Description
Market risk is the risk of losses in positions arising from movements in market prices.
Mikael Bask
Measuring potential market risk
Bank of Finland Research Discussion Papers 20 • 2007
Suomen Pankki Bank of Finland PO Box 160 FI-00101 HELSINKI Finland + 358 10 8311 http://www.bof.fi
Bank of Finland Research Discussion Papers 20 • 2007
Mikael Bask*
Measuring potential market risk
The views expressed are those of the author and do not necessarily reflect the views of the Bank of Finland. * E-mail: [email protected]
Esa Jokivuolle is gratefully acknowledged for giving comments on an earlier draft of this paper. The usual disclaimer applies.
This paper can be downloaded without charge from http://www.bof.fi or from the Social Science Research Network electronic library at http://ssrn.com/abstract_id=1029958.
http://www.bof.fi ISBN 978-952-462-388-9 ISSN 0785-3572 (print) ISBN 978-952-462-389-6 ISSN 1456-6184 (online) Helsinki 2007
Measuring potential market risk
Bank of Finland Research Discussion Papers 20/2007
Mikael Bask Monetary Policy and Research Department
Abstract
The difference between market risk and potential market risk is emphasized and a measure of the latter risk is proposed. Specifically, it is argued that the spectrum of smooth Lyapunov exponents can be utilized in what we call (?, ?2)-analysis, which is a method to monitor the aforementioned risk measures. 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 returns (?2). Keywords: market risk, potential market risk, smooth Lyapunov exponents, stochastic dynamic system, value-at-risk JEL classification number: G11
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Menetelmä potentiaalisen markkinariskin määrittämiseksi
Suomen Pankin keskustelualoitteita 20/2007
Mikael Bask Rahapolitiikka- ja tutkimusosasto
Tiivistelmä
Tutkimuksessa tarkastellaan markkinariskin ja potentiaalisen markkinariskin eroa ja tarkasteluissa esitellään keino jälkimmäisen mittaamiseksi. Täsmällisesti ottaen työssä esitetään, että sileiden Lyapunovin eksponenttien spektriä voidaan hyödyntää ns. (?, ?2)-analyysissa, jota menetelmää käytetään markkinariskin ja potentiaalisen markkinariskin tarkkailussa. Analyysin soveltuvuus näiden riskien seurantaan johtuu siitä, että sileät Lyapunovin eksponentit korostavat rahoitusvaateiden tuottoja synnyttävän stokastisen dynaamisen järjestelmän vakausominaisuuksia (?), kun perinteisissä riskimittareissa, kuten value-at-risk, kyse on sen sijaan tuottojen jakaumasta (?2). Avainsanat: markkinariski, potentiaalinen markkinariski, sileät Lyapunovin eksponentit, stokastinen dynaaminen järjestelmä, value-at-risk JEL-luokittelu: G11
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Contents
Abstract....................................................................................................................3 Tiivistelmä (abstract in Finnish) ..............................................................................4 1 Measuring market risk ....................................................................................7 2 ?: a measure of potential market risk ............................................................8 3 Testing for a change in ? ...............................................................................10 4 (?, ?2)-analysis ................................................................................................13 5 Concluding remarks.......................................................................................14 References..............................................................................................................15
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1 Measuring market risk
Financial market risk re?ects the chance that the actual return on an asset or a portfolio of assets may be very di?erent than the expected return. For this reason, a measure of market risk is necessary to carry through a successful risk management. Nowadays, ?nancial 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 returns.1 Of course, the probability distribution of returns is not constant since asset returns depend on the underlying economic structure.2 To be more precise, quantities such as moneys and interest rates interact with each other through time, and, therefore, constitute a dynamic system, meaning that the stability of the system generating asset returns is crucial for the variance of these returns. That is, a less stable dynamic system is associated with more variable asset returns, meaning that an asset is potentially more risky than another asset, if the returns of the former asset is generated by a less stable system. To clarify this further, let ?2 denote the conditional variance of asset returns, and (for reasons explained below) let ? denote the stability of the dynamic system generating these returns. Then, ?2 = ?2 (?, ?) (1.1)
where ? is exogenous shocks to the dynamic system, meaning that the conditional variance (?2 ) is not only a?ected by the system’s stability (?), it is also a?ected by shocks to the system (?). Speci?cally, the conditional variance of asset returns increases when the dynamic system is less stable, but also when the variance of the shocks increases. Thus, because of shocks to the system, there is no one-to-one correspondence between the conditional variance of asset returns and the stability of the dynamic system generating these returns, meaning that ? is not a measure of market risk. Instead, ? is a measure of potential market risk, while ? 2 is a measure of market risk. In other words, a change in an asset’s potential market risk may or may not change its market risk since it depends on how much the variance of the shocks to the dynamic system generating asset returns has changed, if there has been any change at all. Thus, the variance of the shocks distinguishes
The importance of value-at-risk as a measure of ?nancial market risk is emphasized by the fact that the Basel Committee on Banking Supervision at the Bank for International Settlements imposes ?nancial institutions to meet capital requirements based on value-at-risk. The widespread use of value-at-risk as a measure of market risk also owes much to Dennis Weatherstone, former chairman of JP Morgan & Co., who demanded to know the market risk of the company at 4:15 P.M. every day. Weatherstone’s request was met with a daily value-at-risk report. 2 For this reason, Engle’s (1982) ARCH model and subsequent developments of the model are invaluable tools since they can be used to estimate and predict conditional moments characterizing the probability distribution of returns.
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between an asset’s market risk and its potential market risk. Therefore, the stability of the dynamic system generating asset returns should be contrasted with the volatility of these returns, and this is accomplished in what we call (?, ? 2 )-analysis.
