A Loan Portfolio Model Subject to Random Liabilities and Systemic Jump Risk
AA Loan Portfolio Model Subject to RandomLiabilities and Systemic Jump Risk
Luis H. R. Alvarez ∗ and Jani T. Sainio † June 7, 2010
Abstract
We extend the Vasiˇcek loan portfolio model to a setting where lia-bilities fluctuate randomly and asset values may be subject to systemicjump risk. We derive the probability distribution of the percentage lossof a uniform portfolio and analyze its properties. We find that the impactof liability risk is ambiguous and depends on the correlation between thecontinuous aggregate factor and the asset-liability ratio as well as on thedefault intensity. We also find that systemic jump risk has a significantimpact on the upper percentiles of the loss distribution and, therefore, onboth the VaR-measure as well as on the expected shortfall. ∗ Turku School of Economics, Department of Accounting and Finance, FIN-20500 Turku,Finland, e-mail: luis.alvarez@tse.fi † Turku School of Economics, Department of Accounting and Finance, FIN-20500 Turku,Finland, e-mail: Jani.T.Sainio@tse.fi a r X i v : . [ q -f i n . R M ] J un Introduction
Modeling correlation of defaults plays naturally a central role in the liter-ature on the management of loan portfolios and valuation of credit deriva-tives. The reason for focusing on modeling the correlation of defaults isobvious from a credit risk management perspective: since default prob-abilities are relatively low, the (tail) dependence of different risks mayplay a dominant role in the joint probability of defaults. The role of thisdependence is naturally pronounced during financial crises and, therefore,incorporating a factor (or factors) taking into account such relatively rarephenomena is clearly of interest.In light of the fact that financial crises occur regularly and share sev-eral similar characteristics both in their precedents as well as in theirnegative impact on the overall economy (for excellent recent empiricalstudies on financial crises see Reinhart and Rogoff (2008a, 2008b, 2009)),we extend the classical Vasiˇcek loan portfolio model to a setting wherethe liabilities are subject to random fluctuations and asset values mayface systematic asymmetric jump risk. We follow a factorized approach(for a well-written introduction of this credit risk modeling approach, seeSch¨onbucher (2001)) and model the liabilities as geometric Brownian mo-tions driven by two continuous factors; one aggregate level factor affectingall assets and liabilities and one statistically independent idiosyncratic fac-tor. In this way liabilities become conditionally independent of each othergiven the aggregate factor dynamics. On the other hand, we assume thatthe value of the assets constitute exponential jump diffusions potentiallydriven by three factors. As in the case of liabilities, the continuous partof the fluctuations are determined by an aggregate and an idiosyncraticfactor. In addition to these factors, the assets may also be subjected to adiscontinuous and spectrally negative risk component modeled as a com-pensated Poisson process with only positive jumps. This risk factor isassumed to be systemic in the sense that it affects all assets and, oncerealized, may significantly decrease their values (see, for example, Eber- ein and Madan (2009) for a recent study where assets and liabilities aremodeled as exponential Levy processes). This assumption allows the anal-ysis of rare but potentially dramatic collapses in the values of the assetsbacking up liabilities. Instead of considering valuation issues, we focus onthe determination of the probability distribution of the percentage loss ofa uniform portfolio and investigate how the interplay between differentrisk components affect this distribution.It is worth emphasizing that our model is related to the pioneeringwork by Zhou (1997) and Zhou (2001). Zhou (1997) (see also Zhou (2001);for a recent application of Zhou’s model within consumer credit setting, seede Andrade and Thomas (2007)) presents a factorized model of credit riskwhere the value of the shares of the firm are assumed to evolve accordingto a geometric jump diffusion and the threshold value increases at a knownconstant exponential rate. Thus, liability risk is not