Extreme Risk Modeling for Economic Capital by jyi37927


									Extreme Risk Modeling for Economic Capital


                     YU Ziyou

                Associate Professor
         Department of Finance and Insurance
                Lingnan University
                 Hong Kong, China

                     WU Jianjun

                 Research Assistant
                  School of Finance
     Shanghai University of Finance and Economics
                  Shanghai, China
       Email Address: wayne2010@gmail.com


                   Dr. YU Ziyou
         Department of Finance and Insurance
                Lingnan University
               Tuen Mun, Hong Kong

                Tel: (852) 26168303
                Fax: (852) 24670982
             E-mail: lngpzzyu@ln.edu.hk

                     May 2007

               Extreme Risk Modeling for Economic Capital


Economic Capital is generally accepted as the amount of capital that is required to
absorb unexpected losses during a time horizon at a portfolio manager’s level of
confidence. This paper illustrates the limitations of the VaR-based EC .Then explains
how extreme value theory can applied to tail modeling and explorers the connection
between EVT and EC. Based on the EVT and the concept of spectral risk measure, this
paper further defines and employs EVT approach to calculate the risk-neutral EC as a
reference frame to adjust the VaR-based EC. Finally, three-layer framework of EC is
proposed which sheds light on the sequence of EC allocating.

KEY WORDS: Economic Capital, Extreme Value Theory, Expected Shortfall,
Spectral Risk

     Economic Capital (EC hereafter), generally accepted as the amount of capital that
is required to absorb unexpected losses(UL hereafter) during a time horizon at a
portfolio manager’s level of confidence, is becoming a common currency for all such
types of risks into a single enterprise-wide measure of aggregated risk. Practitioners
commonly employ Value-at-Risk (VaR) as the standard risk measure for calculations in
most aspects of EC. However, VaR does not satisfy the criteria of coherent risk measure
(Atzner, 1997, 1999). Specifically, it may not have the sub-additive property which
requires that the combined VaR of two sub-portfolios should not be greater than the sum
of their respective VaR treated separately. Moreover, the amount of EC calculated by
VaR merely tells us that a firm will exhaust the available capital when the upper bound
of loss is reached rather than how much the additional capital that can cushion loses that
exceeds it. In other words, the tail risk will disturb the EC impact on the unexpected
losses and will further affect the rate of return that the shareholders demand on the EC.

     Therefore, one such coherent risk measure called expected shortfall (ES hereafter)
or conditional expectations shed lights on the alternative measure for calculating the
economic capital. This measure, based on the extreme value theory (EVT hereafter),
concentrates on the tail behavior of loss distribution. The main limitation of the method
is in the selection process of the threshold. If a high value of threshold is selected, less
observation is left for the estimation of the parameters of the tail distribution function.
Thus, the simulation technique such as the Monte Carlo simulations can be used to rich
the tail region. However, on the other hand, the tail region of the loss distribution
represents the high severity low frequency events. Calculating EC on this area tends to
overestimate EC and will further affect the efficiency of EC allocation.

     One purpose of this paper is to define a risk measure for EC in a perspective of
“full range of risk”. To achieve this goal, we utilize the concept of spectral risk measure
and the original meaning of EC. Spectral risk measure is likewise a coherent risk
measure with a highlight on its consideration of a user’s risk aversion. This measure
employs a weighting function to estimate the weighted averages of the quantiles of a
certain loss distribution. Therefore the more risk aversion of the user, the more EC will
be allocated for the UL. On the other hand, the risk lover tends to allocate less EC for
the same UL. However, according to the original meaning of EC, over-allocating EC
will affect the return of the capital while under-estimating EC will suffer from

insolvency. A proper EC should keep a balance for the risk and return. In this
background, only a risk neutral user tends to allocate a suitable amount. Considering the
fact that ES is a tail-risk neutral risk measure, we extend the “tail” threshold to the point
of expected loss of the loss distribution, and ES for this loss region represents a risk
neutral measure in a perspective of “full range of risk”. Therefore, we can obtain the
risk-neutral EC, which is the embedded EC for the loss distribution.

     The rest of the paper is organized as follows: Section 2 presents the VaR-based
method to calculate EC and discuses its limitations via samples. Section 3 introduces
the EVT and conducts an empirical analysis to calculate EC based on ES model. Section
4 discuses the concept of spectral risk measure, by which we define the risk-neutral EC
and likewise employ EVT models to estimate the EC in the risk-neutral perspective.
Finally, section 5 offers the main conclusions of the paper.

