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Pricing In General Insurance 2nd Edition Pietro Parodi - Solutions
A company X (whose main offices are in London) has three business units: one in Gloucestershire, one in Essex, and one in Yorkshire. The company buys property insurance through a captive. Each business unit has a deductible of £10,000 and the captive has a deductible of £100,000. All the rest is
(For some readers, this problem might be better tackled after reading Chapter 22.)Petrochemical company X insures its single oil refinery with insurer Y and its property damage insurance has just been renewed and will incept 1 May 2023 (2023 policy year). Traditionally Y’s pricing guidelines
(*) The Venn diagram below shows symbolically a portfolio at two successive underwriting years. The shaded area (intersection) ℘′ ∩℘ is made of the contracts that have been renewed; the area ℘−℘′ is made of the contracts that have lapsed;℘’−℘ is made of new business.i. Prove
You are monitoring a small portfolio of marine hull policies. The following table gives a snapshot of the portfolio for contracts written during 2019 (left) and 2020(right).Year 2019 Year 2020 Contract ID Technical Premium Premium charged Technical premium Premium charged 1 10,000 12,000 11,000
Suppose that your gross aggregate loss distribution for a contract can be well approximated by a lognormal distribution with µ = 13, σ = 2 (outputs in USD). The contract covers losses to the layer USD 9M (in aggregate) xs USD 1M (in aggregate).a. Using Monte Carlo simulation, estimate the value
Prove that TVaR is a coherent risk measure if the underlying probability density function of the loss distribution is continuous.
You are asked to help your underwriter with an actuarial quote for an aviation liability policy. Your actuarial model produces yearly expected losses of £1.2M.To produce an estimate of the technical premium, you use the Cost Plus methodology. You want to take these elements into account: expenses,
As a result of a pricing exercise, you produce the following estimates for a public liability risk:a. Expected losses = £850kb. Allowance for uncertainty = Nonec. Other costs = £150kd. Commissions = Nonee. Profit = 12% of premiumf. Mean weighted payment time = 5.5 years Using current publicly
As the consulting actuary for WWW, you have produced the following gross loss model for their public liability (PL) risks. The model is based on 1000 groundup losses (min = £150, max = £1,280,000) over the last 10 years (700 of which are fully paid losses, and 300 are reserve estimates), and is a
(*) Write down Equations 17.22–17.25 for the case where the frequency model is Poisson and the severity is a spliced Lognormal/Pareto or a spliced Lognormal/GPD.
(*) Write down Equations 17.24 and 17.25 for the Poisson/Lognormal case.
(*) Prove Equation 17.21. Hint: write g0 = Pr(S = 0) as an infinite sum Pr P S Nr P n X r X X N n n( = ) = = ( ) ( + + + = N = ) =∞0 0 ∑0 1 2 | and note that for a sum of positive-definite random variables to be zero, all of them must be simultaneously zero.
R-based question. Your aggregate loss model is a compound Poisson/lognormal model with parameters λ = 5 (Poisson rate) and μ = 10.2, σ = 1.2. Using R and the FFT algorithm given in Appendix B, you want to produce aggregate loss models.Choose values of M and h so that (i) the running time is less
R-based question. Modify the R code in Appendix A so that it is able to deal with(a) a negative binomial distribution for the frequency; (b) a spliced lognormal/GPD severity distribution; (c) a generic severity distribution, entered as a vector of N N N NN percentiles , 1 12+ + 1 1 +
You are a quantitative analyst for Company X. Company X sells cheap digital cameras, which are sold at €50 in the European market and have a production cost of €20. The cameras are offered with a 1-year warranty that offers a free replacement for failures during the warranty period. A new model
You are an actuary for a commercial lines insurance company. One of your jobs is to set the annual aggregate deductible (hence ‘the aggregate’) for commercial policies at a level that is satisfactory for clients and is breached with a likelihood of less than or equal to 5%. The policy you have
Gaussian approximation (either computer-based or using numerical tables). Suppose that losses can be modelled according to the frequency/severity model described in Question 2 (frequency: Poi(4.4), severity: LogN(11.2,1.5)). (a) Using the Gaussian approximation model, produce an estimate for the
Monte Carlo simulation (computer-based exercise). Assume that losses related to public liability are generated according to this process: next year, the number of losses will come from a Poisson distribution with rate = 4.4 and severity coming from a lognormal distribution with μ = 11.2 and σ =
Basic statistics for the collective risk model. An actuary has analysed company X’s motor third-party liability losses for next year by modelling the number of losses as a negative binomial distribution with rate equal to 208.8 and a variance-to-mean ratio of 2.5, and the severity of losses as a
(*) Effect of claims inflation on various distributions. Show that applying an inflation of r% to all the losses (i.e. X X ’ = + (1 r)) modelled by:• a Lognormal distribution with parameters µ and σ leads to a Lognormal distribution with parameters µ µ ′ ′ = + log , (1+ r) σ σ = ;•
(*) Show that the single-parameter Pareto, F x x ( ) = − θ α, and the Type-II Pareto,, are both special cases of the GPD, F x x ( ) = − + −.