considered as in ourstudy. Moreover, in both Zhou (1997) and Zhou (2001) the jumps in thevalue of shares may have both signs implying that even though the firmis subject to sudden large drops, it may also face unexpected significantincreases in its value. In this way, his model does not take into account therealization of rare but potentially significant negative outcomes erodingthe value of assets. However, in contrast to our study, Zhou (1997) derivesalso the arbitrage free bond prices both in the case where default occurswhenever the value falls short the known threshold value at a given fixedmaturity as well as in the technically demanding first passage time settingwhere default occurs whenever the value falls short the threshold valueprior expiry. He also studies the term structure of credit spreads anddemonstrates that the considered class of models can generate a varietyof yield spread curves as well as marginal default rate curves.Our findings indicate that the impact of liability risk on the distri-bution of the percentage loss is ambiguous and depends, among others,on the correlation between the continuous aggregate factor and the asset-liability ratio as well as on the default intensity. If liabilities are subject o purely idiosyncratic risk and are unaffected by the aggregate marketfactor, then the resulting probability distribution is similar but yet notidentical with the limiting distribution in the standard Vasiˇcek setting.The main reason for this is that in the present setting also the idiosyn-cratic risk factor affects the total volatility of the percentage growth rateof the asset-liability ratio. However, in contrast with the standard Vasiˇcekmodel, our results indicate that there are circumstances under which thepercentage loss converges to the known default probability of an individ-ual loan. Such a case arises when the continuous aggregate factor affectsequally strong both assets and liabilities. Such a case might potentiallyappear in situations where both assets and liabilities depend on a welldiversified portfolio (for example, in the case of unit linked products).The impact of the systemic risk component is more pronounced and ourfindings show that its presence may have a radical impact on the limitingdistribution. First, the limiting probability distribution may have morethan two modes; a phenomenon which does not arise in the standard con-tinuous setting. Second, the systemic risk term has a significant impacton the tail probabilities and tends to increase both the upper percentilesas well as the expected shortfall associated to these percentiles even whenthe realization intensity is low.The contents of this study are as follows. In section two we presentthe basic continuous model and state our main findings on the probabilitydistribution of the percentage loss of a uniform portfolio. In section threewe introduce the discontinuous systemic risk component and analyze itsimpact on the probability distribution of the percentage loss. Finally,section four concludes our study. Our main objective is to investigate the percentage loss distribution ofa loan portfolio within a conditionally independent factor model along he lines indicated by the pioneering work in Vasicek (1987) (see alsoVasicek (1991) and Vasicek (2002)). To this end, we first assume that theasset values evolve according to the random dynamics characterized bythe stochastic differential equation dA it = µ i A it dt + σ i A it dW it , A i = A i , i = 1 , . . . , n, (1)where both the drift coefficient µ i as well as the volatility coefficient σ i are exogenously given and W it is standard Brownian motion. In order tomodel the statistical dependence of the various asset values, we assumethat the driving Brownian motions can be decomposed into the form W it = √ ρ i Y t + (cid:112) − ρ i X it , (2)where Y t , X t , . . . , X nt are a family of independent driving Brownian mo-tions and ρ i ∈ [0 , , i = 1 , . . . , n measures the correlation between theunderlying driving factor dynamics. The factor Y t is a joint aggregate riskfactor (market risk) affecting all the driving processes and the X it ’s are idiosyncratic risk factors associated to the particular asset value.The basic Vasicek loan portfolio model assumes that