Analytical Value-at-Risk and Expected Shortfall under Regime Switching *

Transcription

1 Working Paper 9-4 Departamento de Economía Economic Series 14) Universidad Carlos III de Madrid March 9 Calle Madrid, Getafe Spain) Fax 34) Analytical Vale-at-Risk and Expected Shortfall nder Regime Switching * Abderrahim Taamoti Universidad Carlos III de Madrid March 3, 9 Abstract It is well known that the se of Gassian models to assess financial risk leads to an nderestimation of risk. The reason is becase these models are nable to captre some important facts sch as heavy tails and volatility clstering which indicate the presence of large flctations in retrns. An alternative way is to se regime-switching models, the latter are able to captre the previos facts. Using regime-switching model, we propose an analytical approximation for mlti-horizon conditional Vale-at-Risk and a closed-form soltion for conditional Expected Shortfall. By comparing the Vale-at- Risks and Expected Shortfalls calclated analytically and sing simlations, we find that the both approaches lead to almost the same reslt. Frther, the analytical approach is less time and compter intensive compared to simlations, which are typically sed in risk management. JEL Classification: C, C19, G11, G19 Keywords: Regime switching; probability distribtion; Vale-at-Risk; Expected Shortfall; analytical approximation; closed-form soltion; simlation; mlti-horizon. * The athor thank Tao.k Boezmarni, Jean-Marie Dfor, Xiaodong Lin, Nor Meddahi, Jeroen Rombots, Chen Ying, and the participants to the Conference on Volatility and High Freqency Data at Chicago 7) for their helpfl comments. This paper was a part of paper entitled.risk Measres and Portfolio Optimization nder Regime Switching Models. Financial spport from the Spanish Ministry of Edcation throgh grants SEJ is acknowledged. Economics Department, Universidad Carlos III de Madrid. Adresse and Departamento de Economía Universidad Carlos III de Madrid Calle Madrid, Getafe Madrid) España. TEL: ; FAX: ; ataamot@eco.c3m.es.

2 1 Introdction Since the seminal work of Hamilton 1989, Econometrica), Markov switching models have been increasingly sed in financial time-series econometrics becase of their ability to captre some key featres, sch as heavy tails, volatility clstering, and mean reversion in asset retrns [see Cecchetti and al. 199), Pagan and Schwert 199), Trner and al. 1989), Gray 1996), Hamilton 1988), and Timmermann ), among others]. In this paper, we se these poplar models to provide an analytical approximation for mlti-horizon conditional Vale-at-Risk hereafter VaR) and derive a closed-form soltion for Expected Shortfall hereafter ES). VaR has become the most widely sed techniqe to measre and control market risk. It is a qantile measre that qantifies risk for financial instittions and measres the worst expected loss over a given horizon typically a day or a week) at a given statistical confidence level typically 1% or 5%). Different methods exist to calclate VaR nder different models of risk factors. Generally, there is a trade-off between the simplicity of the calclation method and realism of the assmptions in the risk factor model: As we allow the latter to captre more stylized effects, the calclation method becomes more complex. Under the assmption that retrns follow elliptical conditional distribtion, one can show that the VaR is given by a simple analytical formla [see for example RiskMetrics 1995) and Baer )]. However, when we relax this assmption, the analytical calclation of VaR becomes complicated and people tend to se compter intensive simlation based methods. An alternative measre of financial risk is ES defined by the conditional expectation of loss given that the loss is beyond the VaR level. This paper proposes an analytical approximation for conditional Vale-at-Risk and a closed-form soltion for conditional Expected Shortfall nder more realistic assmptions sing regime-switching. The isse of VaR calclation nder regime-switching has been considered at least by Billio and Pelizzon ) and Gidolin and Timmermann 6). Billio and Pelizzon ) se a switching volatility model to forecast the distribtion of retrns and calclate the VaR of both single assets and linear portfolios. Comparing the calclated VaR with the variance-covariance approach and GARCH1, 1) models, they find that VaR vales nder switching regime models are preferable to both approaches. Gidolin and Timmermann 6) examine the term strctre of VaR nder different econometric approaches, inclding mltivariate regime switching, and they find that bootstrap and regime switching models are best overall for VaR levels of 5% and 1%, respectively. However, to or knowledge, no analytical method has been proposed to calclate the VaR and ES nder regime-switching. In this paper, we derive a closed-form soltion for mlti-horizon conditional ES nder regime switching model. We se the same approach as Cardenas et al. 1997), Rovinez 1997), and Dffie and Pan 1) to provide an analytical approximation for mlti-horizon conditional VaR. We first

