Download Severity, Frequency, and Aggregate Loss

Part I
Severity, Frequency, and
Aggregate Loss
1
Basic Probability
Raw moments
Central moments
Skewness
Kurtosis
Coefficient of variation
Covariance
Correlation
MGF
PGF
Moments via MGF
Moments via PGF
Conditional mean
2
µ0n = E[X n ]
µn = E[(X − µ)n ]
γ1 = µ3 /σ 3
γ2 = µ4 /σ 4
CV = σ/µ
Cov[X, Y ] = E[(X − µX )(Y − µY )] = E[XY ] − E[X] E[Y ]
ρXY = Cov[X, Y ]/(σX σY )
MX (t) = E[etX ]
PX (t) = E[tX ] = MX (log t)
(n)
MX (0) = E[X n ]
(n)
PX (1) = E[X(X − 1) . . . (X − n + 1)]
E[X] = EY [EX [X|Y ]]
Variance
Mixtures
#
n
Var[X]
1X
Xi =
Var[X̄] = Var
n i=1
n
n
X
wi FXi (x),
w1 + w2 + · · · + wn = 1
F (x) =
Bernoulli shortcut
Var[Y ] = (a − b)2 q(1 − q)
"
Sample variance
i=1
3
Conditional Variance
Conditional variance
Var[X] = Var[E[X|I]] + E[Var[X|I]]
1
4
Expected Values
Z
Payment per loss with
deductible
E[(X − d)+ ] =
Payment per loss with
claims limit/Limited expected value
Decomposition relation
E[X ∧ d] =
LEV higher moments
Z
(x − d)f (x) dx =
∞
S(x) dx
d
Z
Payment per payment
event/mean residual life
∞
d
d
Z
d
xf (x) dx + dS(d) =
0
S(x) dx
0
E[X] = E[(X − d)+ ] + E[X ∧ d]
E[(X − d)+ ]
e(d) = E[X − d|X > d] =
1 − F (d)
Z d
E[(X ∧ d)k ] =
kxk−1 S(x) dx
0
Deductible + Limit
(max. payment = u − d)
5
Parametric Distributions
Tail weight:
1. Compare moments
2. Density ratios
3. Hazard rate
4. Mean residual life
6
E[Y L ] = E[X ∧ u] − E[X ∧ d]
More moments =⇒ less tail weight
Low ratio =⇒ numerator has less tail weight
Increasing hazard rate =⇒ less tail weight
Decreasing MRL =⇒ less tail weight
Lognormal Distribution
Continuously compounded growth rate
Continuously compounded dividend return
Volatility
Lognormal parameters
Asset price at time t
Strike price
European call (option to buy)
European put (option to sell)
American options
Black-Scholes
Cumulative distribution
Limited expected value
2
α
δ
σv
µ = (α√
− δ − 21 σv2 )t
σ = σv t
St = S0 exp(µ + Zσ)
K
C = max{0, ST − K}
C = max{0, K − ST }
exercise at any time up to T
−dˆ1 = (log(K/S0 ) − µ − σ 2 )/σ
−dˆ2 = (log(K/S0 ) − µ)/σ
Pr[St < K] = Φ(−dˆ2 )
Φ(−dˆ1 )
E[St |St < K] = S0 e(α−δ)t
Φ(−dˆ2 )
Φ(dˆ1 )
E[St |St > K] = S0 e(α−δ)t
Φ(dˆ2 )
7
Deductibles, LER, Inflation
Loss elimination ratio
Inflation if Y = (1 + r)X
8
LER(d) = E[X ∧ d]/ hE[X]
E[Y ∧ d] = (1 + r) E X ∧
d
1+r
i
Other Coverage Modifications
With deductible d, maximum covered loss u, coinsurance α, policy limit/maximum
payment L = α(u − d):
Payment per loss
E[Y L ] = α(E[X ∧ u] − E[X ∧ d])
Second moment of per-loss E[(Y L )2 ] = E[(X ∧ u)2 ] − E[(X ∧ d)2 ] − 2d E[Y L ]
9
Bonuses