the liabilitiesof the company are constant. However, this assumption is not alwayssatisfied and liabilities may actually depend on the aggregate risk factorthrough the investment policy of the corporation (for an approach basedon exponentially increasing but deterministic liabilities see Zhou (2001)).Such a circumstance arises quite naturally, for example, in the case ofunit linked insurance contracts. In order to introduce liability risk, weassume that the liabilities B it evolve according to the random dynamicscharacterized by the stochastic differential equation dB it = α i B it dt + β i B it ( √ θ i dY t + √ − θ i dZ it ) , B i = B i , (3)where both the drift coefficient α i as well as the volatility coefficient β i are exogenously given, θ i ∈ [0 ,
1] is a coefficient measuring correlationbetween different liabilities, and Z t , . . . , Z nt are a family of independentdriving Brownian motions independent of the aggregate risk factor Y t and he asset-specific idiosyncratic risks X t , . . . , X nt . Thus, the liabilities areassumed to fluctuate in a similar, yet not necessarily identical, fashionwith the assets.As usually, we assume that default occurs whenever the assets do notmeet the liabilities at a given date T . Since both the asset values aswell as the liabilities follow two ordinary potentially correlated geometricBrownian motions, a standard application of Itˆo’s lemma yields P [ A iT ≤ B iT ] = P (cid:20) A iT B iT ≤ (cid:21) = Φ (cid:18) Ξ i Σ i √ T (cid:19) , where Σ i = σ i + β i − σ i β i √ ρ i θ i measures the variance of the differenceof the driving factors andΞ i = ln (cid:18) B i A i (cid:19) − (cid:18) µ i − α i − (cid:0) σ i − β i (cid:1)(cid:19) T. For simplicity, we assume that the recovery rate from defaulted loansin the portfolio is zero. Under this assumption, the loss L i of the i th loancan be defined as the random variable L i = 1 (0 , ( A iT /B iT ) = L = 1 n n (cid:88) i =1 L i . Straightforward computation shows that the probability of default condi-tional on the aggregate factor Y now reads as p i ( Y ) = P [ L i = 1 | Y ] = P (cid:20) A iT B iT < (cid:12)(cid:12)(cid:12) Y (cid:21) = Φ (cid:18) Ξ i ζ i √ T − Λ i ζ i Y (cid:19) , where ζ i = σ i (1 − ρ i ) + β i (1 − θ i )measures the variance of the difference of the idiosyncratic risk factors,and Λ i = σ i √ ρ i − β i √ θ i denotes the volatility multiplier of the aggregatefactor Y in the dynamics of the asset-liability ratio A iT /B iT . In contrastto the standard Vasicek loan portfolio model subject to deterministically volving liabilities, we now observe that the losses given default are inde-pendent random variables whenever the volatility multiplier Λ i is identi-cally zero for all the loans in the portfolio. As intuitively is clear, thatcase arises when the aggregate factor dynamics affects both assets as wellas liabilities in a similar fashion. Otherwise, the losses are statisticallydependent due to the joint dependence on the aggregate market factor.Moreover, applying the law of total probability shows that E [ L i L j ] = (cid:90) ∞−∞ E [ L i L j | Y T ] P [ Y T ∈ dy ] = (cid:90) ∞−∞ p i ( √ T y ) p j ( √ T y )Φ (cid:48) ( y ) dy implying that the covariance of the loss given default reads ascov[ L i , L j ] = (cid:90) ∞−∞ p i ( √ T y ) p j ( √ T y )Φ (cid:48) ( y ) dy − Φ (cid:18) Ξ i Σ i √ T (cid:19) Φ (cid:18) Ξ j Σ j √ T (cid:19) . Along the lines of our observations above, we find that if Λ i = Λ j = 0then cov[ L i , L j ] = 0. It is worth emphasizing that these covariances (and,therefore, default correlations) are typically very sensitive with respect tochanges in the maturity T of the loans.In order to investigate the probability distribution of the percentageloss of a loan portfolio, let us now assume that the portfolio is formed by n identical contracts and denote the unconditional probability of defaultof an individual loan as p . In that case we observe that the probability ofdefault conditional on the aggregate factor Y can be expressed as p ( Y ) = Φ (cid:18) ζ (cid:0) ΣΦ − ( p ) − Λ Y (cid:1)(cid:19) . (4)Given this expression, denote now as ¯ L = lim n →∞ L the limiting loanportfolio percentage loss of a infinitely large portfolio. We can now estab-lish the following: Proposition 2.1.
