STAM Formula Sheet
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Formula Sheet for STAM...
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Exam STAM Adapt to Your Exam SEVERITY, FREQUENCY & SEVERITY, FREQUENCY & AGGREGATE AGGREGATE MODELS MODELS MODELS Basic CDFs, Survival Functions, and Hazard Functions
() = Pr(Pr( ≤ )) = *//- () d () = Pr(( ) > ) = *- () d ℎ() = (- ) () = */ℎ() d = − ln(() ; () = ((-) E[( )] = */// () ⋅ () d ( ) ( ) = * ′ ⋅ ( d H H D D = =E[ E[(] ; −−=)] M = M M [ ] VarVar[ Var[ = Var[( )] )] = E[( ) ] − E[E[( )]M Moments
raw moment: central moment:
( ) [ ] [ ] [ ] Cov( Cov , = E ] − E E Coefficient of vari_ ation: = Skewness = _ ; Kurtosis = (()()0 =) =E[E[ ]] () D (()()1 =) =E[E[ ]( − 1)1) ⋯ ( − + 1)] () D Pr(Pr( ∣ ) = Pr(rPr( (∩))()=) Pr(r(Pr∣ () Pr()Pr( ) ∣∣() = Pr(Pr( +) Paret ( ) Exponent i a l Uniform( rm (0, − ))
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á = 1 −−1 1, forfor = 1,1, 2,⋯ E[(á)] = 1 −− E[] ä = 11−−− ä 1 −−, forfoär = 1,1,2,⋯, ⋯ E[(ä)] = 1 − E[] (,,0) = + , forfor = 1,1,2,⋯, ⋯ Prbabiobabbiabilitytiylit=y =1 − Var[ =] ã =,, (Proba Pro−−Prob )M(1 −− ) ∣ ∼∼ Neg.Bi PoiPoisson( sonBinomi()al(l ( =,∼Gamma( Gamma Neg. = )) (,, ) ℎ(( )∣ ) =[ ⋅(()]) ( ) - ( ) = ô − , where where = */ d ö E[ö( =ö)∧] = = E=[ (ù ,,∧ )) VaRæ ( )y = VaRæ( ) + VaRæ ( )y TVaRæ( ) æ √ƒ + ¿¬ 1− E[ ] ⋅ ¿Φ¬1 −−æ√ƒ ( ) ( +=) =⋅ ( ( )) + () ( + ) ≤ ( ) +() () ≤ () Pr( ≤ ) = 1 Tail-Value-at-Risk (TVaR)
(,,0)
Choosing from Class Two methods to fit data to an distributions: Method 1: Compare and • •
Distribution Poisson Binomial
Normal
Lognormal
Coherence is coherent if it satisfies the properties below: Translation invariance: Positive homogeneity: Subadditivity: Monotonicity: , if VaR is not coherent because it fails subaddivity. TVaR is coherent. • • • •
Tail Weight 1. Fewer positive raw moments
À(-) = ∞ lim ÕÀ(-) = ∞⟹ lim -→ Ã(-) -→ ÕÃ(-) ℎ() ⟹ () ⟹
2. If
or
heavier tail
, then numerator
has a heavier tail. 3. decreases with 4. increases with
heavy tail heavy tail
CONSTRUCTION AND SELECTION OF CONSTRUCTION AND SELECTION OF PARAMETRIC MODELS PARAMETRIC MODELS Maximum Likelihood Estimators Steps to Calculating MLE 1. 3. 2. 4.
