Sensitivity Analysis:
A weighting approach for ranking the order of model inputs
Vaishno Devi Makam

This dissertation is submitted as part of the requirements for the awardof the M.Sc in Actuarial Science.

Under the supervision of Dr. Andreas Tsanakas
Cass Business School
August, 20181

Abstract
A new approach to sensitivity analysis is introduced in this paper to aid
understanding of the nuances of model behavior. The main objective is to identify
the most influential inputs in a risk model and rank the inputs based on thesensitivity levels that they exhibit. The framework is closely related to theapproach taken by Pesenti et al. (2018).
The methodology adopted consists of two steps to obtain the sensitivities of
the inputs. First, different scenarios (states of the world) are weighted using achange of probability measure. Different sets of weights are derived from each
input, such that adverse scenarios of that input are assigned higher emphasis. In
that sense, each set of weights produces a stress to the distribution of an input.
The weights are derived using the normal transform, which has been defined in
the context of distortion risk measures. Second, the change in the distribution of
the output is quantified, for such a stress applied to each input. The sensitivity
to an input, is assessed by comparing standard tail risk measures of the output,
such as Value-at-Risk and Expected Shortfall, before and after stressing that
input. Further, Monte-Carlo simulation is used to quantify the response of themodel when the value of a certain input is varied.
The approach taken in this paper results in computational ease as the calculations
can be performed on a single set of simulated input/output scenarios. In addition,
it is a coherent approach since all inputs are varied in a consistent way. We use
the same distortion function for obtaining all the different sets of weights andeach set of weights produced exhibit similar characteristics i.e., higher valued
outcomes are given higher emphasis and lower valued outcomes, lower emphasis.
A numerical example is presented to elucidate the effectiveness of the proposed
approach.
Keywords: Sensitivity analysis, Value-at-Risk, Expected Shortfall, Distortion
Risk Measures, Normal Transform.

2

Acknowledgements
I would like to express my sincere gratitude to my advisor Dr. Andreas Tsanakas
for the continuous support and guidance that he had offered me for my dissertation. I would also like to thank my family for encouraging me throughout thisperiod of study and offering me their support whenever I needed. This project
would not have been possible without them. Thank you.
With regards,
Vaishno Devi Makam

3

CONTENTS

CONTENTS

Contents
1 Introduction

6

2 Risk Measures

7

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Properties of Risk Measures . . . . . . . . . . . . . . . . . . . . .

10

2.3

Some Important Risk Measures . . . . . . . . . . . . . . . . . . .

14

2.3.1

Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.2

Expected Shortfall . . . . . . . . . . . . . . . . . . . . . .

16

Distortion Risk Measures . . . . . . . . . . . . . . . . . . . . . .

18

2.4.1

Definition and Interpretation . . . . . . . . . . . . . . . .

18

2.4.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4

3 Sensitivity Analysis
3.1

3.2

21

Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.1.1

Why Sensitivity Analysis is used?

. . . . . . . . . . . . .

21

3.1.2

Local and Global Measures . . . . . . . . . . . . . . . . .

22

3.1.3

Reverse Sensitivity Testing . . . . . . . . . . . . . . . . .

23

A New Sensitivity Analysis Method. . . . . . . . . . . . . . . .

24

3.2.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2.2

Setup and Notation . . . . . . . . . . . . . . . . . . . . .

25

3.2.3

Implementation in a Monte-Carlo Setting . . . . . . . . .

27

4 Examples
4.1

28

Benchmark Model . . . . . . . . . . . . . . . . . . . . . . . . . .

28

4.1.1

Model Specification. . . . . . . . . . . . . . . . . . . . .

28

4.1.2

Analysis and Results . . . . . . . . . . . . . . . . . . . . .

28

4.2

Variations of the example . . . . . . . . . . . . . . . . . . . . . .

34

4.3

Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

5 Conclusion

39

4

LIST OF FIGURESLIST OF TABLES

6 Appendix

40

References

41

List of Figures
1

Samples of W1 , W2 , W3 , W4 against X1 , X2 , X3 , X4 respectively,for λ = 0.5, 1, 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Distorted probability distributions of Y obtained using W1 , W2 ,W3 , W4 for λ = 0.5, 1.0, 1.5 . . . . . . . . . . . . . . . . . . . . .

3

35

Sensitivity levels of X1 , X2 , X3 , X4 when d changes from 300 to
600 units for p = 0.95. . . . . . . . . . . . . . . . . . . . . . . . .

7

34

Sensitivity levels of X1 , X2 , X3 , X4 when the correlation between
inputs change from 0 to 1 for p = 0.95. . . . . . . . . . . . . . . .

633

Distorted probability distributions for X1 , X2 , X3 , X4 obtainedusing W4 for λ = 0.5, 1.0, 1.5 . . . . . . . . . . . . . . . . . . . . .

530

Distorted probability distributions for X1 , X2 , X3 , X4 obtainedusing W1 for λ = 0.5, 1.0, 1.5 . . . . . . . . . . . . . . . . . . . . .

4

29

36

Sensitivity levels of X1 , X2 , X3 , X4 when the standard deviationof X2 changes from 10 to 100 units for p = 0.95.

. . . . . . . . .

37

List of Tables
1

Percentage increase in VaRp and ESp for H Q1 with respect to thebaseline model, H. . . . . . . . . . . . . . . . . . . . . . . . . . .

2Percentage increase in VaRp and ESp for H Q2 with respect to thebaseline model, H. . . . . . . . . . . . . . . . . . . . . . . . . . .

3

31Percentage increase in VaRp and ESp for H Q3 with respect to thebaseline model, H. . . . . . . . . . . . . . . . . . . . . . . . . . .

4

31

32

Percentage increase in VaRp and ESp for H Q4 with respect to thebaseline model, H. . . . . . . . . . . . . . . . . . . . . . . . . . .

532

1 INTRODUCTION

1Introduction

Managing the risk that an institution faces in the short and long run is foremost
the greatest responsibility of a risk manager. As part of the solvency II requirements, it is mandatory for the insurance companies to keep aside enough capitalso that the largest loss can be dealt with at a 99.5% confidence level. Companies
are in charge of their own risk and solvency assessment (ORSA), rather thanfollowing a perspective approach to risk quantification and management. Depending on the risk profiles of the insurance businesses, the capital requirements
would change and therefore building their own internal models to assess the risk
leads to more appropriate calculations (Ralph, 2016).
Numerous factors must be taken into account for model building. Due to theincreasing complexities that arise in running a business, complicated models arebeing integrated into insurance operations. Specifically for an insurance company, all the business lines have to incorporated into the model along with factors
such as inflation, interest rate, and expected premium growth. These numerous
variables in a model makes it impossible for the analyst to base any judgments onmere intuition and therefore reliance is placed on various simulation techniques.
By quantifying portfolio risk and by performing sensitivity analysis, it becomes
tractable to pinpoint a particular scenario or a combination of events as catastrophic to the company’s performance. It is therefore essential that the analysts
understand the model behavior before proposing any changes in the businessstrategy.
Sensitivity analysis helps aid understanding the relationships between the inputs and output and is an essential component of model building. By runningthe model several times, it allows us to identify the set of inputs which havethe greatest impact on the model output. The inputs which affect the output
significantly are said to be highly sensitive. It therefore allows an analyst to focus more on the highly sensitive inputs rather than on inputs which exhibit low
sensitivities. Due to process being time-consuming as well as expensive, reverse

