MAART 200752THE RISK MARGIN UNDER SOLVENCY IITEKST JELMER LANTINGA 1 [608419]

DE ACTUARIS
MAART 200752THE RISK MARGIN UNDER SOLVENCY IITEKST JELMER LANTINGA 1
1 Jelmer Lantinga studies actuarial
sciences at the University of
Amsterdam and wrote this paper in
cooperation with Pricewaterhouse-
Coopers. This paper is a summary
of his master thesis written to fina-
lise the actuarial studies. For a
copy of the master thesis we refer
to the university. The view expres-
sed is the one of the author and
does not necessarily reflect the
view of PricewaterhouseCoopers.In the development of Solvency II there is a lot of discus-
sion about the method used to calculate the risk margin
of the insurance liabilities. The main purpose of the risk
margin is to have the possibility to transfer the liabilities
to a third party. It is a reflection of the non-hedgeable
risks which the insurer is bearing.
Several possible approaches are available to calculate
the risk margin, but two methods are most common
nowadays. These are the cost of capital and the percen-
tile method. Both methods are recommended for the
future calculation of the risk margin in the EU, the first
by the Comité Européen des Assurances (CEA) and Chief
Risk Officer (CRO) Forum and the second by the Euro-
pean Commission. There are many papers written about
the pros and cons of both methods, however the rela-
tionship is still underexposed.
The main objective of this article is to obtain a mathe-
matical relationship between the cost of capital and per-
centile approach, with the final goal of being able to
quantify the difference between the two approaches for
life as well as non-life insurers.
TWO APPROACHES
One of the possible approaches to determine the risk
margin is the percentile approach. This method takes
the perspective that the insurer must be able to meet its
liability with some probability. The method focuses on
the volatility of the distribution of the liabilities and is
currently used in Australia. After identifying the distri-
bution of the liabilities, the risk margin is computed by
subtracting the best estimate from a predefined critical
percentile value. In Australia this critical percentile is
set to 75%. This leads to the following general formula
for the risk margin under the percentile approach:
Risk Margin Percentile = zcritical.ótotal (1)
(1 + iduration)duration
The second approach to calculate the risk margin is the
cost of capital approach. The cost of capital approach
takes the perspective that sufficient capital is needed to
be able to run-off the business. The method focuses on
the costs involved with providing solvency capital (SCR)
to support the business-in-force until run-off. This isexactly the cost that is needed to transfer the liability to
another party. The amount of solvency capital that a firm
holds is invested in risk free investments and conse-
quently yields a low profit. Because the investors
demand a certain return higher than the risk free rate
on all capital, the company is making a cost by holding
the extra amount of capital. This is called the cost of
capital.
There are different methods to calculate the risk margin
based on the cost of capital approach. One of them is the
Swiss Solvency Test (SST) which was set to be the
benchmark cost of capital approach.
The formula for the risk margin under the cost of capital
method is:
run-off-period SCRi(t)
Risk Margin Cost of Capital = CoC /H9018(2)
t=l (1 + it)t
For both approaches the purpose of the risk margin is
exactly the same, that is to say having the possibility to
transfer the liabilities to another party. When the pur-
pose of both calculations is the same it is obvious that
the results should also be approximately the same.
When this assumption is made, it is possible to express
the critical percentile value in a cost of capital percen-
tage and the other way around which leads to:
run-off-period duration
CoC /H9018SCR(O) BE(t) (1 + iduration )
t=lBE(O) (1 + it)t
zcritical = (3)
ótotal
ASSUMPTIONS
The insurance companies in Europe all have different
client files, what makes it difficult to make generalisa-
tions. The estimation of future cash flows is directly lin-
ked to the present client file and is necessary to identify,
because the risk margin is based on this information. It
is however not unrealistic to approximate the distribu-
tion of future cash flows by an exponential distribution.
An exponential distribution is characterized by a high
peak at the beginning of the probability distribution. The
peak is followed by a fast decrease, but the probabilityscriptie
The prudential regulation framework of insurance businesses across the European Union (EU) will be radically
transformed in the near future. Solvency II will replace the present framework, Solvency I, and is expected to
come into force around 2010. Solvency II presumably represents a big step forward since it is a more transparent
and risk oriented framework than Solvency I.