With earned premium P , losses X, proportion of premiums r:
Bonus B = c(rP − X)+
10
Discrete Distributions
pk
(a, b, 0) recursion
zero-truncated relation
pk−1
pT0 = 0,
zero-modified relation
pM
n =
11
b
k
pTn =
=a+
pn
1 − p0
1 − pM
0
pn
1 − p0
Poisson/Gamma
If S = X1 + X2 + · · · + XN , where X ∼ Gamma(α, θ), N ∼ Poisson(λ) where λ
varies by X, then S ∼ NegBinomial(r = α, β = θ).
12
Frequency Distributions—Exposure and Coverage Modifications
Model
Exposure
Pr[X > 0]
Poisson
Binomial
Neg. Binomial
Original
n1
1
λ
m, q
r, β
Exposure mod.
n2
1
(n2 /n1 )λ
(n2 /n1 )m, q
(n2 /n1 )r, β
3
Coverage mod.
n1
υ
υλ
m, υq
r, υβ
13
Aggregate Loss Models: Approximating Distribution
Compound variance
14
Aggregate Loss Models: Recursive Formula
Frequency
Severity
Aggregate loss
(a, b, 0) recursion
15
Var[S] = Var[X] E[N ] + E[X]2 Var[N ]
pn = Pr[N = n]
fn = Pr[X = n]
gn = Pr[S = n] = fS (n)
k X
1
bj
gk =
fj gk−j
a+
1 − af0 j=1
k
Aggregate Losses—Aggregate Deductible
dd/he−1
E[S ∧ d] = d(1 − F (d)) +
X
hjghj
j=0
16
Aggregate Losses—Misc. Topics
If X ∼ Exponential(θ), N ∼ Geometric(β), then FS (x) =
β
0
1+β FX (x),
0
where X ∼ Exponential(θ(1 + β)).
If S = X1 + · · · + Xn , then S ∼ Gamma(α = n, θ).
Method of rounding: pk = FX (k + 1/2) − FX (k − 1/2)
17
Ruin Theory
Ruin probability, discrete, finite horizon
Survival probability, continuous, infinite horizon
4
ψ̃(u, t)
φ(u)
1
1+β [x
= 0] +
Part II
Empirical Models
18
Review of Mathematical Statistics
Bias
Consistency
Biasθ̂ (θ) = E[θ̂ − θ|θ]
lim Pr[|θ̂n − θ| < δ] = 1, ∀δ > 0
Mean square error
MSEθ̂ (θ) = E[(θ̂ − θ)2 |θ]
n
1 X
s2 =
(xi − x̄)2
n−1
k=1
σ 2 /n
MSEθ̂ (θ) = Var[θ̂] + Biasθ̂ (θ)2
Sample variance
Variance of s2
MSE/Bias relation
19
n→∞
Empirical Distribution for Complete Data
Total number of observations
Observations in j-th interval
Width of j-th interval
Empirical density
20
Variance of Empirical Estimators with Complete Data
Empirical variance
21
n
nj
cj − cj−1
nj
fn (x) =
n(cj − cj−1 )
d n (x)] = Sn (x)(1 − Sn (x))/n = nx (n − nx )/n3
Var[S
Kaplan-Meier and Nelson Åalen Estimators
Risk set at time yj
Loss events at time yj
rj
sj
Kaplan-Meier product-limit estimator
Sn (t) =
Nelson-Åalen cumulative hazard
Ĥ(t) =
j−1
Y
i=1
j−1
X
i=1
22
1−
si
,
ri
si
ri
,
yj−1 ≤ t < yj
yj−1 ≤ t < yj
Estimation of Related Quantities
Exponential extrapolation: fit Sn (yk ) = exp(−yk /θ), and solve for the parameter θ.
5
23
Variance of Kaplan-Meier and Nelson-Åalen
Estimators
Greenwood’s approximation
for KM
Greenwood’s approximation
for NÅ
d n (yj )] = Sn (yj )2
Var[S
j
X
i=1
si
ri (ri − si )
j
X
si
d Ĥ(yj )] =
Var[
r2
i=1 i
q