6

2 RISK MEASURES

sensitivity testing introduced by Pesenti et al. (2018) is an alternative method
as it requires only a single set of input/output scenarios.
The structure of the paper is as follows: we discuss risk measures in section
2 placing emphasis on certain risk measures such as Value-at-Risk, Expected
Shortfall and Distortion Risk Measures. In section 3, we provide an introduction to sensitivity analysis and the various methods available in the literature.
Recent methods such as reverse sensitivity analysis method is acknowledged. In
section 4, we propose a new sensitivity analysis method to analyze and compare
the sensitivities of the inputs of the model. We use the concept of distortion risk
measures to increase the severity of the assumptions considered in the model.
We then use Monte-Carlo simulation as a tool to investigate the model behavior.
Finally, we set out three variations of the model for the reader to appreciate theusefulness of the method.

2

Risk Measures

2.1

Introduction

The term risk in a layman’s context indicates any kind of uncertainty that he/she
may be facing. Where decisions concerning money have to be taken, there is anatural tendency to quantify the risk involved in the decision. Extending theconcept of risk to an insurance company, the process of quantifying risk that it
faces is an important task for identifying and to differentiate between businesslines which perform well against those that perform poorly. One can view risk
measures as a tool which allows companies to control and limit the amount ofrisk they are willing to take (McNeil et al., 2015).
According to Denuit et al. (2005), a risk measure is a “functional ρ mapping arisk X to a non-negative real number ρ(X), possibly infinite, representing theextra cash which has to be added to X to make it acceptable” to an internal orexternal controller.

72.1 Introduction

2 RISK MEASURESThe above definition implies that the interpretation and analysis of the risk inherent in an investment is based on a real value which is obtained as the output.
The most obvious relationship between ρ(X) and the risk is a direct relationshipie., as ρ(X) increases, the riskiness of the portfolio under consideration increases.
In principle, we can view the underlying concept of risk measures as being analogous to the premium calculations in the insurance sector and this was studied
extensively by Bühlmann (2007) and Goovaerts et al. (2001). The basic idea ofsuch a comparability emerges from the viewpoint that the premium calculationreflects the minimum amount that an insurer is willing to accept in order to bear
the risk of proceeding with an insurance contract (Tsanakas and Desli, 2003).
Therefore, the insurer would charge a higher premium if there is a higher risk
involved in the contract. As aptly put, “both insurance premium and the priceof a financial product can be regarded as a measure of risk involved in the financial transaction between the buyer of the product and the seller in the market(Goovaerts et al., 2001). For a more comprehensive disquisition, see Denuit et al.
(2005) and Goovaerts et al. (2001)).
Quantifying financial risks plays an important role in safeguarding the interests
and to ensure smooth functioning of the company. According to Denuit et al.
(2005), there are two key approaches to decision making under risk and are as
follows:
1. Classical Expected Utility Theory, which was axiomatized by von Neumann
and Morgenstern in 1947 (for further discussion, see Kaas et al. (2008,
chapter 1)).
2. Yaari’s (1987) Dual Theory for Choice under risk.

Yaari’s Dual Theory of Choice was proposed as a complementary approach to
the Expected Utility Theory in 1987. There were two main reasons behind the
motivation for the Dual Theory. The first being that under the utility theory, risk
aversion and diminishing marginal utility of wealth are in many respects alike.
Risk aversion is the basic characteristic exhibited by an agent. Here, an agent

82.1 Introduction

2 RISK MEASURES

would tend to avoid those deals which involves greater uncertainty. Diminishingmarginal utility implies that the agents’ total wealth relates directly to the sizeof the losses he is willing to accept. Therefore, higher the amount of wealth thatan agent possesses, the higher the level of risk that he is willing to accept. The
second reason behind the motivation as stated by Yaari is that certain behavior patterns that are inconsistent, such as those observed in Allais (1953) and
Kahneman-Tversky (1979) can be explained by the dual theory. Allais questions
the “independence axiom” and from experimental evidence, we note that somepeople prefer certain choices of investment which do not seem rational when Expected Utility theory is considered. Kahneman-Tversky in their seminal paper
Prospect Theory: An analysis of Decision under Risk argue that people tend tochange their behavior patterns depending on how the problem is presented tothem. For instance, if there is a greater emphasis on “gains”, individuals are“risk averse” but when emphasis is placed on “losses”, individuals tend to be
more risk-seeking. Further, individuals prefer to choose those situations wherean outcome is certain as opposed to those where a probability is attached to theoutcome amount, even if the latter is an equally good or a better deal (Kahneman and Tversky, 2013). Hence, the inconsistencies that arise while considering
Expected Utility Theory are enormous and therefore Dual Theory might be agood alternative.
Dual theory itself generates certain paradoxes which can be resolved by the utlity
theory. Thus according to Yaari, the two theories resolve each others paradoxes.
Attempts have been made by Tsanakas and Desli (2003) to combine the aforementioned theories to obtain a new risk measure, Distorted Exponential Premium
Principle, which possesses properties from both theories.

92.2 Properties of Risk Measures

2.2

2 RISK MEASURES

Properties of Risk Measures

There are a few fundamental and important properties of risk measures thatare useful to know, but there exists no such set of properties that an ideal risk
measure should satisfy under all circumstances. It should be left to the discretionof the analyst to decide whether a particular metric which they would like toapply, must satisfy a particular property or not depending on the situation.
A few properties are mentioned below and some of them are simple and selfexplanatory.
Let X, Y be the random variables (r.v. for short) denoting the losses.
1. No-Ripoff
For bounded r.v. X, ρ(X) ≤ max(X) = FX−1 (1)(1)

The no-ripoff property implies that it is inefficient for an institution to holdmore money than the maximum possible loss as this would guarantee 100%solvency. Holding more capital than what is required is uneconomical as
this can be invested in other instruments such as bonds and shares to obtaina higher value.
2. Non-Negative Loading
ρ(X) ≥ E(X)(2)

The least amount of capital that must be kept aside by an institution must
be equal to at least the loss expected to incur during a given time period.
This is because we assume that the rough estimate of the predictive loss isequal to the mean of the past losses.
3. Translation Invarianceρ(X + c) = ρ(X) + c(3)

As we have defined X to be a random variable representing the losses, this
implies that if the precise amount of a loss c is known in advance then10

2.2 Properties of Risk Measures

2 RISK MEASURES

it is equivalent to adding the deterministic cash amount c to the positionin order to comply with the definition of a risk measure mentioned above(Denuit et al., 2005).
4. Constancy
ρ(c) = c(4)