DE ACTUARIS
MAART 200753density function extinguishes very slowly. In the full
paper a gamma distribution of future cash flows is also
examined.
Besides the distribution of the future cash flows it is
also essential to make assumptions on the distribution
of the liability. The expected cost can easily turn out to
be totally different when the calibration of parameters is
incorrect, a wrong functional form is used or a non
expected shock occurs. The normal and the gamma
distribution are examined as possible liability distribu-
tions.
RESULTS
By use of the assumptions as stated it is possible to cal-
culate the risk margin under the cost of capital appro-
ach. The risk margin can be translated into a percentile
value, leading to the same risk margin by using formula
(3). In this paper the Swiss cost of capital factor of 6% is
used. Results are calculated for different durations and
the different distributions for the liability. This leads for
a life insurer with an exponential distribution of future
cash flows to the following graph:
FIGURE 1: EXPONENTIALLY DISTRIBUTED FUTURE
CASH FLOWS FOR A LIFE INSURER WITH A NORMAL
AND GAMMA DISTRIBUTION FOR THE LIABILITIES
This graph shows a converging percentile value when
the duration of the cash flows increases. The percentile
value is more or less 75% if the duration is three and
this percentile slowly decreases to 60%. It is important
to note that the position of the normal curve as well as
the gamma curve highly depend on the ratio of the vol-
atilities of the different risk categories.The volatility increases due to a higher run-off period,
which is the main reason for the decrease of the per-
centile curve for the CoC method. Nonetheless, the
decrease in percentile value becomes smaller and
smaller. This is primarily due to the diminishing in-
crease, or negative second derivative, of the run-off
period. Another important reason for the shape of the
percentile curve of the CoC method is the distribution of
the liability. The difference of the two risk margins in-
creases, but this increase is barely translated into a
smaller percentile due to very small probabilities in the
tail.
Besides the form, the graph also shows comparable
results for the different liability distributions. Except for
the very large durations the gamma distribution gives
similar results to the normal distribution. The difference
is very small because the volatility in proportion to the
mean is small but rising. In a situation with a low dura-
tion the gamma distribution is hardly skewed thus
almost the same as the normal distribution. When we
enlarge the volatility, the difference for long durations
between the gamma and normal distributions becomes
significantly positive. The difference for long durations is
positive since the gamma distribution is skewed to the
right and the cost of capital risk margin consequently
leads to a higher percentile.
For a non-life insurer with an exponential distribution of
future cash flows the following graph results:
FIGURE 2: EXPONENTIALLY DISTRIBUTED FUTURE
CASH FLOWS FOR A NON-LIFE INSURER WITH A
NORMAL AND GAMMA DISTRIBUTION FOR THE
LIABILITIESscriptie
1.30 3.09 4.69 6.01 7.18 8.24 9.16 10.00 10.73 11.40 12.03 12.56 13.05 13.52 13.96 14.3810%
0%20%30%40%50%60%70%80%90%100%
CoC percentile with a Normal distribution for liabilities
CoC percentile under a Gamma distribution for liabilities
Percentile used in Australia
1.30 3.09 4.69 6.01 7.18 8.24 9.16 10.0
010.73 11.40 12.03 12.56 13.05 13.52 13.96 14.3810%
0%20%30%40%50%60%70%80%90%100%
CoC percentile with a Normal distribution for liabilities
CoC percentile under a Gamma distribution for liabilities
Percentile used in Australia

DE ACTUARIS
MAART 200754The graph shows a similar result as the results already
acquired for the life insurer; the appropriate percentile
decreases when the duration increases. However, the
percentile curve for the non-life insurer is somewhat
less steep. The percentile curve of the CoC method cros-
ses the 75th percentile at a larger duration, because of
the less negative derivative with respect to the duration.
The percentile method leads under these circumstances
to a lower risk margin than the cost of capital method,
when the duration is smaller than about 5 years. A lar-
ger volatility leads to a larger difference between the
gamma and exponential percentile curve, a similar
result as for the life insurer.
Another important variable for a non-life insurer is the
size of the total liability. The size of the liability is impor-
tant because there is a non-linear relationship between
the premium risk volatility and its solvency requirement.
The unit of measurement becomes relevant in this situ-
ation, which is a very unwelcome consequence of the
non-linear relationship. In this respect CEIOPS might
want to reconsider the non-linear relationship and
change this relation into a linear relation between the
premium volatility and the appropriate solvency capital.
Under the non-linear relationship the percentile curve of
the CoC method shifts upward when the liability increa-
ses. Large non-life insurance companies with large
amounts of future cash flows do therefore hold on to too
much capital as a risk margin under the cost of capital
method.
A CORRECTION TERM
The duration of the cash flows and in case of a non-life
insurer the size of the liability, are the variables with a
major influence on the appropriate percentile value cor-
responding to the cost of capital risk margin. The diffe-
rence in every situation can easily be derived by subtrac-
ting the cost of capital risk margin from the risk margin
stemming from the percentile approach. By carrying out
a multivariate regression it is possible to define a for-
mula which approximates the difference between the
calculated risk margins.FIGURE 3: REGRESSION FOR A LIFE INSURER
WITH AN EXPONENTIAL DISTRIBUTION OF FUTURE
CASH FLOWS
For a life insurer the final equation for the correction
term based on figure 3 is:
Correction life,exp = (1.6543duration – 2.1876 ) .size (4)
where size is the ratio of the liability of the insurer which
correction term needs to be identified and the liability
based on a fixed sum of cash flows used in this paper.
For a non-life insurer the following correction formula
results:
The correction term leads to much more comparable
results and as a consequence is a good quantification of
the difference between the risk margin calculated under
the percentile and cost of capital approach.
It is important to note that the model used in this paper
is a simplification of reality. The assumptions made in
this article are certainly not applicable to every insurer,
even though the assumptions are a best estimate of the
insurance market. Nonetheless, this article gives a use-
ful insight in the mathematical relationship between the
cost of capital and percentile approach and moreover
presents the quantification of the difference between the
two approaches for the average life or non-life insurer.scriptie
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.0010.00–
15.00–5.00––10.00
5.0015.0020.0025.00
DurationCorrection2y =1.6543x – 2.1876
R = 0.9049
CoC percentile with a Normal distribution for liabilities
CoC percentile under a Gamma distribution for liabilities
Percentile used in Australia
Correction non-life,exp = (6.2379size – 0.6533 )duration – 6 0.174 size + 8.1589 (5)