car insurance
- Contents insurance
Under Quadratic loss we need the ….
mean
Under All-or-nothing loss we use the…
mode
9 Characteristics of insurable risks
- Policyholder has an interest in the risk
- Risk is of a financial nature and reasonably quantifiable
Ideally:
- Independence of risks
- probability of event is relatively small
- pool large numbers of potentially similar risks
- ultimate limit on liability of insurer
- moral hazards eliminated as far as possible
- claim amount must bear some relationship to financial loss
- sufficient data to reasonably estimate extent of risk / likelihood of occurence
6 Key characteristics of a short term insurance contract
- policy lasts for a fixed term
- option (but no obligation) to renew policy
- claims are not of fixed amounts
- the existence of a claim and its amount have to be proven before being settled
- claim does not bring policy to an end
- claims that take a long time to settle are known as long-tailed (& short -> short tailed)
Give an advantage of the Box-Muller algorithm relative to the Polar method
- Generated a pair of u1 and u2 - no possibility of rejection
Give a disadvantage of the Box-Muller algorithm relative to the polar method
- Requires calculation of sin and cos functions which is computationally intensive
Describe characteristics of Pearson and deviance residuals
Pearson residuals are often skewed for non-normal data which makes the interpretation of residual plots difficult.
Deviance residuals are usually more likely to be symmetrically distributed and are preferred for actuarial applications.
Why do insurance companies make use of run-off triangles?
- There is normally a delay between incidents leading to claim and the insurance pay out.
- Insurance companies need to estimate future claims for their reserve.
- It makes sense to use historical data to infer future patterns of claims.
3 Main components of a generalised linear model
- Distribution for the data (Poisson, exponential, gamma, normal or binomial)
- A linear predictor (function of the covariates that is linear in the parameters)
- Link function (links the mean of the response variable to the linear predictor)
What is meant by a saturated model?
Discuss if it is useful in practice.
Saturated model has as many parameters as there are data points and is therefore a perfect fit to the data.
It is not useful from a predictive point of view which is why it is not used in practice.
It is, however, a useful benchmark against which to compare the fit of other models.
3 Common assumptions underlying a runoff triangle
- Number of claims relating to each development year is a constant proportion of the total claim numbers from the relevant accident year.
- Claim amounts for each development year are a constant proportion of the total claim amount for the relevant accident year.
- Claims are fully run off after the last listed development year.
Disadvantage of using truly random numbers
- Generating truly random numbers is not a trivial problem, and may require expensive hardware.
- It may take a lot of memory to store a large set of randomly generated numbers.
- The pseudo-random numbers generated by LCG are reproducible, one only need initialize the algorithm with the same seed. This is helpful for comparing the results of different models.
4 Methods for generating random variates
- Linear congruential generator
- Inverse transform method
- Acceptance-rejection method
- Box-Muller algorithm or the Polar algorithm for generating normal variates.
Bayesian estimator for the loss function: squared error (quadratic loss)
Mean
Bayesian estimator for the loss function:
absolute error
Median
Bayesian estimator for the loss function:
Zero-one (all or nothing)
Mode
Proportional reinsurance
Under proportional reinsurance, the insurer and reinsurer split the claim in pre-defined proportions.
Individual excess of loss
The insurer will pay any claim in full up to amount M, the retention level.
Any amount above M will be met by the reinsurer.
Suitable prior for p, with data ~ Binomial(m,p)?
Posterior?
p ~ Beta(ɑ, β)
p | X ~ Beta(ɑ + Σx, β + mn - Σx)
Suitable prior for λ, with data ~ Poisson(λ)?
Posterior?
λ ~ Gamma(ɑ, β)
λ | X ~ Gamma(ɑ + Σx, β + n)
Suitable prior for p, with data ~ Binomial(m,p)?
Posterior?
p ~ Beta(ɑ, β)
p | X ~ Beta(ɑ + Σx, β + mn - Σx)
Assumptions of EBCT Model 1
- For each risk i, the distribution of Xij depends on a parameter θi whose value is the same for each j but is unknown.
- {Xij | θi} are i.i.d. random variables
- {θi} are i.i.d. random variables
- For i≠k, the pairs (Xij, θi) and (Xkm, θk) are i.i.d. random variables.
Advantages of pseudo-random numbers over truly random numbers:
- reproducibility
- storage (only a seed and a single routine is needed)
- efficiency (it is very quick to generate several billions of random numbers).
Assumptions of EBCT Model 2
- For each risk i, the distribution of Xij depends on a parameter θi whose value is the same for each j but is unknown.