We had specified c to be a deterministic loss and hence for a company toremain solvent, we would need to set aside an amount c. This is not thecase that we generally expect to observe in the real world situations but,
if we broaden X to denote liabilities, then we can expect to have somedeterministic liabilities in the future (Denuit et al., 2005). This property
does not take into consideration the interest rate.
5. Sub-additivityρ(X + Y ) ≤ ρ(X) + ρ(Y ) for all r.v.s X and Y(5)

The central idea of sub-additivity was summed up by Artzner et al. (1999)
that a merger does not create extra risk. This introduces the effect of diversification and for sub-additive risk measures, the resultant effect is positive(Denuit et al., 2005). Diversification effect is defined by McNeil et al. (2015)
as the “difference between the sum of the risk measures of stand-alone risks
and the risk measure of all risks taken together.”Sub-additivity is a much debated concept in respect to the necessity of arisk measure satisfying the property as we will see further on that Value-atRisk, which is a commonly used risk measure, does not satisfy the property.
However, there have been attempts to justify its necessity as according to
McNeil et al. (2015), a financial institution can reduce its capital requirements by breaking it up into various subsidiaries if a non-subadditive risk
measure is used. Artzner et al. (1999) and Wang et al. (1997) also claimthat risk measures would satisfy sub-additivity for any dependence struc-112.2 Properties of Risk Measures

2 RISK MEASURES

ture, as risks that exhibit positive dependence also provide diversification
to a certain level (Tsanakas and Desli, 2003).
6. Co-monotonic Additivity
Firstly, co-monotonicity is defined as the following by Denuit et al. (2005):
A random vector X is co-monotonic if and only if there exist a r.v. Z and
non-decreasing functions t1 , t2 , ..., tn , such that :X =d (t1 (Z), t2 (Z), ..., tn (Z))t(6)(X1c , X2c , ..., Xnc ) denotes a co-monotonic random vector and for any two
r.v.s X and Y if FX ≡ FY , then X =d Y (Denuit et al., 2005).
Co-monotonic r.v.s are also referred to as undiversifiable r.v.s. It implies
that the variables are increasing functions of one another. A more precise
definition can be found in McNeil et al. (2015).
The property is stated as follows :ρ(X + Y ) = ρ(X) + ρ(Y ) for all r.v.s X and Y(7)

From the above, we can notice that the combined risk can never decrease
with respect to their stand-alone risks. This can be viewed as a special caseof the sub-additivity property where there is no effect of diversification.
Therefore, it is a no-hedge condition (Denuit et al., 2005).
7. Positive Homogeneity
ρ(cX) = cρ(X), for all random variables X and c ∈ R+

(8)This is a similar concept to the one seen above and it was proposed by
Artzner et al. (1999). From the equation, it is clear that if we increase theportfolio size, then the capital requirements would also increase proportionally. However, it does fail to take into consideration the extra increase in
the liquidity risk that the agent would face if he increases his portfolio size.

122.2 Properties of Risk Measures

2 RISK MEASURES

By increasing the magnitude of the portfolio, the agent is now faced witha higher risk of experiencing larger losses which in turn makes it harder tocope up with the losses (Tsanakas and Desli, 2003).
8. Monotonicity
If X ≤ Y almost surely ⇒ ρ(X) ≤ ρ(Y )(9)

If loss Y is at least equal to or greater than loss X in most cases, thenit is rational to allocate proportionally more capital for Y than X to deal
with the losses appropriately. In other words, the greater the returns of aportfolio, the less risky it is considered to be.
9. Law of Invariance / Objectivity
X =d Y ⇒ ρ(X) = ρ(Y )(10)This property implies that a risk measure is ‘objective’ if ρ(X) depends on
X only through its cumulative distribution function FX (x).
Objectivity plays an important role for the applicability of a risk measure.
If a measure is observed to satisfy this property, then the risk measure canbe estimated using the empirical data (Denuit et al., 2005). Therefore, it isa desirable property as the riskiness of X can be measured through FX (x).

10. Convexity
Convexity is a more general property and we see below that the risk measureis sub-additive when the property of convexity is combined with positive
homogeneity.
ρ(λX + (1 − λ)Y ≤ λρ(X) + (1 − λ)ρ(Y )(11)Further, a risk measure is classified as a coherent risk measure if it exhibits four properties: translation invariance, positive homogeneous, subadditivity and monotonicity (Artzner et al., 1999). However according to13

2.3 Some Important Risk Measures

2 RISK MEASURES

Danielsson et al. (2005), coherence is not an appealing mathematical property from a financial point because it is a restrictive condition and alternative risk measures which are coherent can not be widely applied due tothe complexities that arise while back-testing. Priority must be given to
economic justification rather than mathematical tractability (Dhaene et al.,2003).

2.3

Some Important Risk Measures

Risk Measures can be classified and studied in a number of ways and wenote that the classification need not be exclusive. In other words, there
may be instances where a risk measure may be classified under more than
one category. Without dwelling deep into the different classifications thatexist, we would like to specifically look at Value-at-Risk, Expected Shortfall,
Distortion Risk Measures and their important properties.

2.3.1

Value at Risk

According to Denuit et al. (2005),
VaRp (X) = FX−1 (p),(12)

for a given risk X and p ∈ (0, 1) and where,
FX−1 (p) = inf{x ∈ R|FX (x) ≥ p}(13)

p is also referred to as the confidence level.
The origins of VaR can be traced back to the early twentieth century where
actuaries used the concept of VaR implicitly for calculating the initial reserve in classical ruin problems. The measure was used by the financial
institutions as a means of forecasting and aggregating risks but, as firms
started performing more complex operations, it became increasingly more
important to understand the interactions between various variables in context (Dowd and Blake, 2006). The interpretation of VaR as given in McNeil

14

2.3 Some Important Risk Measures

2 RISK MEASURESet al. (2015) is that it is the “maximum loss that is not exceeded with agiven high probability.” It is clear from the definition, that VaR does nottake into account the tail of the distribution. In other words, it is blind to
the shape of the distribution beyond the value p and therefore it is indifferent to the magnitude of the damage beyond the agreed threshold p. Hence,
it is advantageous to use a measure which can cope with dependencies andalso takes into account the tail of the distribution (Goovaerts et al., 2001).
One such measure which takes into account the tail of the distribution is the
Expected Shortfall (also called as the Tail Value-at-Risk) which is discussed
in section 2.3.2. Furthermore, as VaR is a quantile based risk measure, theuser must be aware of the existent discontinuities and intervals in the distribution function as this can have a significant effect on the output.
According to Acerbi and Tasche (2002), VaR is in fact not suitable to be labeled as a risk measure as it does not satisfy the property of sub-additivity.
They place a more stringent emphasis on the need for a measure to becoherent and even though there may exist alternative means of defining
coherence, they stress on the importance of sub-additivity as a mandatory
requirement. They argue that it is easy to show that different portfolios
with varying levels of risk can have the same VaR. One such example canbe found in Danielsson et al. (2005). To summarize the flaws of VaR, westate the two reasons from Artzner et al. (1999) to reject VaR as a measureof risk :(a) value at risk does not behave nicely with respect to the addition of risks,even independent ones, thereby creating severe aggregation problems.
(b) the use of value at risk does not encourage and, indeed, sometimes prohibits diversification because value at risk does not take into accountthe economic consequences of the events, the probabilities of which it
controls.