                   VaR-based Economic Capital Calculation

     Economic capital acts as a buffer that provides protection against all the credit,
market, operational and business risks faced by an institution. General speaking, EC is
set at a confidence level that is less than 100% (e.g., 95 %), since it would be too costly
to operate at the 100% level. The confidence interval is chosen as a balance between
providing high returns on capital for shareholders and providing protection to the debt
holders (and achieving a desired rating) as well as confidence to other claim holders,
such as depositors. Sample 1 below contributes our understanding of the Economic

     Sample 1: Denote by At and Dt the market values (at time t) of the assets and
liabilities, respectively. The available capital Ct for the current time, t = 0 ,and at the end
of one year, t = 1, can be expressed as:

                                         C0 = A0 -D0,

                                         C1 = A1-D1.

     If the nominal returns on the assets and liabilities are equal to rA and rD,
respectively, then a worst-case loss from all sources, l (i.e., when C1 = 0 for a given
confidence interval), would result in the value of assets at t = 1 just being sufficient to
cover the value of debt in t = 1. Then

                          C1 = 0 = A0 (1 + rA)(1- L ) D0(1 + rD).

     Thus the maximum amount of debt allowable to sustain solvency under the
worst-case scenario cannot exceed

                             D0 = A0 (1 + rA) (1- L )/(1 + rD).

     Since EC0 is the minimum amount of capital required to sustain such a loss, it is
given by:

                         EC0 = A0 (1 - [(1 + rA) (1-L )/(1 + rD)]).

     Hence the minimum amount of EC a financial institution must take on in order to
avoid insolvency increases as the level of the worse-case loss l increases.

     For ease of presentation, consider the case where credit risk is the sole source of
business risk to which the firm is exposed. Therefore under the simplifying assumption
that the spread between the nominal return on the assets and the return on the liabilities
is roughly equal to the expected default loss, u, then,

                       EC0 = A0 {1-(1 + rD) (1 + u) (1-L )/(1 + rD)}

                                 = A0 {1-(1 + u) (1- L )}.

     By then ignoring second-order effects, equation (III.0.5) simplifies to the following
more familiar expression for economic capital:

                                     EC0≈ A0(L- u)

     This relationship shows EC is defined to absorb only UL up to a certain confidence
level (i.e., A0(L- u)). Figure 1 shows the definition. Here, A0 and u is predetermined for
a given loss distribution, and only L need to be estimated. In practical, such
quantile-based risk measure as VaR is used to calculate the worst case loss at a specific
confidence level. Therefore, the relative VaR is defined as the risk measure for EC
calculation. The rapid rise of VaR was due to large part to the VaR having certain
characteristics. For instance, the VaR provides a common measure of risk across
different positions and risk factors. VaR is probabilistic, and gives a risk manager useful
information on the probabilities associated with specified loss amounts. And VaR is
expressed in the simplest and most easily understood unit of measure, namely, ‘lost
money’. Many other measures are expressed in less transparent units.

                                                                                   Loss Distribution
                                            EC covers the UL within the
                       UL out of the         risk tolerance at the time
                     tolerance at the                  horizon
                       time horizon

                                             Expected Loss
         Risk tolerance

                    Loss                Ⅱ                                   Ⅰ         0

                                         Figure 1 Concept of EC

      However, the VaR also suffers from some serious limitations. One limitation is
that the VaR only tells us the most we can lose in good states where a tail event does not
occur; it tells us nothing about what we can lose in ‘bad’ states where a tail event does
occur (i.e., where we suffer a loss in excess of the VaR). VaR’s failure to consider tail
losses can then create some perverse outcomes. This is especially true when we employ
relative VaR as the risk measure for EC estimation. For instance, given two loan
portfolios with same amount, they may have different expected loss rate and different
worst case loss rate at the same risk tolerance. However, when calculating EC, the
results may close to each other. In other words, a relative VaR-based decision calculus
might suggest that two loan portfolios with different tail risk can obtain the same EC
allocation at the specific confidence level. This can be illustrated more clearly in the
following stimulations.

      In order to obtain the two loan portfolio with EC allocation close to each other at
the same risk tolerance. We adopt Monte Carlo simulations technique and follow the
steps below. First of all we employ the log normal distribution as the loan portfolio loss
distribution, because the credit risk generally presents a fat-tailed behavior. Second, we
define 21 groups of loan portfolio with the expected loss rate range from 0.02 to 0.04,
and standard deviation rate from 0.01 to 0.03 accordingly1. Third, we conduct 10000
trials for the simulation and estimate the 95% relative VaR (EC) for each loss
distribution. Finally, we compare the each of the two portfolios to find the closest gap of
EC, if the lowest gap is around 0.001%, we approximately consider the two portfolio
are of same EC at 95% confidence level. Otherwise, another 10000 trials will continue.