3 se Forier inversion method to compte the probability distribtion fnction of mlti-horizon portfolio retrns. Thereafter, we se an efficient nmerical integration step, designed by Davies 1973, 198), to approximate the infinite integral in the inversion formla and make the calclation of VaR feasible. To accont for conditional information and compte the conditional VaR and ES, we se the Hamilton filter. However, or derivations of VaR and ES are made nder the assmption that the error terms in the Markov switching model are i.i.d. Conseqently, the dependence in or framework is de to the mean and volatility dependence, ths we ignore the dependence of highorder moments. This cold be a limitation relative to simlations based calclation of VaR and ES, which may allow for non-i.i.d. errors. By comparing the Vale-at-Risks and Expected Shortfalls calclated analytically and sing simlations, we find that the both approaches lead to almost the same reslts. Frther, the analytical approach is less time and compter intensive compared to simlations, which are typically sed in risk management. The remainder of this paper is organized as follows. In Section, we introdce some notations and define or model. In Section 3, we derive the mlti-horizon conditional VaR and ES of linear portfolio. In Section 4, we compare the simlation and analytical calclations of VaR and ES. We conclde in Section 5. Framework We assme that there are n risky assets in the economy. We denote by r t = r 1t, r t,..., r nt ) the vector of risky assets retrns. We consider the following notations: 1,,,..., ) when s t = 1, 1,,..., ) ζ t = when s t =...,,,..., 1) when s t = N where s t is a stationary and homogenos Markov chain with N states, and we define the information sets J t = σr τ, ζ τ, τ t) = σr τ, s τ, τ t), I t = σr τ, τ t). It is well known that [see, e.g., Hamilton 1994, page 679)] where P is a transition probability matrix E[ζ t+h J t ] = P h ζ t, h 1, 1) P = [p ij ] 1 i,j N, p ij = P rs t+1 = j s t = i), 3 N p ij = 1, i {1,..., N}. ) j=1

4 The probability p ij of which regime is in operation at time t+1 depends on the past only throgh the most recent vale s t. 1 We assme that the Markov chain is stationary with an ergodic distribtion Π, Π R N, i.e. E[ζ t ] = Π = π 1,..., π N ). 3) Observe that P h Π = Π, h. 4) In what follows, we sppose that r t follows a mltivariate Markov switching model r t+1 = µζ t + Σζ t )ε t+1, ε t+1 i.i.d. N, I n ), 5) ] E [Σζ t )ε t+1 ) ε t+1 Σζ t )) J t = Σζ t )I n Σζ t ) = Ωζ t ), where I n is an n n identity matrix and µ 11 µ 1... µ 1N ω11 ζ t ω1 ζ t... ω1n ζ t µ 1 µ... µ N ω1 ζ t ω ζ t... ωn ζ t µ = , Ωζ t ) = µ n1 µ n... µ nn ωn1 ζ t ωn ζ t... ωnnζ t µ ij, for i = 1,..., n and j = 1,..., N, is the mean retrn of an asset i at state j, and ω il, for i, l = 1,..., n, is the vector of covariances between assets i and l at the N states. The processes {s t } and {ε t } are assmed jointly independent. Finally, we adopt the notations, U = 1,,..., N ) R N,, AU) = Diag exp 1 ), exp ),..., exp N )) P. 6) 3 Vale-at-Risk and Expected Shortfall nder regime switching In this section, we propose an analytical approximation for conditional VaR nder regime switching. We se the conditional characteristic fnction and a standard Forier-inversion formla [see for example Gil-Pelaez 1951)] to derive the conditional distribtion fnction, from which the conditional VaR is immediate. Frther, we provide a closed-form soltion for conditional ES. 3.1 Simple retrns We consider the portfolio allocation between the n risky assets. The portfolio s retrn at time t + h is given by: r p,t+h = n i h r it+h = h r t+h, 7) i=1 1 The assmption of fixed transition probabilities p ij cold be relaxed. Many models with time-varying transition probabilities have been considered [see Diebold, Rdebsch and Sichel 1993), Diebold, Ohanian and Berkowitz 1994), Filardo 1994), Lahiri and Wang 1994), Drland and McCrdy 1994), among others]. 4