152.3 Some Important Risk Measures

2 RISK MEASURES

Despite its limitations and weaknesses, VaR is the most commonly used risk
metric today by banks and regulators and its importance in Basel II and Solvency
II is likely to remain unchanged. This may be because it is a convenient measure
to estimate and implement and also according to Danielsson et al. (2005), subadditivity violations are a serious concern only if the tail of the distribution issuper fat (ie., 1st moment does not exist). We note here for the benefit of thereader that the tail index measures the thickness of the tail and for most asset
classes, the tail index is observed to be between 3 and 5 (Danielsson et al., 2005),
which is not considered as super fat. Another important practical advantage of
VaR is that back-testing VaR estimates are easier than other types of estimates.

2.3.2

Expected Shortfall

The definition given by Denuit et al. (2005) is as follows :1
ESp (X) =1−p

Z 1
VaRp (X)dq,

(14)

p

for a given risk X and a probability level p ∈ (0, 1).
Corresponding changes in the formula can be made if we consider a discrete loss
distribution rather than a continuous one.
The interpretation given by Denuit et al. (2005) for the expected shortfall (ESfor short) is that it is the “average loss in the worst 100(1 − p)% cases” or simply
the average of the losses exceeding the VaR at the confidence level p.
An alternative representation of ES when FX (x) is continuous is as follows:
ESp (X) = E(X|X > VaRp (X))(15)

ES is also referred to as the tail conditional expectation, worst conditional expectation, conditional VaR and so on in other literature papers. These inconsistencies arise as different authors use different terminology for the same definition orotherwise the definitions vary slightly. However, all the mathematical formulasof the various alternatives lead to same results if a continuous loss distribution is16

2.3 Some Important Risk Measures

2 RISK MEASURES

considered, while differences emerge when we either use a discrete loss distribution or a distribution that has discontinuities (Acerbi and Tasche (2002); Dowd
and Blake (2006); McNeil et al. (2015)).
ES at p can also be understood as an adjusted value of VaR at a confidence level
greater than p. From the above, it is clear that ES is at least equal to or greaterthan VaR (Rockafellar et al., 2000). Hence, ES is better suited to measure thetail as compared to VaR and unlike VaR, it is always sub-additive. We note herethat ES is a coherent risk measure as it satisfies all the four required properties
needed for it to be classified as a coherent risk measure whereas, VaR is notcoherent in general as it fails to fulfill sub-additivity in some cases.
A table is presented below which shows the properties satisfied by the two risk
measures - VaR and ES. We use check marks to denote if a particular property
is satisfied, whereas a cross denotes that the particular property is not satisfied
by the corresponding risk measure at all times.

Property

Value-at-Risk

Expected Shortfall

No-Ripoff

3

3

Non-Negative Loading

7

3

Translation Invariance33Constancy

33Sub-additivity

7

3

Co-monotonic Additivity

3

3

Positive Homogeneity33

Monotonicity

33

Objectivity

3

3

Convexity

7

3

17

2.4 Distortion Risk Measures

2.4

2 RISK MEASURESDistortion Risk Measures

We have earlier seen the motivation behind the dual theory of choice under risk.
In Yaari (1987), he mentions that his proposed theory requires independencewith respect to direct mixing of payments of risky prospects and not the independence with respect to probability mixtures of risky prospects.
Distortion risk measures (DRM for short) are widely applied for calculating insurance premiums, risk management and in importance sampling for the Monte
Carlo simulations (McLeish and Reesor, 2003). ES for example is a coherent
DRM while VaR is a DRM but not coherent.

2.4.1

Definition and Interpretation

Dowd and Blake (2006) define distortion risk measure as the expected loss undera transformation of the cumulative distribution function.
Z 0
Z ∞ρg (X) = −(1 − g(F X (x)))dx +g(F X (x))dx
−∞

(16)

0

When X ≥ 0, we get
Z ∞g(F X (x))dx,

ρg (X) =(17)0

where, F X (x) = P (X > x) and g is a distribution function such thatg : [0,1] → [0,1] satisfying g(0) = 0 and g(1) = 1. g must be increasing andright-continuous (McLeish and Reesor, 2003).
FX (x) is a tail function and for g as defined above, g(F X (x)) is also a tail function.
ρg is thus the expectation with respect to the distorted tail function g(F X ) andin particular if g(s) = s, ρg (Y ) = E(Y ).
g is referred to as the distortion function as it distorts F X (x) before the generalized expected value is calculated (Denuit et al., 2005).
The above function ρ possesses the following properties: no-ripoff, positive homogeneity, translation invariance, monotonicity, co-monotonic additivity.

18

2.4 Distortion Risk Measures

2 RISK MEASURES

ρ is coherent if and only if g is concave (refer to the appendix for the proof).
If g is differentiable,Z ∞g(F X (x))dx

ρg (X) =0= [xg(F X (x)]∞0 −Z ∞xg0(F X (x))
0dF X (x)dxdx

Z ∞xg0(F X (x))dFX (x)=0= E[Xg0(F X (X))]= E[FX−1 (U )g0(1 − U )]where, U ∼ U (0, 1).

We notice that g0(F X (x)) is a re-weighting of the loss distribution and for concave g, higher weights are assigned to larger values of X (Tsanakas and Desli,2003). Therefore, the distorted risk measure is in fact a weighted average ofpercentiles.
We can understand the inherent nature of the agent in terms of the risk he iswilling to accept by observing the behavior of the distortion and this is done bylooking at the difference between the original and the distorted preferences. A
risk averse agent would tend to be more cautious while estimating the probabilities to take as little risk as possible. In such a case, it is natural and prudent for
the agent to overestimate the extreme events and therefore the tail probabilities.
According to Denuit et al. (2005), for a risk averse agent the following would
always hold:g(q) ≥ q =⇒ g(F X (x)) ≥ F X (x) , x  R.
From the above, we can say that ρg (X) ≥ E(X).
For a risk neutral agent, the distortion and the original preferences would matchat each point and therefore, g(q) = q =⇒ ρg (X) = E(X).