  First portfolio with the EL and SD rate (0.02, 0.01), second (0.021, 0.011), third (0.022, 0.012)… next to last (0.039,
0.029), and last (0.04, 0.03).

Figures 2 and Figure 3 below illustrate the stimulation results.

                                     Figure 2 Portfolio 1
               Log-normal distribution with mean rate= 0.038 and SD rate=0.028

                                     Figure 3 Portfolio 2
               Log-normal distribution with mean rate= 0.039 and SD rate=0.029

     As indicated in the above figures, EC rate for the two portfolios at 95% risk
tolerance are approximately 5.27%, gap equals 0.0012%. Given the same time horizon
and amount, the EC of the two portfolios at 95% confidence level are equivalent
regardless of the negligible gap. In view of the two figures, in the perspective of risk,
when the market move to bad situation (i.e move to 99% level), the portfolio with
higher tail risk (portfolio 1) may exhaust EC faster that that of the lower one (portfolio
2). In the perspective of investor, the same amount EC does not mean the same risk
embedded for the portfolio, the higher tail risk portfolio indicates higher risk premium
for the invertors. This phenomenon disturb the original meaning of EC that the same EC
should bear the same risk, which is different from the meaning of capital that the same
capital allocation may yield different risk.Therefore, two issues remains discussed. First
is how to calculate the EC in a bad situation. Second is how to adjust the EC in a normal
situation to reflect its embedded risk. Section 3 and Section 4 below offer the answers.

                    Tail-based Economic Capital Calculation

     More light was shed on the limits of VaR by some important theoretical work by
Artzner, Delbaen, Eber and Heath in the 1990s (Artzner (1997, 1999)). Their starting
point is that although we all have an intuitive sense of what financial risk entails, it is
difficult to give a good assessment of financial risk unless we specify what a measure of
financial risk actually means. To clarify these issues, Artzner proposed a set of
risk-measure axioms – the axioms of coherence. The risk measure ρ (.) is said to be
coherent if it satisfies the following properties:

     Monotonicity: V(Y) ≥ V(X ) ⇒ ρ (Y) ≤ ρ (X ) .
     Subadditivity: ρ (X + Y) ≤ ρ (X ) + ρ (Y) .
     Positive homogeneity: ρ (hX ) = hρ (X ) for h > 0.
     Translational invariance: ρ (X + n) = ρ (X ) − n for some certain amount n.
     It then follows that the VaR cannot be a qualified measure in this sense, because
VaR is not subadditive. In fact, VaR is only subadditive in the restrictive case where the
loss distribution is elliptically distributed, and this is of limited consolation because
most real-world loss distributions are not elliptical ones. One promising coherent
candidate is the Expected Shortfall (ES), which is the average of the worst 1−α losses.
In the case of a continuous loss distribution, the ES is given by:

                                   ESα =       ∫ q p dp
                                           1−α α

     This measure is defined in terms of a probability threshold. The other is its
quantile-delimited cousin, the average of losses exceeding VaR, i.e., E[X | X> qα(X)].
The two measures will always coincide when the loss distribution is continuous.
However, this latter measure can be ambiguous and incoherent when the loss
distribution is discrete (Acerbi ,2004), whereas the ES is always unique and coherent.
Following we will employ ES as the risk measure to calculate the tail-based EC.

     As we are particularly interested in the tail region of the portfolio in this section,
we resort a more recent innovation in financial data modeling that uses EVT, a branch
of statistics that deals with the extreme deviations from the mean of probability
distributions. Generally, two approaches, Block maxima and peak to threshold (POT),
are applied for EVT models. Among which, Generalized Pareto distribution (GPD)

based on POT is easier to implement as compared to other EVT models. It requires
obtaining simple parametric formulas by data-fitting techniques such as maximum
likelihood estimation (MLE). The GPD is given by:

                                ⎧ ⎛
                                                   −1 ξ

                                ⎪           x⎞
                  Gξ ,β ( x ) = ⎨1 − ⎜1 + ξ β ⎟ ξ ≠ 0
                                     ⎜        ⎟
                                     ⎝        ⎠                                        (2)
                                ⎪1 − exp(− x β ) ξ = 0
     Where ξ is tail index and accounts for the shape, while β represents the scaling
parameter for dispersion measurement. If ξ >0 then the GPD is characterized by fat
tails. This approach focuses on the realisations of a random variable X over a high
threshold u. if the exceedance (X-u) has the distribution function F (e), then the
distribution of excess losses over a threshold u is defined as:

                                                          Fu (e + u ) − Fu (u )
                Fu (e ) = P ( X − u ≤ e X > u ) =
                                                              1 − Fu (u )

     Where e = X − u stands for the excess losses and Fu (e ) converges to a

Generalized Pareto distribution. Furthermore, if we define T as the total number of
observations and N as the observations above threshold u, (2) can be transformed as:

                              Fu (e + u ) − Fu (u )      T (1 − u − e )
                  Fu (e ) =                         = 1−
                                  1 − Fu (u )
                           ˆ     ˆ
     Given the parameters ξ and β estimated by fitting the data using maximum
likelihood method, VaRq beyond the threshold u can be calculated by:

                         (                           )
                                                ⎛ β ⎞⎛ T (1 − q ) −ξ ⎞
                ⎛β ⎞                               ˆ
     VaRq = u + ⎜ ⎟ (1 − Fu (q − u )) − 1 = u + ⎜ ⎟⎜                  − 1⎟
                ⎜ξ ⎟
                  ˆ                             ⎜ ξ ⎟⎜ N
                                                   ˆ                     ⎟             (5)
                ⎝ ⎠                             ⎝ ⎠⎝                     ⎠
     Thus, the expected shortfall of EC is calculated as the mean excess distribution of
the unexpected loss beyond VaRq minus the Expected Loss and is derived as follows:

                           ⎛ VaRq − EL ⎞ ⎛ β − ξ (u − EL ) ⎞
                   EC Eq = ⎜
                           ⎜ 1− ξ ⎟ ⎜  ⎟+⎜                 ⎟
                                   ˆ ⎠         1− ξ ˆ      ⎟                           (6)
                           ⎝             ⎝                 ⎠
     By far, there is no robust method for selecting u with satisfactory performance.
However, graphical data exploration techniques, such as sample mean excess plot or
Hill plot, help to visually select the threshold u. The sample mean excess function is
defined as:

                                 f (u ) = E (X − u X > u )                                  (7)

     A significant upward trend beyond the certain percentile in the plot indicates a
heavy-tailed behavior. Therefore, this percentile can be selected as a threshold. Note
that the plot is erratic for large u, when the averaging is over very few excesses. It is
better to omit these from the plot. Continue the samples in section 2, we illustrate the
sample mean excess plot of the two portfolio.

              Figure 4 Mean Excess Plot of                   Figure 5 Mean Excess Plot of
                       Portfolio 1                                    Portfolio 2

     As is shown in the figure, the positive gradient of the plot both appeared at the
early stage, indicating the obvious fat-tail characteristic of the Log-normal distribution.
Accept any of the thresholds on the upward trend, the distribution of exceedance over
the threshold tends to GPD. Considering our purpose is on the tail behavior and the
requirement for a rich tail region for fitness, we provide the 90% percentile as the
threshold. Therefore, the cumulative distribution of excess of losses can be estimated in
terms of equation 3.

           Figure 6 Distribution of Excess                       Figure 7 Distribution of Excess
              Loss Rate of Portfolio 1                              Loss Rate of Portfolio 2

     Figure 6 and Figure 7 displays a goodness of fit (R-square>99.95%) for a
Generalized Pareto distribution. The shape and scaling parameters are accordingly
obtained with significance (p-value all below 0.001) and the VaR corresponding to the
quantiles can be estimated by inversing the Excess loss distribution function as referred
in equation 5. Thus, the Expected Shortfall for EC at 95% percentile can be calculated
and shown in Table 1.

    Table 1 Gap between the two approaches for calculating EC rate with 95 percent percentile

          Portfolio      VaR-based EC            Tail-based EC           Gap       Ratio
         Portfolio 1        0.0527                   0.1013             0.0486     1.92
         Portfolio 2         0.0527                 0.0878              0.0351      1.67