2 ?: a measure of potential market risk
The purpose of this section is twofold: (i) to de?ne the Lyapunov exponents of a stochastic dynamic system; and (ii) to motivate why these exponents provide a measure of a system’s stability, meaning that they also provide a measure of potential market risk. De?nition of ? Bask and de Luna (2002) argue that the spectrum of smooth Lyapunov exponents can be used in the determination of the stability of a stochastic dynamic system. Speci?cally, assume that the dynamic system, f : Rn ? Rn , generating asset returns is St+1 = f (St ) + ?s t+1 (2.1)
where St and ?s t are the state of the system and a shock to the system, respectively, both at time t ? [1, 2, . . . , ?]. For an n-dimensional system as in (2.1), there are n Lyapunov exponents that are ranked from the largest to the smallest exponent ?1 ? ?2 ? . . . ? ?n (2.2)
and it is these exponents that provide information on the stability properties of the system f . Then, how are the Lyapunov exponents de?ned? Assume temporarily that there are no shocks to the system f , and consider how it ampli?es a small 0 di?erence between the initial states S0 and S0
0 0 0 Sj ? Sj = f j (S0 ) ? f j (S0 ) ' Df j (S0 ) (S0 ? S0 )
(2.3)
where f j (S0 ) = f (· · · f (f (S0 )) · · · ) denotes j successive iterations of the system starting at state S0 , and Df is the Jacobian of the system Df j (S0 ) = Df (Sj ?1 ) Df (Sj ?2 ) · · · Df (S0 ) (2.4)
Then, associated with each Lyapunov exponent, ?i , i ? [1, 2, . . . , n], there are nested subspaces U i ? Rn of dimension n + 1 ? i with the property that ?i ? lim loge kDf j (S0 )k 1X = lim loge kDf (Sk )k j ?? j ?? j j k=0
j ?1
(2.5)
for all S0 ? U i ? U i+1 . Due to Oseledec’s multiplicative ergodic theorem, the limits in (2.5) exist and are independent of S0 almost surely with respect to 8
3 the measure induced by the process {St }? Then, allow for shocks to the t=1 . system f , meaning that the measure is induced by a stochastic process. In this case, the Lyapunov exponents have been named smooth Lyapunov exponents in the literature.
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 di?erent starting values of a system, where the di?erence is an exogenous shock at time t = 0. The largest smooth Lyapunov exponent, ?1 , measures the slowest exponential rate of convergence of two trajectories of the dynamic system starting at these di?erent starting values at time t = 0, but with identical exogenous shocks at times t > 0.4 In fact, ?1 measures the convergence of a shock in the direction de?ned by the eigenvector corresponding to this exponent. However, if the di?erence between the two starting values lies in another direction of Rn , then the convergence is faster. Thus, ?1 measures a ‘worst case scenario’. The average of the smooth Lyapunov exponents 1X ?i ?? n i=1
n
(2.6)
measures the exponential rate of convergence in a geometrical average direction. That is, the convergence of two trajectories of the dynamic system in the geometrical average of the directions de?ned by the eigenvectors corresponding to the di?erent exponents. Thus, ? measures an ‘average scenario’. We can, therefore, compare the stability of two stochastic dynamic systems via the smooth Lyapunov exponents since a one-time shock has a smaller e?ect on the dynamic system with a smaller ? than for the system with a larger ?. Thus, since we are dealing with dissipative systems, meaning that ? < 0 by de?nition, a dynamic system is more stable than another system, if ? is more negative. An extensive discussion of the spectrum of smooth Lyapunov exponents as a measure of the stability of a stochastic dynamic system is provided in Bask and de Luna (2002). As an illustration, it is shown therein that the decrease in volatility of the exchange rates between the Swedish Krona and the ECU/Euro, after the launch of the Euro, is due to a decrease in the volatility of the shocks to the dynamic system generating these exchange rates and not to a more stable system. Thus, one can say that the market risk decreased, but that the potential market risk was unchanged.
See Guckenheimer and Holmes (1983) for a careful de?nition of the Lyapunov exponents and their properties. 4 When ?1 > 0, the trajectories diverge from each other, and for a bounded stochastic dynamic system, this is an operational de?nition of chaotic dynamics.
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3 Testing for a change in ?
Contrary to risk measures like value-at-risk, potential market risk does not have a straightforward economic interpretation. However, it is not level of potential market risk that is of interest. Instead, it is the change in this risk, ??, since we are interested in the potential change in market risk. The purpose of this section is, therefore, 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. In other words, 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. 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 the system. However, it is possible to reconstruct the dynamics of the system using only a scalar time series, and, thereafter, to measure the stability of this reconstructed system. Therefore, associate the system f with an observer function, g : Rn ? R, that generates observed asset returns st = g (St ) + ?m t (3.1)
where st ? St and ?m are the asset return and a measurement error, t respectively, both at time t. Thus, (3.1) means that the asset return series {st }N t=1 (3.2)
is observed, which is used to reconstruct the dynamics of the system f , where N is the number of consecutive returns in the time series. Speci?cally, the observations in a scalar time series, like the asset return series in (3.2), contain information about unobserved state variables that can be used to de?ne a state in present time. Therefore, let T = (T1 , T2 , . . . , TM )0 (3.3)
be the reconstructed trajectory, where Tt is the reconstructed state at time t and M is the number of states on the reconstructed trajectory. Each Tt is given by Tt = {st , st+1 , . . . , st+m?1 } (3.4)
where m is the embedding dimension and time t ? [1, 2, . . . , N ? m + 1]. 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 © ¡ ¢ ¡ ¢ ¡ ¢ª ? (St ) = g f 0 (St ) , g f 1 (St ) , . . . , g f m?1 (St ) (3.5) 10
which maps the n-dimensional state St onto the m-dimensional state Tt , is an embedding if m > 2n.5 This means that the map is a smooth map that performs a one-to-one coordinate transformation and has a smooth inverse. A map that is an embedding preserves topological information about the unknown dynamic system, like the smooth Lyapunov exponents, and, in particular, the map induces a function, h : Rm ? Rm , on the reconstructed trajectory Tt+1 = h (Tt ) which is topologically conjugate to the unknown system f . That is hj (Tt ) = ? ? f j ? ??1 (Tt ) (3.7) (3.6)
An intuitive explanation of Takens’ (1981) embedding theorem may be in place due to its importance in the estimation of ?. For the sake of the argument, assume that M1 ? M and M2 ? M are two subspaces of dimension n1 and n2 , respectively, where M ? Rm is an m-dimensional manifold representing phase space for the reconstructed dynamic system. In general, two subspaces intersect in a subspace of dimension n1 + n2 ? m, meaning that when this expression is negative, there is no intersection of the two 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 Takens’, 1981, theorem). A problem is that the dimension of the ‘true’ dynamic system is not known, meaning that the required embedding dimension is not either known. This problem can, however, be solved indirectly by making use of a generic property of a proper reconstruction, namely, that the dynamics in original phase space must be completely unfolded in reconstructed phase space. In other words, if the embedding dimension is too low, the dynamics is not completely unfolded, meaning that distant states in original phase space are close states in reconstructed phase space, and, therefore, are named false neighbors in phase space. There are at least two methods to calculate the required embedding dimension from an observed time series: (i) false nearest neighbors; and (ii) the saturation of invariants on the reconstructed dynamics such as the saturation of the Lyapunov exponents. The ?rst method is based on the aforementioned generic property of a proper reconstruction, meaning that by increasing the embedding dimension, the dynamics is completely unfolded when there are no false neighbors in reconstructed phase space (see Kennel et al, 1992). The second method, the saturation of invariants on the reconstructed dynamics, is based on the fact that when the dynamics is completely unfolded, the Lyapunov exponents and other invariants such as entropy and fractal dimension are independent of the embedding dimension. If, however, the dynamics is not completely unfolded in reconstructed phase space, these invariants depend on the embedding dimension. Therefore, by increasing the embedding dimension, the dynamics is completely unfolded when the value of an invariant stops changing (see Fernández-Rodríguez et al, 2005, for an example regarding the largest Lyapunov exponent and a statistical test for chaotic dynamics). 6 Since the m-dimensional system h has a larger dimension than the n-dimensional system f , the number of smooth Lyapunov exponents that are spurious is m ? n. This issue is discussed in Dechert and Gencay (1996)—(2000) and Gencay and Dechert (1996).
