Journal of Financial Stability

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1 Journal of Financial Stability 6 (2010) Contents lists available at ScienceDirect Journal of Financial Stability journal homepage: wwwelseviercom/locate/jfstabil Measuring potential market risk Mikael Bask Department of Economics and Statistics, Åbo Akademi University, FIN Turku, Finland article info abstract Article history: Received 23 September 2007 Received in revised form 19 November 2008 Accepted 16 July 2009 Available online 19 August 2009 JEL classification: G11 We argue herein that there is a fundamental and an important difference between the market risk and the potential market risk in financial markets We also argue that the spectrum of smooth Lyapunov exponents can be used in (, 2 )-analysis, which is a method to measure and monitor these risks The reason is that these exponents focus on the stability properties () of the stochastic dynamic system generating asset returns, while more traditional risk measures such as value-at-risk are concerned with the distribution of asset returns ( 2 ) 2009 Elsevier BV All rights reserved Keywords: Market risk Potential market risk Smooth Lyapunov exponents Stochastic dynamic system 1 Market risk and potential market risk Certainly, it is of uttermost importance to understand the mechanisms behind the variability of asset returns since they are associated with financial market risk Think, for example, of the financial turmoil that started in the US in the fall 2008 and thereafter was spread to the rest of the industrialized world with devastating effects In fact, the essence of financial market risk is that it reflects the chance that the actual return on a portfolio of assets may be very different than the expected return It is also for this reason a measure of financial market risk is crucial to carry through a successful risk management Today, financial investors often use value-at-risk to assess the market risk in their portfolio since they would like to ensure that the value of the portfolio does not fall below some minimum level that would expose the investor to insolvency The value-at-risk is the level of loss on a portfolio that is expected to be equaled or exceeded with a given small probability This risk measure can therefore be seen as a forecast of a given percentile, usually in the lower tail, of the probability distribution of portfolio returns Esa Jokivuolle and two anonymous referees are gratefully acknowledged for giving comments on an earlier draft of this paper Further on, I am grateful to the Bank of Finland and Stanford University for giving me the opportunity to write this paper during two longer research visits The usual disclaimer applies addresses: mikaelbask@abofi, mikaelbask@gmailcom The importance of value-at-risk as a measure of financial market risk is emphasized by the fact that the Basel Committee on Banking Supervision at the Bank for International Settlements imposes financial institutions to meet capital requirements based on this risk measure The widespread use of value-at-risk owes also much to Dennis Weatherstone, former chairman of JP Morgan & Co, who demanded to know the market risk of the company at 4:15 PM every day Weatherstone s request was met with a daily value-atrisk report Obviously, the probability distribution of asset and portfolio returns are time-dependent since they depend on the underlying economic structure To be more precise, quantities such as moneys and interest rates interact with each other through time and therefore constitute a dynamic system This means that the stability of this system is crucial for the variability of asset and portfolio returns In fact, everything else equal, these returns become more variable when the dynamic system decreases in stability and they become less variable when the same system increases in stability To write it slightly more formally, let 2 denote the conditional variance of asset or portfolio returns and (for reasons that will be clear below) let denote the stability of the dynamic system generating these returns Then, 2 = 2 (, ε), (1) where ε is shocks to the dynamic system This means that the conditional variance is not only affected by the stability of the system, it is also affected by the shocks to the system As we forcefully will /$ see front matter 2009 Elsevier BV All rights reserved doi:101016/jjfs

