Loss Models
Basic distributional quantities
Katrien Antonio
Faculty of Economics and Business
KU Leuven
katrien.antonio@kuleuven.be
October 3, 2023
Reading list and study material
▶ Klugman et al. (2012) book, Sections 3.1-3.5 (see TOLEDO).
▶ Including exercises from the book.
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Today’s agenda
▶ This Chapter covers:
•
moments, including the excess loss, the left censored and shifted, the limited loss variable
•
percentiles
•
generating functions and sums of r.v.’s
•
risk measures.
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Moments
▶ Interesting calculations using our models?
▶ Some examples:
•
average amount paid on a claim that is subject to a deductible or policy limit
•
the average remaining lifetime of a person age 40.
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Moments
▶ Definition 3.1:
The kth raw moment of a random variable is the expected (average) value of the kth
′
power of the variable, provided it exists. It is denoted by E [X k ] or by µk . The first raw
moment is called the mean of the random variable and is usually denoted by µ.
▶ Formulas for continuous respectively discrete r.v.’s:
Z +∞
E [X k ] =
x k · f (x)dx continuous
−∞
X
=
xjk · P(X = xj ) discrete.
j
▶ It is possible that integral or sum will not converge ⇒ moment does not exist.
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Moments
▶ For mixed-type distributions:
evaluate by integrating wrt pdf where the r.v. is continuous, and by summing wrt pf where
the r.v. is discrete
add the results.
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Moments
▶ For Model 1:
Z
EX
100
x · (0.01)dx = 50
=
0
EX
2
Z
=
100
x 2 · (0.01)dx = 3 333.33.
0
▶ For Model 4:
Z
EX
= 0 · (0.7) +
| {z }
discrete
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|0
∞
x · (0.000003) · exp (−0.00001x)dx = 30 000.
{z
}
continuous
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Moments
▶ Definition 3.2:
The kth central moment of a random variable is the expected value of the kth power of
the deviation of the variable from its mean. It is denoted by E [(X − µ)k ] or by µk .
The second central moment is usually called the variance and denoted σ 2 or Var(X ), and
its square root, σ, is called the standard deviation.
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Moments
▶ Continuous formula:
µk
= E [(X − µ)k ]
Z +∞
=
(x − µ)k · f (x)dx.
−∞
▶ Discrete formula:
µk
= E [(X − µ)k ]
X
=
(xj − µ)k · P(X = xj ).
j
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Moments
▶ Standard deviation:
measure of how much the probability is spread out over the random variable’s possible
values.
▶ Coefficient of variation:
measures the spread relative to the mean; it is the ratio of the standard deviation to the
mean.
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Moments
▶ Link between raw and central moments: (for continuous r.v.’s)
Z +∞
(x − µ)2 · f (x)dx
Var(X ) = µ2 =
−∞
+∞
Z
(x 2 − 2 · x · µ + µ2 ) · f (x)dx
=
−∞
2
= E (X ) − 2 · µ · E (X ) + µ2
= E (X 2 ) − (EX )2
′
= µ2 − µ2 .
This result applies to all random variables.
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Moments
The ratio of the third central moment to the cube of the standard deviation, γ1 = µ3 /σ 3 ,
is called the skewness.
The ratio of the fourth central moment to the fourth power of the standard deviation,
γ2 = µ4 /σ 4 , is called the kurtosis.
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Moments
▶ The skewness is a measure of asymmetry.
A symmetric distribution has a skewness of zero, while a positive skewness indicates a
longer and heavier right tail than left tail.
▶ The kurtosis measures flatness of the distribution relative to a normal distribution (which
has a kurtosis of 3).
Kurtosis values above 3 indicate that (keeping the standard deviation constant), relative to
a normal distribution, more probability tends to be at points away from the mean than at
points near the mean.
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Moments
▶ Exercise:
The density function of the Gamma distribution appears to be positively skewed.
Use parametrization in Appendix A of Klugman et al. (2012):
f (x) =
with Γ(α) =
R∞
0
(x/θ)α · e −x/θ
, x >0
x · Γ(α)
t α−1 · e −t dt the Gamma function.