d n (t)]
Var[S

(Sn (t)1/U , Sn (t)U ), U = exp 
Sn (t) log Sn (t)
q


d Ĥ(t)]
zα/2 Var[

(Ĥ(t)/U, Ĥ(t)U ), U = exp 
Ĥ(t)

100(1−α)% log-transformed
confidence interval for KM
100(1−α)% log-transformed
confidence interval for NÅ
24
zα/2
Kernel Smoothing
Uniform kernel density
Uniform kernel CDF
Triangular kernel density
Empirical probability at yi
Fitted density
Fitted distribution
1
ky (x) = 2b
, y − b ≤ x ≤ y + b
x<y−b
 0,
x−(y−b)
Ky (x) =
,
y−b≤x≤y =b
2b

1,
y+b<x
height = 1/b, base = 2b
pn (yi )
n
X
fˆ(x) =
pn (yi )kyi (x)
F̂ (x) =
i=1
n
X
pn (yi )Kyi (x)
i=1
Use conditional expectation formulas to find moments of kernel-smoothed
distributions; condition on the empirical distribution.
25
Approximations for Large Data Sets
Right/upper endpoint of j-th interval
Number of new entrants in [cj , cj+1 )
Number of withdrawals in (cj , cj+1 ]
Number of events in (cj , cj+1 ]
Risk set for the interval (cj , cj+1 ]
Conditional mortality rate in (cj , cj+1 ]
cj
dj
uj
sj
rj
qj
Population at time cj
Pj =
j−1
X
di − ui − si
i=0
rj = Pj + αdj − βuj
α = β = 1/2
Generalized relation
UD of entrants/withdrawals
6
Part III
Parametric Models
26
Method of Moments
For a k-parameter distribution, match the first k empirical moments to the fitted
distribution:
E[X m ] =
n
X
(xi − x̄)m
i=1
27
Percentile Matching
Interpolated k-th order statistic
Smoothed empirical 100p-th
percentile
28
xk+w = (1 − w)xk + wxk+1 ,
πp = xp(n+1)
0<w<1
Maximum Likelihood Estimators
n
Y
Likelihood function
~ =
L(θ)
Loglikelihood
l = log L
~
Pr[X ∈ Xi |θ]
i=1
Xi are the observed events—each is a subset of the sample space. Maximize l
∂l
= 0 for each parameter in the fitted distribution.
by finding θ~ such that
∂θi
29
MLEs—Special Techniques
Exponential
Gamma (fixed α)
Normal
Poisson
Neg. Binomial
Lognormal
MLE = sample mean
MLE = method of moments
MLE(µ) = sample mean, MLE(σ 2 ) = population variance
MLE = sample mean
MLE(rβ) = sample mean
take logs of sample, then use Normal shortcut
Censored exponential MLE–Take each observation (including censored ones)
and subtract the deductible; sum the result and divide by the number of uncensored observations.
7
30
Estimating Parameters of a Lognormal Distribution
31
Variance of MLEs
For n estimated parameters θ~ = (θ1 , θ2 , . . . , θn ), the estimated variance of a
function of MLEs is computed using the delta method:
 2