2 M Bask / Journal of Financial Stability 6 (2010) argue below, the conditional variance of asset returns increases when the dynamic system decreases in stability, but also when the variance of the shocks increases Further on, because of the shocks (ε) to the dynamic system, there is no one-to-one correspondence between the conditional variance ( 2 ) of asset and portfolio returns and the stability () of the system generating these returns This means that is not a measure of market risk Instead, is a measure of potential market risk, while 2 is a measure of market risk The rest of this paper is organized as follows: in Section 2, we give some intuition and background to (, 2 )-analysis, especially when it comes to our proposed measure of potential market risk, In Section 3, we formally define and further motivate it as a measure of potential market risk In Section 4, the estimation and inference of are in focus In Section 5, two variations and applications of (, 2 )-analysis are discussed In Section 6, the paper is finally concluded with a short sum up of (, 2 )-analysis 2 (, 2 )-analysis: some intuition and background This is the message in the paper: a change in an asset s potential market risk () may or may not change its market risk ( 2 ) since it depends on how much the variance of the shocks (ε) to the dynamic system generating asset returns has changed, if there has been any change at all Thus, the variance of the shocks distinguishes between an asset s market risk and its potential market risk For this reason should the stability of the dynamic system generating asset returns be contrasted with the variability of these returns This is accomplished in what we call (, 2 )-analysis As was claimed in the previous section, because of the presence of ε in (1), there is no one-to-one correspondence between 2 and, which also motivates (, 2 )-analysis To better understand this claim, imagine a very simple model of how the stock return, s, is determined at the stock market: s t = ˇs t 1 + ε t, (2) where ε is independently and identically distributed shocks with zero mean and finite variance Thus, the model in (2) is a linear autoregression of order one If we focus on the deterministic part of the model, it is explosive when ˇ > 1, conservative when ˇ =1 and stable when ˇ <1 Further on, if we assume that the model in (2) is stable, we can decompose the variability of stock returns as follows (see Bask and de Luna, 2002): 2 var(s t ) = var(ε t) 1 ˇ2 var(ε t ) 1 (exp()) 2 (3) Thus, stock returns are more variable when the shocks are more variable, but also when the model is less stable Be aware that for a stable model, < 0 since log ˇ This means that approaches zero from below when the model in (2) decreases in stability The stability concept just outlined can in a straightforward way be generalized to linear autoregressions of higher orders than one (see Bask and de Luna, 2002, for a careful demonstration and discussion of this claim) If the model is a linear autoregression of order n, one can characterize the stability of the model by looking at the modulus of the n eigenvalues that solve the eigenvalue problem for the model and this is because the eigenvectors of the model are orthogonal by definition In fact, if the autoregression is stable, the contraction of it in n-dimensional space is dominated by the largest eigenvalue This means that the largest eigenvalue can be used when comparing the stability of two linear autoregressions of order n This, however, is not the only way to compare the stability of two autoregressions An alternative way is to look at the product of the modulus of the n eigenvalues since it describes the rate of contraction of an n-dimensional volume in n-dimensional space Thus, the autoregression with the faster contraction is more stable Since we cannot escape the fact that linearity is a special case of non-linearity, we do not want to restrict the stability analysis to linear systems such as linear autoregressions We would instead like to have a tool that is able to determine the stability of nonlinear systems Unfortunately, this means that we cannot use the eigenvalues in the stability analysis since they are defined for linear systems But, fortunately, Bask and de Luna (2002) argue that the Lyapunov exponents can be used in the stability analysis of nonlinear dynamic systems To be more precise, in the same way as the product of the modulus of the n eigenvalues for an n-dimensional linear system describes the rate of contraction of an n-dimensional volume, the average of the n Lyapunov exponents for an n-dimensional nonlinear system describes the rate of contraction of an n-dimensional volume The reason is that the Lyapunov exponents for a linear system are the logarithms of this system s eigenvalues Moreover, the Lyapunov exponents for a non-linear system are the non-linear counterparts of the logarithms of the eigenvalues for a linear system To sum up the discussion so far: assets in financial markets are