5 where r t+h follows mltivariate Markov switching model 5) and h = h 1, h,..., h n) is the vector representing the weights attribted to each risky asset in the portfolio. To compte the conditional VaR of the portfolio s retrn r p,t+h we proceed in two steps. Firstly, we derive the conditional distribtion fnction of r p,t+h by inverting its conditional characteristic fnction sing a standard Forier-inversion formla [see Gil-Pelaez 1951)]. Secondly, we compte the conditional VaR of r p,t+h by inverting nmerically its conditional distribtion fnction sing an efficient nmerical integration step designed by Davies 198); for related work, see also Imhof 1961), Bohmann 1961, 197), and Davies 1973). We can show [see Taamoti 8)] that the conditional characteristic fnction of simple retrns r p,t+h is given by, R and h, E [expir p,t+h ) J t ] = e A iµ h 1 l 1,l n l h 1 l h ω l1 l P h ζ t, 8) where i = 1, the matrix A.) is defined in 6), and e denotes the N 1 vector whose components are all eqal to one. The characteristic fnction 8) depends on the state variable ζ t. In practice, the crrent state variable ζ t is not observable. Using the observable information set I t, the law of iterated expectations yields where E [expir p,t+h ) I t ] = e A iµ h 1 l 1,l n Π t = P r s t = 1 I t ),..., P r s t = N I t )). l h 1 l h ω l1 l P h Π t, R 9) Notice that, Eqation 9) is a complex fnction and sing Eler s formla it can be written as follows: E [exp ir p,t+h ) I t ] = e [A 1 ) + i A )] P h 1 Π t, R 1) where, for any R ) ) exp h Ω 1 h cos h µ 1))... A 1 ) = ) )... exp h Ω N h cos h µ N)) ) ) exp h Ω 1 h sin h µ 1))... A ) = ) )... exp h Ω N h sin h µ )) N,. 11) 5

6 µ j and Ω j, for j = 1,..., N, are the vector of mean and variance-covariance matrix of retrns at state j, respectively. Now, we derive the conditional probability distribtion fnction of r p,t+h. Given the conditional characteristic fnction of r p,t+h from 9), a standard Forier-inversion formla [see Gil-Pelaez 1951)] implies that, for r p R, P r r p,t+h < r p I t ) = 1 1 π Im E [exp ir p,t+h ) I t ] exp ir p )) d, 1) where Im.) denotes the imaginary part of a complex nmber. Using eqations 1)-1), we can show that the conditional distribtion fnction of r p,t+h, evalated at r p, for r p R, is given by: P r r p,t+h < r p I t ) = 1 1 π e Ā, r p ) d P h 1 Π t, 13) where, for any R ) ) exp h Ω 1 h sin h µ 1 r p ) )... Ā, r p ) = ) )... exp h Ω N h sin h µ N r p ) ) 14) VaR is a qantile measre that qantifies risk and measres the worst expected loss over a given horizon h typically a day or a week) at a given statistical confidence level typically 1% or 5%). Considering that conditional Vale-at-Risk, say V ar t r p,t+h ), is a positive qantity, we have. P r r p,t+h < V ar t r p,t+h ) I t ) = 1 1 π e Ā, V art r p,t+h )) d P h 1 Π t, 15) where Ā, V ar t r p,t+h )) is defined by the right-hand side of Eqation 14) where we replace the constant r p with V ar t r p,t+h ). The conditional VaR of r p,t+h can be calclated by inverting the conditional distribtion fnction 15). However, inverting analytically 15) is not feasible, for reasons explained below, and a nmerical soltion hereafter analytical approximation) is reqired. Proposition 1 The conditional VaR of a portfolio s simple retrns r p,t+h with coverage probability, denoted V ar t r p,t+h ), is the soltion of the following eqation: e Ā, V art r p,t+h )) d P h 1 Π t 1 )π =, 16) where Ā, V ar t r p,t+h )) is defined by the right-hand side of Eqation 14) where we replace the constant r p with V ar t r p,t+h ). 6

7 Proposition 1 reslts from Eqation 15). Ths, the conditional VaR of r p,t+h can be obtained by solving nmerically the eqation: fv ar t r p,t+h )) = e Ā, V art r p,t+h )) d P h 1 Π t 1 )π =. 17) The fnction fv ar t r p,t+h )) can be rewritten in the following form: f V ar t r p,t+h )) = π [P r r p,t+h < V ar t r p,t+h ) I t ) ]. 18) Using the properties of the probability distribtion fnction [monotonically increasing, lim P r r p,t+h < x) = x, and lim P r r p,t+h < x) = 1] we can show that 17) admits a niqe soltion. Another way to x + calclate the conditional VaR of r p,t+h is to consider the following optimization problem: V ar t r p,t+h) = arg min [e V art r p,t+h) Ā, V art r p,t+h )) d P h 1 Π t 1 ] )π. 19) The following algorithm shows how to estimate the conditional portfolio s VaR sing Hamilton filter: 1. Estimate the vector of nknown parameters θ = vecµ), vechω 1 ),..., vechω N ), vecp ) ) ) sing the maximm-likelihood method [see Hamilton 1994, pages )]. In Eqation ), vec denotes the colmn stacking operator and vech is the colmn stacking operator that stacks the elements on and below the diagonal only.. Estimate the conditional probabilities of regimes, Π t = ˆξ t t = P rs t = 1 I t ),..., P rs t = N I t )), by iterating on the following pair of eqations [see Hamilton 1994)]: ) ˆξt t 1 η t ˆξ t t = e ˆξt t 1 η t ), 1) ˆξ t+1 t = P ˆξ t t, ) where, for t = 1,..., T, η t = { } π) T Ω exp r t µ 1 ) Ω 1 1 r t µ 1 ) { } π) T Ω 1 1 exp r t µ ) Ω 1 r t µ ) {... } π) T Ω 1 N 1 exp r t µ N ) Ω 1 N r t µ N ) 7,