(()) == l∏n (())
H() H = –—– () Set () = 0
Incomplete Data
Left-truncated at Right-censored at Grouped data on interval
(,]
()⁄() () Pr( < ≤ )
Special Cases Distribution Gamma, fixed
Normal
Lognormal
Poisson Binomial, fixed
Shortcuts
” = ̅ ̂g= ̅ ÷ = ∑¥∂I ¥ −̂ ̂ = ∑g¥∂Iln¥ ÷ = ∑g¥∂I(ln¥) − ̂ ◊ = ̅ ÷ = ̅ ◊ = ̅
E[ á] ̅ E[Bä ä] ̅ (0,) ” = max(I,,…,g) Neg. Binomial, fixed
Zero-Truncated Distribution: Match to Zero-Modified Distribution: Match to the proportion of zero observations Match to Uniform Distribution on : •
• •
•
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(,,0) ̅ Gg‹ Method 2: Observe the slope of g‹›À Neg. Binomial
class
Method 1
̅ = ̅ > ̅ <
( )
Plot Graph the difference between empirical CDF and fitted CDF
Method 2 0 Negative Positive
Variance of MLE Fisher’s Information One Parameter:
[′′()] () Var”=y =−E [()].I (,) = −E ¿ fi,HfiHH—((,,)) Var[÷] [(,)].I = flCov÷, ”y
() = g¬ √− ∗¬∗√ () = g¬.I√− ¬√ Coordinate: Óg¬ √,∗¬√Ô where g¬√ = + 1 Peak: Valley:
Two Parameters:
fi,HHH— (,) ƒ — (,”) Cov÷,y‡ Var”y
- Plot
Hypothesis Tests: Chi-Square Goodness-of-Fit Chi-Square Goodness-of-Fit Test
Delta Approximation One-Variable:
Var¬”√y ≈ Æ ()Ø Var”y Var¬÷,”√y ≈ (fiH)VarH [÷] + 2” fiH—HCov÷,”y +(—) Vary ” ± (I•æ)/‰ VÂar ”y B I B > Two-Variable:
Confidence Interval
Hypothesis Tests : null hypothesis : alternative hypothesis Reject when test statistic
critical value
is true
Reject
Fail to reject
is false
Type I Error
Correct Decision
Correct Decision
Type II Error
Hypothesis Tests: Kolmogorov-Smirnov Empirical Distribution Equal probability for each observation
≤ g () = # of observations Test statistic: = max ÍÎÎ ∗ y = max¬Ïg ¬√− ¬√Ï,Ïg ¬.I√− ∗¬√Ï√ ( ) ( ) − ∗() = 1 − () ,for ≥ Kolmogorov-Smirnov Test
where
If data is truncated at , then
Kolmogorov-Smirnov Test Properties Individual data only Continuous fit only Lower critical value for censored data If parameters are estimated, critical value should be adjusted Lower critical value if sample size is large No discretion Uniform weight on all parts of distribution • • •
G ¬ − √ Test statistic: = Ò where ∂I ::#expected of groups # of observations in group : actual # of observations in group •
•
•
= − 1 −
Degrees of freedom where : # of estimated parameters •
Chi-Square Goodness-of-Fit Test Properties Individual and grouped data Continuous and discrete fit No adjustments to critical value for censored data If parameters are estimated, critical value is automatically adjusted via degrees of freedom No change for critical value if s ample size is large Data needs to be grouped according to More weights on intervals with poor fit • • •
•
•
•
•
Hypothesis Tests: Likelihood Ratio
)] Test statistic: = 2[( (Bparameters = #−I) of#−free of free parametersin inI B Degrees of freedom
Score-Based Approaches Two types of criteria: Schwarz Bayesian Criterion (SBC), a.k.a. Bayesian Information Criterion (BIC) Akaike Information Criterion (AIC) •
•
SBC/BIC AIC
− 2 ln −
where log-likelihood
::# of estimated parameters : sample size
Select model with the highest SBC or AIC value.