associated with financial market risk and this is not only because asset returns are volatile, but also that the chance that the actual returns may be very different than the expected returns It is therefore important to understand the mechanisms behind volatile asset returns We have also argued that it is the stability of the dynamic system generating asset returns that is the key for this understanding and that the stability can be measured with the Lyapunov exponents Now, after having motivated the use of (, 2 )-analysis by giving some intuition and background to it, let us move on and formally define our stability measure,, and further motivate why it is a measure of the stability of a dynamic system 3 : a measure of potential market risk This section has a twofold purpose: (i) to formally define the Lyapunov exponents of a stochastic dynamic system and (ii) to further motivate why these exponents provide a measure of the stability of a dynamic system That is, to further motivate why the Lyapunov exponents provide a measure of the potential market risk in a single asset or in a portfolio of assets 31 Definition of As we already have pointed out in the previous section, Bask and de Luna (2002) argue that the Lyapunov exponents can be used in the determination of the stability of a dynamic system Specifically, assume that the dynamic system, f : R n R n, generates S t+1 = f (S t ) + ε s t+1, (4) where S t and ε s t are the state of the system and a shock to the system, respectively For an n-dimensional system as in (4), there are n Lyapunov exponents that are ranked from the largest to the smallest exponent: 1 2 n, (5) and it is these quantities that provide information on the stability properties of the dynamic system f Assume temporarily that there are no shocks and consider how the dynamic system f amplifies a small difference between the

3 182 M Bask / Journal of Financial Stability 6 (2010) initial states S 0 and S 0 : S j S j = f j (S 0 ) f j (S 0 ) = Df j (S 0 )(S 0 S 0 ), (6) where f j (S 0 )=f(f(f(s 0 )))isj successive iterations of this system starting at state S 0 and Df is the Jacobian of the system: Df j (S 0 ) = Df (S j 1 )Df (S j 2 ) Df(S 0 ) (7) Then, associated with each Lyapunov exponent, i, i [1, 2,, n], there are nested subspaces U i R n of dimension n +1 i with the property that log e Df j (S 0 ) j 1 1 i lim j = lim j j log j e Df (Sk ), (8) k=0 for all S 0 U i U i+1 Due to Oseledec s multiplicative ergodic theorem, the limits in (8) exist and are independent of S 0 almost surely with respect to the measure induced by the process {S t } t=1 (see Guckenheimer and Holmes, 1983, for a careful definition of the Lyapunov exponents and their properties) Allow thereafter for shocks to the dynamic system, which means that the aforementioned measure is induced by a stochastic process In this case, the Lyapunov exponents have been named smooth Lyapunov exponents in the literature (see Bask and de Luna, 2002, but also McCaffrey et al, 1992; Nychka et al, 1992, even though they have not named these Lyapunov exponents smooth) 32 Motivation of The reason why the spectrum of smooth Lyapunov exponents provides information on the stability properties of a stochastic dynamic system may be seen by considering two starting values of a dynamic system, where the difference in starting values can be interpreted as a shock to the dynamic system at time t = 0 Moreover, the shocks to the system are identical at times t > 0 (see Bask and de Luna, 2002, for the origin of this discussion) The largest smooth Lyapunov exponent, 1, measures the slowest exponential rate of convergence of two trajectories of a stochastic dynamic system starting at the two starting values at time t = 0 In fact, 1 measures the convergence of a shock in the direction defined by the eigenvector corresponding to this Lyapunov exponent If the difference between the two starting values lies in another direction of R n, then the convergence is faster If 1 > 0, the trajectories diverge from each other and for a bounded dynamic system, this is an operational definition of chaotic dynamics The characterizing feature of a chaotic system is the property of sensitive dependence on initial conditions : any two trajectories with arbitrarily close, but not equal, starting values will diverge at exponential rates, even though they will remain within a bounded set This means that time series data generated by a chaotic system have an erratic and a highly irregular behavior 1 The property of sensitive dependence on initial conditions means that the predictive ability of the dynamic system is strongly limited In a way, if asset returns are chaotic, it could be a sign that the asset