In Section 16.4, we have justified the use of the empirical distribution to represent attritional losses on the basis of the Glivenko–Cantelli theorem and the Dvoretzky–Kiefer–Wolfowitz inequality. Why do we not use the empirical distribution to represent large losses as well?
Do the following:• Calibrate (i.e. find the parameters of) the Lognormal distribution based on these losses, pretending you do not know what the original parameters were.• Calculate the mean, the standard deviation and the percentiles 50%, 75%, 80%, 90%, 95%, 98%, 99%, and 99.5% of the severity
The following losses have been generated from a Lognormal distribution with parameters μ = 10 and σ =
Discuss the advantages and disadvantages of a frequency/severity model in comparison to a burning cost approach that includes adjustments for inflation, exposure, etc.
As a result of the overall IBNER analysis, you produce the following cumulative IBNER factors.Development years IBNER (cumulative)1->onwards 2.44 2->onwards 1.32 3->onwards 1.20 4->onwards 0.95 5->onwards 1.00 Development year = 1 is the first year when the loss was reported.Apply the relevant
You need to produce the input losses for a severity model.i. Describe three methods for dealing with IBNER when producing such input.ii. Explain the advantages and disadvantages of each method listed in (i).iii. Calculate the IBNER factor from year of development, 2 to year of development, and 3
(*) Discuss (a) how the relationship p pa bk kk−= +1, which holds for the Panjer class variates, can theoretically be used to determine whether realisations from a count process should be classified as binomial, Poisson or negative binomial, and (b) the practical feasibility of using this as an
The table below (Case 1) shows the number of employers’ liability losses reported to an organisation over the period 2017 to 2024, and the relevant exposure over the same period. Some IBNR factors, based on portfolio estimates, are also provided.a. Based on these figures, advise on a frequency
(*) Prove that the Poisson distribution, the binomial distribution, and the negative binomial distribution all satisfy Equation 14.1 and find the values of a and b in terms of the parameters of these distributions.
(*) Prove that Equation 14.5 holds true if Conditions 1 through 4 in Section 14.2 are satisfied. Also prove that if Conditions 1 and 3 are satisfied and Equation 14.5 is satisfied, then Conditions 2 and 4 must hold true.
Using a spreadsheet, simulate a random number of claims from a Poisson process using the property that the interarrival time of claims from a Poisson process is an exponential distribution. Hint: Use Equation 14.9 and calculate the cumulative time.
(*) Show that when the tail distribution is a Generalised Pareto distribution (see Chapter 16) with null threshold: F x x ( ) = − + a. the tail factor and the projected/ultimate ratio are as follows:(b) the formulae for the exponential and Type II Pareto can be obtained from the formulae above
The following table records all claims that occurred from 1 April 2010 and were reported by 31 October 2013.Loss ID Occurred Reported Delay (days) Delay (years)R01 29/7/2011 21/12/2012 511 1.399 R02 14/2/2013 10/3/2013 24 0.066 R03 17/7/2011 22/1/2012 189 0.517 Loss ID Occurred Reported Delay
You are given the following claim count triangulation, providing the number of claims occurred in each policy year and are reported within a certain number of development years.Development Year 0 1 2 3 4 5 6 7 8 9 Policy Year 2004 4 9 9 10 10 10 10 10 10 10 2005 4 8 9 9 9 9 9 9 9 2006 4 6 7 8 9 9 9
(*) Show how Equation 11.7 can be justified as an approximation of the formula for the total expected losses (S x ) = × f x( )dx∞∫ λ0 where λ is the expected number of claims from the ground up and 0∞∫xf (xd) x is the expected severity. (Hint: split 0∞∫xf (xd) x into two parts,
Show how the burning cost formulae shown in Section 11.1.12:a. simplify in the case where there is no annual aggregate deductible (AAD = ∞);b. should be amended if an annual aggregate limit AAL is present.