Thus, h is a reconstructed dynamic system that has the same smooth Lyapunov exponents as the unknown system f .6 Then, to estimate the smooth Lyapunov exponents of the system f generating asset returns, one must ?rst estimate h. However, since ? ? ? ? st+1 st ? ? ? st+1 ? st+2 ? ? ? ? (3.8) h:? ? ? ?? ? . . . . ? ? ? ? . . st+m?1 v (st , st+1 , . . . , st+m?1 )
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the estimation of h reduces to the estimation of v st+m = v (st , st+1 , . . . , st+m?1 ) Moreover, since the Jacobian of h at the reconstructed state Tt is ? ? 0 1 0 ··· 0 ? 0 ? 0 1 ··· 0 ? ? ? ? 0 0 ··· 0 Dh (Tt ) = ? 0 ? ? . ? . . . . . . . ? . ? . . . ?v ?v ?v ?v · · · ?st+m?1 ?st ?st+1 ?st+2 (3.9)
(3.10)
a feedforward neural network is a natural choice to estimate the above derivatives to be able to calculate the smooth Lyapunov exponents (see Dechert and Gencay, 1992, Gencay and Dechert, 1992, McCa?rey et al, 1992, and Nychka et al, 1992), and 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. Inference of ? Shintani and Linton (2004) derive the asymptotic distribution of a neural network estimator of the smooth Lyapunov exponents ´ ? ³ biM ? ?i =? N (0, Vi ) M ? (3.11)
biM is the estimator of the i:th exponent, based on the M reconstructed where ? states on the trajectory, Vi is the variance of the i:th exponent, and i ? [1, 2, . . . , n].7 When it comes to the average of the smooth Lyapunov exponents, our conjecture P is that asymptotic normality holds for a neural network 1 estimator of n n i=1 ?i since the eigenvectors corresponding to the di?erent exponents are pairwise orthogonal ´ ? ³ bMn ? ? =? N (0, Vn ) M ? (3.12) bMn is the estimator of 1 Pn ?i , based on the M reconstructed states where ? i=1 n Pn 1 on the trajectory, and Vn is the variance of n i=1 ?i . If this conjecture is correct, it is possible to make inference of a change in potential market risk.
It is, therefore, possible to test for the presence of chaotic dynamics in an observed scalar time series since ?1 > 0 is an operational de?nition of chaos (see Bask et al, 2007, for an application using electricity prices, who ?nd evidence of complicated dynamics).
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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. Speci?cally, a parametric model in the form of a polynomial autoregression on a projected space is ?tted to the observed time series, which is utilized to measure the stability and volatility of the variable of interest (see Bask and de Luna, 2002, and de Luna, 1998, for details).8 However, when a successful risk management is in focus, it is necessary to measure the stability of the ‘true’ stochastic dynamic system generating asset returns, and not the stability of the model ?tted 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. Therefore, we argue that a non-parametric approach should be used when estimating the stability of the system, whereas any (good) volatility model may be used when estimating the volatility.9 Applications of (?, ? 2 )-analysis Our belief is that (?, ? 2 )-analysis has at least two di?erent but closely connected applications (i) To monitor the evolution of an asset’s market risk (? 2 ) and its potential market risk (?), meaning that (?, ? 2 )-analysis is used as a tool to detect actual and potential changes in market risk. Think of an asset with an unchanged market risk. That is, the conditional volatility of asset returns is measured in a rolling window, where it is found that there are no statistically signi?cant changes in volatility over some period of time (see Leeves, 2007, for an application using stock prices before and after the Asian crisis). However, during the same period of time, the stability of asset returns has decreased since the average of the smooth Lyapunov exponents has become less negative, meaning that the asset’s potential market risk has increased. Thus, in this case, (?, ? 2 )-analysis gives an early warning that an increase in the asset’s market risk may soon occur.
A large-scale analysis of the European monetary integration, with the creation of the EMU, is carried out in Bask and de Luna (2005) using this methodology. To be more speci?c, changes in the stability and volatility of 16 European currencies and in the volatility of the shocks to these currencies are examined, and the results indicate 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, most currencies became more stable and less volatile, whereas they became less stable and more volatile when the Danish public voted against the treaty. 9 Bask and Widerberg (2007) use this methodology when they examine how the integration process at the Nordic power market has a?ected the stability and volatility of electricity prices. To be more speci?c, the non-parametric approach outlined above is used when estimating the stability, whereas an EGARCH model is used when estimating the volatility. The results indicate that the integration process is associated with more stable electricity prices and a decrease in volatility of these prices, but without having a one-to-one correspondence between the changes in stability and volatility.
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¢ ¡ ?, ? 2 -analysis
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Of course, the same tool can be used to monitor changes in market risk and potential market risk of a portfolio of assets. (i) To compare the market risk and potential market risk of two portfolios of assets. Imagine an investor who is planning to make a portfolio investment, but is unsure about which asset to invest in. Of course, if this investor is using what has been named modern portfolio theory when making investments, it is clear that the potential market risk of di?erent assets should not directly a?ect the composition of the portfolio. On the other hand, due to the fact that a portfolio’s market risk depends on its potential market risk, we believe that one should not neglect the latter risk. Think of a situation in which two di?erent assets give rise to portfolios with the same risk-return pro?les. We argue, in this case, 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 time-varying and that it may be the case that the market risk of the portfolio with the higher potential market risk is unusually low. It is, of course, part of future research to derive a reasonable portfolio theory that supports such a claim.