8 T is the sample size and the symbol denotes element-by-element mltiplication. Given a starting vale ˆξ 1 and the estimator ˆθ MV of the vector θ, one can iterate on 1) and ) to compte the vales of ˆξ t t and ˆξ t+1 t for each date t in the sample. Hamilton 1994, pages ) sggests several options for choosing the starting vale ˆξ 1. One approach is to set ˆξ 1 eqal to the vector of nconditional probabilities of regimes Π. Another option is to set ˆξ 1 = ρ, where ρ is a fixed N 1 vector of nonnegative constants smming to nity, sch as ρ = N 1 e. Alternatively, ρ can be estimated by maximm likelihood, along with θ, sbject to the constraint that e ρ = 1 and ρ j for j = 1,,..., N. 3. Given ˆθ MV and Π t, the portfolio s conditional VaR with coverage probability can be obtained by solving the optimization problem in 19). Notice that, in practice an exact soltion for 19) is not feasible, since the integral Ā,V art r p,t+h)) d is difficlt to evalate. The latter can be approximated sing the reslts from Imhof 1961), Bohmann 1961, 197), and Davies 1973, 198). These athors propose a nmerical approximation for distribtion fnction sing characteristic fnction. The proposed approximation introdces two types of errors: discretization and trncation errors. Davies 1973), provides a criterion to control for discretization error, and Davies 198) sggests three different bonds to control for trncation error. More details abot how to approximate nmerically the distribtion fnction sing the characteristic fnction can be fond in Dffie and Pan 1). An alternative measre to assess financial risk is Expected Shortfall, say ES t r p,t+h ), defined by the conditional expectation of loss given that the loss is beyond the VaR level: ES t r p,t+h ) = E t [r p,t+h r p,t+h V ar t r p,t+h )], where E t [.] defines the expectation conditional on I t. The following proposition provides a closedform soltion for the conditional ES of simple retrns r p,t+h. Proposition The conditional ES of a portfolio s simple retrns r p,t+h with coverage probability, denoted ESt r p,t+h ), is given by: ESt r p,t+h ) = 1 [ e R 1 V art r p,t+h )) 1 ] R V art r p,t+h )) P h 1 Π t, 3) π where R 1 V art r p,t+h )) = Diag h µ 1 Φ V ar t r p,t+h ) + h µ 1,..., h Ω h µ N Φ V ar t r p,t+h ) + h µ N 1 h ) h Ω N h ), R V art r p,t+h )) = Diag h Ω 1 h )exp 1 V ar t r p,t+h ) + h µ ) 1) h Ω,..., h Ω N h )exp 1 V ar t r p,t+h ) + h µ )) N) 1 h ) h Ω, N h ) Φ.) is the standard normal distribtion fnction. See proof in Appendix A. 8

9 The calclation of conditional ES given by Eqation 3) does not reqire any nmerical approximation as in case of VaR. In practice, to compte the conditional ES we need to rn steps 1 and of the above algorithm. Once we get the estimates of θ and Π t and for a known V ar t r p,t+h ), we evalate the standard normal distribtion fnctions Φ.) and plg-in the formla 3) to get an estimate of the conditional ES of r p,t+h. 3. Extension to aggregated retrns We extend the discssion on the analytical calclation of VaR to the case of aggregated retrns. To compte the VaR of aggregated retrns we follow the same steps of Section 3.1 [see Paragraph of Section 3.1]. Unfortnately, a closed-form soltion for the ES of aggregated retrns may not be tractable for reasons explained below. Now, consider h periods ahead aggregated retrns: r t:t+h = h r t+k, 4) k=1 where r t+k follows mltivariate Markov switching model 5). The portfolio s aggregated retrns is given by: r p,t:t+h = ᾱ h r t:t+h, 5) where ᾱ h = ᾱ h 1, ᾱh,..., ᾱh n) is the vector of weights attribted to each risky asset in the portfolio. We can show [see Taamoti 8)] that the conditional characteristic fnction of r p,t:t+h is given by, R and h, E [expir p,t:t+h ) J t ] = e A iµ ᾱ h exp iµ ᾱ h 1 l 1,l n 1 l 1,l n ᾱl h 1 ᾱl h ω l1 l ᾱl h 1 ᾱl h ω l1 l h 1 ζ t ζ t, where the matrix A.) is defined in 6) and e denotes the N 1 vector whose components are all eqal to one. The law of iterated expectations yields, E [expir p,t:t+h ) I t ] = e A iµ ᾱ h where, for any R D) =Diag exp iᾱ h µ 1 1 l 1,l n ᾱl h 1 ᾱl h ω l1 l h 1 D)Π t, R 6) ) ) ) ᾱh Ω 1ᾱ h,...,exp iᾱ )) h µ N ᾱh Ω Nᾱ h. The characteristic fnction 6) is expressed in terms of the observable information set I t. Using Eler s formla, the fnction 6) can be written as follows: E [expir p,t:t+h ) I t ] = e [D 1 ) + i D )] Π t, R 9