Consider a simple loss process where the number of claims follows a Poisson distribution with rate λ = 5 and the individual claim amount follows a lognormal distribution with µ = 11 and σ = 2.Assume that the insurer offers a simple stop loss protection by which the insurer pays all losses in
As part of a pricing exercise for a Commercial Property Risk XL programme incepting on 1 January 2016 and written on a losses occurring basis, you have to calculate the earned premium for a number of policy years and compare it to the claims experience. The underlying policies of the reinsured are
A medium-sized UK-based company (A) is currently insuring its public liability risks with an insurance company (B). You have been asked (as a consulting actuary) to advise company A on whether or not it is paying too much for its insurance.The loss data (which is based on information provided on 30
In Box 11.2, we say that the standard way of calculating the average development ratios in the chain ladder is a weighted average. What weights are used in the definition of D(0→1) (and, for that matter, for all development ratios)?
In Box 11.2, we have seen that in the standard chain ladder method, the weighted average is normally used to calculate the average development ratios between two successive development years.a. Produce a definition of the average development ratios as a simple average.b. Calculate the average
In Box 11.1, we have introduced weighted statistics for the empirical mean, standard deviations, and the percentiles of a data set. Show that by setting all weights to 1, the formulae for the mean, the standard deviation, and the percentiles for the unweighted case are obtained.
The underwriting manager in charge of Energy has asked you to build a model to cover operations all risks (OAR) for Offshore Windfarm. This model will cover property damage (PD) only.As a starting point, you decide to build a simple model in which each windfarm is modelled as a single property,
Explain the rationale behind the four different possible formulations of the total cost of risk given in Equations 30.3 and 30.4, and their relative advantages and disadvantages.
Company X has traditionally purchased public liability insurance from the ground up, but to decrease the insurance premium (currently at £2,500,000) spend and in view of its financial robustness, it is now investigating the opportunity of retaining the portion of each loss, which is below a given
Consider the following list of insurance structures and highlight those that are surely inefficient using VaR@99% as a measure of volatility:a. Structure 1: premium = £1.2M, VaR@99% = £99Mb. Structure 2: premium = £1.7M, VaR@99% = £78Mc. Structure 3: premium = £1.3M, VaR@99% = £105Md.
Monkey business. Assume that you own an establishment with a troop of 120 macaques (on average) that you sell as domestic pets and that the probability of one of your monkeys dying in a normal year is approximately p = 1.1%. However, this probability can increase to approximately p = 15% if there
Calculate the theoretical tail correlation for the copula of Question 4.
Do the same as in Question 3 but combining the two vectors with a t copula with a desired correlation of 0.6 and ν = 1.3.
Using a spreadsheet, generate two random vectors of 1000 elements each, with each element coming from a lognormal distribution. Then combine them with a Gaussian copula, producing a rank correlation of around 0.6. Do the same using R code.
Create two vectors X, Y such that Y = f(X) (and are therefore fully dependent) but have zero linear correlation.a. Can you produce an example for which the linear correlation is equal to 0?b. Can you produce an example for which the rank correlation is also equal to 0?
In the table below, X and Y represent the property damage and bodily injury components of 19 motor third-party liability losses.a. Calculate the linear correlation and the rank correlation (Spearman’s ρ) between X and Y.b. For both linear correlation and rank correlation, determine (if possible)
Compare advantages and disadvantages of analysing a risk (such as motor) by splitting it into its main components (such as own damage, third-party liability for motor).
Explain (a) the difference between categorical and non-categorical factors; (b) the reason why, when the number of levels of a categorical factor becomes large, using GLM may become difficult; and (c) describe possible ways to address this problem.(Note: The answer to part (c) of the question is
A general insurance company that writes Fine Art insurance has decided to use its historical loss data to replace the current model (mostly based on underwriters’experience and judgment) with a generalised linear model calibrated using historical loss data, and you are in charge of this
Your company decided a few years ago to enter the lucrative business of selling alien abduction policies. This business has performed extremely well in the past, with an observed loss ratio consistently at 0% over the last five years.Building on this good fortune, the underwriting manager has
You are the actuary for a reinsurance company. An insurance company, A, wants to buy a layer of excess-of-loss reinsurance (£10M × £5M) for its property portfolio from your company. The underwriter has asked you to produce a price for this product by loading the expected losses to this layer by
(*) Prove that Bayesian credibility estimate – when the number of claims of a client follows a Poisson distribution and the prior distribution for the Poisson rate follows a gamma distribution – can be expressed in the standard format of Equation 26.29.
