Graphics Reference
In-Depth Information
Limited Expected Value Function
6.3.2
helimitedexpectedvaluefunction L ofaclaimsizevariable X,orofthecorrespond-
ing cdf F
x
, is defined as
(
)
∫
x
L
(
x
)=
E
min
(
X, x
) =
ydF
(
y
)+
x
−
F
(
x
)
, x
.
(
.
)
he value of the function L at point x is equal to the expectation of the cdf F
)
truncated at this point. In other words, it represents the expected amount per claim
retainedbytheinsuredonapolicywithafixedamountdeductibleof x.heempirical
estimate is defined as follows:
(
x
x
j
<
x
n
L
n
(
x
)=
x
j
+
x
j
x
x
.
(
.
)
In order to fit the limited expected value function L of an analytical distribution to
the observed data, the estimate L
n
is first constructed. hereater, one attempts to
find a suitable analytical cdf F such that the corresponding limited expected value
function L is as close to the observed L
n
as possible.
he limited expected value function (LEVF) has the following important proper-
ties:
(i) the graph of L is concave, continuous and increasing
(ii) L
(
x
)
E
(
X
)
as x
L
′
,whereL
′
(iii) F
is the derivative of the function L at point x;ifF
is discontinuous at x, then the equality holds true for the right-hand derivative
L
′
x
x
x
(
)=
−
(
)
(
)
.
he limited expected value function is a particularly suitable tool for our purposes
because it represents the claim size distribution in the monetary dimension. For ex-
ample, we have L
x
(
+)
if it exists. he cdf F, on the other hand, operates
on the probability scale, i.e., it takes values of between
and
. herefore, it is usu-
ally di
cult to work out how sensitive the price of the insurance - the premium - is
to changes in the values of F by looking only at F
() =
E
(
X
)
, while the LEVF immediately
showshowdifferentpartsoftheclaimsizecdfcontributetothepremium.Asidefrom
its applicability to curve-fitting, the function L also turns outtobeavery usefulcon-
ceptwhendealingwithdeductiblesBurneckietal.(
).Itisalsoworthmentioning
that there is a connection between the limited expected value function and the mean
excess function:
(
x
)
E
(
X
)=
L
(
x
)+
P
(
X
x
)
e
(
x
)
(
.
)
he limited expected value functions (LEVFs) for all of the distributions considered
in this chapter are given by the following formulae:
exponential distribution:
β
L
(
x
)=
−
exp
(−
βx
)
;