     The gap in table 1 shows the additional capital rate above VaR-based EC that is
required if the extreme tail risk is to be considered. And the ratio presents a relative gap
between Tail-based EC and VaR-based EC. Compare the gap and ratio of the two
portfolios, portfolio 1 should allocate more capital than Portfolio 2 in case of bad
situation. In other word, although both portfolios are of same EC in normal situation,
capital allocated for Portfolio 1 should bear more risk than that of Portfolio two in terms
of the tail risk. Then here comes the question: should we substitute Tail-based EC for
VaR-based EC as our standard approach to calculate EC? The answer is partly true,
because tail-base EC do distinguish the risk of the two portfolios and taken the bad
situation into account. In addition ES is a coherent risk measure superior to VaR.
However, Tail-based EC concern too much about the tail region losses which will lead
addition capital allocation. More capital allocated do good for the solvency but will
reduce the efficiency of the capital and enhance the opportunity cost of the capital. In

addition, the tail region losses are characterized low frequency high severity losses. In
normal situation, the losses are high frequency but low severity. Therefore, tail-based
EC can be used as a stress test or scenario analysis for a given portfolio. An alternative
approach to adjust the VaR-based EC in the normal situation will be discussed in
following section.

                   Risk-neutral Economic Capital Calculation

     Adjustment for the VaR-based EC requires a certain kind of EC as a reference
frame. Under this frame, the same EC calculated by relative VaR of the two loan
portfolios will present their different risk bearing. Tail-based EC could reflect the
different risk bearing for the portfolios while it could not be the reference frame because
of the changeable threshold. The reference frame should be objective. When the loss
distribution is given, the reference frame is accordingly embedded. Then what is
objective EC? The primary guideline is that there should be no subjective issues
included. A risk lover or risk aversion user will not allocate the objective EC for a
certain unexpected losses. Because, a risk lover is likely to allocate less EC while a risk
aversion user tends to allocate more EC. Therefore only the risk neutral user could
allocate the proper EC. This statement coincides with the fundamental meaning of EC,
which contends that EC should balance the risk and return. More capital allocated
means the decline of return while less capital allocated indicates the rise of risk. Then
how to define the risk neutral user? The concept of Spectral Risk measure offers the
insight into this issue.

     Spectral risk measure was proposed by Carlo Acerbi (2002, 2004). It is likewise a
coherent risk measure and employs a weighting function to reflect the user’s risk
attitude. This risk measure is defined as follows:

                                      M φ = ∫ φ ( p )q p dp                             (8)

     Where M φ are the weighted averages of the quantiles of our loss distribution.

φ ( p ) is the weighting function, also known as the risk spectrum or risk aversion
function, remains to be determined. φ ( p ) must satisfied the following properties
(Acerbi,2004) in order to make M φ coherent:

     Non-negativity: φ ( p ) ≥ 0 for all p belong in the range [0,1].


     Normalization:     ∫ φ ( p )dp = 1

     Increasingness: φ ( p1 ) ≤ φ ( p 2 ) for all 0 ≤ p1 ≤ p 2 ≤ 1

     The first condition requires that the weights are non-negative, and the second
requires that the probability-weighted weights should be added up to 1. Both are
obvious. The third condition is more important. This condition is a direct reflection of
risk-attitude, and requires that the weights attached to higher losses should be no less
than the weights attached to lower losses. The message indicates that the key to
coherence is that a risk measure must give higher losses at least the same weight as
lower losses.

     This explains why VaR is not coherent and ES is. VaR places all its weight on a
single quantile that corresponds to a chosen confidence level, and it places no weight on
any others. ES measure implies taking an average of quantiles in which tail quantiles
have an equal weight and non-tail quantiles have a zero weight. It is a special case of
M φ obtained by setting φ ( p ) to the following:

                                          ⎧1 /(1 − α )        p >α
                               φ ( p) = ⎨                if                            (9)
                                          ⎩ 0                 p ≤α

     The fact that the ES gives all tail losses equal weights suggests that a user who
uses this measure is risk-neutral at the tail region between better and worse tail
outcomes (Grootveld and Hallerbach, 2004).Although it is inconsistent with
risk-aversion, it sheds light on the definition of the risk neutral user. The risk neutral
user will allocate the same weight for each unexpected losses in calculate EC. If we
extend the threshold to the expected loss level, ES could be the risk-neutral measure.
Meanwhile, the expected loss level is predetermined for a given loss distribution.
Therefore, EC based on this kind of ES could be the reference frame. Following
presents the equation for the risk-neutral EC.