5 Concluding remarks
The aim of this paper has been to argue in favor of ? as a measure of potential market risk, and to discuss how this measure can be used in what we call (?, ? 2 )-analysis, which is a method to distinguish between market risk and potential market risk. What remains is to derive the asymptotic distribution of a neural network estimator of the average of the smooth Lyapunov exponents, and, thereafter, take the proposed method to ?nancial data to study its merits and possible weaknesses.
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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, 737—750. Bask, M — Liu, T — Widerberg, A (2007) The Stability of Electricity Prices: Estimation and Inference of the Lyapunov Exponents. Physica A, 376, 565—572. Bask, M — Widerberg, A (2007) The Stability and Volatility of Electricity Prices: An Illustration of (?, ? 2 )-Analysis. Göteborg University Working Paper in Economics, No. 267. Dechert, W D — Gencay, R (1992) Lyapunov Exponents as a Nonparametric Diagnostic for Stability Analysis. Journal of Applied Econometrics, 7, S41—S60. Dechert, W D — Gencay, R (1996) The Topological Invariance of Lyapunov Exponents in Embedded Dynamics. Physica D, 90, 40—55. Dechert, W D — Gencay, R (2000) Is the Largest Lyapunov Exponent Preserved in Embedded Dynamics? Physics Letters A, 276, 59—64. de Luna, X (1998) Projected Polynomial Autoregression for Prediction of Stationary Time Series. Journal of Applied Statistics, 25, 763—775. Engle, R F (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom In?ation. Econometrica, 50, 987—1007. Fernández-Rodríguez, F — Sosvilla-Rivero, S — Andrada-Félix, J (2005) Testing Chaotic Dynamics via Lyapunov Exponents. Journal of Applied Econometrics, 20, 911—930. Gencay, R — Dechert, W D (1992) An Algorithm for the n Lyapunov Exponents of an n-Dimensional Unknown Dynamical System. Physica D, 59, 142—157. Gencay, R — Dechert, W D (1996) The Identi?cation of Spurious Lyapunov Exponents in Jacobian Algorithms. Studies in Nonlinear Dynamics and Econometrics, 1 (3) art. 2. Guckenheimer, J — Holmes, P (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. (Applied Mathematical Sciences, Vol. 42), Springer-Verlag: Berlin.
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Hornik, K — Stinchcombe, M — White, H (1990) Universal Approximation of an Unknown Mapping and its Derivatives using Multilayer Feedforward Networks. Neural Networks, 3, 551—560. Kennel, M B — Brown, R — Abarbanel, H D I (1992) Determining Embedding Dimension for Phase-Space Reconstruction using a Geometrical Construction. Physical Review A, 45, 3403—3411. Leeves, G (2007) Asymmetric Volatility of Stock Returns during the Asian Crisis: Evidence from Indonesia. International Review of Economics and Finance, 16, 272—286. McCa?rey, D — Ellner, S — Gallant, A R — Nychka, D (1992) Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression. Journal of the American Statistical Association, 87, 682—695. Nychka, D — Ellner, S — Gallant, A R — McCa?rey, D (1992) Finding Chaos in Noisy Systems. Journal of the Royal Statistical Society B, 54, 399—426. Sauer, T — Yorke, J A — Casdagli, M (1991) Embedology. Journal of Statistical Physics, 65, 579—616. Shintani, M — Linton, O (2004) Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos. Journal of Econometrics, 120, 1—33. Takens, F (1981) Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence (Lecture Notes in Mathematics, Vol. 898) by Rand, D A and Young, L S, eds., Springer-Verlag: Berlin, 366—381.
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10/2007 Juha Kilponen – Torsten Santavirta When do R&D subsidies boost innovation? Revisiting the inverted U-shape. 2007. 30 p. ISBN 978-952-462-368-1, print; ISBN 978-952-462-369-8, online. 11/2007 Karlo Kauko Managers and efficiency in banking. 2007. 34 p. ISBN 978-952-462-370-4, print; ISBN 978-952-462-371-1, online.
12/2007 Helena Holopainen Integration of financial supervision. 2007. 30 p. ISBN 978-952-462-372-8, print; ISBN 978-952-462-373-5, online. 13/2007 Esa Jokivuolle – Timo Vesala Portfolio effects and efficiency of lending under Basel II. 2007. 23 p. ISBN 978-952-462-374-2, print; ISBN 978-952-462-375-9, online. 14/2007 Maritta Paloviita Estimating a small DSGE model under rational and measured expectations: some comparisons. 2007. 30 p. ISBN 978-952-462-376-6, print; ISBN 978-952-462-377-3, online. 15/2007 Jarmo Pesola Financial fragility, macroeconomic shocks and banks’ loan losses: evidence from Europe. 2007. 38 p. ISBN 978-952-462-378-0, print; ISBN 978-952-462-379-7, online. 16/2007 Allen N Berger – Iftekhar Hasan – Mingming Zhou Bank ownership and efficiency in China: what lies ahead in the world’s largest nation? 2007. 47 p. ISBN 978-952-462-380-3, print; ISBN 978-952-462-381-0, online. 17/2007 Jozsef Molnar Pre-emptive horizontal mergers: theory and evidence. 2007. 37 p. ISBN 978-952-462-382-7, print; ISBN 978-952-462-383-4, online. 18/2007 Federico Ravenna – Juha Seppälä Monetary policy, expected inflation and inflation risk premia. 2007. 33 p. ISBN 978-952-462-384-1, print; ISBN 978-952-462-385-8, online. 19/2007 Mikael Bask Long swings and chaos in the exchange rate in a DSGE model with a Taylor rule. 2007. 28 p. ISBN 978-952-462-386-5, print; ISBN 978-952-462-387-2, online. 20/2007 Mikael Bask Measuring potential market risk. 2007. 18 p. ISBN 978-952-462-388-9, print; ISBN 978-952-462-389-6, online.
Suomen Pankki Bank of Finland P.O.Box 160 FI-00101 HELSINKI Finland
doc_957676663.pdf
Market risk is the risk of losses in positions arising from movements in market prices.
Mikael Bask
Measuring potential market risk
Bank of Finland Research Discussion Papers 20 • 2007
Suomen Pankki Bank of Finland PO Box 160 FI-00101 HELSINKI Finland + 358 10 8311 http://www.bof.fi
Bank of Finland Research Discussion Papers 20 • 2007
Mikael Bask*
Measuring potential market risk
The views expressed are those of the author and do not necessarily reflect the views of the Bank of Finland. * E-mail: [email protected]
Esa Jokivuolle is gratefully acknowledged for giving comments on an earlier draft of this paper. The usual disclaimer applies.