- {Xij | θi} are INDEPENDENT (not i.i.d. like M1) random variables
- {θi} are i.i.d. random variables
- For i≠k, the pairs (Xij, θi) and (Xkm, θk) are INDEPENDENT (not i.i.d. like M1) random variables.
4 Methods for generating random variates
- Linear congruential generator
- Inverse transform method
- Acceptance-rejection method
- Box-Muller algorithm or the Polar algorithm for generating normal variates.
Disadvantage of using truly random numbers
- Generating truly random numbers is not a trivial problem, and may require expensive hardware.
- It may take a lot of memory to store a large set of randomly generated numbers.
- The pseudo-random numbers generated by LCG are reproducible, one only need initialize the algorithm with the same seed. This is helpful for comparing the results of different models.
4 Methods for generating random variates
- Linear congruential generator
- Inverse transform method
- Acceptance-rejection method
- Box-Muller algorithm or the Polar algorithm for generating normal variates.
Collective Risk Model
Aggregate claim amount S, is modelled as the sum of a random number of IID random variables,
S = X1 + X2 + … + XN,
where S is taken to be zero if N=0.
Assumptions underlying the collective risk model
- The random variable N is independent of the random variables Xi.
- The moments of N and Xi are known.
- Claims are settled more or less as soon as they occur.
- Expenses and investment returns are ignored.
Direct Data
Data from the risk under consideration
Collateral data
Data from other similar, but not necessarily identical, risks.
Interpretation of E[X2(θ)] (EBCT)
Represents the average variability of claim numbers from year to year for a single risk.
Interpretation of V[m(θ)] (EBCT)
Represents the variability of average claim numbers for different risks.
Explain how the value of the credibility factor Z depends on E[X2(θ)] and V[m(θ)].
If E[X2(θ)] is higher relative to V[m(θ)], this means there is more variability from year to year than from risk to risk.
More credibility can be placed on the data from other risks leading to a lower value of Z.
On the other hand, if V[m(θ)] is relatively higher this means there is greater variation from risk to risk, so that we can place less reliance on the data as a whole leading to a higher value of Z.
Loss Function
A measure of the “loss” incurred when g(X) is used as an estimator of θ.
Conjugate prior
For a given likelihood, a prior distribution yielding a posterior distribution of the same family as the prior distribution.
Uninformative prior
Assumes that the parameter is equally likely to take any possible value.
Prior distribution
The prior distribution of θ represents the knowledge available about the possible values of θ before the collection of any sample data.
Posterior distribution
The PDF of the prior distribution and the likelihood function combined.
Likelihood function
A likelihood function is the same as a joint density function of X1,X2,…Xn | θ.
It is however considered to be a function of θ | X, rather than X | θ.
Ruin Theory: Effect of changing λ
An increase in λ will NOT affect the probability of ultimate ruin since the expected aggregate claims, the variance of aggregate claims and the premium rate all increase proportionately in line with λ.
However, it will reduce the time it takes for ruin to occur.
Ruin Theory: Effect of changing var(X)
An increase in var(X) will INCREASE the probability of ruin.
It will increase the uncertainty associated with the aggregate claims process without any corresponding increase in premium.
Ruin Theory: Effect of changing E(X)
An increase in E(X) will INCREASE the probability of ruin.
The expected aggregate claims and the premium rate both increase proportionately in line with E(X), however, the variance of the aggregate claims amount increases disproportionately
(it increases proportional to E(X)^2).
Ruin Theory: Effect of changing θ
Probability of ultimate ruin DECREASES if θ, the premium loading, is increased.
Ruin Theory: Effect of changing U
Probability of ultimate ruin DECREASES if the initial surplus U is increased.
Time to the first claim & time between claims
If number of claims ~ Poisson( λ )
time between 2 successive claims ~ Exp(λ)
Variable
A variable is a type of covariate whose real numerical value enters the predictor directly. (e.g. age)
Factor
A type of covariate that takes categorical values to which we need to assign numerical values for the purpose of the linear predictor. (e.g. sex)
Residual
A measure of the difference between the observed values yi and the fitted values ˆµi.
Monte Carlo method
Involves carrying out a large number of simulations of variates of a distribution, which can then be used to estimate probabilities and moments of the distribution.
Explain the main differences between a pure Bayesian estimate and an estimate based on EBCT1
- The pure Bayesian approach makes use of prior information about the distribution of θ1 and EBCT1 does not.
- The pure Bayesian approach uses only information from the category itself to produce a posterior estimate, whereas the approach under EBCT1 assumes that information from the other categories can give some information about the category in question.
- The pure Bayesian approach makes precise distributional assumptions about the number of claims, whereas the EBCT1 approach makes no such assumptions.
Write down the general form of a statistical model for a claims run-off triangle.