(*) Using the Bayesian approach to credibility, prove that the posterior distribution of Θ given s (see Section 26.3.2 for the meaning of these parameters) is itself a normal distribution with these parameters:Θ Θ | ~s N , n s
A large industrial company X wants to insure its properties against the risk of earthquakes, because many of its plants are in earthquake-prone regions.a. Explain why a simple burning cost analysis or a frequency/severity analysis over the last 10 years of experience may very well not provide a
Credit risk Modify the algorithm in Section 24.3.2 (b) to take into account changes in the value of the economic factor over the K periods. Assume the values of ψ over the period, ψt, are provided by an economic scenario generator. Discuss what effect this might have on the number of simulations
Personal accident i. Describe the main coverage of personal accident insurance and give an example of a benefit schedule.ii. Mention two composite personal lines product to which personal accident is normally attached and explain why personal accident is relevant to these products.iii. Explain why
Extended warranty (and common shocks)You are a quantitative analyst for Company X, which sells cheap digital cameras at €50 in the European market at a production cost of €20. The cameras are offered with a 1-year warranty that comes with a free replacement for failures during the warranty
Answer the following questions about periodic payment orders (PPOs).i Explain what a PPO is.ii Explain why they were introduced, and who is ultimately responsible for deciding whether a PPO rather than a lump sum should be granted.iii Explain the main advantages and disadvantages of PPOs compared
Professional indemnity, claims-made policies i. Explain the difference between a claims-made basis policy and an occurrencebasis policy in direct insurance.X is a law firm that buys professional indemnity. Assume that of the claims originating from advice given in year y,• 25% of the claims are
Your company uses the following table of ILFs.Policy limit (£) ILF 95,238 0.953 100,000 1.000 105,000 1.046 476,190 1.773 500,000 1.785 525,000 1.796 952,381 2.281 1,000,000 2.285 1,050,000 2.289a. Define ILFs, and show (with a formula) how they are related to the underlying loss severity
(*) Prove Equation 23.9 directly for the case of the lognormal distribution, starting from Equation 23.2.
(*) Prove that the ILF curves corresponding to the following severity models: (a)lognormal, (b) lognormal/Pareto, (c) lognormal/GPD, (d) empirical/GPD are as in Sections 23.4.1.1–4.
(*) Prove that the CDF of the severity curve above b (conditional on being above b)corresponding to an ILF curve with basis point b is F x b b ( ) = −1 ILF (x b ) ILFb ( ) ’ ’ / .Hint: Start from the conditional survival distribution above b.
Assume that the correct severity distribution for a public liability portfolio is a lognormal with μ = 10.5 and σ = 1.2. Derive the analytical form of the corresponding ILF curve above £100,000 and draw it using a spreadsheet.
Consider the same food manufacturing plant as in Question 12.The underwriting guidelines for food manufacturing plants are also as before, with the following exceptions:• The exposure curves in use are the Swiss Re curve with c=3.8 for PD, c=3.1 for BI. In both cases, these are defined from the
Consider a food manufacturing plant with the following characteristics:• TIV (PD) = £200M, TIV (BI) = £200M, MPL (PD) = £130M, MPL (BI) = £200M• Local deductible (LD) for PD = £1M, LD(BI) = 45d• Indemnity period (IP) = 12 months (from the accident date)The underwriting guidelines for
(*) Calculate the exposure curve corresponding to the cumulative distribution function that incorporates MPL uncertainty (Equation 22.39).
(*) Adapt the algorithm in Section 22.8.2.3 for the simulation of losses for a single location to the case where the exposure curve is defined in excess of the underlying deductible.