in focus is a risky investment However, even though the predictive ability of the system is strongly limited, it does not mean that the forecast errors necessarily must be large Moreover, it does not either mean that asset returns necessarily must be highly variable This is because the dynamic system is bounded Putting it differently, the sign of the largest smooth Lyapunov exponent 1 In the 1980s, a stream of papers started to emerge that searched for chaotic does not really catch our proposed definition of potential market risk 2 Recall our definition of this risk, but now formulated somewhat differently: a shock or impulse is hitting a dynamic system generating asset returns, which thereafter is propagated in the system so it has persistence over time Then, if the persistence of a shock with a certain amplitude for some reason increases in size, it means that even a small shock could have a large effect on asset returns Thus, the market risk has become potentially larger in this asset, even though the actual risk may be unaffected since the amplitudes of the shocks have decreased What really is needed is some kind of measure of the strength in the aforementioned impulse-propagation mechanism Recall that we argued in the previous section that in the same way as the product of the modulus of the n eigenvalues for an n-dimensional linear system describes the rate of contraction of an n-dimensional volume, the average of the n (smooth) Lyapunov exponents for an n-dimensional non-linear system describes the rate of contraction of an n-dimensional volume In other words, the average of the (smooth) Lyapunov exponents is catching our proposed definition of potential market risk since it measures the strength in the aforementioned impulse-propagation mechanism Specifically, the average of the smooth Lyapunov exponents, n 1 n i, (9) i=1 measures the exponential rate of convergence of two trajectories of a stochastic dynamic system in a geometrical average direction In fact, measures the convergence of a shock in the geometrical average of the directions defined by the eigenvectors corresponding to the different Lyapunov exponents Consequently, we can compare the stability of two stochastic dynamic systems via the smooth Lyapunov exponents since a shock has a smaller effect on the dynamic system with a smaller than for the system with a larger Since we are dealing with dissipative systems, which means that < 0 by definition, a dynamic system is more stable than another system if is more negative dynamics in time series data In the beginning, data were collected in the sciences, but later on time series data in economics and finance were also used (see Urbach, 1999, for an extensive although not complete list of papers) Two things made this stream of papers possible First, the dynamics that govern the evolution of a variable must be reconstructed since it may not be known In 1981, Takens (1981) proved that it is possible to reconstruct the unknown dynamics using only a scalar time series Remarkably, the reconstructed dynamics and the unknown dynamics hidden in the black box are equivalent in the sense that the reconstruction preserves topological information about the dynamic system such as the Lyapunov exponents Unfortunately, even though Takens (1981) reconstruction theorem have made an impact on the sciences, it is still rather unknown in economics and finance However, more was needed than a reconstruction theorem Specifically, a method to calculate the Lyapunov exponents from time series data was necessary and the first such method was presented in 1985 by Wolf et al (1985) Several other methods have thereafter been proposed in the literature Some of the methods only calculate the largest Lyapunov exponent, whereas other methods calculate the whole spectrum of Lyapunov exponents If we turn to the findings, most of the early papers demonstrated that time series data in economics and finance often were chaotic However, two limitations were later on revealed in these studies The first was the lack of statistical inference and the second was that some of the methods to calculate the Lyapunov exponents suffered from upward bias This also meant that the number of new studies diminished over time However, as will be clear in Section 4, both limitations have been resolved in the literature 2 We should, however, make it clear here that the sign of the largest smooth Lyapunov exponent is not at all uninteresting from the perspective of a successful risk management This is because the erratic and highly irregular behavior of chaotic time series data Typically, volatility is strongly time-dependent and there are also large jumps in data In other words, 1 > 0 calls for a careful analysis of the further development of asset returns