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Moments
▶ Exercise:
From Appendix A:
EX = α · θ, EX 2 = α · (α + 1) · θ2 and EX 3 = α · (α + 1) · (α + 2) · θ3 .
Variance is α · θ2 .
Third central moment is 2 · α · θ3 .
Skewness is 2 · α−1/2 .
Consider (on the next page) a Gamma distribution with α = 0.5 and θ = 100 and another
one with α = 5 and θ = 10. Same mean, but skewness is 2.83 versus 0.89.
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Moments
Gamma density: f(x) =
x
x
x × G(a) q
q
1
( )a exp (- )
a = shape = 0.5 & q = scale = 100
Gamma density: Gamma( a = a , q = q )
a = shape = 5
0
50
& q = scale = 10
100
150
x
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Excess loss variable
▶ Definition 3.3: For a given value of d with P(X > d) > 0 the excess loss variable is
Y P = X − d given that X > d. Its expected value
eX (d) = e(d) = E (Y P )
= E (X − d|X > d),
is called the mean excess loss function.
▶ The variable could be called left truncated and shifted.
•
Left truncated: observations below d are discarded.
•
Shifted: d is subtracted from remaining values.
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Excess loss variable
▶ The ‘P’ in notation Y P is for ‘per payment made’.
▶ This variable exists only when a payment is made.
▶ Left truncation?
•
with truncation: individuals/objects of a subset of the population of interest don’t appear in
the sample
•
with left-truncation: if Y is below a threshold - here - d then Y is not observed
•
fX |X >d (x) =
fX (x)
1−FX (d)
with x > d, a conditional pdf
this conditional pdf is a true pdf, since
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R∞
d
fX |X >d (x)dx =
Moments
R∞
d
fX (x)
1−FX (d) dx
=
1−FX (d)
1−FX (d)
= 1.
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Excess loss variable
▶ With X a payment variable, the mean excess loss is:
the expected amount paid given that there has been a payment in excess of a deductible of
d.
▶ kth moment of the excess loss variable is
R +∞
(x − d)k · f (x)dx
(k)
continuous
eX (d) = d
1 − F (d)
P
k
xj >d (xj − d) · p(xj )
=
discrete,
1 − F (d)
which is defined if and only if the integral or sum converges.
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Excess loss variable
▶ Consider for example
eX (d) = E (X − d|X > d)
R +∞
(x − d) · f (x)dx
.
= d
1 − F (d)
Reasoning: we use 2 steps
R
•
E [g (X )] =
•
fX |X >d (x) =
g (x)fX (x)dx
fX (x)
1−FX (d)
with x > d, a conditional pdf
this conditional pdf is a true pdf, since
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R∞
d
fX |X >d (x)dx =
Moments
R∞
d
fX (x)
1−FX (d) dx
=
1−FX (d)
1−FX (d)
= 1.
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Excess loss variable
▶ A demonstration of calculating the first moment eX (d):
R∞
(x − d) · f (x)dx
eX (d) = d
1 − F (d)
R∞
−(x − d) · S(x)|∞
d + d S(x)dx
=
S(d)
R∞
S(x)dx
= d
,
S(d)
using (recall: S(x) = 1 − F (x))
lim x · S(x) = 0.
x→∞
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Excess loss variable
▶ Why does limx→∞ x · S(x) = 0 hold?
▶ Use the following argument
Z
0 ≤ b · (1 − FX (b)) = b
Z
≤
∞
Z
b
∞
∞
b · fX (x)dx
fX (x)dx =
b
x · fX (x)dx.
b
R∞
Because
0 x · fX (x)dx < ∞ (i.e. the first moment exists) we have that
R∞
b x · fX (x)dx → 0 as b → ∞. Consequently,
0 ≤ b · (1 − FX (b)) → 0,
as b → ∞.
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Left censored and shifted variable
▶ Definition 3.4: the left censored and shifted variable is
(
0,
X <d
Y L = (X − d)+ =
X − d, X ≥ d.
▶ Left censored and shifted since values below d are not ignored but set equal to 0.
▶ With censoring we have some information about the dependent variable but do not always
know the dependent variable itself.