10 where, for any R D 1 ) =Re A iµ )) ) h 1 ᾱ h 1 l 1,l n ᾱh l 1 ᾱl h ω l1 l D), D ) =Im A iµ )) ) h 1 ᾱ h 1 l 1,l n ᾱh l 1 ᾱl h ω l1 l D). Re.) and Im.) denote the real and imaginary parts of a complex matrix, respectively. Using Gil-Pelaez s 1951) inversion formla, the conditional distribtion fnction of r p,t:t+h, evalated at r p, for r p R, is given by: P r r p,t:t+h < r p I t ) = 1 1 π e where, for any R D, r p )=Im exp i r p ) A iµ ᾱ h 1 l 1,l n D, r p ) d Π t, ᾱl h 1 ᾱl h ω l1 l h 1 D). 7) An explicit expression for matrix D, r p ) is not easy to compte when the horizon h is large and this makes it hard to get a closed-form soltion for ES. However, for a given short horizon h, one can calclate this expression and get an analytical formla for ES. Proposition 3 The conditional VaR of a portfolio s aggregated retrns r p,t:t+h with coverage probability, denoted V ar t r p,t:t+h ), is the soltion of the following eqation: e D, V art r p,t:t+h )) d Π t 1 )π =, where D, V ar t r p,t:t+h )) is defined by the right-hand side of Eqation 7) where we replace the constant r p with V ar t r p,t:t+h ). In practice, to calclate the conditional VaR of r p,t:t+h, we follow the same algorithm described in section 3.1). Given an estimator ˆθ MV of the vector θ = vecµ), vechω 1 ),..., vechω N ), vecp ) ) 8) and the conditional probabilities of regimes Π t, the conditional VaR of r p,t:t+h with coverage probability can be obtained by solving the following optimization problem: V ar t r D p,t:t+h) = arg min [e, V art r p,t:t+h )) d Π t 1 ] V art r p,t:t+h) )π, 9) where the integral D,V art r p,t:t+h)) d can be approximated sing the reslts from Imhof 1961), Bohmann 1961, 197), and Davies 1973, 198) [see Section 3.1)]. In the next section, we compare the analytical and simlation calclations of VaR and ES sing a mltivariate regime switching model estimated from a real data. 1

11 4 Analytical verss simlation calclations of VaR and ES In this section, we compare the analytical and simlation calclations of VaR and ES nder regime switching. Or comparison is based on the following mltivariate regime switching model: ) ) State 1: r t+1 = + Σ.1 1 ε t+1 V ar [Σ 1 ε t+1 ] = Ω 1 =, State : r t+1 =.3 ) Σ ε t+1 V ar [Σ ε t+1 ] = Ω = where ε t+1 N, I), and the transition probability matrix is given by: P = [ ), 3) ]. 31) The parameter vales in the model 3)-31) are obtained by estimating a two-regime Markov switching model sing a real data. The latter consists of monthly observations on S&P composite index and 1-years Government Bond from Janary 1958 to December 6 for a total of 588 observations. The retrns are compted sing the standard continos componding formla. We applied the Hamilton filter to recperate the conditional probabilities Π t [see Section 3.1] and compte the conditional VaR and ES. To compte analytically the conditional VaR, we apply the algorithm described in Section We calclate the VaR and ES for different portfolios constrcted by considering a nmber of different investment strategies: A) 1% stock; B) 75% stock and 5% bond; C) 5% stock and 5% bond; D) 5% stock and 75% bond; and E) 1% bond. Using model 3)-31), the analytical and simlated vales of VaR of aggregated retrns 5) and ES of simple retrns 7) at horizons h = 1,, 3, 4, 5 are presented in Tables 1 and, respectively. The second colmn of Table 1 shows the 1% VaR calclated sing 1 simlations and the third and forth colmns present the 1% VaR calclated sing an analytical approximation nder two different approximation errors 1 3 and 1 5, respectively. From this table we draw the following conclsions. First, there is almost no difference between the analytical and simlated vales of 1% VaR at horizons nder consideration; this is tre for all the investment strategies that we consider. Second, it seems that investing only in stock 1% stock) or bond 1% bond) is more risky, since for these portfolios the VaRs are higher. Finally, the best investment strategy, in terms of redcing risk, is the one that corresponds to 5% stock and 5% bond. We also consider other nivariate models and or conclsion, that there is almost no difference between simlation and analytical soltions of VaR, does not change. The reslts are available from the athor pon reqest. 3 The details abot how to control for the discretization and trncation errors in the nmerical approximation of the integral D,V art r p,t:t+h)) d and a GAUSS code for calclation of ES and VaR are available from the athor pon reqest. 11