19

2.4 Distortion Risk Measures

2.4.2

2 RISK MEASURES

Examples

1. We now demonstrate that ES is a DRM:
ES is obtained as a DRM using the distortion function q−α , if q ≥ α
1−αg(q) =
0,
if q < α
ES is coherent since g(q) is concave.
The distortion of ES is obtained by adjusting one parameter and therefore,
they are also referred to as the “one parameter distortion functions.” ES
is also used in Basel III for determining minimum capital requirements toensure capital adequacy (Schumacher, 2018).
2. Normal Transform Risk Measure:
A very important concept is the Normal Transform risk measure introducedby Wang (2000), which will be used in later sections of this paper as well.
The distortion function is given by
g(q) = Φ(Φ−1 (q) + λ)(18)

Φ represents the cumulative distribution function of a standard normal
distribution and q ∈ [0, 1]. We refer to the risk measures derived from thegiven distortion function as normal transform risk measures (Denuit et al.,2005).
As noted in section 2.3.1, VaR is just a point on the distribution whereas,
ES considers the tail of the distribution beyond VaR. The normal transformon the other hand from equation (18), takes into consideration the entire
distribution of the porftolio and in that sense, it may be regarded as asuperior measure to ES and VaR (Dowd and Blake, 2006).
3. VaR can be obtained using the piecewise function :
1 if q ≥ αg(q) =

0 if q < α
VaR is not coherent as g(q) is not concave.

20

3 SENSITIVITY ANALYSIS

3

Sensitivity Analysis

3.1

Literature Review

3.1.1

Why Sensitivity Analysis is used?

Saltelli (2002) defines sensitivity analysis as the study of how the uncertainty inthe output of a model (numerical or otherwise) can be apportioned to different
sources of uncertainty in its inputs. An insurance company needs to take a rigorous approach before amending its existing business strategy, as any change has
to be validated to confirm that it enhances the value of the business. Sensitivity
analysis is an important tool for substantiating such changes. Another reason for
the extensive use of sensitivity analysis by the insurance companies is because
an analyst can not foresee the variations in the outputs by mere intuition as themodels used are complicated (Borgonovo and Plischke, 2016).
Sensitivity analysis is a flexible approach where each input in the model canbe altered and by performing simulations, the corresponding output can be obtained. Therefore, the user is able to detect the response of the model in termsof magnitude of the impact and the direction of the change (Borgonovo and Plischke, 2016). Consequently, comparisons of the output obtained before and after
the change in the value of an input can be used as a litmus test before taking adecision (Pritchard et al., 2014). Altering other parameter values one at a timein a similar way would result in many outputs and the procedure can be repeatedfor each input in the model. Referred to as the One-at-a-time (OAT) method, itis the simplest method that can be applied on a model. Various analyses can be
performed on the different outputs obtained to understand the vulnerability ofthe model and conclusions can be made by ranking the degree of the influenceof each input (Tsanakas and Millossovich, 2016).
Sensitivity analysis is thus used when the values of the input parameters are
uncertain and using VaR and ES when carrying out sensitivity analysis is particularly helpful for decision making (Tsanakas and Millossovich, 2016). Though213.1 Literature Review

3 SENSITIVITY ANALYSIS

sensitivity analysis can be applied for both quantitative and qualitative assessments (Pritchard et al., 2014), in this paper we will focus on the quantitative
assessments.

3.1.2

Local and Global Measures

Sensitivity analysis methods can be broadly classified into local and global methods (Borgonovo and Plischke, 2016). Local methods are put into practice when
probability distributions are not specified to the inputs (deterministic framework)
and when the user is required to make analysis around “a point of interest in themodel input space” (Borgonovo and Plischke, 2016). The local includes methods
such as Tornado diagrams, One-way sensitivity functions, Differentiation-based
methods and Scenario decomposition through finite change sensitivity indiceswhile methods such as sequential bifurcation, Morris method, Variance-based,
Moment-Independent are examples of global measures.
A global measure “reflects the model behavior over the whole of the input range”(Tsanakas and Millossovich, 2016). ” Both local and global measures are usefuldepending on the information available to the analyst. Variance-Based methods
are used for apportioning the uncertainty of an output to different inputs and theinteractions between them (Marzban, 2013). Here, the output variance is split
up into two parts as the ‘explained variance’ resulting from the dependence on
the inputs and the ‘residual variance’ (Borgonovo and Plischke, 2016). MomentIndependent methods can be used where the model inputs are allowed to varythus making it possible to obtain the unconditional output densities. By using
this method, the sensitivities can be assessed by considering entire distribution(Borgonovo and Plischke, 2016). For further discussion about the different typesof sensitivity methods, refer to Borgonovo and Plischke (2016).

22

3.1 Literature Review

3.1.3

3 SENSITIVITY ANALYSIS

Reverse Sensitivity TestingIn addition to the existing local and global methods, Pesenti et al. (2018) haveintroduced a new sensitivity analysis method, referred to as the Reverse Sensitivity Testing, in an attempt to determine the most influential factor which drives
the model, as it could be the basis under which a model can fall apart. It is a
global measure as it is used when probability distributions of inputs are known.
The application of reverse sensitivity testing can bring about a few advantages.
Firstly, it is computationally less arduous, as Monte-Carlo simulations can be
performed on a single set of input/output scenarios. Secondly, factor prioritisation, which is the process of identifying and ranking the most influential factors,
is based on the changes in the output distribution rather than considering an
output state. Furthermore, an input is proclaimed to be highly sensitive fromthe changes observed in its distribution. Therefore, the one with the most noticeable change is termed to be the most sensitive. The user should be able todecide whether the highly sensitive input(s) can ‘break the model’.
We state the three steps required in order to perform the analysis presented by
Pesenti et al. (2018).
1. An output stress is specified, corresponding to an increase in the risk measure(s) of the output.
The objective of deriving the stresses is to intensify the problem faced by
the agent and therefore obtaining larger losses. The risk measure VaR wasused to obtain the stress and by doing so there may arise a situation where
there are inadequate resources to meet the regulatory requirements. This
would allow to test the model under a more severe set of conditions.
2. A stressed probability measure is derived by minimizing the KullbackLeibler divergence with respect to the baseline probability, under constraints generated by the stress on the output risk measure.

23

3.2 A New Sensitivity Analysis Method

3 SENSITIVITY ANALYSIS

Due to the manner in which the stresses are derived, they have obtained
a constant weight which is applied to claims below a certain value and alarger weight which is applied to claims above a certain value to derive thedistortion probability distributions.
3. changes in the distributions of input factors are evaluated under the stressed
conditions to assess the sensitivities. An input is said to be highly sensitive if there is a considerable change in its distribution under the stressed
conditions.

3.2

A New Sensitivity Analysis Method

3.2.1

Motivation

A modified approach of the reverse sensitivity testing is presented in this paper.
Pesenti et al. (2018) focused on distorting the output in order to observe thechanges in the inputs. However, in this paper, importance is given to the input
to output relationship. In other words, we distort the inputs to observe changes
in the output. As a consequence of the methodology, distorting one input would
lead to the distortion of other correlated inputs of the model.