▶ Here: all observations of X that are smaller than d are put equal to d, then we apply a
shift (i.e. subtract d).
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Left censored and shifted variable
▶ Distinction with excess loss variable (for dollar events):
•
excess loss variable is per payment
exists only when a payment is made
•
left censored and shifted is per loss
takes value 0 whenever a loss produces no payment.
▶ See later on in this course.
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Left censored and shifted variable
▶ The moments can be calculated from
E [(X − d)k+ ] =
Z
∞
(x − d)k · f (x)dx
continuous
d
=
X
(xj − d)k · p(xj )
discrete.
xj >d
A such, we have
E [(X − d)k+ ] = e k (d) · [1 − F (d)].
▶ Exercise: construct graphs to illustrate the difference between the excess loss variable and
the left censored and shifted variable.
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Left censored and shifted variable
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Left censored and shifted variable
▶ Example 3.4:
An automobile insurance policy with no coverage modifications has the following possible
losses, with probabilities in parentheses: 100 (0.4), 500 (0.2), 1 000 (0.2), 2 500 (0.1), and
10 000 (0.1). Determine the probability mass functions and expected values for the excess
loss and left censored and shifted variables, where the deductible is set at 750.
For the excess loss variable:
- possible values:
250 and 1 750 and 9 250
- with probabilities:
0.2
0.4
and
0.1
0.4
and
0.1
0.4
thus, expected value is 250 · (0.5) + 1 750 · (0.25) + 9 250 · (0.25) = 2 875.
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Left censored and shifted variable
▶ Example 3.4:
An automobile insurance policy with no coverage modifications has the following possible
losses, with probabilities in parentheses: 100 (0.4), 500 (0.2), 1,000 (0.2), 2,500 (0.1), and
10,000 (0.1). Determine the probability mass functions and expected values for the excess
loss and left censored and shifted variables, where the deductible is set at 750.
For the left censored and shifted variable:
probability mass function: 0 (0.6), 250 (0.2), 1 750 (0.1), 9 250 (0.1);
expected value is then 0 · 0.6 + 250 · 0.2 + 1 750 · 0.1 + 9 250 · 0.1 = 1 150.
The ratio of the two expected values is 1 − F (750) = 0.4.
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Limited loss variable
▶ Definition 3.5: the limited loss variable is
(
Y
= X ∧u =
X , X < u,
u, X ≥ u.
▶ Its expected value, E [X ∧ u], is called the limited expected value.
▶ This variable could also be called the right censored variable.
▶ Right censored because values above u are set equal to u.
▶ Related insurance phenomenon: (cfr. chapter on ‘Policy modifications’ further on in the
book): existence of a policy limit.
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Limited loss variable
▶ Note that
(X − d)+ + (X ∧ d) = X .
Thus,
buying one policy with a deductible of d and another one with a limit of d is equivalent to
buying full coverage.
▶ Reasoning:
•
if X ≤ d then (X − d)+ takes value 0 and (X ∧ d) becomes X
•
if X > d then (X − d)+ becomes (X − d) and (X ∧ d) becomes d
•
in both cases: X results.
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Limited loss variable
▶ kth moment of the limited loss variable
Z u
k
x k · f (x)dx + u k · [1 − F (u)] continuous
E [(X ∧ u) ] =
−∞
X
=
xjk · p(xj ) + u k · [1 − F (u)] discrete.
xj ≤u
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Limited loss variable
▶ Furthermore, we have
k
Z
0
E [(X ∧ u) ] =
u
Z
k
x k · f (x)dx + u k · [1 − F (u)]
x · f (x)dx +
−∞
0
= x k · F (x)|0−∞ −
Z
0
k · x k−1 · F (x)dx
−∞
Z
Z
u
k · x k−1 · S(x)dx + u k · S(u)
0
Z u
k−1
k ·x
· F (x)dx +
k · x k−1 · S(x)dx.
−x k · S(x)u0 +
0
= −
−∞
For k = 1 we have E [X ∧ u] = −
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R0
−∞ F (x)dx
0
+
Ru
0
S(x)dx.