This paper can be downloaded without charge from http://www.bof.fi or from the Social Science Research Network electronic library at http://ssrn.com/abstract_id=1029958.
http://www.bof.fi ISBN 978-952-462-388-9 ISSN 0785-3572 (print) ISBN 978-952-462-389-6 ISSN 1456-6184 (online) Helsinki 2007
Measuring potential market risk
Bank of Finland Research Discussion Papers 20/2007
Mikael Bask Monetary Policy and Research Department
Abstract
The difference between market risk and potential market risk is emphasized and a measure of the latter risk is proposed. Specifically, it is argued that the spectrum of smooth Lyapunov exponents can be utilized in what we call (?, ?2)-analysis, which is a method to monitor the aforementioned risk measures. 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 returns (?2). Keywords: market risk, potential market risk, smooth Lyapunov exponents, stochastic dynamic system, value-at-risk JEL classification number: G11
3
Menetelmä potentiaalisen markkinariskin määrittämiseksi
Suomen Pankin keskustelualoitteita 20/2007
Mikael Bask Rahapolitiikka- ja tutkimusosasto
Tiivistelmä
Tutkimuksessa tarkastellaan markkinariskin ja potentiaalisen markkinariskin eroa ja tarkasteluissa esitellään keino jälkimmäisen mittaamiseksi. Täsmällisesti ottaen työssä esitetään, että sileiden Lyapunovin eksponenttien spektriä voidaan hyödyntää ns. (?, ?2)-analyysissa, jota menetelmää käytetään markkinariskin ja potentiaalisen markkinariskin tarkkailussa. Analyysin soveltuvuus näiden riskien seurantaan johtuu siitä, että sileät Lyapunovin eksponentit korostavat rahoitusvaateiden tuottoja synnyttävän stokastisen dynaamisen järjestelmän vakausominaisuuksia (?), kun perinteisissä riskimittareissa, kuten value-at-risk, kyse on sen sijaan tuottojen jakaumasta (?2). Avainsanat: markkinariski, potentiaalinen markkinariski, sileät Lyapunovin eksponentit, stokastinen dynaaminen järjestelmä, value-at-risk JEL-luokittelu: G11
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Contents
Abstract....................................................................................................................3 Tiivistelmä (abstract in Finnish) ..............................................................................4 1 Measuring market risk ....................................................................................7 2 ?: a measure of potential market risk ............................................................8 3 Testing for a change in ? ...............................................................................10 4 (?, ?2)-analysis ................................................................................................13 5 Concluding remarks.......................................................................................14 References..............................................................................................................15
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1 Measuring market risk
Financial market risk re?ects the chance that the actual return on an asset or a portfolio of assets may be very di?erent than the expected return. For this reason, a measure of market risk is necessary to carry through a successful risk management. Nowadays, ?nancial 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 returns.1 Of course, the probability distribution of returns is not constant since asset returns depend on the underlying economic structure.2 To be more precise, quantities such as moneys and interest rates interact with each other through time, and, therefore, constitute a dynamic system, meaning that the stability of the system generating asset returns is crucial for the variance of these returns. That is, a less stable dynamic system is associated with more variable asset returns, meaning that an asset is potentially more risky than another asset, if the returns of the former asset is generated by a less stable system. To clarify this further, let ?2 denote the conditional variance of asset returns, and (for reasons explained below) let ? denote the stability of the dynamic system generating these returns. Then, ?2 = ?2 (?, ?) (1.1)
where ? is exogenous shocks to the dynamic system, meaning that the conditional variance (?2 ) is not only a?ected by the system’s stability (?), it is also a?ected by shocks to the system (?). Speci?cally, the conditional variance of asset returns increases when the dynamic system is less stable, but also when the variance of the shocks increases. Thus, because of shocks to the system, there is no one-to-one correspondence between the conditional variance of asset returns and the stability of the dynamic system generating these returns, meaning that ? is not a measure of market risk. Instead, ? is a measure of potential market risk, while ? 2 is a measure of market risk. In other words, a change in an asset’s potential market risk may or may not change its market risk since it depends on how much the variance of the shocks to the dynamic system generating asset returns has changed, if there has been any change at all. Thus, the variance of the shocks distinguishes
The importance of value-at-risk as a measure of ?nancial market risk is emphasized by the fact that the Basel Committee on Banking Supervision at the Bank for International Settlements imposes ?nancial institutions to meet capital requirements based on value-at-risk. The widespread use of value-at-risk as a measure of market risk also owes much to Dennis Weatherstone, former chairman of JP Morgan & Co., who demanded to know the market risk of the company at 4:15 P.M. every day. Weatherstone’s request was met with a daily value-at-risk report. 2 For this reason, Engle’s (1982) ARCH model and subsequent developments of the model are invaluable tools since they can be used to estimate and predict conditional moments characterizing the probability distribution of returns.
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between an asset’s market risk and its potential market risk. Therefore, the stability of the dynamic system generating asset returns should be contrasted with the volatility of these returns, and this is accomplished in what we call (?, ? 2 )-analysis.
2 ?: a measure of potential market risk
The purpose of this section is twofold: (i) to de?ne the Lyapunov exponents of a stochastic dynamic system; and (ii) to motivate why these exponents provide a measure of a system’s stability, meaning that they also provide a measure of potential market risk. De?nition of ? Bask and de Luna (2002) argue that the spectrum of smooth Lyapunov exponents can be used in the determination of the stability of a stochastic dynamic system. Speci?cally, assume that the dynamic system, f : Rn ? Rn , generating asset returns is St+1 = f (St ) + ?s t+1 (2.1)
where St and ?s t are the state of the system and a shock to the system, respectively, both at time t ? [1, 2, . . . , ?]. For an n-dimensional system as in (2.1), there are n Lyapunov exponents that are ranked from the largest to the smallest exponent ?1 ? ?2 ? . . . ? ?n (2.2)
and it is these exponents that provide information on the stability properties of the system f . Then, how are the Lyapunov exponents de?ned? Assume temporarily that there are no shocks to the system f , and consider how it ampli?es a small 0 di?erence between the initial states S0 and S0
0 0 0 Sj ? Sj = f j (S0 ) ? f j (S0 ) ' Df j (S0 ) (S0 ? S0 )
(2.3)
where f j (S0 ) = f (· · · f (f (S0 )) · · · ) denotes j successive iterations of the system starting at state S0 , and Df is the Jacobian of the system Df j (S0 ) = Df (Sj ?1 ) Df (Sj ?2 ) · · · Df (S0 ) (2.4)
Then, associated with each Lyapunov exponent, ?i , i ? [1, 2, . . . , n], there are nested subspaces U i ? Rn of dimension n + 1 ? i with the property that ?i ? lim loge kDf j (S0 )k 1X = lim loge kDf (Sk )k j ?? j ?? j j k=0
j ?1
(2.5)
for all S0 ? U i ? U i+1 . Due to Oseledec’s multiplicative ergodic theorem, the limits in (2.5) exist and are independent of S0 almost surely with respect to 8
3 the measure induced by the process {St }? Then, allow for shocks to the t=1 . system f , meaning that the measure is induced by a stochastic process. In this case, the Lyapunov exponents have been named smooth Lyapunov exponents in the literature.