4 M Bask / Journal of Financial Stability 6 (2010) After we have formally defined our stability measure,, and further motivated why it is a measure of the stability of a stochastic dynamic system and also therefore a measure of the potential market risk, let us move on to the estimation and inference of 4 Testing for a change in The purpose of this section is twofold: (i) to show how the smooth Lyapunov exponents can be estimated from time series data and (ii) to discuss how hypothesis tests of these exponents can be constructed Thus, the purpose is to show how an asset s potential market risk can be estimated from an asset return series and to discuss how to test for a change in this risk 41 Estimation of Since the actual form of the dynamic system f is not known, it may seem like an impossible task to determine the stability of this system Fortunately, it is possible to reconstruct the dynamics using only a scalar time series and thereafter measure the stability of the reconstructed dynamic system Therefore, associate the dynamic system f with an observer function, g : R n R, that generates asset returns: s t = g(s t ) + ε m t, (10) where s t S t and ε m t are the asset return and a measurement error, respectively Thus (10) means that the asset return series {s t } N t=1 (11) is observed, where N is the number of consecutive returns in the time series Specifically, the observations in a scalar time series, like the asset return series in (11), contain information about unobserved state variables that can be used to define a state in present time Therefore, let T = (T 1,T 2,,T M ) (12) be the reconstructed trajectory, where T t is the reconstructed state and M is the number of states on the reconstructed trajectory Moreover, each T t is given by T t ={s t,s t+1,,s t+m 1 }, (13) where m is the embedding dimension Thus, T is an M m matrix and the constants M, m and N are related as M = N m +1 Takens (1981) proved that the map (S t ) ={g(f 0 (S t )),g(f 1 (s t )),,g(f m 1 (S t ))}, (14) which maps the n-dimensional state S t onto the m-dimensional state T t, is an embedding when m >2n This means that the map is a smooth map that performs a one-to-one coordinate transformation and has a smooth inverse Be aware that the condition m >2n is a sufficient but not a necessary condition to have an embedding (see Sauer et al, 1991) A map that is an embedding preserves topological information about the unknown dynamic system f such as the smooth Lyapunov exponents In particular, the map induces a function, h : R m R m, on the reconstructed trajectory, T t+1 = h(t t ), (15) which is topologically conjugate to the unknown dynamic system f That is, h j (T t ) = f j 1 (T t ) (16) Thus, h is a reconstructed dynamic system that has the same smooth Lyapunov exponents as the unknown dynamic system f Then, to be able to estimate the smooth Lyapunov exponents of the dynamic system f generating asset returns, one must first estimate h However, since s t s t+1 h : s t+m 1 s t+1 s t+2 v(s t,s t+1,,s t+m 1 ) the estimation of h reduces to the estimation of v:, (17) s t+m = v(s t,s t+1,,s t+m 1 ), (18) which essentially is a non-linear autoregression of order m (see the discussion in Section 2) Moreover, since the Jacobian of h at the reconstructed state T t is Dh(T t ) = s t s t+1 s t+2 s t+m 1, (19) a feedforward neural network is recommended to estimate the above derivatives to be able to estimate the smooth Lyapunov exponents (see Dechert and Gencay, 1992; Gencay and Dechert, 1992; McCaffrey et al, 1992; Nychka et al, 1992) This is because Hornik et al (1990) have shown that a map and its derivatives of any unknown functional form can be approximated arbitrarily accurately by such a network After we have shown how to estimate, an intuitive explanation of Takens (1981) embedding theorem may be in place due to its central importance For the sake of the argument, assume that M 1 M and M 2 M are two subspaces of dimension n 1 and n 2, respectively, where M R m is an m-dimensional manifold that surrounds the reconstructed dynamics In general, the two subspaces intersect in a subspace of dimension n 1 + n 2 m Thus, when this expression is negative, there is no intersection of the subspaces Therefore, and of greater interest, the self-intersection of an n-dimensional manifold with itself fails to occur when m >2n (see Sauer et al, 1991, for generalizations of this theorem) A difficulty here is that the dimension, n, of the true dynamics in (4) is not known, which means that the required embedding dimension, m, in(14) is not either known This problem can, however, be solved indirectly