σ1 σ12 · · · σ1n
 σ21 σ22 · · · σ2n 

~ =
Covariance matrix
Σ(θ)
 ..
..
.. 
..
 .
.
.
. 
Delta method
Fisher’s information
Covariance-information relation
32
σn1 σn2 · · · σn2
> X ∂g
~
~ ∂g
Var[g(θ)] =
(θ)
~
∂ θ~
"∂ θ
#
2 ~
∂ l(θ)
I(θrs ) = − E
∂θs ∂θr
~ θ)
~ = In
Σ(θ)I(
Fitting Discrete Distributions
To choose which (a, b, 0) distribution to fit to a set of data, compute the empirical
mean and variance. Then note
Binomial
Poisson
Negative Binomial
33
E[N ] > Var[N ]
E[N ] = Var[N ]
E[N ] < Var[N ]
Cox Proportional Hazards Model
Hazard class i/Covariate
logarithm of proportionality constant for class i
Proportionality constant/relative risk
Baseline hazard
Hazard relation
34
zi
βi
c = exp(β1 z1 + β2 z2 + · · · + βn zn )
H0 (t)
H(t|z1 , . . . , zn ) = H0 (t)c
Cox Proportional Hazards Model: Partial
Likelihood
Number at risk at time y
Proportionality constants of members at risk
Failures at time y
Breslow’s partial likelihood
k
{c1 , c2 , . . . , ck }
{j1 , j2 , . . . , jd }
!−d
d
k
Y
X
exp(βji )
exp(βi )
i=1
8
i=1
35
Cox Proportional Hazards Model: Estimating Baseline Survival
Risk set at time yj
Proportionality constants for the
members of risk set R(yj )
Baseline hazard rate
R(yj )
ci
Ĥ0 (t) =
sj
X
P
yj ≤t
i∈R(yj ) ci
36
The Generalized Linear Model
37
Hypothesis Tests: Graphic Comparison
f (x)
1 − F (d)
F (x) − F (d)
∗
F (x) =
1 − F (d)
D(x) = Fn (x) − F ∗ (x)
x1 , x2 , . . . , xn
(Fn (xj ), F ∗ (xj ))
(xj , F ∗−1 (Fn (x)))
f ∗ (x) =
Adjusted fitted density
Adjusted fitted distribution
D(x) plot
empirical observations
p-p plot
Normal probability plot
Where the p-p plot has slope > 1, then the fitted distribution has more weight
than the empirical distribution; where the slope < 1, the fitted distribution has
less weight.
38
Hypothesis Tests: Kolmogorov-Smirnov
Komolgorov-Smirnov statistic
D = max |Fn (x) − F ∗ (x; θ̂)|
KS-statistic is the largest absolute difference between the fitted and empirical
distribution. Should be used on individual data, but bounds on KS can be
established with grouped data. Fitted distribution must be continuous. Uniform
weight across distribution. Lower critical value for fitted parameters and for
more samples.
39
Hypothesis Tests: Anderson-Darling
Anderson-Darling statistic
A2 = n
Z
t
u
(Fn (x) − F ∗ (x))2 ∗
f (x) dx
F ∗ (x)(1 − F ∗ (x))
AS-statistic used only on individual data. Heavier weight on tails of distribution.
Critical value independent of sample size, but decreases for fitted parameters.
9
40
Hypothesis Tests: Chi-square
Total number of observations
Hypothetical probability X is in j-th group
Number of observations in j-th group
n
pj
nj
Chi-square statistic
Q=
k
X
(nj − Ej )2
Ej
j=1
Ej = n j pj
Degrees of freedom df = total number of groups, minus number of estimated
parameters, minus 1 if n is predetermined
41
Likelihood Ratio Algorithm, Schwarz Bayesian
Criterion
Likelihood Ratio—compute loglikelihood for each parametric model. Twice the
difference of the loglikelihoods must be greater than 100(1 − α)% percentile
of chi-square with df = difference in the number of parameters between the
compared models.
Schwarz Bayesian Criterion—Compute loglikelihoods and subtract 2r log n,
where r is the number of estimated parameters in the model and n is the sample
size of each model.
Part IV
Credibility
42
Limited Fluctuation Credibility—Poisson Frequency
Poisson frequency of claims
Margin of acceptable fluctuation
Confidence of fluctuation being within k
Severity CV
n0
Credibility for
Exposure units eF
Number of claims nF
Aggregate losses sF
Frequency
n0
λ
n0
n0 µs
λ
k
P
CVs2 = σs2 /µ2s
−1 1+P 2
Φ
2
n0 =
k
Severity
n0
CVs2
λ
n0 CVs2
n0 µs CVs2
10
Aggregate
n0
1 + CVs2
λ
n0 (1 + CVs2 )
n0 µs (1 + CVs2 )
43
Limited Fluctuation Credibility: Non-Poisson
Frequency
Credibility for
Exposure units eF
Frequency
σf2
n0 2
µf
σf2
µf
Number of claims nF
n0
Aggregate losses sF
n0 µs
σf2
µf
Severity
σ2
n0 s 2
µf µs
n0
σs2
µ2s
n0
σs2
µs
Aggregate !
n0 σf2
σ2
+ s2
µf µf
µs
!
2
σf
σs2
n0
+ 2
µf
µs
!
2
σf
σs2
n0 µs
+ 2
µf
µs
Poisson group frequency is the special case µf = σf2 = λ. If a compound Poisson
frequency model is used, you cannot use the Poisson formula—you must use the
mixed distribution (e.g., Poisson/Gamma mixture is Negative Binomial).
44
Limited Fluctuation Credibility: Partial Credibility
Credibility factor
Manual premium (presumed
value before observations)
Observed premium
Credibility premium