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Limited loss variable
▶ Due to the relation
(X − d)+ + (X ∧ d) = X ,
the limited expected value represents the expected dollar saving per incident when a
deductible is imposed.
Indeed, it gives the difference between X and (X − d)+ .
▶ The kth limited moment of many common distributions is listed in Appendix A; see
actuar package in R for functions to calculate these.
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Limited loss variable
▶ Example 3.5: (Example 3.4 continued)
Calculate the probability function and the expected value of the limited loss variable with a
limit of 750. Then show that the sum of the expected values of the limited loss and left
censored and shifted random variables is equal to the expected value of the original
random variable.
All possible values at or above 750 are assigned a value of 750, and their probabilities
summed.
Thus, probability function 100 (0.4), 500 (0.2), and 750 (0.4),
with expected value 100 · (0.4) + 500 · (0.2) + 750 · (0.4) = 440.
Expected value of original r.v. is
100 · (0.4) + 500 · (0.2) + 1, 000 · (0.2) + 2, 500 · (0.1) + 10, 000 · (0.1) = 1, 590.
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Percentiles
▶ Definition 3.6: The 100pth percentile of a r.v. is any value πp such that
F (πp −) ≤ p ≤ F (πp ).
The 50th percentile, π0.5 , is called the median.
▶ Also called a quantile.
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Percentiles
▶ Note (see page 12 in the book): F (b−) = limx↗b F (x).
▶ If the cdf has a value of p for one and only one x value, then the percentile is uniquely
defined.
▶ If the cdf jumps from a value below p to a value above p, then the percentile is at the
location of the jump.
▶ Percentile is not uniquely defined when the cdf is constant at a value of p over a range of
values of the r.v.
Any value in that range can then be used as a percentile.
▶ We illustrate this with some graphs.
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Percentiles
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Percentiles
▶ Mind the connection with the inverse distribution function (or quantile function) F −1 (.).
▶ Definition (the inverse cdf FX−1 ):
FX−1 (p) = inf (x ∈ R|FX (x) ≥ p),
p ∈ [0, 1].
For a continuous r.v. we know: FX (FX−1 (x)) = x.
▶ This definition coincides with Definition 3.6, except for the case where the cdf is constant
at a value of p over a range of values of the r.v..
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Sums of random variables
▶ Consider a portfolio of insurance risks covered by policies issued by an insurance company.
▶ Total claims paid by the insurance company on all policies is, say:
Sk
= X1 + . . . + Xk .
Therefore, it is useful to study properties of this r.v.
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Sums of random variables
▶ Theorem 3.7:
For a r.v. Sk = X1 + . . . + Xk , E (Sk ) = E (X1 ) + . . . + E (Xk ). Also, if X1 , . . . , Xk are
independent, Var(Sk ) = Var(X1 ) + . . . + Var(Xk ). If the random variables X1 , X2 , . . . are
independent and their first two moments meet certain conditions,
Sk − E (Sk )
lim p
is N(0, 1).
k→∞
Var(Sk )
▶ The limit used in this Theorem is convergence in distribution. The limit result is known as
the Central Limit Theorem (CLT).
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Sums of random variables
▶ Obtaining the exact distribution or density function of Sk is usually very difficult.
▶ The key to simplicity is the generating function.
▶ Definition 3.8: for a r.v. X the moment generating function (mgf) is MX (z) = E (e zX ) for
all z for which the expected value exists. The probability generating function (pgf) is
PX (z) = E (z X ) for all z for which the expectation exists.
▶ Note that: MX (z) = PX (e z ) and PX (z) = MX (ln z). Often the mgf is used for continuous
r.v.’s and the pgf for discrete r.v.’s.
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Sums of random variables
▶ For us, the value of these functions is not so much that they generate moments or
probabilities (as their names suggest).
▶ What matters: a one-to-one correspondence between a r.v.’s distribution function and its
mgf and pgf.
▶ Thus, two r.v.’s with different distribution functions cannot have the same mgf or pgf.