by making use of a generic property of a proper reconstruction Namely, the dynamics in original phase space must be completely unfolded in reconstructed phase space Thus, if m is too low, distant states in original phase space are close states in reconstructed phase space and are therefore false neighbors There are two closely related methods to calculate the required m from time series data to have an embedding: (i) false nearest neighbors and (ii) the saturation of invariants on the reconstructed dynamics such as the Lyapunov exponents The first method is based on the aforementioned generic property of a proper reconstruction, which means that by increasing m, the dynamics is completely unfolded when there are no false neighbors in reconstructed phase space (see Kennel et al, 1992) The second method utilizes the fact that when the dynamics is completely unfolded, the Lyapunov exponents and other invariants such as the fractal dimension are independent of m If, however, the dynamics is not unfolded, these invariants depend on m Therefore, by increasing m, we have an embedding when the value of an

5 184 M Bask / Journal of Financial Stability 6 (2010) invariant stops changing (see Fernández-Rodríguez et al, 2005, for a statistical test for chaotic dynamics based on this observation) Finally, since the m-dimensional reconstructed dynamic system h in (15) almost always has a larger dimension than the n-dimensional dynamic system f in (4), the number of spurious Lyapunov exponents is m n Of course, if the map ( ) in(14) is an embedding when m = n, there are no such exponents It is an open question how to identify the n non-spurious or true Lyapunov exponents in the spectrum of m Lyapunov exponents A key, however, to this identification is that the true Lyapunov exponents by definition are associated with the derivatives in Dh( ) in (19) that are in the n-dimensional tangent space of the image of the dynamics under the m-dimensional embedding in (14) In other words, it is necessary to identify this tangent space in embedding space to be able to identify the true Lyapunov exponents (see Dechert and Gencay, 1996, 2000; Gencay and Dechert, 1996) 42 Inference of Shintani and Linton (2004) derive the asymptotic distribution of a neural network estimator of the smooth Lyapunov exponents: M( ˆiM i ) N(0,V i ), (20) where ˆ im is the estimator of the ith Lyapunov exponent, based on the M reconstructed states on the trajectory in (12), V i is the variance of the ith Lyapunov exponent and i [1, 2,, n] If we turn to the average of the smooth Lyapunov exponents, our conjecture is that asymptotic normality holds for a neural network estimator of 1 n n i=1 i since the eigenvectors corresponding to the different Lyapunov exponents are pairwise orthogonal: M( ˆM ) N(0,V), (21) where ˆ M is the estimator of 1 n n i=1 i, based on the M reconstructed states on the trajectory in (12), and V is the variance of 1 n n i=1 i If our conjecture in (21) is correct, it is possible to make inference of a change in potential market risk, More on this in the next section An alternative to Shintani and Linton (2004) is to derive an empirical distribution for via bootstrapping (see Bask and Gencay, 1998; Gencay, 1996, for the case of 1 ) However, as is demonstrated in Ziehmann et al (1999), this approach is not straightforward for multiplicative ergodic statistics since the limits in (8) may not exist under bootstrapping 5 (, 2 )-analysis: variations and applications The first we do herein is to shortly present previous studies in which (, 2 )-analysis has been used, even though the analysis may not have been named this in these studies In fact, two variations of (, 2 )-analysis are presented Thereafter, two applications of (, 2 )-analysis are discussed that both relate to a successful risk management 51 Two variations of (, 2 )-analysis The origin of (, 2 )-analysis is found in Bask and de Luna (2002) since it is argued therein that when the volatility of a variable modelled is of interest, one should also consider the stability properties of the same model Specifically, a parametric model in the form of a polynomial autoregression on a projected space is fitted to the time series, which thereafter is utilized to measure the stability and volatility of the variable of interest (see Bask and de Luna, 2002; de Luna, 1998) A large-scale analysis of the European monetary integration, with the creation of the EMU, is carried out in Bask and de Luna (2005) utilizing the methodology in Bask and de Luna (2002) and daily exchange rate data from the 1990s and early 2000s Specifically, changes in the stability and volatility of 16 currencies and in the volatility of the shocks to