For the steps mentioned in section 3.1.3, we use different techniques for ourmodified version.
In the first step, instead of minimizing the Kullback-Leibler divergence, we use
the normal transform risk measure to obtain the stresses on the inputs. Pesenti
et al. (2018) derived a Radon-Nikodym density corresponding to weights on particular states of the world. The weights obtained are piecewise constant andtherefore, a low value is used to stress the low values of Y and a higher emphasis is applied to higher values of Y . However, by using a normal transform, we
obtain a continuously increasing weights to distort the output Y . This perhaps
is more suitable as it is effective to obtain greater distortions for Y .

24

3.2 A New Sensitivity Analysis Method

3 SENSITIVITY ANALYSIS

For the second step, we use the risk measures VaR and ES as a means of comparison between the baseline and the stressed models. We calculate the percentage
increase in VaR and ES for the stressed models with respect to the baseline toanalyze the sensitivity levels of the input factors.
Another possibility is to alternatively use mean and standard deviation to get aquick glance of the model behavior.

3.2.2

Setup and Notation

Let X1 , X2 ,..., Xn denote n random variables representing the input factors ofthe model under consideration and we indicate these n input factors by a vector
X = (X1 , X2 , . . . Xn ). Let ψ be the aggregation function that maps the inputs
to an output. We write, ψ : Rn → R. We apply the function ψ on the inputs toobtain the one-dimensional random output Y = ψ(X).
Let Xi follow a distribution function denoted by Fi , Xi ∼ Fi , and let Y follow a
distribution denoted by H, Y ∼ H.
Let P denote the set of all the probability measures on a measurable space(Ω, A). We denote the baseline model as (X, ψ, P ) with baseline probability
measure P ∈ P.
Define the random variable Wi by Wi = g 0 (1 − Ui ), where Ui ∼ U nif orm(0, 1)
such that Wi is co-monotonic to Xi . g is the normal transform given by
g(q) = Φ(Φ−1 (q) + λ).
i
Define a new measure Qi by dQ
dP = Wi . This is the Radon-Nikodym density that

operates as weights on different scenarios. Thus, we can write the expectationof a random variable, Z, under the new probability measure Qi as follows:
E Qi (Z) = E(Wi Z)
The main objective of this paper is to observe the distortions caused to the distribution Y using the stresses obtained from the normal transform risk measure.
Therefore, in the plots and tables that we produce in section 4.1.2, we aim to25

3.2 A New Sensitivity Analysis Method

3 SENSITIVITY ANALYSIS

compare the distribution function and risk measures (VaR and ES) of the output
Y under the baseline model (P ) with the distribution function and risk measures
under the new probability measure Qi . Hence, a significant change in the output
distribution under the new measure Qi would indicate that the output is particularly sensitive to the input Xi .
We derive g 0 (q) as follows:dΦ(Φ−1 (q) + λ)dqd
= Φ0 (Φ−1 (q) + λ) (Φ−1 (q) + λ)dq

g 0 (q) == Φ0 (Φ−1 (q) + λ)d −1
(Φ (q))dq= Φ0 (Φ−1 (q) + λ)1
Φ0 (Φ−1 (q))

Evaluating it at (1 − Ui ), we arrive at the following which is used in our example
to generate the weights for the distortion.
Wi = gp0 (1 − Ui ) = Φ0 (Φ−1 (1 − Ui ) + λ)1
; i = 1, 2, ..., nΦ0 (Φ−1 (1 − Ui ))(19)

We denote Y ∼ H Qi under the new measure Qi . Any distribution function canbe written as an expectation using the indicator function, we can write the new
distorted distributions obtained under Qi as follows:
H Qi (t) = Qi (Y ≤ t) = E Qi (1{Y ≤t} ) = E(1{Y ≤t} Wi )Qi at p, that is, the value such thati
It holds that: VaRQ
p (Y ) is the inverse of Hi
H Qi (VaRQ
p (Y )) = p.

Thus, a change in the measure is a simple method to put different weights fordifferent outcomes.

263.2 A New Sensitivity Analysis Method

3 SENSITIVITY ANALYSISFor Expected Shortfall we have:
Qi
Qi
i
ESQ
p (Y ) = E (Y |Y > VaRp (Y ))

=



1
Qi
E
Y
·
1
Qi
{Y >VaRp (Y )}i
Qi (Y > VaRQ
p (Y ))=


1
E Wi · Y · 1{Y >VaRQi (Y )}
p
1−p

(20)

Percentage increase in VaR and ES for each input is calculated for each distortion case with respect to the baseline model. This value is assigned to be thesensitivity of the input.

3.2.3

Implementation in a Monte-Carlo Setting

As mentioned above, H is the distribution of Y under P and H Qi is the distrii
bution of Y under measure Qi , where Wi = dQ
dP . Let us for simplicity assume

that both distributions H and H Qi are continuous. Assume that we have an
iid simulated sample from Y obtained under P (in other words, drawn from the
distribution H) and denote that sample by y (1) , . . . , y (m) , where m is the numberof simulations performed. Furthermore, we have corresponding samples from the
(1)

(m)

Radon-Nikodym density Wi ; these are denoted by wi , . . . , wi

. We are inter-Qiiested in calculating VaRQ
p (Y ) and ESp (Y ) from that simulated sample.
i
We can estimate H Qi and VaRQ
p (Y )) by1d
Qi (t) =
H
m

m
X

(k)

wi 1{y(k) ≤t}

k=1

\
Qi
Qi
VaR
p (Y ) = inf{t ∈ R : H (t) ≥ p}\
Qi
The above expression for VaR
p (Y ) reflects the fact that the empirical distribud
Qi (t) is actually discrete. Now, turning attention to the Expected Shorttion H
fall, we can estimate equation (20) by
m

[
Qi
ES
p (Y ) =X (k)1wi y (k) 1\Qm(1 − p){y (k) >V aRp i (Y )}k=127

4 EXAMPLES

4Examples4.1

Benchmark Model

4.1.1

Model Specification

Consider an insurance portfolio consisting of the input factors X1 , X2 , X3 andX4 with the same specifications as in Pesenti et al. (2018). X1 represents a
set of claims that follow a Log-normal distribution with mean 150 and standard
deviation 35 and X2 represents the set of claims that follow a Gamma distribution
with mean 200 and standard deviation 20. We do not in this paper discuss thevalidity of making such assumptions.
Define L = X1 X3 + X2 X3 , where X3 is the inflation factor and it follows a
Log-normal distribution with mean 1.05 and standard deviation 0.02. We makea further assumption that X1 , X2 and X3 are independent random variables.
The model takes into consideration the reinsurance. Let X4 model the proportionof the amount that is lost by the insurer due to the failure of payment by the reinsurer. Assume that X4 follows a Beta distribution with mean 0.1 and standard
deviation 0.2. We assume that X4 is related to L through a Gaussian copula andhas a correlation of 0.6.
We can now represent the total liability (Y ) as
Y = L − (1 − X4 ) min{(L − d)+ , l}(21)

where, d is the deductible and l is the limit. Assign d = 380 and l = 30.