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Sums of random variables
▶ Theorem 3.9: let Sk = X1 + . . . + Xk , where the r.v.’s in the sum are independent. Then
Q
Q
MSk (z) = kj=1 MXj (z) and PSk (z) = kj=1 PXj (z) provided all component mgfs and pgfs
exist.
▶ Proof: we use
MSk (z) = E e z·Sk = E e z·(X1 +...+Xk )
=
k
Y
E e z·Xj
j=1
=
k
Y
MXj (z).
j=1
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Sums of random variables
▶ Example 3.7: sum of independent gamma r.v.’s, each with the same value of β, has a
gamma distribution.
▶ Mgf of a gamma r.v. X ∼ Γ(α, β) is
Z ∞
β α α−1 −βx
tX
E e
x
e
dx
=
e tx
Γ(α)
Z0 ∞ α
β
=
x α−1 e −(β−t)x dx
Γ(α)
0
Z ∞
βα
(β − t)α α−1 −(β−t)x
=
x
e
dx
(β − t)α 0
Γ(α)
βα
=
, β > t.
(β − t)α
Note: compare this to parametrization used in the book (i.e. β = 1/θ).
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Sums of random variables
▶ Example 3.7: sum of independent gamma r.v.’s, each with the same value of β, has a
gamma distribution.
▶ Let Xj have a gamma distribution with parameters αj and β. Then,
"
MSk (t) = E
=
#
e tXi
t=1
k
Y
i=1
=
k
Y
αi
β
β−t
Pki=1 αi
β
β−t
,
which is mgf of a gamma distribution with parameters α1 + . . . + αk and β.
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Tails of distributions
▶ The tail of a distribution (more specifically: the right tail) is the portion of the distribution
corresponding to large values of the r.v.
▶ Understanding large possible loss values is important because they have the greatest
impact on the total of losses.
▶ R.v.’s that tend to assign higher probabilities to larger values are said to be heavier tailed.
▶ When choosing models, tail weight can help narrow choices or can confirm a choice for a
model.
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Classification based on moments
▶ One way of classifying distributions:
do all moments exist, or not?
▶ The existence of all positive moments indicates a (relatively) light right tail.
▶ The existence of only positive moments up to a certain value indicates a heavy right tail.
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Classification based on moments
▶ Example 3.9: demonstrate that for the gamma distribution all positive moments exist but
for the Pareto distribution they do not.
▶ For the gamma distribution
′
µk
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x α−1 e −x/θ
dx
Γ(α)θα
0
Z +∞
(y θ)α−1 e −y
=
(y θ)k ·
θdy
(y = x/θ)
Γ(α)θα
0
θk
=
· Γ(α + k) < +∞ for all k > 0.
Γ(α)
Z
=
+∞
xk ·
Tails of distributions
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Classification based on moments
▶ For the Pareto distribution
′
µk
+∞
αθα
dx
(x + θ)α+1
0
Z +∞
αθα
=
(y − θ)k · α+1 dy
(y = x + θ)
y
θ
Z ∞X
k k
α
= α·θ
y j−α−1 (−θ)k−j dy ,
j
θ
Z
=
xk ·
j=0
for integer values of k (use polynomial expansion of binomial power).
This integral exists only if
R∞
θ
y j−α−1 dy =
y j−α ∞
j−α θ
exists.
Therefore, j − α < 0, or j − α − 1 < −1 for all j. Or, equivalently, k < α.
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Tails of distributions
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Classification based on moments
▶ Pareto is said to have a heavy tail, and gamma has a light tail.
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Comparison based on limiting tail behavior
▶ Consider two distributions with the same mean.
▶ If ratio of S1 (.) and S2 (.) diverges to infinity, then distribution 1 has a heavier tail than
distribution 2.
▶ Divergence implies that the numerator (=teller ) distribution puts more probability on large
values.
▶ Thus, we examine (with SX (x) = 1 − FX (x))
S1 (x)
lim
x→+∞ S2 (x)
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′
S (x)
= lim 1′
x→∞ S (x)
2
f1 (x)
−f1 (x)
=
lim
= lim
.
x→∞ f2 (x)
x→+∞ −f2 (x)
Tails of distributions
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Comparison based on limiting tail behavior
▶ Example 3.10: demonstrate that Pareto distribution has a heavier tail than the gamma
distribution using the limit of the ratio of their density functions.