           ⎛ VaRq − EL ⎞ ⎛ β − ξ (u − EL ) ⎞ VaRq =u = EL
                                ˆ                          ⎛     ⎞
      EC = ⎜           ⎟+⎜                 ⎟ ⎯⎯ ⎯⎯→ EC N = ⎜ β ⎟
           ⎝ 1− ξ  ˆ ⎟ ⎜
                       ⎠ ⎝     1− ξ ˆ      ⎟               ⎜1− ξ ⎟
                                                           ⎝    ˆ⎠                    (10)

     Equation 10 suggests that the risk neutral EC is only dependant on the shape factor
and the scale factor. These two parameters are predetermined from the loss distribution
beyond EL. Thus it is an embedded EC for a give loss distribution. Continue with the
samples above, we further estimate the risk neutral EC of two portfolios. Owing to fact
that there is no need to select the threshold in this case, only fitness of the distribution of
excess loss rate and significance of the parameters may affect the results.

              Figure8 Distribution of Excess                      Figure9 Distribution of Excess
                 Loss Rate of Portfolio 1                            Loss Rate of Portfolio 1

     Figure 8 and Figure 9 displays a goodness of fit (R-square>99.97%) for a
Generalized Pareto distribution. The shape and scaling parameters are accordingly
obtained with significance (p-value all below 0.001). Then the risk-neutral EC can be
calculated with equation 10. Table 2 below shows the results

                                     Table 2 Risk-neutral EC rate

 Portfolio      VaR based       Scale Factor          Shape Factor       Risk-neutral     GAP with
                   EC                                                        EC         VaR-based EC
Portfolio 1      0.0527           0.024866             0.064402           0.026577        0.026123
Portfolio 2       0.0527          0.024795             0.062349           0.026444        0.026256

     As risk neutral EC indicated a full-range of risk of the loss distribution. Portfolio 1
on the whole will suffer more risk than portfolio 2. Therefore an adjustment on the EC
of portfolio 1 can be made by adding a multiplier (0.026577/0.026444=1.005). Or if we
still hold same EC for the two portfolios, than the multiplier should be add on the return
of portfolio 1. In other words, we can unitize the Risk-neural EC to balance the risk and
return of the portfolio. It is also useful to note that risk-neural EC is far below the
VaR-based EC. This is because the financial intuitions are normally risk aversion. The
EC allocation in practice will surpass the risk-neural EC. The gap between them
indicates the degree of risk aversion. In addition, we propose the three-layer framework

for EC. Figure 10 below shows the idea.

                                      Three Layers of EC

                                                           Tail risk adding
                   0.08                                    capital
                                                           Risk Aversion
                                                           adding Capital
                   0.04                                    Risk-neutral EC
                                1              2

                               Figure 10 Three-layer framework for EC

     Bottom layer represents the Risk-neutral EC for the portfolio. It is the embedded
EC for a given portfolio and indicates full-range of risk of the loss distribution. The
second layer shows the risk aversion adding capital in normal situation. The amount of
the adding capital is adjusted by the Risk-neutral EC in terms of rule that return and risk
should be balanced. The top layer focuses on the tail risk in bad situation. They are used
for stress test or scenario analysis. Furthermore, we can allocate different kind of capital
to the corresponding layers. For instance, in terms of the capital classification in New
Basel Accord, we can allocate tire 1 to risk-neutral EC, then tire 2 for risk aversion
adding capital and tire 3 for tail risk adding capital.


     1. The same relative VaR-based EC does not mean the same risk embedded for the
portfolios. VaR is criticized failing to consider the tail risk. This is also true when
adopting relative VaR to calculate EC. Chances are that we may obtain the same EC of
two portfolios at a specific risk tolerance. Owing to the regardless of tail risk, same EC
may experience different risk.

     2. The tail-based EC derived by EVT theory clearly presents the addition EC
needed if the normal situation moves to bad status. However, Tail-based EC tend to
allocate more capital for the portfolio which will lower the efficiency of the capital and
enhance the opportunity cost of the capital. Considering the tail region losses are
characterized low frequency high severity losses, tail-based EC can be used as a stress
test or scenario analysis for a given portfolio.

     3. Based on the concept of spectral risk measure, we proposed the concept of risk
neutral EC. Risk neutral EC gives all unexpected losses equal weights. In addition, the
risk neutral EC is only dependant on the shape factor and the scale factor, which are
predetermined from the loss distribution beyond EL. Thus it is an embedded EC for a
give loss distribution and could be the reference frame to adjust the VaR-based EC.

     4. Three-layer framework of EC sheds light on the sequence of EC allocating. The
bottom layer, the embedded EC, is of prime importance which reflects the whole risk
spectral of the portfolio. Therefore, such capital as tier 1 that referred in New Basel
Accord should be allocated to the bottom layer in first order. Then tire 2 for risk
aversion adding capital and tire 3 for tail risk adding capital accordingly.


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