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 di?erent starting values of a system, where the di?erence is an exogenous shock at time t = 0. The largest smooth Lyapunov exponent, ?1 , measures the slowest exponential rate of convergence of two trajectories of the dynamic system starting at these di?erent starting values at time t = 0, but with identical exogenous shocks at times t > 0.4 In fact, ?1 measures the convergence of a shock in the direction de?ned by the eigenvector corresponding to this exponent. However, if the di?erence between the two starting values lies in another direction of Rn , then the convergence is faster. Thus, ?1 measures a ‘worst case scenario’. The average of the smooth Lyapunov exponents 1X ?i ?? n i=1
n
(2.6)
measures the exponential rate of convergence in a geometrical average direction. That is, the convergence of two trajectories of the dynamic system in the geometrical average of the directions de?ned by the eigenvectors corresponding to the di?erent exponents. Thus, ? measures an ‘average scenario’. We can, therefore, compare the stability of two stochastic dynamic systems via the smooth Lyapunov exponents since a one-time shock has a smaller e?ect on the dynamic system with a smaller ? than for the system with a larger ?. Thus, since we are dealing with dissipative systems, meaning that ? < 0 by de?nition, a dynamic system is more stable than another system, if ? is more negative. An extensive discussion of the spectrum of smooth Lyapunov exponents as a measure of the stability of a stochastic dynamic system is provided in Bask and de Luna (2002). As an illustration, it is shown therein that the decrease in volatility of the exchange rates between the Swedish Krona and the ECU/Euro, after the launch of the Euro, is due to a decrease in the volatility of the shocks to the dynamic system generating these exchange rates and not to a more stable system. Thus, one can say that the market risk decreased, but that the potential market risk was unchanged.
See Guckenheimer and Holmes (1983) for a careful de?nition of the Lyapunov exponents and their properties. 4 When ?1 > 0, the trajectories diverge from each other, and for a bounded stochastic dynamic system, this is an operational de?nition of chaotic dynamics.
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3 Testing for a change in ?
Contrary to risk measures like value-at-risk, potential market risk does not have a straightforward economic interpretation. However, it is not level of potential market risk that is of interest. Instead, it is the change in this risk, ??, since we are interested in the potential change in market risk. The purpose of this section is, therefore, 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. In other words, 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. 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 the system. However, it is possible to reconstruct the dynamics of the system using only a scalar time series, and, thereafter, to measure the stability of this reconstructed system. Therefore, associate the system f with an observer function, g : Rn ? R, that generates observed asset returns st = g (St ) + ?m t (3.1)
where st ? St and ?m are the asset return and a measurement error, t respectively, both at time t. Thus, (3.1) means that the asset return series {st }N t=1 (3.2)
is observed, which is used to reconstruct the dynamics of the system f , where N is the number of consecutive returns in the time series. Speci?cally, the observations in a scalar time series, like the asset return series in (3.2), contain information about unobserved state variables that can be used to de?ne a state in present time. Therefore, let T = (T1 , T2 , . . . , TM )0 (3.3)
be the reconstructed trajectory, where Tt is the reconstructed state at time t and M is the number of states on the reconstructed trajectory. Each Tt is given by Tt = {st , st+1 , . . . , st+m?1 } (3.4)
where m is the embedding dimension and time t ? [1, 2, . . . , N ? m + 1]. 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 © ¡ ¢ ¡ ¢ ¡ ¢ª ? (St ) = g f 0 (St ) , g f 1 (St ) , . . . , g f m?1 (St ) (3.5) 10
which maps the n-dimensional state St onto the m-dimensional state Tt , is an embedding if m > 2n.5 This means that the map is a smooth map that performs a one-to-one coordinate transformation and has a smooth inverse. A map that is an embedding preserves topological information about the unknown dynamic system, like the smooth Lyapunov exponents, and, in particular, the map induces a function, h : Rm ? Rm , on the reconstructed trajectory Tt+1 = h (Tt ) which is topologically conjugate to the unknown system f . That is hj (Tt ) = ? ? f j ? ??1 (Tt ) (3.7) (3.6)
An intuitive explanation of Takens’ (1981) embedding theorem may be in place due to its importance in the estimation of ?. For the sake of the argument, assume that M1 ? M and M2 ? M are two subspaces of dimension n1 and n2 , respectively, where M ? Rm is an m-dimensional manifold representing phase space for the reconstructed dynamic system. In general, two subspaces intersect in a subspace of dimension n1 + n2 ? m, meaning that when this expression is negative, there is no intersection of the two 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 Takens’, 1981, theorem). A problem is that the dimension of the ‘true’ dynamic system is not known, meaning that the required embedding dimension is not either known. This problem can, however, be solved indirectly by making use of a generic property of a proper reconstruction, namely, that the dynamics in original phase space must be completely unfolded in reconstructed phase space. In other words, if the embedding dimension is too low, the dynamics is not completely unfolded, meaning that distant states in original phase space are close states in reconstructed phase space, and, therefore, are named false neighbors in phase space. There are at least two methods to calculate the required embedding dimension from an observed time series: (i) false nearest neighbors; and (ii) the saturation of invariants on the reconstructed dynamics such as the saturation of the Lyapunov exponents. The ?rst method is based on the aforementioned generic property of a proper reconstruction, meaning that by increasing the embedding dimension, the dynamics is completely unfolded when there are no false neighbors in reconstructed phase space (see Kennel et al, 1992). The second method, the saturation of invariants on the reconstructed dynamics, is based on the fact that when the dynamics is completely unfolded, the Lyapunov exponents and other invariants such as entropy and fractal dimension are independent of the embedding dimension. If, however, the dynamics is not completely unfolded in reconstructed phase space, these invariants depend on the embedding dimension. Therefore, by increasing the embedding dimension, the dynamics is completely unfolded when the value of an invariant stops changing (see Fernández-Rodríguez et al, 2005, for an example regarding the largest Lyapunov exponent and a statistical test for chaotic dynamics). 6 Since the m-dimensional system h has a larger dimension than the n-dimensional system f , the number of smooth Lyapunov exponents that are spurious is m ? n. This issue is discussed in Dechert and Gencay (1996)—(2000) and Gencay and Dechert (1996).