45
Z=
M
p
p
p
n/nF = e/eF = s/sF
X̄
PC = Z X̄ + (1 − Z)M
Bayesian Estimation and Credibility—Discrete
Prior
Constructing a table: First row is the prior probability, the chance of membership in a particular class before any observations are made. Second row is the
likelihood function of the observation(s) given the hypothesis of membership in
that particular class. Third row is the joint probability, the product of Rows
1 and 2. Row 4 is the posterior probability, which is Row 3 divided by the
sum of Row 3. Row 5 is the hypothetical mean or conditional probability, the
expectation or probability of the desired outcome given that the observations
belong that that class. Row 6 is the expectation, Bayesian premium, or expected probability, the desired result given the observations, and is the sum of
the products of Rows 4 and 5.
11
46
Bayesian Estimation and Credibility—Continuous
Prior
Observations
Prior density
Model density
Joint density
~x = (x1 , x2 , . . . , xn )
π(θ)
f (~x|θ)
f (~x, θ) =
Z f (~x|θ)π(θ)
Unconditional density
f (~x) =
f (~x, θ) dθ
f (~x, θ)
π(θ|x1 , . . . , xn ) =
f (~x)
Z
f (xn+1 |~x) = f (xn+1 |θ)π(θ|~x) dθ
Posterior density
Predictive density
Loss function minimizing MSE
Loss function minimizing absolute error
Zero-one loss function
posterior mean E[Θ|~x]
posterior median
posterior mode
A conjugate prior is the prior distribution when the prior and posterior distributions belong to the same parametric family.
47
Bayesian Credibility: Poisson/Gamma
With Poisson frequency with mean λ, where λ is Gamma distributed with parameters α, θ,
Number of claims
Number of exposures
Average claims per exposure
Conjugate prior parameters
Posterior parameters
Credibility premium
48
x
n
x̄ = x/n
α, γ = 1/θ
α∗ = α + x
γ∗ = γ + n
PC = α∗ /γ∗
Bayesian Credibility: Normal/Normal
With Normal frequency with mean θ and fixed variance v, where θ is Normal
with mean µ and variance a,
vµ + nax̄
Posterior parameters µ∗ =
v + na
va
Posterior variance
a∗ =
v + na
n
Credibility factor
Z=
n + v/a
Credibility premium µ∗
12
49
Bayesian Credibility: Binomial/Beta
With Binomial frequency with parameters M , q, where q is Beta with parameters a, b,
Number of trials
Number of claims in m trials
Posterior parameters
Credibility premium
50
m
k
a∗ = a + k
b∗ = b + m − k
PC = a∗ /(a∗ + b∗ )
Bayesian Credibility: Exponential/Inverse Gamma
With exponential severity with mean Θ, where Θ is inverse Gamma with parameters α, β,
Posterior parameters
51
α∗ = α + n
β∗ = β + nx̄
Bühlmann Credibility: Basics
Expected hypothetical mean
Variance of hypothetical mean (VHM)
Expected value of process variance (EPV)
Bühlmann’s k
Bühlmann credibility factor
Bühlmann credibility premium
52
µ = E[E[X|θ]]
a = Var[E[X|θ]]
v = E[Var[X|θ]]
k = v/a
Z = n/(n + k)
PC = Z X̄ + (1 − Z)µ
Bühlmann Credibility: Discrete Prior
No additional formulas
53
Bühlmann Credibility: Continuous Prior
No additional formulas
54
Bühlmann-Straub Credibility
55
Exact Credibility
Bühlmann equals Bayesian credibility when the model distribution is a member
of the linear exponential family and the conjugate prior is used.
13
Frequency/Severity
Poisson/Gamma
Normal/Normal
Binomial/Beta
56
Bühlmann’s k
k = 1/θ = γ
k = v/a
k =a+b
Bühlmann As Least Squares Estimate of Bayes
Variance of observations
Bayesian estimates
Covariance
Mean relationship
regression slope/Bühlmann credibility estimate
regression intercept
Bühlmann predictions
57
P
Var[X] = pi Xi2 − X̄ 2
Yi
P
Cov[X, Y ] =
pi Xi Yi − X̄ Ȳ
E[X] = E[Y ] = E[Ŷ ]
Cov[X, Y ]
b=Z=
Var[X]
a = (1 − Z) E[X]
Ŷi = a + bXi
Empirical Bayes Non-Parametric Methods
For uniform exposures,
Number of exposures/years data
Number of classes/groups
Observation of of group i, year j
Unbiased manual premium
Unbiased EPV
Unbiased MHV
Bühlmann credibility factor
58
n
r
xij
r
n
1 XX
xij
rn i=1 j=1
r
n
1X 1 X
v̂ =
(xij − x̄i )2
r i=1 n − 1 j=1
r
v̂
1 X
(x̄i − x̄)2 −
â =
r − 1 i=1
n
n
Z=
n + k̂
µ̂ = x̄ =
Empirical Bayes Semi-Parametric Methods
When the model is Poisson, v = µ, and we have
EPV
VHM
v̂ = µ
â = Var[S] − v̂
Sample variance
Var[S] = σ 2 =
X (xi − x̄)2
n−1
14
Part V
Simulation
59
Simulation—Inversion Method
Random number
Inversion relationship
Method
u ∈ [0, 1]
Pr[F −1 (u) ≤ x] = Pr[F (u) ≤ F (x)] = F (x)
xi = F −1 (u)
Simply take the generated uniform random number u and compute the inverse
CDF of u to obtain the corresponding simulated xi .
15