▶ We consider (with c a constant)
fPareto (x)
x→+∞ fgamma (x)
lim
=
αθα (x + θ)−α−1
x→+∞ x τ −1 e −x/λ λ−τ Γ(τ )−1
lim
e x/λ
x→+∞ (x + θ)α+1 x τ −1
= +∞
= c lim
Exponentials go to infinity faster than polynomials, thus the limit is infinity.
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Comparison based on limiting tail behavior
4e-04
Example 3.10: pdf of Pareto with α = 3 and θ = 10, versus Gamma with α =
1
3
and θ = 15.
Gamma
f(x)
0e+00
1e-04
2e-04
3e-04
Pareto
50
100
150
200
x
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Tails of distributions
▶ We don’t cover: Section 3.4.3 - 3.4.6.
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Measures of risk
▶ The level of exposure to risk is often described by one number, or a small set of numbers.
▶ These are called ‘key risk indicators’.
▶ Informative about the degree to which a company is subject to particular aspects of risks.
▶ We briefly mention:
•
VaR: Value-at-Risk
•
TVaR: Tail Value-at-Risk.
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Measures of risk
▶ A risk measure (say, ρ(X )) is a mapping from the r.v. X representing the loss associated
with the risks to the real line.
▶ A risk measure gives a single number that is intended to quantify the risk exposure.
▶ For example, the standard deviation is a risk measure. It is a measure of uncertainty.
▶ Think of ρ(X ) as the amount of assets required to protect against adverse outcomes of the
risk X .
▶ For an in depth study of risk measures see Risk Management in Financial Institutions and
Foundations of Quantitative Risk Measurement.
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Measures of risk
▶ Study of risk measures in academic literature has proposed some desirable properties of
risk measures.
▶ Definition 3.10: a coherent risk measure is a risk measure ρ(X ) that has the following
properties for any two loss r.v.’s X and Y :
1. Subadditivity: ρ(X + Y ) ≤ ρ(X ) + ρ(Y )
2. Monotonicity: if X ≤ Y for all possible outcomes, then ρ(X ) ≤ ρ(Y )
3. Positive homogeneity: for any positive constant c, ρ(cX ) = cρ(X )
4. Translation invariance: for any positive constant c, ρ(X + c) = ρ(X ) + c.
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Measures of risk
▶ Study of risk measures in academic literature has proposed some desirable properties of
risk measures.
▶ Definition 3.10: a coherent risk measure is a risk measure ρ(X ) that has the following
properties for any two loss r.v.’s X and Y :
1. Subadditivity: diversification benefit from combining risks.
2. Monotonicity: if one risk always has greater losses than another, it requires more capital.
3. Positive homogeneity: doubling exposure to certain risk implies that capital should be
doubled.
4. Translation invariance: no additional risk (thus, no additional capital) for an additional risk for
which there is no additional uncertainty.
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Value-at-Risk
▶ Say FX (x) represents the cdf of outcomes over a fixed period of time, e.g. one year, of a
portfolio of risks.
▶ We consider positive values of X as losses.
▶ Definition 3.11: let X denote a loss r.v., then the Value-at-Risk of X at the 100p% level,
denoted VaRp (X ) or πp , is the 100p percentile (or quantile) of the distribution of X .
▶ For continuous distributions we have
P(X > πp ) = 1 − p.
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Value-at-Risk
▶ VaR has become the standard risk measure used to evaluate exposure to risk.
▶ VaR is the amount of capital required to ensure, with a high degree of certainty, that the
enterprise does not become technically insolvent.
▶ Which degree of certainty?
•
95%?
•
in Solvency II 99.5% (or: ruin probability of 1 in 200).
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Value-at-Risk
▶ VaR is not subadditive.
▶ Example: let X and Y be i.i.d. r.v.’s which are Bern(0.02) distributed.
Then, P(X ≤ 0) = 0.98 and P(Y ≤ 0) = 0.98. Thus, FX−1 (0.975) = FY−1 (0.975) = 0.