12 Table compares the analytical and simlation calclations of 1% ES for different investment strategies that we discssed above. The second colmn shows the 1% ES calclated sing 1 simlations and the third colmn presents the 1% ES calclated sing the analytical formla of Proposition. From this table we draw the following conclsions. First, there is almost no difference between the analytical and simlated vales of 1% ES. Second, as we find for VaR, it is more risky to invest only in stock or bond. Frther, the best investment strategy, in terms of redcing risk, is the one that corresponds to 5% stock and 5% bond. Table 1: Simlation and analytical approximation of 1% VaR of a portfolio s aggregated retrns r p,t:t+h Horizon Simlation Analytical Approximation Approx Error = 1 3 Approx Error = 1 5 A) 1% Stock B) 75% Stock 5% Bond C) 5% Stock 5% Bond D) 5% Stock 75% Bond E) 1% Bond Note: This table presents the simlated and analytical approximation vales of 1% VaR of aggregated retrns given by Eqation 5) nder model 3)-31) and at horizons h = 1,, 3, 4, 5, where the horizon lengths are in monthly nits. The calclated 1% VaRs correspond to different portfolios constrcted by considering a nmber of different investment strategies: A) 1% stock; B) 75% stock and 5% bond; C) 5% stock and 5% bond; D) 5% stock and 75% bond; and E) 1% bond. The nmber of simlations sed to compte the 1% VaRs is 1. 1

13 Table : Analytical and simlation calclations of 1% ES of a portfolio s simple retrns r p,t+h Horizon Simlation Analytical A) 1% Stock B) 75% Stock 5% Bond C) 5% Stock 5% Bond D) 5% Stock 75% Bond E) 1% Bond Note: This table presents the analytical and simlated vales of 1% ES of simple retrns given by Eqation 7) nder model 3)-31) and at horizons h = 1,, 3, 4, 5, where the horizon lengths are in monthly nits. The calclated 1% ES correspond to different portfolios constrcted by considering a nmber of different investment strategies: A) 1% stock; B) 75% stock and 5% bond; C) 5% stock and 5% bond; D) 5% stock and 75% bond; and E) 1% bond. The nmber of simlations sed to compte the 1% ES is 1. In Figre 1, we calclate 1 vales of 1% ES sing the analytical formla in Proposition and 1 and 1 simlations. In this figre, we assme that we are at time t and we compte the 1% ES of simple retrns given by Eqation 7) at horizons t + 1, t +,..., t + 1, where the horizon lengths are in monthly nits. To get a conditional ES we se the conditional probabilities Π t [see Section 3.1]. The comptational time of compting these vales is presented in Table We se GAUSS for the analytical and simlation calclations of VaR and ES. Some characteristics of the compter hardware employed are: 1) Memory RAM): 358 MB; ) Processor: intelr) Core TM) Do CPU GHz; 3) System type: 3-bit Operating System. 13