(*) Show that it is possible for the KS distance between the CDF of two cumulative distribution functions F x( ) and F x * ( ) with domain [0,1] can be made arbitrarily close to 1 while at the same time making their corresponding exposure curves G x( )and G x * ( ) arbitrarily close.Hint: Assume
(*) Show the connection between the following probability distributions from statistical mechanics: f (ε ε ) = A k / / ( ) exp 1 ( T) − (Bose–Einstein), f (ε ε ) = A k / / ( ) exp 1 ( T) +(Fermi–Dirac), f (ε ε ) = − A k exp / ( T) (Maxwell–Boltzmann) and the Bernegger distribution,
Given the exposure curve tabulated in Question 6, and knowing that it is a Swiss Re curve with c = 3.0:i. Using Bernegger’s formulae in Section 22.5, verify that the values of the exposure curve for percentage of sum insured equal to 26%, 58% and 74% are as shown in the table.ii. Calculate the
As the actuary for a Lloyd’s syndicate specialising in property reinsurance, you are asked to quote a £3M xs £2M layer for a Risk XL policy, which is covering a portfolio mostly of large commercial properties. You are given the following risk profile for an insurer who insures its property
Your house is insured for £300,000 and you pay £500 for your household insurance, which mainly insures your house against fire events. On average, the insurer expects to achieve a loss ratio of 60% on its portfolio of houses. Your insurance company goes to the reinsurer to buy facultative
Prove that an exposure curve G(x) is concave in [0,1] in the general case (the one in which F(x) is not necessarily continuous and with defined first derivative over[0,1]).
Consider the curve y xx x = − + 3 2 4 34 32 . Can this be an exposure curve? If so, calculate: (a) the probability of a total loss and (b) the average loss.
According to Equation 22.15, the average loss only depends on the derivative at point 0, regardless of what the curve does afterwards. Therefore, very different exposure curves will give the same average loss as long as they start in the same way. How do you explain that?
Assume that you have a gunpowder factory which is so badly protected against fire that, once any fire starts, all the gunpowder blows up. (i) Draw the severity curve and the exposure curve. (ii) If you were to represent this as a Swiss Re curve with parameterc, what value of c would you need to
An insurance company X has been writing a mixture of EL and PL business (roughly in the same proportion) over the last 10 years. The summary statistics (for the last 6 years only) are as follows:Policy year(incepting at 1/1)Premium written during the year (all annual policies)Rate change with
You want to adapt the risk-costing process introduced in Section 6.2 so that it works for experience rating of excess-of-loss reinsurance of a liability portfolio, such as employers’ liabilitya. Which feature would you need to add so that this process is adequate to deal with the indexation
Company C has purchased risk excess of loss (XL) cover for its property portfolio.A reinsurer R offers a £3M xs £5M layer of reinsurance with these characteristics:a. Losses occurringb. One-year policy incepting 1 April 2011c. Reinsurance premium = £400kd. Two reinstatements only, the first is a
Modify the algorithm in Section 21.6.2 for the Monte Carlo simulation for a Risk XL contract layer so that it works for a European Indexation Clause.
(*) Prove that the variance of losses to a layer for the case of a GPD severity with a generic claim count distribution is as follows. The formula is quite complex so we’ll need to define a few symbol for shorthand purposes:A x x ξ ξ µσ ( ) = + −λ µ µ : ; = E V (N v ) = (N )It’s
(*) Prove Equation 21.13 and generalise it to the case where (a) the severity distribution is Pareto but the claim count distribution is not Poisson; (b) the severity distribution is a GPD and the claim count distribution is Poisson; (c) the severity distribution is a GPD and the claim count
(*) Generalise Equation 21.12 to the case where instead of a single parameter Pareto distribution above θ your distribution is made of a spliced Pareto distribution, i.e. a distribution with this cumulative distribution function: x θ’Also, prove that this generalisation reduces back to
(*) Prove analytically Equations 21.11, and prove that it reduces to Equation 21.12 in the case of a single-parameter Pareto.
Simplify the risk-costing process for Risk XL reinsurance so that it can be used for costing property reinsurance, stripping out all the features that are necessary only for liability reinsurance.
(*) Prove the formulae given in Box 21.1 to extrapolate back the GPD. By remembering that the single-parameter Pareto distribution and the exponential distribution are but special cases of the GPD, adapt these formulae so that they can be used straightforwardly for these two distributions.
(*) Re-write Equations 3.4 for the case where the policy structure includes an EEL deductible, a limit L, an annual aggregate deductible AAD, and an annual aggregate limit AL.
Three common bases for policies in treaty reinsurance are losses occurring during(LOD), risk attaching during (RAD), and claims-made.i. Explain how these bases work.An annual excess of loss reinsurance policy with inception date = 1 October 2012 is underwritten for the layer £1.5M xs £0.5M. The
A medium-sized UK-based insurance company underwrites mainly commercial and industrial property and motor and liability insurance. Outline, with reasons, the types of reinsurance it is likely to buy for each line of business and (if suitable)for the combined portfolio of risks.
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