For the sum, X + Y , we have P[X + Y = 0] = 0.98 · 0.98 = 0.9604. Thus,
FX−1
+Y (0.975) > 0.
VaR is not subadditive, since VaR(X + Y ) in this case is larger than VaR(X ) + VaR(Y ).
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Value-at-Risk
▶ VaR is not subadditive.
Subadditivity (i.e. diversification benefit) of a risk measure ρ(.) requires
ρ(X + Y ) ≤ ρ(X ) + ρ(Y ).
Intuition behind subadditivity:
combining risks is less riskier than holding them separately, i.e. a diversification benefit
results from combining risks.
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Value-at-Risk
▶ Another drawback of VaR:
•
it is a single quantile risk measure of a predetermined level p
•
no information about the thickness of the upper tail of the distribution function from VaRp on
•
whereas stakeholders are interested in both frequency and severity of default.
▶ Therefore: study other risk measures, e.g. Tail Value at Risk (TVaR).
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Tail-Value-at-Risk
▶ Definition 3.12: let X denote a loss r.v., then the Tail-Value-at-Risk of X at the 100p%
security level, TVaR(p), is the expected loss given that the loss exceeds the 100p
percentile (or: quantile) of the distribution of X .
▶ The Loss Models book restricts consideration to continuous distributions to avoid
ambiguity about definition of TVaR.
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Tail-Value-at-Risk
▶ We have (assume continuous distribution)
TVaRp (X ) = E (X |X > πp )
R∞
πp x · f (x)dx
=
.
1 − F (πp )
▶ We can rewrite this as
[the usual definition of TVaR]
R∞
πp xdFX (x)
TVaRp (X ) =
1−p
R1
p VaRu (X )du
=
,
1−p
using the substitution FX (x) = u and thus x = FX−1 (u) = VaRu (X ).
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Tail-Value-at-Risk
▶ From the definition
R1
TVaRp (X ) =
p
VaRu (X )du
1−p
,
we understand
•
TVaR is the arithmetic average of the quantiles of X , from level p on
•
TVaR is ‘averaging high level VaRs’
•
TVaR tells us much more about the tail of the distribution than does VaR alone.
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Tail-Value-at-Risk
▶ Finally, TVaR can also be written as
TVaRp (X ) = E (X |X > πp )
R∞
πp x · f (x)dx
=
1−p
R∞
π (x − πp ) · f (x)dx
= πp + p
1−p
= VaRp (X ) + e(πp ),
with e(πp ) the mean excess loss function evaluated at the 100pth percentile (see
Definition 3.3).
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Tail-Value-at-Risk
▶ TVaR is a coherent risk measure, see e.g. Foundations of Quantitative Risk Measurement
course.
▶ Thus, TVaR(X + Y ) ≤ TVaR(X ) + TVaR(Y ).
▶ When using this risk measure, we never encounter a situation where combining risks is
viewed as being riskier than keeping them separate.
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Tail-Value-at-Risk
▶ Examples 3.15, 3.16 and 3.17 are DIYs.
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Tail-Value-at-Risk
▶ Example 3.18 (Tail comparisons) Consider three loss distributions for an insurance
company. Losses for the next year are estimated to be on average 100 million with
standard deviation 223.607 million. You are interested in finding high quantiles of the
distribution of losses. Using the normal, Pareto, and Weibull distributions, obtain the VaR
at the 90%, 99%, and 99.9% security levels.
▶ Solution
Normal distribution has a lighter tail than the others, and thus smaller quantiles.
Pareto and Weibull with τ < 1 have heavy tails, and thus relatively larger extreme
quantiles.
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Tail-Value-at-Risk
▶ Example 3.18 (Tail comparisons) Consider three loss distributions for an insurance
company. Losses for the next year are estimated to be on average 100 million with
standard deviation 223.607 million. You are interested in finding high quantiles of the
distribution of losses. Using the normal, Pareto, and Weibull distributions, obtain the VaR
at the 90%, 99%, and 99.9% security levels.