C = r.s.x + e
- r is the development factor for year j, independent of the origin year i, representing the proportion of claims paid by development year j.
- s is a parameter varying by origin year, representing the exposure.
- x is a parameter varying by calendar year representing inflation.
- e is an error term.
Explain what is meant by a sequence of independent, identically
distributed (I.I.D.) random variables.
Each realisation of the variable is unaffected by previous outcomes and
in turn does not affect future outcomes.
The variables all come from the same distribution with the same
parameters.
Explain why an insurance company might purchase reinsurance
To protect itself from the risk of large claims.
2 Types of reinsurance
- Excess of loss
- Proportional
Excess of loss reinsurance (short)
Reinsurer pays any amount of a claim above the retention.
Proportional reinsurance (short)
Reinsurer pays a fixed proportion of any claim.
3 steps in the Box-Jenkins approach to fitting an ARIMA time series
- model identification
- parameter estimation
- diagnostic checking
Explain what is meant by a two player zero-sum game
A game with 2 players where whatever one player loses in the game, the other player wins and vice versa
Normal distribution:
Natural parameter
θ = μ
Normal distribution:
Scale parameter
ϕ=σ²
Poisson Distribution:
Natural parameter
θ = log(μ)
Poisson Distribution:
Scale parameter
ϕ=1
Poisson distribution:
b(θ)
b(θ) = exp(θ)
Binomial Distribution:
Transform to exponential family
Z ~ Bin
Let Y = Z/n
So that Z =Yn
Binomial Distribution:
Natural parameter
θ = log{ μ/(1-μ) }
Binomial Distribution:
Scale parameter
ϕ=n
Binomial Distribution:
b(θ)
b(θ) = log{ 1 + exp(θ) }
Gamma Distribution:
Natural parameter
θ = -1/μ
Where μ = α/λ
Gamma Distribution:
Scale parameter
ϕ= α
Gamma Distribution:
Transform to exponential family
Change the parameters from
α and λ to:
α and μ = α/λ
Scaled deviance for a particular model M
Let ls be the log-likelihood for the saturated model.
Let lm be the log-likelihood for the model
SD = 2(ls - lm)
The deviance for the current model
Dm, defined such that:
Scaled deviance = Dm / ϕ
Where ϕ is the scale parameter.
Lundberg’s Inequality
ψ(U) <= exp{ -RU }
Adjustment coefficient, R
Defined as the unique positive root of:
λ Mx(r) - λ - cr = 0
Where
- c is the rate of premium income,
- M is the moment generating function of the of individual claim amounts
- λ is the parameter of the Poisson process for the claims process
2 equations for solving the adjustment coefficient, R
λ Mx(r) = λ + cr
Or:
Mx(r) = 1 + (1+θ)m1r
Notice we’ve written c = (1+θ)λm1
3 main components of a generalised linear model
- Distribution of the response variable
- a linear predictor of the covariates
- link function between the response variable and the linear predictor
3 Conditions in order to model aggregate claims as a Binomial(n,p)
- The risk is a constant p for each policy
- there can only be one claim per policy per year
- the risk of flood damage is independent from building to building
3 Possible causes of non-stationarity
- a deterministic trend (eg exponential or linear growth)
- a deterministic cycle (eg seasonal effects)
- a time series is integrated
Condition for stationarity using the characteristic polynomial of the autoregression
the roots of the characteristic polynomial are all greater than 1.
Define the surplus process U(t)
Let S(t) denote the total claims up to time t and suppose individual claim amounts follow a distribution X.
The U(t) = U + λt(1 + θ) E(X) − S(t)
Define ψ(U, t)
ψ(U, t) = P( U(s) < 0 for some s ∈ [0, t])
Define ψ(U)
ψ(U) = P( U(t) < 0 for some t > 0)
Explain how ψ(U, t) depends on λ
The probability of ruin by time t will increase as λ increases.
This is because claims and premiums arrive at a faster rate, so that if ruin occurs, it will occur earlier, which leads to an increase in ψ(U, t).
Explain how ψ(U) depends on λ
The probability of ultimate ruin does not depend on how quickly the claims arrive.
We are not interested in the time when ruin occurs as we are looking over an infinite time horizon.
Give 2 examples of exercises where Monte-Carlo simulation should be performed using the same choice of random numbers, explaining your reasoning in each case.
- Testing sensitivity to parameter estimation
- Performance evaluation
Why should
[testing sensitivity to parameter estimation]
using Monte-Carlo simulation be performed using the same set of random numbers?
We want the results to change as a result of changes to the parameter, not as a result of variations in the random numbers.
Why should
[Performance evaluation]
using Monte-Carlo simulation be performed using the same set of random numbers?