these currencies are examined The results show that when most of the currencies became more/less stable, a majority of them also became less/more volatile For example, following the agreement of the Maastricht Treaty in December 1991, most currencies became more stable and less volatile, whereas they became less stable and more volatile when the Danish public in June 1992 voted against the treaty When, however, a successful risk management is in focus, it is necessary to measure the stability of the true stochastic dynamic system generating asset or portfolio returns and not the stability of the model fitted to these returns The reason is that there is no guarantee that the smooth Lyapunov exponents for the true system and the model selected to measure volatility coincide with each other In other words, the market risk and the potential market risk should not be estimated using the same model We therefore argue that the non-parametric approach outlined above should be used when estimating the potential market risk in an asset or in a portfolio of assets, whereas a parametric model should be used when estimating the asset s or the portfolio s market risk Even though Bask and Widerberg (2009) do not focus on the market risk and the potential market risk in the Nordic power market, they utilize the aforementioned methodology when examining how the integration process at this market has affected the stability and volatility of electricity prices Specifically, the non-parametric approach outlined above is used when estimating the stability of electricity prices, whereas four different ARCH models are used when estimating the volatility of the same prices Using the daily average of hourly electricity prices at Nord Pool between 1993 and 2005, it is demonstrated that electricity prices most of the time have increased in stability and decreased in volatility when the Nordic power market has expanded and the degree of competition has increased (see Bask et al, forthcoming, for the competition result) 3 Therefore, if economic models of the energy sector are developed in an attempt to understand the behavior of energy prices and their dependence on the market structure, the stability of the dynamic system generating energy prices should be added as an important dimension in which model and data should be matched Thus, that electricity prices most of the time have increased in stability and decreased in volatility when the Nord Pool area has expanded and the degree of competition has increased should be properties of a well-formulated model of the Nordic power market Last but not least, Potter (2000) should be mentioned who argues (in a footnote) that 1 can be used not only to categorize a time series as stable or unstable (in the sense it is a chaotic time series), but also that it provides a measure of the convergence speed to equilibrium (see Shintani, 2006, who use 1 when looking at convergence speeds of exchange rates toward purchasing power parity) 52 Two applications of (, 2 )-analysis There are two closely related applications of (, 2 )-analysis that both relate to a successful risk management: 3 Bask et al (2007) find chaotic dynamics in electricity prices in one of the time series since the null hypothesis of a non-positive largest smooth Lyapunov exponent, or non-chaotic dynamics, is rejected at the 1 per cent level in this specific case

6 M Bask / Journal of Financial Stability 6 (2010) (i) To monitor the evolution of an asset s or a portfolio s market risk ( 2 ) and its potential market risk () This means that (, 2 )- analysis is used as a tool to detect changes in the actual and potential market risks Think of an asset with an unchanged market risk Specifically, the volatility of asset returns is measured in a rolling window, where it is found that there are no significant changes in volatility during some period of time (see Leeves, 2007, who examine stock prices before and after the Asian crisis in 1997) However, during the same period of time, the stability of asset returns has decreased This means that the asset s potential market risk has increased In this case, (, 2 )-analysis is able to give an early warning that an increase in the asset s actual market risk may occur To write this more formally, let [t {s t p+1,s t p+2,, s t }] denote the stability of asset returns at time t conditioned on the asset return series for the period in the window [t p +1,t] Thus, there are p consecutive returns in this window, which is rolling Therefore, let [t +1 {s t p+2, s t p+3,, s t+1 }] denote the stability of asset returns at time t + 1 conditioned on the asset return series in the window [t p +2, t + 1] Then, to test for a change in the