Thus, h is a reconstructed dynamic system that has the same smooth Lyapunov exponents as the unknown system f .6 Then, to estimate the smooth Lyapunov exponents of the system f generating asset returns, one must ?rst estimate h. However, since ? ? ? ? st+1 st ? ? ? st+1 ? st+2 ? ? ? ? (3.8) h:? ? ? ?? ? . . . . ? ? ? ? . . st+m?1 v (st , st+1 , . . . , st+m?1 )
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the estimation of h reduces to the estimation of v st+m = v (st , st+1 , . . . , st+m?1 ) Moreover, since the Jacobian of h at the reconstructed state Tt is ? ? 0 1 0 ··· 0 ? 0 ? 0 1 ··· 0 ? ? ? ? 0 0 ··· 0 Dh (Tt ) = ? 0 ? ? . ? . . . . . . . ? . ? . . . ?v ?v ?v ?v · · · ?st+m?1 ?st ?st+1 ?st+2 (3.9)
(3.10)
a feedforward neural network is a natural choice to estimate the above derivatives to be able to calculate the smooth Lyapunov exponents (see Dechert and Gencay, 1992, Gencay and Dechert, 1992, McCa?rey et al, 1992, and Nychka et al, 1992), and 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. Inference of ? Shintani and Linton (2004) derive the asymptotic distribution of a neural network estimator of the smooth Lyapunov exponents ´ ? ³ biM ? ?i =? N (0, Vi ) M ? (3.11)
biM is the estimator of the i:th exponent, based on the M reconstructed where ? states on the trajectory, Vi is the variance of the i:th exponent, and i ? [1, 2, . . . , n].7 When it comes to the average of the smooth Lyapunov exponents, our conjecture P is that asymptotic normality holds for a neural network 1 estimator of n n i=1 ?i since the eigenvectors corresponding to the di?erent exponents are pairwise orthogonal ´ ? ³ bMn ? ? =? N (0, Vn ) M ? (3.12) bMn is the estimator of 1 Pn ?i , based on the M reconstructed states where ? i=1 n Pn 1 on the trajectory, and Vn is the variance of n i=1 ?i . If this conjecture is correct, it is possible to make inference of a change in potential market risk.
It is, therefore, possible to test for the presence of chaotic dynamics in an observed scalar time series since ?1 > 0 is an operational de?nition of chaos (see Bask et al, 2007, for an application using electricity prices, who ?nd evidence of complicated dynamics).
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4
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. Speci?cally, a parametric model in the form of a polynomial autoregression on a projected space is ?tted to the observed time series, which is utilized to measure the stability and volatility of the variable of interest (see Bask and de Luna, 2002, and de Luna, 1998, for details).8 However, when a successful risk management is in focus, it is necessary to measure the stability of the ‘true’ stochastic dynamic system generating asset returns, and not the stability of the model ?tted 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. Therefore, we argue that a non-parametric approach should be used when estimating the stability of the system, whereas any (good) volatility model may be used when estimating the volatility.9 Applications of (?, ? 2 )-analysis Our belief is that (?, ? 2 )-analysis has at least two di?erent but closely connected applications (i) To monitor the evolution of an asset’s market risk (? 2 ) and its potential market risk (?), meaning that (?, ? 2 )-analysis is used as a tool to detect actual and potential changes in market risk. Think of an asset with an unchanged market risk. That is, the conditional volatility of asset returns is measured in a rolling window, where it is found that there are no statistically signi?cant changes in volatility over some period of time (see Leeves, 2007, for an application using stock prices before and after the Asian crisis). However, during the same period of time, the stability of asset returns has decreased since the average of the smooth Lyapunov exponents has become less negative, meaning that the asset’s potential market risk has increased. Thus, in this case, (?, ? 2 )-analysis gives an early warning that an increase in the asset’s market risk may soon occur.
A large-scale analysis of the European monetary integration, with the creation of the EMU, is carried out in Bask and de Luna (2005) using this methodology. To be more speci?c, changes in the stability and volatility of 16 European currencies and in the volatility of the shocks to these currencies are examined, and the results indicate 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, most currencies became more stable and less volatile, whereas they became less stable and more volatile when the Danish public voted against the treaty. 9 Bask and Widerberg (2007) use this methodology when they examine how the integration process at the Nordic power market has a?ected the stability and volatility of electricity prices. To be more speci?c, the non-parametric approach outlined above is used when estimating the stability, whereas an EGARCH model is used when estimating the volatility. The results indicate that the integration process is associated with more stable electricity prices and a decrease in volatility of these prices, but without having a one-to-one correspondence between the changes in stability and volatility.
8
¢ ¡ ?, ? 2 -analysis
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Of course, the same tool can be used to monitor changes in market risk and potential market risk of a portfolio of assets. (i) To compare the market risk and potential market risk of two portfolios of assets. Imagine an investor who is planning to make a portfolio investment, but is unsure about which asset to invest in. Of course, if this investor is using what has been named modern portfolio theory when making investments, it is clear that the potential market risk of di?erent assets should not directly a?ect the composition of the portfolio. On the other hand, due to the fact that a portfolio’s market risk depends on its potential market risk, we believe that one should not neglect the latter risk. Think of a situation in which two di?erent assets give rise to portfolios with the same risk-return pro?les. We argue, in this case, 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 time-varying and that it may be the case that the market risk of the portfolio with the higher potential market risk is unusually low. It is, of course, part of future research to derive a reasonable portfolio theory that supports such a claim.
5 Concluding remarks
The aim of this paper has been to argue in favor of ? as a measure of potential market risk, and to discuss how this measure can be used in what we call (?, ? 2 )-analysis, which is a method to distinguish between market risk and potential market risk. What remains is to derive the asymptotic distribution of a neural network estimator of the average of the smooth Lyapunov exponents, and, thereafter, take the proposed method to ?nancial data to study its merits and possible weaknesses.