When comparing two or more schemes which might be adopted we want differences in results to arise from differences between the schemes rather than as a result of variations in the random numbers.
Under EBCT1, give a brief interpretation of:
E[S2(θ)]
E[S2(θ)] represents the average variability of claim numbers from year to year for a single risk.
Under EBCT1, give a brief interpretation of:
V(m(θ))
V(m(θ)) represents the variability of the average claim numbers for different risks, i.e. the variability of the means from risk to risk.
Essential characteristic of liability insurance
To provide indemnity, where the insured - owing to some form of negligence - is legally liable to pay compensation to a 3rd party.
5 Examples of liability insurance
- Employer’s liability
- Motor 3rd party liability
- Public liability
- Product liability
- Professional liability
3 Typical perils under Employer’s liability insurance
- Accidents caused by employer negligence
- Exposure to harmful substances
- Exposure to harmful working conditions
Typical peril under Motor 3rd party liability insurance
Road traffic accidents
2 Typical perils under product liability insurance
- Faulty design, manufacture or packaging of product
- Incorrect or misleading instructions
Typical peril under professional indemnity insurance
Wrong medical diagnosis, error in medical operation, etc.
Describe the difference between STRICTLY stationary processes and WEAKLY stationary processes
Strictly stationary processes have the property that the distribution of (Xt+1, …, Xt+k) is the same as that of (Xt+s+1, …, Xt+s+k) for each t, s and k.
For weak stationarity, only the first 2 moments are needed to satisfy: E(Xt) = μ ∀t and cov(Xt X(t+s) = γ(s) ∀t, s.
List 3 statistical tests that you should apply to the residuals after fitting a model to time series data
- the turning point’s test
- the “portmanteau” Ljung-Box χ2 test
- inspection of the values of the SACF based on their 95% confidence intervals under the white noise null hypothesis.
2 CONDITIONS for a risk to be insurable
- The policyholder must have an interest in the risk being insured, to distinguish between insurance and a wager
- The risk must be of a financial and reasonably quantifiable nature.
X ~ Pareto(α,λ)
What’s the distribution of:
Z = X - M | X > M
(where M is a constant)
Z ~ Pareto( α, λ+M )
Prove by using g(z) = fx(z + M)/Fx(M)
X ~ Pareto(α,λ)
What’s the distribution of:
Z = kX
(with k >1)
Z ~ Pareto( α, λk )
The acceptance-rejection method:
C
C = max{ h(x)/f(x) }
h(x) being the distribution you want to simulate from.
The acceptance-rejection method:
w(x)
W(x) = h(x) / { C f(x) }
h(x) being the distribution you want to simulate from.
The acceptance-rejection method:
Steps
1: Generate u ~ U(0,1)
2: Set x = F-1(u)
3: Generate v ~ U(0,1)
4: If v > w(x) return to step 1 else return x.
The acceptance-rejection method:
The proportion of pseudo-random numbers we accept.
We accept a proportion 1/C
PDF of the reinsurer’s conditional claims distribution
W = X - M | X > M
g(w) = fx(w+M) / { 1 - Fx(M) }
Coefficient of skewness
coeff of skew(Y) = skew(Y) / [Var(Y)]^1.5
Credibility factor for the Poisson/Gamma model
Z = n/(n+β)
Credibility factor for the Normal/Normal model
Z = n/{ n + σ²/σ²_0 }
Credibility factor for the Binomial/Beta model
Z = mn/(ɑ + β + mn)
Explain what is meant by a randomised strategy
A randomised strategy is where the player randomly chooses between different strategies, rather than adopting a fixed approach.
Positive Skew
Mean > Median
4 factors that affect ψ(U, t) for a given t
θ, μ, λ, σ² and initial surplus U
Advantage of using discrete, rather than continuous time in probability of ruin
Easier to measure, more useful for reporting
Disadvantage of using discrete, rather than continuous time in probability of ruin
Less information, artificial, can miss time when ruin occurs.
Effect of θ on ψ(U, t)
Higher θ reduces ψ(U, t) as premiums increase at a quicker rate, so more of a buffer
Effect of μ on ψ(U, t)
Higher μ increases ψ(U, t) as claims are larger relative to the surplus held.
Effect of λ on ψ(U, t)
Higher λ increases ψ(U, t) as the process is faster - claims and premiums come in quicker
Effect of σ² on ψ(U, t)
Higher σ² will typically increase ψ(U, t), assuming that expected premiums are higher than expected claims, since the likelihood of more extreme claims increase.
[but may reduce ψ(U, t) if expected claims are higher than expected returns]
Effect of U on ψ(U, t)
Higher U reduces ψ(U, t) as there is more of a buffer to withstand claims.