The probability of k defaults in the loan portfolio per-centage loss reads as P (cid:20) L = kn (cid:21) = (cid:32) nk (cid:33) (cid:90) ∞−∞ p k ( √ T y )(1 − p ( √ T y )) n − k Φ (cid:48) ( y ) dy, (5) where p ( y ) is given in (4) . If Λ = 0 then loan portfolio percentage lossconverges almost certainly to the deterministic limit ¯ L = p . However, f Λ (cid:54) = 0 then the limiting loan portfolio percentage loss is distributedaccording to the probability distribution P [ ¯ L ≤ x ] = Φ (cid:18) | Λ | (cid:0) ζ Φ − ( x ) − ΣΦ − ( p ) (cid:1)(cid:19) (6) with density f ( x ) = ζ | Λ | Φ (cid:48) (cid:16) | Λ | (cid:0) ζ Φ − ( x ) − ΣΦ − ( p ) (cid:1)(cid:17) Φ (cid:48) (Φ − ( x )) . (7) Proof.
The binomial formula (5) is a direct implication of the law of totalprobability and the binomial nature of the loss given default (see, for ex-ample, chapter 9 in Lando (2004) and chapter 8 in McNeil et al (2005)).On the other hand, since the losses given default are conditionally inde-pendent, we observe that the conditions of the strong law of large numbers(SLLN) are satisfied and, therefore, that the percentage loss conditionalon the aggregate factor converges to its expectation which, in the presentcase, reads as in (4) when Λ (cid:54) = 0 and as p when Λ = 0. Equation (6) thenfollows by computing the probability P [ p ( Y ) ≤ x ]. The density can thenbe derived by ordinary differentiation.Proposition 2.1 extends the results of the standard Vasicek loan port-folio model to the case where also liabilities are subject to random fluc-tuations. The main difference with the standard model is that now thevolatility multiplier of the aggregate market factor in the dynamics of theasset-liability ratio can be zero even in the case where the factor affectsboth assets as well as liabilities. If this multiplier is zero, then the lossesare IID random variables and the probability distribution can be directlyanalyzed in terms of constant binomial probabilities. In that case, thepercentage loss converges almost everywhere to the known binomial prob-ability. However, if the multiplier is not zero, then the percentage lossconverges towards a random variable with known distribution (6) whichresembles, but is not identical, with the limiting distribution in the caseof constant liabilities. traightforward computations show that the probability density func-tion is bimodal when Λ > ζ , monotone when Λ = ζ , and unimodalwith mode at ¯ L M = Φ (cid:18) ζ Σ ζ − Λ Φ − ( p ) (cid:19) when Λ < ζ . Consequently, along the original observations by Vasicekwe find that depending on the precise parametrization of the model, thedistribution may be either unimodal or bimodal and it can also be veryskewed. We illustrate the loss density in Figure 1 for various correlationsunder the assumptions that θ = 0 . σ = 0 . β = 0 . µ = 0 . α = 0 . T = 1, B = 1, and A = 1 . x f (cid:72) x (cid:76) Ρ(cid:61) (cid:72) bimodal (cid:76)
Ρ(cid:187) (cid:72) monotone (cid:76)
Ρ(cid:61) (cid:72) unimodal (cid:76)
Figure 1:
Loss densitiesIn the present setting the ν -percentile L ν satisfying the identity P [ ¯ L ≤ L ν ] = ν is L ν = Φ (cid:18) ζ (cid:0) ΣΦ − ( p ) + | Λ | Φ − ( ν ) (cid:1)(cid:19) . The percentile L ν depends, among others, on the volatility β of the liabil-ities. Unfortunately, it is not monotonic as a function of β and, therefore,the impact of liability risk on the percentiles is ambiguous. The 95%percentile L . is illustrated as a function of β in Figure 2 under the ssumptions θ = 0 . σ = 0 . µ = 0 . α = 0 . T = 1, B = 1, and A = 1 . Β L Ρ(cid:61)
Ρ(cid:187)
Ρ(cid:61)
Figure 2:
The Impact of Liability Risk on the Percentile L . Having considered the impact of liability risk on the limiting probabilitydistribution of the loan portfolio percentage loss, we follow the originalstudy by Zhou (1997) (see also Zhou (2001)) and extend our basic modelto the case where the assets backing up liabilities are subject to unex-pected random jumps modeled as a compound Poisson process. In con-trast with Zhou (1997), we assume that these unexpected jumps are onlyone-sided (downward jumps) and occur at the aggregate level. Therefore,the driving compound process is a common factor affecting all assets; anassumption permitting the analysis of the the impact of rare but poten-tially significant collapses (i.e. realization of systemic risk) in the assetvalues to the limiting default intensity in a large loan portfolio.In line with these arguments, we now assume that the asset valuesevolve according to the dynamics A it = A i e ( µ i + λ (1 − E [ e − ξ ]) − σ i ) t + σ i ( √ ρ i Y t + √ − ρ i X it ) − J t , (8) here J t = N t (cid:88) k =0 ξ k (9)is a compound Poisson process independent of the continuous aggregatefactor Y . In (9), we assume N t is a standard Poisson process with intensity λ , { ξ k } k ≥ is a sequence of nonnegative iid random variables with knowndistribution, and ξ = 0. In equation (8) λ (1 − E [ e − ξ ]) is a compensationterm needed to guarantee that the asset value is expected to grow atthe same rate as in the absence of jumps. If this compensation term isnot taken into account then the proposed asset value model is almostsurely lower and has a smaller expected value than the model consideredin the previous section (due to the nonnegativity of the jumps and themonotonicity of the driving Poisson process). Especially, we observe that(8) can be expressed as dA it = µ i A it dt + σ i A it dW it + A it (cid:90) R ( e − z − d ˜ N ( dt, dz ) , (10)where ˜ N ( dt, dz ) denotes the Poisson random measure associated to theunderlying compensated Poisson process (cf. Chapter 2 in Kyprianou(2006)).It is worth pointing out that the stated specification results into anasset value which coincides in the mean but is more volatile than themodel in the absence of systemic jumps. More precisely, it is clear thatnow that for all t it holds E [ A it ] = A i e µ i t andvar[ A it ] = A i e µ i t (cid:16) e σ i t + λt E [(1 − e − ξ ) ] − (cid:17) > A i e µ i t (cid:16) e σ i t − (cid:17) . In this way the considered process can be interpreted as a mean preservingspread of the continuous asset value dynamics considered in the previoussection.Applying an analogous conditioning argument as in the previous sec-tion, we now find that the probability of default given the aggregate factors Y and J is P [ A iT ≤ B iT | Y, J ] = Φ (cid:18) ˜Ξ ζ i √ T − Λ i ζ i √ T Y T + J T ζ i √ T (cid:19) , here ˜Ξ i = Ξ − λ (1 − E [ e − ξ ]) T. As intuitively is clear, the positivity ofthe jump component J T implies that the probability of default is in thissetting higher that in the absence of unexpected downward jumps in thevalue of the assets. However, it is not beforehand clear how significant theeffect of the Poisson component on the default probability is, and how thiseffect depends on both the intensity of the driving Poisson process andthe precise nature of the jump size distribution. Moreover, in the presentsetting the loans are statistically dependent even when the volatility mul-tiplier of the aggregate market factor in the dynamics of the asset-liabilityratio is zero (i.e. Λ i = 0 for all i ). The reason for this is naturally thepresence of the systemic jump risk component affecting all assets.In order to be able to analyze the limiting probability distributionof the percentage portfolio loss, we now again assume that we have aportfolio of n approximately identical contracts. In that case we find thatthe conditional probability of default given the aggregate factors reads as p ( Y, J ) = Φ (cid:18) Σ ζ (cid:18) Φ − (˜ p ) − ΛΣ √ T Y T + J T Σ √ T (cid:19)(cid:19) , where ˜ p = Φ (cid:18) Ξ − λ (1 − E [ e − ξ ]) T Σ √ T (cid:19) . We can now establish the following result:
Proposition 3.1. If Λ = 0 then the limiting loan portfolio percentageloss is distributed according to the probability distribution P [ ¯ L ≤ x ] = e − λT χ [ p, ( x ) + ∞ (cid:88) k =1 e − λT ( λT ) k k ! (cid:90) M T P [ S k ∈ du ] , (11) where M T = Σ √ T (cid:0) Φ − ( x ) − Φ − (˜ p ) (cid:1) and P [ S k ∈ du ] = P (cid:34) k (cid:88) j =1 ξ j ∈ du (cid:35) = ( g ∗ · · · ∗ g )( u ) du is the k -fold convolution of the density g ( u ) of the random jump-size. If,however, Λ (cid:54) = 0 then P [ ¯ L ≤ x ] = e − λT Φ ( H ( x, ∞ (cid:88) k =1 e − λT ( λT ) k k ! (cid:90) ∞ Φ ( H ( x, u )) P [ S k ∈ du ] , (12) here H ( x, u ) = 1 | Λ | (cid:18) ζ Φ − ( x ) − ΣΦ − (˜ p ) − u √ T (cid:19) . In this case, the density of the loan portfolio percentage loss reads as ˆ f ( x ) = ζ | Λ | (cid:90) ∞ ∞ (cid:88) k =1 e − λT ( λT ) k k ! Φ (cid:48) ( H ( x, u ))Φ (cid:48) (Φ − ( x )) P [ S k ∈ du ]+ e − λT ζ | Λ | Φ (cid:48) ( H ( x, (cid:48) (Φ − ( x )) (13) Proof.
As in Proposition 2.1, the losses given default are conditionallyindependent given the aggregate factors and satisfy the conditions ofthe SLLN. The probability distributions (11) and (12) follow directlyby invoking the law of total probability in computing the probability P [ p ( Y, J ) ≤ x ]. The density (13) can then be derived with ordinary dif-ferentiation.Proposition 3.1 states the limiting probability distribution and its den-sity for a sufficiently large loan portfolio percentage loss. Unfortunately,the distribution is in this case very complicated (being a mixture; for acomprehensive treatment of mixtures within credit risk management ap-plications, see Chapter 8 in McNeil et al (2005)) and identifying thepercentiles explicitly is extremely demanding, if possible at all. However,it is worth emphasizing that in contrast to the case subject to continuousfactor dynamics, the distribution may now be multimodal. The reasonfor this observation is that now the density ˆ f ( x ) is a probability weightedsum of potentially bimodal densities. More precisely, sinceΦ (cid:48) ( H ( x, u ))Φ (cid:48) (Φ − ( x )) = e Φ − ( x ) − (cid:16) ζ Φ − ( x ) − ΣΦ − (˜ p ) − u √ T (cid:17) is bimodal whenever Λ > ζ , we notice that the limiting distribution maybe multimodal depending on the jump size distribution. For example,when the jump size is a known constant, the limiting distribution mayhave more modes than just two.In order to investigate numerically the impact of jumps on the limitingdistribution of the loan portfolio percentage loss, we now consider the spe- ial case where the jump size is exponentially distributed with parameter γ . It is well-known that in this case the series S n = n (cid:88) k =1 ξ is Gamma-distributed according to the density P [ S n ∈ du ] = γe − γu ( γu ) n − ( n − du. In this case the density of the loan portfolio percentage loss reads asˆ f ( x ) = ζ | Λ | (cid:90) ∞ ∞ (cid:88) k =1 e − λT ( λT ) k k ! Φ (cid:48) ( H ( x, u ))Φ (cid:48) (Φ − ( x )) γe − γu ( γu ) k − ( k − du + e − λT ζ | Λ | Φ (cid:48) ( H ( x, (cid:48) (Φ − ( x )) . We illustrate this density in the three different cases arising in the absenceof jump risk. Figure 3 illustrates the case where the limiting distributionis unimodal under the assumptions that θ = ρ = 0 . σ = 0 . β = 0 . µ = 0 . α = 0 . T = 1, λ = 0 . B = 1, and A = 1 .