14 From the latter, we see that sing the analytical formla reqires less than 1 second, whereas if we se 1 simlations the comptational time is abot 3 hors, 3 mintes, and 7 seconds. For 1 simlations, the comptational time decreases to 1 mintes and 18 seconds. However, Figre shows that sing 1 simlations may overestimate the vales of the 1% ES. 5 Ths, we need a very large nmber of simlations in order to get a good approximation for the tails of the distribtion of retrns and a large nmber of simlations reqires several hors Expected Shortfall Simlated ES N=1) Simlated ES N=1) Analytical ES Tre ES) Horizon Monthly) Figre 1: This figre presents the analytical and simlated 1 vales of 1% ES of simple retrns given by Eqation 7) nder model 3)-31) and sing the analytical formla in Proposition and different nmber of simlations N). To compte these vales, we assme that we are at time t and we compte the 1% ES of simple retrns at horizons t + 1, t +,..., t + 1, where the horizon lengths are in monthly nits. The calclated vales of 1% ES correspond to the investment strategy 5% stock and 5% bond. Table 3: Comptational time of 1 vales of 1% ES sing analytical and simlation methods 1 Simlated 1% ES 1 Analytical 1% ES N = 1 N = 1 Comptational Time Less than 1 sec 1 min 18 sec 3 h 3 min 7 sec Note: This table presents the comptational time of compting 1 vales of 1% ES of simple retrns 7) nder model 3)-31) and sing the analytical and simlation methods. The calclated vales of 1% ES correspond to the investment strategy 5% stock and 5% bond. N represents the nmber of simlations. 5 Similarly to Figre 1, in Figre we calclate 5 vales of 1% ES sing the analytical formla in Proposition and 1, 5, 1, 5, 1, and 3 simlations. In this figre, we assme that we are at time t and we compte the 1% ES of simple retrns given by Eqation 7) at horizons t + 1, t +,..., t + 5, where the horizon lengths are in monthly nits. 14

15 Expected Shortfall Analytical ES Tre ES) Simlated ES N=1) Simlated ES N=5) Simlated ES N=1) Simlated ES N=5) Simlated ES N=1) Simlated ES N=3) Horizon Monthly) Figre : This figre presents the analytical and simlated 5 vales of 1% ES of simple retrns given by Eqation 7) nder model 3)-31) and sing the analytical formla in Proposition and different nmber of simlations N). To compte these vales, we assme that we are at time t and we compte the 1% ES of simple retrns at horizons t + 1, t +,..., t + 5, where the horizon lengths are in monthly nits. The calclated vales of 1% ES correspond to the investment strategy 5% stock and 5% bond. 5 Conclsion We consider a regime switching model to captre important featres sch as heavy tails, persistence, and nonlinear dynamics in retrns. These featres are crcial to assess financial risk. Using Forier inversion method, we propose an analytical approximation for mlti-horizon Vale-at-Risk and a closed-form soltion for Expected Shortfall nder regime-switching. By comparing the Vale-at- Risks and Expected Shortfalls calclated analytically and sing simlations, we find that the both approaches lead to almost the same reslt. Frther, the analytical approach is less time and compter intensive compared to simlations, which are typically sed in risk management. 15

16 References [1] Billio, M., Pelizzon, L.,. Vale-at-Risk: a mltivariate switching regime approach. Jornal of Empirical Finance 7, [] Baer, C.,. Vale-at-Risk sing hyperbolic distribtions. Jornal of Economics and Bsiness 5, [3] Bohmann, H., Approximate Forier analysis of distribtion fnctions. Arkiv für Matematik 4, [4] Bohmann, H., 197. A method to calclate the distribtion fnction when the characteristic fnction is known. Nordisk Tidskr. Informationsbehandling BIT) 1, [5] Cardenas, J., Frchard, E., Koehler, E., Michel, C., Thomazea, I., VAR: One step beyond. Risk 1 1), [6] Cecchetti, S.G., Lam, P., Mark, N.C., 199. Mean reversion in eqilibrim asset prices. American Economic Review 8, [7] Davies, R.B., Nmerical inversion of a characteristic fnction. Biometrika 6 ), [8] Davies, R.B., 198. The distribtion of a linear combination of χ random variables. Applied Statistics 9, [9] Diebold, F.X., Rdebsch, G.D., Sichel, D.E., Frther evidence on bsiness cycle dration dependence. In Stock, J., and Watson, M. eds.), Bsiness Cycles, Indicators, and Forecasting, Chicago: University of Chicago Press and NBER. [1] Diebold, F.X., Ohanian, L.E., Berkowitz, J., Dynamic eqilibrim economies: A framework for comparing models and data. Review of Economic Stdies 65, [11] Dffie, D., and Pan, J., 1. Analytical Vale-at-Risk with Jmps and Credit Risk. Finance and Stochastics 5 ), [1] Drland, J.M., McCrdy, T.H., Dration dependent transitions in a Markov model of U.S. GNP growth. Jornal of Bsiness and Economic Statistics 1, [13] Gil-Pelaez, J., Note on the inversion theorem. Biometrika 38, [14] Gray, S.F., Modeling the conditional distribtion of interest rates as a regime switching process. Jornal of Financial Economics 4, 7-6. [15] Gidolin, M., Timmermann, A., 6. Term strctre of risk nder alternative econometric specifications. Jornal of Econometrics 131, [16] Feller, W., An introdction to probability theory and its applications. New York: Wiley. [17] Filardo, A.J., Bsiness cycle phases and their transitional dynamics. Jornal of Bsiness and Economic Statistics 1, [18] Hamilton, J.D., A new approach to the economic analysis of nonstationary time series and the bsiness cycle. Econometrica 57 ),

