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Extreme Risk Modeling for Economic Capital By 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 Address: 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 1 Extreme Risk Modeling for Economic Capital Abstract 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 2 Introduction 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 3 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 Capital. 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 4 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. 5 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. 1 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). 6 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. 7 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: 1 1 ESα = ∫ q p dp 1−α α (1) 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) 8 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 ) (3) 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 ) (4) N 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 ⎞ ˆ ˆ ⎛β ⎞ ˆ VaRq = u + ⎜ ⎟ (1 − Fu (q − u )) − 1 = u + ⎜ ⎟⎜ − 1⎟ −ξˆ ⎜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 ) ⎞ ˆ EC Eq = ⎜ ⎜ 1− ξ ⎟ ⎜ ⎟+⎜ ⎟ ˆ ⎠ 1− ξ ˆ ⎟ (6) ⎝ ⎝ ⎠ 9 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. 10 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 11 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: 1 M φ = ∫ φ ( p )q p dp (8) 0 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: 12 Non-negativity: φ ( p ) ≥ 0 for all p belong in the range [0,1]. 1 Normalization: ∫ φ ( p )dp = 1 0 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) ⎠ 13 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 14 for EC. Figure 10 below shows the idea. Three Layers of EC 0.12 0.1 Tail risk adding 0.08 capital Risk Aversion 0.06 adding Capital 0.04 Risk-neutral EC 0.02 0 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. Conclusion 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. 15 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. 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