Unformatted text preview: BA3202
Actuarial Statistics
Lecture 4 Summary:
 Credibility theory BA3203
L5 Credibility Theory Lower • Proposed Approach: Weighted average of and ∗ + 1 − , 0 ≤ ≤ 1 Quality of
Collateral data
Higher • : the credibility factor; reflects how much “trust” is placed in the data
from the risk itself () compared with the data from the larger group () Actuarial Science NTU Quantity of
Collateral data • Poisson/Gamma model
• ~ • The prior distribution for : ~Γ , • =∗ =1 + 1− = ∗ + 1 − , where = + • Normal/normal model
• ~ , 12
• The prior distribution of : ~ , 22
• = = 12 +22 12 +22 22
12 +22 = 12 12 +22 + 22 12 +22 = + 1 − , where Jade Nie [email protected] • Bayesian credibility 2
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L5 Estimator for ,+1 1− + 1− = =1 =1 ∗ + 2 / =1 2 =1 − =1
−1 − −1 − =1 =1 + 2 =1 2 − =1
−1
∗ + =1 =1 =1 EBCT2 2 =1 =1 =1 2 / − =1
−1 2 − =1 ∗ ∗−1 − =1 − =1
−1
∗ 2 2 Jade Nie [email protected] EBCT1 Actuarial Science NTU EBCT1 Vs EBCT2 3
BA3203
L5 BA3202
Actuarial Statistics
Lecture 5:
 Risk models BA3203
L5 3. Understand the criteria for insurability
Construct models appropriate for short term insurance contracts
in terms of the numbers of claims and the amounts of individual
claims.
The collective risk model
• The assumptions • The moments and MGF of aggregate claims
• Understand the compound distributions, in particular when has a binomial, Poisson, or negative binomial distribution.
4. Derive the mean and variance for aggregate loss distributions for
both the insurer and the reinsurer after the operation of simple
forms of proportional and excess of loss reinsurance.
5. Derive formulae for aggregate claims under the individual risk
model Actuarial Science NTU 1.
2. Jade Nie [email protected] Objectives 5
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L5 • The party selling the risk has to have an incentive to avoid the risk (insurance
interests)
• The risk should be measurable and easily transformed into an agreeable financial
amount
• The probability of occurring a loss should be small
• Events should be independent of each other (or of very low correlation)
• There should be enough policies in place to allow for a reduction in the variance of
losses
• A limit on the exposure has to be set
• Moral hazard should be very low or absent
• Adverse selection should be very low or absent In reality, insurance is provided in some additional circumstances:
• Contract period is typically one year after which there no obligation by either party to
continue.
• Severity of claim is not fixed (i.e. it depends on the extent of the damage)
• A policy can have more than one claims.
• A claim can occur any time during the contract period.