17 [19] Hamilton, J.D., Rational expectations econometric analysis of changes in regime: An investigation of the term strctre of interest rates. Jornal of Economic Dynamics and Control 1, [] Hamilton, J.D., Time series analysis. Princeton University Press. [1] Imhof, J.P., Compting the distribtion of qadratic forms in normal variables. Biometrika, 48, [] Lahiri, K., Wang, J.G., Predicting cyclical trning points with leading index in a Markov switching model. Jornal of Forecasting 13, [3] Pagan, A., Schwert, G.W., 199. Alternative models for conditional stock volatility. Jornal of Econometrics 45, [4] RiskMetrics, Technical docment, JP Morgan. New York, USA. [5] Rovinez, C., Going greek with VaR, Risk 1 ): [6] Taamoti, A., 8. The risk-retrn trade-off nder regime switching. Working Paper, Universidad Carlos III de Madrid. [7] Timmermann, A.,. Moments of Markov switching models. Jornal of Econometrics 96, [8] Trner, C.M, Startz, R., Nelson, C.R., A Markov model of heteroskedasticity, risk, and learning in the stock market. Jornal of Financial Economics 5, 3-. A Appendix: Proof of Propositions Proof of Proposition. ES t r p,t+h) = E t [r p,t+h r p,t+h V ar t r p,t+h )] = V ar t r p,t+h) V ar t r p,t+h) = r p r p f t r p r p V ar t r p,t+h )) dr p N j=1 P r s 1 t+h 1 = j I t ) exp 1 π h Ω j h ) P r r p V ar t r p,t+h ) I t ) ) r p h µj) h Ω j h ) dr p. 17

18 ) Since P r r p V ar t r p,t+h ) I t =, we get ES t r p,t+h) = 1 N π j=1 P r s t+h 1 = j I t ) V ar t r p,t+h) r p exp 1 h Ω j h ) = 1 [ N π j=1 P r s t+h 1 = j I t ) h Ω j h ) V ar t r p,t+h) 1 [ N π j=1 P r s t+h 1 = j I t ) h Ω j h ) V ar t r p,t+h) + 1 [ N π j=1 P r s t+h 1 = j I t ) h Ω j h ) V ar t r p,t+h) = 1 [ N π j=1 P r s t+h 1 = j I t ) h Ω V ar j h ) + 1 N j=1 P r s t+h 1 = j I t ) h µ j Notice that V ar t r p,t+h) and V ar t r p,t+h) rp + h µ ) j h Ω exp 1 j h ) 1 π h Ω j h ) ) [ V ar t r p,t+h) t r p,t+h) ) r p h µ j) h Ω dr j h ) p r p h Ω j h ) exp 1 h µj h Ω j h ) exp 1 h µj h Ω j h ) exp 1 ) rp + h µ j h Ω exp 1 j h ) 1 π h Ω j h ) exp 1 ) ] r p h µ j) h Ω dr j h ) p ) ] r p h µ j) h Ω dr j h ) p ) ] r p h µ j) h Ω dr j h ) p ) ] r p h µ j) h Ω dr j h ) p ) ] r p h µj) h Ω dr j h ) p. rp h µ ) j) h Ω dr p =exp 1 V ar t r p,t+h ) + h µ ) j) j h ) h Ω j h ) exp 1 rp h µ ) j) h Ω dr p = Φ j h ) V ar t r p,t+h ) + h µ j h Ω j h ) 3), 33) where Φ.) is the standard normal distribtion fnction. Ths, given 3) and 33), we have ES t r p,t+h) = 1 N P r s t+h 1 = j I t ) h π j h ) exp 1 V ar t r p,t+h ) + h µ ) j) j=1 h Ω j h ) + 1 N P r s t+h 1 = j I t ) h ) µ j Φ V ar t r p,t+h ) + h µ j. j=1 h Ω j h ) ESt r p,t+h ) can be written as follow ESt r p,t+h ) = 1 [ e R 1 V art r p,t+h )) 1 ] R V art r p,t+h )) P h 1 Π t, π where R 1 V art r p,t+h )) = Diag h µ 1 Φ V ar t r p,t+h ) + h µ 1,..., h Ω h µ N Φ V ar t r p,t+h ) + h µ N 1 h ) h Ω N h ), R V art r p,t+h )) = Diag h Ω 1 h )exp 1 V ar t r p,t+h ) + h µ ) 1) h Ω,..., h Ω N h )exp 1 V ar t r p,t+h ) + h µ )) N) 1 h ) h Ω, N h ) Φ.) is the standard normal distribtion fnction. See proof in Appendix. 18