4.1.2

Analysis and Results

As discussed in section 3.2.1, the weights derived from the normal transform risk
measure are used to obtain the stressed models. We assume three different valuesfor λ thus, generating three sets of weights or three sets of distorted probability
measures for each random variable separately in the model considered.
We use the values: λ = 0.5, 1.0 and 1.5.

28

4.1 Benchmark Model

4 EXAMPLES

In figure 1, we plot samples of the random variables W1 ,. . . ,W4 against X1 ,. . . ,X4
respectively.

Figure 1: Samples of W1 , W2 , W3 , W4 against X1 , X2 , X3 , X4 respectively,for λ = 0.5, 1, 1.5.

From figure 1(a), we note that for claims below 200 units the weights, W1 , do
not vary significantly for different λ values. However, for claims greater than 200
units, heavier weights are assigned as λ increases. Similar observations can bemade from figure 1(b), 1(c) and 1(d). Hence, it is reasonable to say that a higher
λ focuses more on the higher valued claims rather than lower valued claims. By
increasing the λ, we are intensifying the severity of the conditions assumed andhence, we can presume that a highly risk averse agent would prefer assigning ahigher λ value to carry out the sensitivity analysis.

294.1 Benchmark Model

4 EXAMPLES(a) Distortions of Y obtained using W1(b) Distortions of Y obtained using W2(c) Distortions of Y obtained using W3(d) Distortions of Y obtained using W4

Figure 2: Distorted probability distributions of Y obtained using W1 , W2 , W3 , W4 for
λ = 0.5, 1.0, 1.5

Figure 2 shows H Qi (Y ) using Wi , i =1,2,3,4 for the λ values alongside thebaseline probability distribution, H. As the distortions for each λ value in figure2(a) and 2(d) are more pronounced than in figures 2(b) and 2(c), X1 and X4
are the most sensitive inputs for Y . This is validated from the values obtainedin Tables 1-4 as the tables show the percentage increase in VaR and ES withrespect to the VaR calculated for the benchmark model for Y at different confidence intervals.

30

4.1 Benchmark Model

4 EXAMPLES

The r.v.s X1 and X2 are the claims modeled by a Log-normal and a Gamma
distribution respectively. As Log-normal is heavier tailed than a Gamma distribution, the percentage increase in VaR and ES of Y obtained for different valuesof λ in Tables 1 and 2 vary substantially. For instance in Table 1, the percentage
increase in VaR at p = 0.99 for λ = 1.5 is 24.08 whereas, the corresponding valuein Table 2 is only 8.38. Further, the percentage increase in VaR and ES at eachconfidence level in Table 2 is lower than the corresponding value in Table 1. This
is attributable to the fact that a higher λ impacts the Log-normal to a greater
extent than the Gamma distribution. Hence, we say that X1 is more sensitive
than X2 . Comparing the tables in a similar manner results in the ordering of
inputs’ sensitivities as X1 , X4 , X2 and X3 .
Table 1: Percentage increase in VaRp and ESp for H Q1 with respect to the baselinemodel, H.

% increase in VaRp
p =0.75 p=0.85 p= 0.95

% increase in ESp
p=0.99p =0.75 p=0.85 p= 0.95

p=0.99λ = 0.5

3.06

6.35

6.94

7.51

6.15

6.92

7.11

7.68

λ = 1.0

10.11

13.75

14.46

15.48

13.72

14.63

15.31

16.59

λ = 1.5

17.92

21.74

23.56

24.08

22.06

23.23

23.92

25.05

Table 2: Percentage increase in VaRp and ESp for H Q2 with respect to the baselinemodel, H.

% increase in VaRp
p =0.75 p=0.85 p= 0.95

% increase in ESp
p=0.99p =0.75 p=0.85 p= 0.95

p=0.99λ = 0.5

0.90

2.87

3.10

2.73

2.63

3.04

2.88

2.53

λ = 1.0

3.28

6.34

6.33

5.43

5.78

6.25

5.84

5.03

λ = 1.5

6.86

9.71

9.64

8.38

9.19

9.56

8.78

7.23

31

4.1 Benchmark Model

4 EXAMPLES

Table 3: Percentage increase in VaRp and ESp for H Q3 with respect to the baselinemodel, H.

% increase in VaRp
p =0.75 p=0.85 p= 0.95

% increase in ESp
p=0.99p =0.75 p=0.85 p= 0.95

p=0.99λ = 0.5

0.15

0.88

1.07

1.04

0.86

1.06

1.11

0.93

λ = 1.0

0.42

1.85

2.07

2.01

1.74

2.08

2.08

1.77

λ = 1.5

0.88

2.68

2.97

2.93

2.57

2.99

2.93

2.28

Table 4: Percentage increase in VaRp and ESp for H Q4 with respect to the baselinemodel, H.

% increase in VaRp
p =0.75 p=0.85 p= 0.95

% increase in ESp
p=0.99p =0.75 p=0.85 p= 0.95

p=0.99λ = 0.5

2.19

4.94

5.58

5.43

4.77

5.44

5.42

5.46

λ = 1.0

7.23

10.85

11.10

10.75

10.46

11.16

11.10

12.57

λ = 1.5

13.13

16.66

16.43

18.02

16.35

17.03

17.53

23.43

While our emphasis is on demonstrating how a stress in Xi affects the distribution of Y , it is also the case that a stress on Xi modifies the distributions ofother inputs, if these are correlated with Xi . We exhibit these distorted probability distributions of the input factors obtained when H is stressed by W1 in
figure 3. The probability distributions for X2 and X3 do not change with λ as
X1 , X2 and X3 are assumed to be independent in section 4.1.1. We do however
see distortions in X1 and X4 , as a change in X1 has a cascading effect on X4 .
Therefore, the distortions of the output (Y ) shown in figure 2 occur due to thechanges observed in X1 and X4 in figure 3.

32

4.1 Benchmark Model

4 EXAMPLES

(a)

(b)

(c)

(d)

Figure 3: Distorted probability distributions for X1 , X2 , X3 , X4 obtained using W1 for
λ = 0.5, 1.0, 1.5

Similar distortions can be derived when weights W2 and W3 are used. Since,
the inferences drawn are analogous to the inferences made when weights W1 wereused, we do not provide any more details on this.
In Figure 4, we show the stressed distributions of the input factors obtained
when weights W4 are used. As X4 is related to L, it affects all the probability
distributions of all the other inputs. The cascading effect on the inputs due tothe stress in X4 is seen clearly here.