•
• • •
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2
CREDIBILITY
CREDIBILITY
Classical Credibility a.k.a. Limited Fluctuation Credibility Full Credibility # of exposures needed for full credibility, Full credibility of aggregate claims:
ı
:
ı = ´(•æ) ⁄≠ () ˆ ˆ = ´(•æ) ⁄≠ flµµ + ‡ = 0 à ˜ ¯ Full credibility of cl aˆim severity: set ˘¯ = 0 ˆ = ı ⋅ µ ; ı = µ Credibility premium: ˙ == ̅ ++ ((1̅−−)) Square Root Rule: = ¸ ı = ¸ ′ˆ ′ # of claims needed for ful l credibility, Full credibility of aggregate claims:
•
:
Full credibility of claim frequency: set
•
Exact Credibility
Minimize
•
”--
where : actual # of exposures : actual # of claims
Bayesian Credibility Model Distribution Distribution of model conditioned on a parameter Model density function:
( ∣ ) ( ) () ⋅ ∣ (data) ( data ∣ ( ∣ data) = ∫// (data ∣ ) ⋅ () ) d Prior Distribution Initial distribution of the parameter Prior density function:
• •
•
•
( ∣ data) Predictive Mean = Bayesian Premi um = EE[ ∣ ] = EVar[ ∣ ] = VarE[ ∣ ] Bühlmann : = Bühlmann Credibility Factor: = + ˙ == +̅ + ((1̅ −− )) Bühlmann Credibility Expected Hypothetical Mean (EHM): Expected Process Variance (EPV):
Variance of Hypothetical Mean (VHM):
Bühlmann Credibility Premium:
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where
: Bayesian estimate given : Bühlmann estimate given
• •
• • •
Poisson/Gamma Binomial/Beta Exponential/Inv. Gamma Normal/Normal
Empirical Bayes Non-Parametric Methods Uniform Exposures
̂ = ∑¥∂∑⋅∂ ¥ ÷ = ∑¥∂ ∑∂( ¬− ¥1 )− ̅¥√ ÷ = ∑¥∂(−̅ ¥ −1 ̅) − ÷ Poisson () ∑¥∂ ∑∂ ¥¥ Gamma(,) ̂ = (∗ ∣ data ) ∼ Gamma (∗,∗) ¥¬¥ − ̅¥√ ∑ ∑ ¥∂ ∂ ÷ = ∗ = + ∑¥∂¥ = Ó— + Ô ∗ ∗ ÷ = ∑¥∂ ∑¥¥∂(̅ ¥( −¥ −̅ )1) − ÷( − 1) − ∑¥∂ ¥ Neg.Binomial ( = , = ) ( ∼ Beta(, ∣ ) ∼ Bin,1omi) al (,) Estimate E M as: ̂ = ∑∑¥∂¥∂¥¥̅ ¥ (∗ ∣ data ) ∼ Beta(∗,∗,1) ÷ ∗ == ++ [∑¥∂()¥ − ∑¥∂ ¥] PoiNeg.sson() +̅ ) ̅ ( ) ( B i n omi a l , 1 Gamma (,) ̅ Exponential () ̂ ÷ ( ∼ Inv.∣ )G ∼amma(,) (∗∣ data) ∼ Inv.Gamma(∗,∗) ∗ == ++ ∑¥∂ ¥ Pareto( = ∗, = ∗) ∣ ) ∼ Normal (,) (∼ Normal(,) (∗ ∣ data ) ∼ Normal (∗,∗) ∗ == (1−̅ +) (1− ) Normal( = ∗, = + ∗) form (0, ) ( ∼ ∣ ) ∼ Uni(,) ( ∗ ∣ data ) ∼ (∗,∗) ∗ == max(, + ,…,)
Conjugate Priors Poisson/Gamma Model: Prior:
Posterior
!
!
!
Non-uniform Exposures !
"
!
"
•
!
!
•
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Predictive
Balancing the Estimators
Binomial/Beta Model: Prior: • •
!
H
!
Empirical Bayes Semi-Parametric Methods To estimate :
Posterior
•
Model
•
Predictive
• •
Posterior
%
-
Exponential/Inv. Gamma Model: Prior:
Posterior Distribution Revised distribution of the parameter Posterior density function:
Predictive Distribution Revised unconditional distribution (w.r.t. model) of the model Predictive density function:
∑ÍÎÎ - ´-¬- − ”-√ ≠ = =
Properties of a Bayesian/Bühlmann graph Bühlmann estimates are on a straight line Bayesian estimates are within the range of hypothetical means There are Bayesian estimates above and below the Bühlmann line Bühlmann estimates are between the sample mean and theoretical mean
Partial Credibility
where : manual premium : credibility factor/credibility
Bayesian estimate = Bühlmann estimate
Bühlmann As Least Squares Estimate of Bayesian
To estimate and , use the non-parametric method formulas shown above.