• Claims may take with short time (short‐tail) or long time (long‐tail) to settle Actuarial Science NTU For a risk to be insurable, the following conditions are desirable: Jade Nie [email protected] Introduction 6
BA3203
L5 • One year horizon
• A premium is paid by the policyholder upfront
• The insurance company reimburses policyholders of any losses • Aggregate loss within the contract period • Several models exist for • Collective risk model
• Individual risk model • Notations:
• : number of claims during the contract period
• : i.i.d. RV representing loss amount for each individual claim
• is independent of ’s • = =1 Actuarial Science NTU • E.g.: motor insurance, homeowners etc
• Typical character: Jade Nie [email protected] Shortterm insurance 7
BA3203
L5 • pdf = Pr = • cdf = Pr( ≤ )
• : MGF for • ’s are defined to be +ve
•
•
•
• cdf = Pr( ≤ ) , > 0
pdf is assumed to exist : MGF for ’s : ℎ moment of • The cdf of aggregate loss is denoted as = Pr ≤ Actuarial Science NTU • is a discrete random variable with Jade Nie [email protected] Shortterm insurance • (): MGF for 8
BA3203
L5 ∞
=0 Pr( = ) Pr ≤ = • If the number of claims = is given, then
Pr ≤ = = Pr =1 ≤ = = ∗ ()
• ∗ () is the fold convolution function of ()
• If = 1, then 1∗ = () ∗
• = ∞
=0 ()
• Note that Pr( = 0) = Pr( = 0) = (0) Actuarial Science NTU • = =1 • The cdf of : = Pr ≤ = Jade Nie [email protected] The collective risk model 9
BA3203
L5 Moments of • First moment = = = = 1
= =1 = =1 = ∗ 1 = 1 • Second moment •
•
•
•
• = + = = =1 = =1 = 2 − 12 = 2 − 12 = 2 − 12 = 1 = 12 Hence = 2 + − 12 Jade Nie [email protected] •
•
•
• Actuarial Science NTU The collective risk model 10
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L5 MGF of • Definition of MGF: = = = = =1 • Hence = • = = =1 = = ∗ln = ln • Recall that = • i.e. MGF of is expressed in terms of MGF of and MGF of Jade Nie [email protected] • Actuarial Science NTU The collective risk model 11
BA3203
L5 • Assume that the number of claims ~ • =
• = • = −1 Actuarial Science NTU The compound Poisson
distribution • Notation of (), (), , same as before • The aggregate loss • = 1 = 1
• = 2 + − 12 = 2
• = ln = −1 • An important feature of the compound Poisson distribution: sum of
compound Poisson distributions is also a compound Poisson distribution
• If = =1 ,where ’s are independent compound Poisson distributions
with Poisson parameters = −1 =
=1 =
• MGF
of
: = =1 =1 =1 −1
• We can see that is the same as the MGF for a compound Poisson
distribution with Poisson parameter Λ = =1 Jade Nie [email protected] • No specific assumptions for individual claim sizes distributions 12
BA3203
L5 • Assume that the number of claims ~ , • No specific assumptions for individual claim sizes distributions • Notation of (), (), , same as before • The aggregate loss • = 1 = 1
• = 2 + − 12 = 2 − 2 12
• = ln = + 1 − Jade Nie [email protected] • = • = (1 − )
• = + 1 − Actuarial Science NTU The compound Binomial
distribution 13
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L5 • is a +ve integer, = 1 − +−1 • Pr( = ) = , for = 0,1,2 … (formula book NBtype 2) • = • = 2
• = 1− • No specific assumptions for individual claim sizes distributions • Notation of (), (), , same as before • The aggregate loss • = 1 = 1 /
• = 2 + − 12 =
• = ln = 1− 2 + Actuarial Science NTU • Assume that the number of claims ~ , Jade Nie [email protected] The compound Negative
Binomial distribution 2 2
1
2 14
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L5 Actuarial Science NTU • Poisson distribution: mean=variance
• Binomial: mean>variance
• Negative Binomial: mean<variance Jade Nie [email protected] Comparison 15
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L5 • individual excess of loss reinsurance
•
•
•
• : insurer’s retention level for individual losses : R.V. representing individual losses : R.V. representing the amount paid by the insurer for each : R.V. representing the amount paid by the reinsurer for each • Proportional reinsurance
• : insurer’s retention portion for individual losses
• : R.V. representing individual losses • With the collective risk model, we studied the aggregate loss • Derive the expected value and variance of insurance/reinsurance
losses at aggregate claim level Actuarial Science NTU • Recall in L3, we looked at Jade Nie [email protected] Reinsurance effect 16
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L5 • pdf of ’s: ()
• cdf of ’s: • = min , : amount paid by the insurer for each • =
• 2 = −∞ 2 −∞ + ∞ ∞
+ 2 = −∞ + 1 − • = max 0, − : amount paid by the reinsurer for each • = −∞ 0 ∗ + 1− • 2 = ∞ ∞ − = ∞ − Actuarial Science NTU • : insurer’s retention level for individual losses
• : R.V. representing individual losses Jade Nie [email protected] Excess of loss reinsurance (XL) − 2 • : number of claims incurred over the insured period 17
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L5 =1 : • = aggregate loss for direct insurer • = = • = 2 + − • = =1 : 2 aggregate loss for the reinsurer Actuarial Science NTU Excess of loss reinsurance (XL) • Let = =1 • = −  > ,
• cdf of : = Pr ≤ = Pr( − ≤  > )
= Pr( ≤+ >)
Pr > • pdf of : = = = + − 1− +
1− • is the number of that is larger than • = 1 + 2 + ⋯ + , where
0, ℎ ()
0, ≤ =
1, > 1, ℎ 1 − ()
• is a binomial R.V. Jade Nie [email protected] • Reinsurer may not know the total number of claims , e.g. they may only know the
number of claims that are larger than 18 • = BA3203
L5 • ~ 10 = 10, = 10
• ~ 0,2000 = 1/2000, = /2000, 1 = 1000 • Assume an XL reinsurance with a retention limit of 1600.