33

4.2 Variations of the example

4 EXAMPLES

(a)

(b)

(c)

(d)

Figure 4: Distorted probability distributions for X1 , X2 , X3 , X4 obtained using W4 for
λ = 0.5, 1.0, 1.5

4.2

Variations of the example

In this section, we vary one of the factors of the model and observe the changes
in the model behavior. We are particularly interested in inspecting the changes
in the order of the sensitivities of the inputs. We look at three specific variations:
1. When correlation between X1 , X2 , X3 and X4 is varied between 0 and 1.
2. When d is varied between 300 and 600.
3. (a) When claims modeled by X2 follow an Inverse Gamma distribution.
(b) We also vary the standard deviation of X2 from 10 to 100 units.
Using Monte-Carlo simulation, we measure the percentage increase in the ES
when the distribution of Y is distorted using λ = 1, with respect to the ES34

4.2 Variations of the example

4 EXAMPLES

calculated for the benchmark model. The percentage increase in ES is taken to
be the sensitivity of the input. The sensitivities of the inputs are ordered for
p = 0.95.

Variation 1
We vary the correlation between the inputs from the smallest plausible value (0)to the highest possible value (1). When the correlation is 0, we notice that X1 has
the highest sensitivity followed by X2 . X3 and X4 exhibit very little sensitivities.
As the correlation increases, X1 , X2 and X3 show almost no variation in their
sensitivity levels. X4 is the only input whose sensitivity changes as the correlation
changes. The sensitivity of X4 increases as the correlation between the inputs
increases and its sensitivity level is approximately that of X1 for high values ofcorrelation.
Figure 5 shows the sensitivity level of the inputs when the correlation between
X1 , X2 , X3 and X4 increases from 0 to 1. From figure 5, we observe that X4
starts to become more prevalent than X2 when the correlation is greater than
0.2 approximately.

Figure 5: Sensitivity levels of X1 , X2 , X3 , X4 when the correlation between inputs
change from 0 to 1 for p = 0.95.

35

4.2 Variations of the example

4 EXAMPLES

Variation 2
We vary the value of d from 300 to 600 units to examine the behavior of thesensitivities of the inputs. Figure 6 implies that at all times, X1 exhibits thehighest sensitivity level followed by X4 , X2 and X3 . The sensitivities show
almost no variation as d increases from 300 to 425 units. The sensitivity levelsof X1 and X4 drop a few points upto about 475 units after which they seem toremain roughly constant. The sensitivities of X2 and X3 on the other hand do
not seem to change much.
Hence, a change in d does not impact the order of the sensitivities of the inputs.

Figure 6: Sensitivity levels of X1 , X2 , X3 , X4 when d changes from 300 to 600 units
for p = 0.95.

Variation 3
In this variation, we have two parts. For the first part, the claims represented by
X2 is modeled using an Inverse Gamma distribution with mean 200 and standard
deviation 20. As expected, X1 and X4 exhibit a higher sensitivity than X2 andX3 . We obtain the same order for the sensitivities of the inputs as the benchmark
model.

36

4.2 Variations of the example

4 EXAMPLES

It is in our interest to examine the behavioral change of the sensitivities when
the standard deviation is now varied from 10 to 100 units and this is shown infigure 7. At the lowest standard deviation considered (10 units), the order of thesensitivities are as follows: X1 , X4 , X2 and X3 . As the standard deviation of
X2 increases, the sensitivity of X1 decreases whereas, X2 and X4 increase. The
sensitivity level of X3 remains roughly at the same level.
There is a steep increase in the change of the sensitivity level of X2 as thestandard deviation increases and this is expected. X2 has more than a 60%
sensitivity level and X4 has a 40% approximately when standard deviation is
100 for part 3(b), due to the increased volatility of the claim size.

Figure 7: Sensitivity levels of X1 , X2 , X3 , X4 when the standard deviation of X2changes from 10 to 100 units for p = 0.95.

37

4.3 Comments

4.3

4 EXAMPLES

Comments

The benchmark model indicates that the output has a higher sensitivity to X1
followed by X4 , X2 and X3 . We obtain consistent results with Pesenti et al.
(2018) as they also report a higher sensitivity to X1 and X4 for the same model.
Therefore, it is reasonable to state that X1 and X4 are the most influential factors of the model.

A few aspects in the way the sensitivities were specified to the model inputs
were noted. The first being that the percentage increase in VaR and ES produced in Tables 1-4 contained inconsistent trends as the value of p was altered.
For instance in Table 1, we see that the sensitivity levels for each distortion
increases as p increases. However, in Tables 2-4, the sensitivity levels for eachdistortion increases till p = 0.95 and then decreases at p = 0.99.

Further, VaR and ES are used to rank the inputs based on comparing the sensitivity levels at each p value for the distortions. If an input A has higher percentage
increases for all the corresponding values than input B, then input A exhibits agreater sensitivity than input B. We might have instances where all the percentage increases in VaR and ES for one input are not higher or lower than another
input. So the ranking may not be very obvious and hence, depending on the
purpose for which the sensitivity analysis is used, different values of p may be
more important.

38

5 CONCLUSION

5Conclusion

The proposed framework was effective for understanding the relationships between inputs and the output. We had used distortion risk measures to stressthe probability distributions and also employed VaR and ES for examining the
behavioral changes in the model. The methodology adopted in this paper leads
to consistent results with other sensitivity methods such as those obtained by
Pesenti et al. (2018).
In addition to this, we were able to examine the cascading effects on other inputs
while one input was altered. As a result, we were able to point out those aspectsof the model which needed the most attention. In such a case, it is possible
to spot unexpected relationships between inputs and outputs from running themodel. Other factors such as GDP, interest rates, returns from assets, claims
from adjacent underwriting years can be included in the model for a more realistic application. Nevertheless, the framework would remain unaltered even if
other inputs are being integrated into the model.

39

6 APPENDIX

6

Appendix

According to Denuit et al. (2005), ρg (X) is coherent if, and only if, g is concave.
We provide a simpler proof which is more intuitive.
Let us suppose that ρg (X) = E(g0(1−U )FX−1 (U )), U is any uniform distribution.
This naturally satisfies positive homogeneity, translation invariance, monotonicity. For ρg (X) to be coherent, it has to satisfy the property of sub-additivity.
We know that concavity and sub-additivity are concomitant.
Let us choose the uniform to be UX such that it is co-monotonic to X.
Hence,
since FX−1 (UX ) = X

ρg (X) = E(g0(1 − UX )X)
Now, letρg (X + Y ) = E[g0(1 − UX+Y )(X + Y )]= E[g0(1 − UX+Y )X] + E[g0(1 − UX+Y )Y ]≤ E[g0(1 − UX )X] + E[g0(1 − UY )Y ]= ρg (X) + ρg (Y )The inequality follows from the result that E[A.B] ≤ E[A∗ .B ∗ ] if A∗ and B ∗ areco-monotonic and have the same marginal distributions as A and B, for any two
random variables A and B (Kaas et al., 2002).
Hence, as X and UX are co-monotonic, E[g0(1 − UX )X] ≥ E[g0(1 − UX+Y )X].
Similarly as Y and UY are co-monotonic, E[g0(1 − UY )Y ] ≥ E[g0(1 − UX+Y )Y ].
Further, g0(1 − UX ) can only be increasing in UX if g is concave.
Hence, ρg is sub-additive and therefore coherent.

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