• •
Predictive
Normal/Normal Model: Prior: • •
Posterior
• •
Predictive
Uniform/S-P Pareto Model: Prior: S-P Pareto • •
S-P Pareto
Posterior
• •
Predictive
-
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3
Variable Expense Ratio: =ç ä FiPerxedmiExpens e Rat i o : = ssiëble Loss Ratio: PLR = 1− −ë
SHORT-TERM INSURANCES
Expenses and Profit
Insurance Coverages Homeowners Coinsurance
Compensation: = /mimin,n(, ⋅),7, ≥< , ≤ − = 0,−7, ≤ 0, K=M. =KM. − ⋅S KUMUM.. == ∏U,ZXWYZ[\ ⋅ UM.W = KM. − S =KMK. M. 1 − 1M.7 where AlterMnat. ively, = ⋅ + (1 −) ⋅ where = 1M. K=M. =KMk. M−. ⋅SKM. UU,W,W == k̂Wq.k,UM,. − U,W\z |U,W= = ∑U[U,WWÄ ⋅ |UU,,WW
where
Disappearing Deductible Deductible decreases linearly over a specific range:
Premium
Premium
Claim Payment:
Loss Reserving Expected Loss Ratio Method 1. 2.
Uí = Uì − U + Uí\
Extension of Exposures Method Recalculates the premiums of historical policies under the current rate l evel Parallelogram Method Calculates average factors to be a pplied to the aggregate historical premiums to make them on-level
2. 3.
Bornhuetter-Ferguson Method
Ratemaking Loss Ratio Method
is calculated based on the expected loss ratio method is calculated based on the c hain-ladder method
Frequency-Severity Method Alternate Method:
1. Apply the chain-ladder method to frequency and severity separately 2. 3. Closure Method:
Frequency 1. 2. Aggregate 1. 2.
, where is the valuation CY
Data Preparation Losses
Aggregation
Develop to Ultimate • Loss Development Factors
Projected Losses
U U
ÉU = SU + U − U\ ÉU = SU + U
Incurred losses for CY : where is the reserves at the end of CY
Incurred losses for AY or PY : where is the reserves as of the valuation date
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Indicated Avg. Rate Change = 1 − +− ë − 1 Indicated RelativityU = Current RelativityU ⋅1+ IòôöõnUdicated Avg.Rate Change Indicated Base Rate = InCurdicratentedBasAvg.e RatReleat⋅ivity Off-Balance Factor Off-Balance Factor = CurrentAvg. Relativity † † † † + ç Indicated Avg.Rat eA=vg.1R−ate − ë Avg.Rela†tivityU = Base RateUU Adj.†U = Avg.RelativityUU ⋅ Expos† u†reU Indicated RelativityU = Adj I Adjndi.†.còôöõatU†ed Avg.Rate Indicated Base Rate = Indicated Avg.Relativity New Relativity = (Indicated Relativity) + (1 −)(Current Relativity) = (()+ )+ ß® = ß ⋅ Indicated Base Rate ) ̅ − ()() © = ̅(− = (1− ©)⋅ Indicated Base Rate Pure Premium Method
Credibility-Weighted Relativities
Losses
• Calendar Year (CY) • Accident Year (AY) • Policy Year (PY)
Current Rate Level • Extension of Exposures Method • Parallelogram Method
Unearned premium for CY :
1.
•
Aggregation • Calendar Year (CY) • Policy Year (PY)
Premium at Current Rates
Chain-Ladder Method a.k.a. Loss Development Triangle Method
•
, is the target profit and contingencies ratio
Trending • Trend Period • Trend Factor
Other Topics Increased Limit Factor
: original limit : increased limit Rate of policy variation with limit
•
•
Loss Elimination Ratio
: original deductible : increased deductible Rate of policy variation with deductible
• •
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