• What is the mean and variance of aggregate losses for both the
insurance and reinsurance company?
Answer:
• = =1 • = = 10 ∗ 1000 = 1000
• = 2 = 10∗1,333,333.3=13,333,333 Actuarial Science NTU Question:
• Assume an annual aggregate claim amount , which is generated by
a compound Poisson distribution with Jade Nie [email protected] Excess of loss reinsurance 19
BA3203
L5 Answer: =1 • = ; = 2 from compound Poisson
• = • 2 = 1600 0
2000 1600 + 1600 ∗ 1 − 2000 = 960 1600 2 0
2000 1600 + 16002 ∗ 1 − 2000 = 1,194,666.7 • = 10 ∗ = 9600; = 10 ∗ 2 = 11,946,667 • Reinsurer: = =1 • = − = 10000 − 9600 = 400
• 2000 −1600 2
= 1600 2000 = 10,666.7 = 10 ∗ 2 = 106,667 2 • • Note: = + but = + + 2 , ≠ + because and are correlated Jade Nie [email protected] • Insurer: = Actuarial Science NTU Excess of loss reinsurance 20
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L5 • Direct insurer’s loss: • Reinsurer’s loss: 1 − • : the aggregate loss for the insured risks
• Direct insurer’s loss: • Reinsurer’s loss: 1 − Actuarial Science NTU • : the retention level for the insurer
• : individual claim amounts Jade Nie [email protected] Proportional reinsurance 21
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L5 • Number of risks is a known number and remains the same during
the claim period • : amount of claim generated from each risk
• Risks under consideration are independent
• are independent but not identically distributed
• Note that for some values of , can be of value 0 • = =1 • Comparison with collective risk model = =1 • is random, representing no. of claims
• > 0, representing the amount of claims and are i.i.d. Actuarial Science NTU • : aggregate claim amount
• : number of risks being considered Jade Nie [email protected] Individual risk model 22
BA3203
L5 • : the number of claims from risk and
0, ℎ 1 − =
1, ℎ • Hence ~ 1, • : the claim amount for risk , given that = 1
• cdf of : • = • = 2 • has a compound Binomial distribution
• = Actuarial Science NTU • For each risk in the risks that are being considered Jade Nie [email protected] Individual risk model • = 2 + 1 − 2
0 , ℎ 1 − • = , ℎ 23
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L5 =1 = =1[ 2 • = =1 + 1 − 2 ] due to
independence
• When are also identically distributed
• = , = , = for all • = • = 2 + 1 − 2 Actuarial Science NTU • = =1 • = =1 = Jade Nie [email protected] Individual risk model 24
BA3203
L5 ...
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 Spring '16
 Actuarial Science, Poisson Distribution, Binomial distribution, Reinsurance, Jade Nie cynie

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