potential market risk in this asset, we consider the following null and alternative hypotheses: H 0 : [t + 1] = [t], H 1 : [t + 1] /= [t], (22) or, equivalently, H 0 : [t + 1] = 0, H 1 : [t + 1] /= 0 (23) What is needed is a statistical theory for this test (possibly based on the conjecture in (21)) An alternative approach to monitor an asset s potential market risk, which is in the spirit of the studies in Bask and de Luna (2005) and Bask and Widerberg (2009), isthe following approach: let [t {st p0 +1,s t p0 +2,,s t } ] denote the stability of asset returns at time t conditioned on the asset return series for the period [t p 0 +1, t] Thus, there are p 0 consecutive returns in this period Moreover, let [t + p {st+1 1,s t+2,,s t+p1 } ] denote the stability of asset returns at time t + p 1 conditioned on the asset return series for the period [t +1,t + p 1 ] Thus, there are p 1 consecutive returns in this period Then, to test for a change in the potential market risk in this asset between the two periods, we consider the following null and alternative hypotheses: H 0 : [t + p 1 ] = [t], H 1 : [t + p 1 ] /= [t] (24) What again is needed is a statistical theory for this test (possibly based on the conjecture in (21)) The fundamental difference between the two approaches above is that the two periods in the latter approach are not overlapping Of course, both approaches may also be used to monitor changes in the actual and potential market risks in a portfolio of assets, even though the former approach is more suitable in a real-time monitoring of these risks (either in a single asset or in a portfolio of assets) (ii) To compare the market risk and potential market risk in two portfolios of assets Imagine an investor who is planning to make a portfolio investment, but is unsure about which asset to invest in It goes without saying that if the investor is using mean-variance analysis when building the portfolio, the potential market risk in different assets should not directly affect the composition of the portfolio On the other hand, since a portfolio s actual market risk depends on its potential market risk, 2 (), one should not neglect the latter risk Think of a situation in which two different assets would give rise to portfolios with the same risk-return profile In other words, it does not matter which of the two assets the investor invests in since mean-variance analysis cannot distinguish between the two investments We argue that the investor should invest in the asset that gives rise to the portfolio with the smaller potential market risk since the market risk is timevarying, 2 ([t]), and that it may be the case that the market risk in the portfolio with the higher potential market risk is unusually low 6 Sum up of (, 2 )-analysis We have argued in this paper in favor of as a measure of potential market risk and also discussed how this measure can be used in what we have called (, 2 )-analysis, which is a tool in which the actual and potential market risks in a single asset or in a portfolio of assets can be distinguished The former risk the actual market risk is concerned with the distribution of asset or portfolio returns, whereas the latter risk the potential market risk is concerned with the stability of the stochastic dynamic system generating these returns We have also argued that a non-parametric approach should be used when estimating, whereas a parametric model (most likely) is used when estimating 2 This is because it is necessary to measure the stability of the true stochastic dynamic system generating asset or portfolio returns and not the stability of the volatility model fitted to these returns This is because there is no guarantee that the smooth Lyapunov exponents for the true system and the model selected to measure volatility coincide with each other Finally, we are the first to admit that (, 2 )-analysis at this stage is an incomplete tool from a statistical point of view What remains to be done is to derive the asymptotic distribution of a neural network estimator of, also including the case when the noise is not restricted to additive noise as in (4), and thereafter take this tool to financial data to study its merits and possible weaknesses References Bask, M, de Luna, X, 2002 Characterizing the degree of stability of non-linear dynamic models Studies in Nonlinear Dynamics and Econometrics 6 (1) (art 3) Bask, M, de Luna, X, 2005 EMU and the stability and volatility of foreign exchange: some empirical evidence Chaos, Solitons and Fractals 25, Bask, M, Gencay, R, 1998 Testing chaotic dynamics via Lyapunov exponents Physica D 114, 1 2 Bask, M, Liu, T, Widerberg, 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