> qnorm(c(0.9, 0.99, 0.999),mu,sigma)
[1] 386.5639 620.1877 790.9976
> qpareto(c(0.9, 0.99, 0.999),alpha,s)
[1] 226.7830 796.4362 2227.3411
> qweibull(c(0.9, 0.99, 0.999),tau,theta)
[1] 265.0949 1060.3796 2385.8541
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Tail-Value-at-Risk
▶ We learn from Example 3.18 that results vary widely depending on the choice of
distribution.
▶ Thus, the selection of an appropriate loss model is highly important.
▶ To obtain numerical values of VaR or TVaR:
•
estimate from the data directly
•
or use distributional formulas, and plug in parameter estimates.
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Tail-Value-at-Risk
▶ When estimating VaR directly from the data:
•
use R to get quantile from the empirical distribution
•
R has 9 ways to estimate a VaR at level p from a sample of size n, differing in the way how
the interpolation works between order statistics close to n · p (see MART book) (page 132).
▶ When estimating TVaR directly from the data:
•
take average of all observations that exceed the threshold (i.e. πp ).
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Tail-Value-at-Risk
▶ Caution: we need a large number of observations (and a large number of observations
> πp ) in order to get reliable estimates.
▶ When not many observations in excess of the threshold are available:
•
construct a loss model
•
calculate values of VaR and TVaR directly from the fitted distribution.
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Tail-Value-at-Risk
▶ Example of getting TVaR directly from a given distribution:
=
=
=
=
TVaRp (X ) = E (X |X > πp )
R∞
π (x − πp )f (x)dx
πp + p
1−p
R∞
R πp
(x
−
π
)f
(x)dx−
p
−∞ (x − πp )f (x)dx
πp + −∞
1−p
R πp
E (X ) − −∞
xf (x)dx − πp (1 − F (πp ))
πp +
1−p
E (X ) − E [min (X , πp )]
E (X ) − E (X ∧ πp )
πp +
= πp +
,
1−p
1−p
see Appendix A for those expressions.
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That’s a wrap!
▶ You can calculate moments of distributions, in particular: the mean excess-of-loss, the
expected value of the left censored and shifted variable and the expected limited loss.
▶ You master the calculation of and the use of percentiles (or: quantiles) of distributions.
▶ You understand different ways to approximate or to derive analytically the distribution of a
sum of independent r.v.’s.
▶ You grasp the relevance of appropriately modelling the tail of a distribution, and can use
tools to compare the behavior in the tail of different distributions.
▶ You can work with a selection of risk measures, in particular: VaR and TVaR.
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August 2019, question 2
Insurance agent James B. will receive no annual bonus if the ratio of incurred losses to
earned premiums for his book of business is 60% or more for the year. If the ratio is less
than 60% James’ bonus will be a percentage of his earned premium equal to 15% of the
difference between 60% and his ratio. James’ annual earned premium is 800 000. Incurred
losses are distributed according to the Pareto distribution, with θ = 500 000 and α = 2.
Calculate the expected value of James’ bonus.
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August 2019, question 2
Let L be the incurred losses and P the earned premium. The realized value of P is 800 000.
The bonus can then be expressed as:
L
0.15 · 0.6 −
· P = 0.15 · (0.6 · P − L)
P
= 0.15 · (480, 000 − L) if positive.
This can be written as 0.15 · [480000 − (L ∧ 480 000)], with expected value
0.15 · [480000 − E (L ∧ 480 000)]
= 0.15 · [480 000 − [500 000 · (1 − (500 000/(480 000 + 500 000)))]
= 35 265,
where the formula from the Klugman Appendix for E [X ∧ x] with X Pareto distributed, was
used.
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August 2019, question 2
Extras
With L ∼ Pa(α, θ) we have
E [L ∧ d] =
"
α−1 #
θ
θ
1−
.
α−1
d +θ
Check the screencast video on TOLEDO for a derivation of this expression.
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January 2020, question 1
A random loss variable has density function f (x) = θ−1 · e −x/θ , for x ≥ 0 and θ > 0.
(a) Calculate the density of the per-payment variable when there is an ordinary
deductible d.
(b) Determine eX (d), the mean excess loss function evaluated at ordinary deductible d.
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