1. As is clear
0. 0.2 0.4 0.6 0.8 1. x f (cid:96) (cid:72) x (cid:76) Γ(cid:61)
Γ(cid:61)
Figure 3:
Loss densitiesfrom Figure 3, the presence of downward jump risk has a pronouncedimpact on the upper tail of the limiting density and, therefore, on thepercentiles of the distribution. These percentiles are numerically illus-trated in the following table. As Table 1 clearly illustrates the differencebetween the percentiles is significant for sufficiently high percentiles. Forexample, in the absence of the systemic jump component the percentage γ → ∞ γ = 1 56.5 59.32 62.61 66.6 71.81 80.01 γ = 0 . Percentiles in the Case of Figure 3loss exceeds 73.97% with probability 2.5%. In the presence of the systemicjump component this percentile is radically changed and the percentageloss is expected to exceed 80.01% (81.02%) with the same probability. Theexpected shortfalls associated with the percentiles appearing on Table 1are illustrated on Table 2. As Table 2 shows, the impact of the systemic ν γ → ∞ γ = 1 72.7 75.31 78.39 82.17 87.09 94.05 γ = 0 . Expected Shortfall ES ν jump component on the expected shortfalls is significant as well. Interest-ingly, the difference becomes higher as the confidence limit increases. Thereason for this observation is the skewness of the density towards higherrealizations in the presence of the systemic jump component.For the sake of comparison, the case where the limiting density ismonotone in the absence of jump risk is illustrated in Figure 4 under theassumptions that θ = 0 . ρ ≈ . σ = 0 . β = 0 . µ = 0 . α = 0 . T = 1, λ = 0 . B = 1, and A = 1 . . 0.2 0.4 0.6 0.8 1. x f (cid:96) (cid:72) x (cid:76) Γ(cid:61)
Γ(cid:61)
Figure 4:
Loss densities ν γ → ∞ γ = 1 65.94 69.71 73.91 78.7 84.38 91.69 γ = 0 . Percentiles in the Case of Figure 4(92.84%). The expected shortfalls associated with the percentiles appear-ing on Table 3 are now, in turn, illustrated on Table 4. ν γ → ∞ γ = 1 82.26 84.81 87.61 90.7 94.18 97.98 γ = 0 . Expected Shortfall ES ν We considered the impact of liability risk on the percentage loss distri-bution of a large uniform loan portfolio both in the presence and in theabsence of discontinuous systemic risk. As our findings show, the impact of iability risk is ambiguous and it may increase or decrease the percentilesdepending on the precise parametrization of the considered model and,especially, on the strength of the dependence between the asset-liability-ratio and the the driving continuous aggregate factor. The discontinuousjump factor capturing the systemic risk has a more pronounced impacton the limiting percentage loss distribution since it affects all the asset-liability-ratios through the asset values. Our results seem to indicate thatits impact becomes more significant at the tails of the distribution, whichare found to be bimodal in the exponential case. According to our find-ings, the presence of systemic risk affects in a relatively significant waythe expected shortfall associated to the upper tail probabilities even whenthe realization of the risk is assumed to be rare.There are several directions towards which our model could be gen-eralized. First, the considered loan portfolio is assumed to be large anduniform, thus overlooking the potentially significant effect of the gran-ularity of a loan portfolio. Second, assuming that there is no recoveryonce default has occurred is another simplifying assumption which couldrelaxed. Third, our analysis focuses solely on the distribution of the per-centage loss distribution and overlooks the pricing of bonds within theconsidered setting. All these interesting questions are left for future re-search. Acknowledgements : The authors are grateful to the deputy managingdirector at Federation of Finnish Financial Services
Esko Kivisaari forproposing this research subject and to
Teppo Rakkolainen for insightfulcomments on the contents of the study. The financial support from the
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