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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, 737—750. Bask, M — Liu, T — Widerberg, A (2007) The Stability of Electricity Prices: Estimation and Inference of the Lyapunov Exponents. Physica A, 376, 565—572. Bask, M — Widerberg, A (2007) The Stability and Volatility of Electricity Prices: An Illustration of (?, ? 2 )-Analysis. Göteborg University Working Paper in Economics, No. 267. Dechert, W D — Gencay, R (1992) Lyapunov Exponents as a Nonparametric Diagnostic for Stability Analysis. Journal of Applied Econometrics, 7, S41—S60. Dechert, W D — Gencay, R (1996) The Topological Invariance of Lyapunov Exponents in Embedded Dynamics. Physica D, 90, 40—55. Dechert, W D — Gencay, R (2000) Is the Largest Lyapunov Exponent Preserved in Embedded Dynamics? Physics Letters A, 276, 59—64. de Luna, X (1998) Projected Polynomial Autoregression for Prediction of Stationary Time Series. Journal of Applied Statistics, 25, 763—775. Engle, R F (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom In?ation. Econometrica, 50, 987—1007. Fernández-Rodríguez, F — Sosvilla-Rivero, S — Andrada-Félix, J (2005) Testing Chaotic Dynamics via Lyapunov Exponents. Journal of Applied Econometrics, 20, 911—930. Gencay, R — Dechert, W D (1992) An Algorithm for the n Lyapunov Exponents of an n-Dimensional Unknown Dynamical System. Physica D, 59, 142—157. Gencay, R — Dechert, W D (1996) The Identi?cation of Spurious Lyapunov Exponents in Jacobian Algorithms. Studies in Nonlinear Dynamics and Econometrics, 1 (3) art. 2. Guckenheimer, J — Holmes, P (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. (Applied Mathematical Sciences, Vol. 42), Springer-Verlag: Berlin.
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Hornik, K — Stinchcombe, M — White, H (1990) Universal Approximation of an Unknown Mapping and its Derivatives using Multilayer Feedforward Networks. Neural Networks, 3, 551—560. Kennel, M B — Brown, R — Abarbanel, H D I (1992) Determining Embedding Dimension for Phase-Space Reconstruction using a Geometrical Construction. Physical Review A, 45, 3403—3411. Leeves, G (2007) Asymmetric Volatility of Stock Returns during the Asian Crisis: Evidence from Indonesia. International Review of Economics and Finance, 16, 272—286. McCa?rey, D — Ellner, S — Gallant, A R — Nychka, D (1992) Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression. Journal of the American Statistical Association, 87, 682—695. Nychka, D — Ellner, S — Gallant, A R — McCa?rey, D (1992) Finding Chaos in Noisy Systems. Journal of the Royal Statistical Society B, 54, 399—426. Sauer, T — Yorke, J A — Casdagli, M (1991) Embedology. Journal of Statistical Physics, 65, 579—616. Shintani, M — Linton, O (2004) Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos. Journal of Econometrics, 120, 1—33. Takens, F (1981) Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence (Lecture Notes in Mathematics, Vol. 898) by Rand, D A and Young, L S, eds., Springer-Verlag: Berlin, 366—381.
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BANK OF FINLAND RESEARCH DISCUSSION PAPERS ISSN 0785-3572, print; ISSN 1456-6184, online 1/2007 Timo Korkeamäki – Yrjö Koskinen – Tuomas Takalo Phoenix rising: Legal reforms and changes in valuations in Finland during the economic crisis. 2007. 39 p. ISBN 978-952-462-346-9, print; ISBN 978-952-462-347-6, online. Aaron Mehrotra A note on the national contributions to euro area M3. 2007. 25 p. ISBN 978-952-462-348-3, print; ISBN 978-952-462-349-0, online. Ilmo Pyyhtiä Why is Europe lagging behind? 2007. 41 p. ISBN 978-952-462350-6, print; ISBN 978-952-462-351-3, online. Benedikt Goderis – Ian W Marsh – Judit Vall Castello – Wolf Wagner Bank behaviour with access to credit risk transfer markets. 2007. 28 p. ISBN 978-952-462-352-0, print; ISBN 978-952-462-353-7, online. Risto Herrala – Karlo Kauko Household loan loss risk in Finland – estimations and simulations with micro data. 2007. 44 p. ISBN 978-952-462-354-4, print; ISBN 978-952-462-355-1, online. Mikael Bask – Carina Selander Robust Taylor rules in an open economy with heterogeneous expectations and least squares learning. 2007. 54 p. ISBN 978-952-462-356-8, print; ISBN 978-952-462-357-5, online. David G Mayes – Maria J Nieto – Larry Wall Multiple safety net regulators and agency problems in the EU: is Prompt Corrective Action a partial solution? 2007. 39 p. ISBN 978-952-462-358-2, print; ISBN 978-952-462-359-9, online. Juha Kilponen – Kai Leitemo Discretion and the transmission lags of monetary policy. 2007. 24 p. ISBN 978-952-462-362-9, print; ISBN 978-952-462-363-6, online. Mika Kortelainen Adjustment of the US current account deficit. 2007. 35 p. ISBN 978-952-462-366-7, print; ISBN 978-952-462-367-4, online.
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6/2007
7/2007
8/2007
9/2007
10/2007 Juha Kilponen – Torsten Santavirta When do R&D subsidies boost innovation? Revisiting the inverted U-shape. 2007. 30 p. ISBN 978-952-462-368-1, print; ISBN 978-952-462-369-8, online. 11/2007 Karlo Kauko Managers and efficiency in banking. 2007. 34 p. ISBN 978-952-462-370-4, print; ISBN 978-952-462-371-1, online.
12/2007 Helena Holopainen Integration of financial supervision. 2007. 30 p. ISBN 978-952-462-372-8, print; ISBN 978-952-462-373-5, online. 13/2007 Esa Jokivuolle – Timo Vesala Portfolio effects and efficiency of lending under Basel II. 2007. 23 p. ISBN 978-952-462-374-2, print; ISBN 978-952-462-375-9, online. 14/2007 Maritta Paloviita Estimating a small DSGE model under rational and measured expectations: some comparisons. 2007. 30 p. ISBN 978-952-462-376-6, print; ISBN 978-952-462-377-3, online. 15/2007 Jarmo Pesola Financial fragility, macroeconomic shocks and banks’ loan losses: evidence from Europe. 2007. 38 p. ISBN 978-952-462-378-0, print; ISBN 978-952-462-379-7, online. 16/2007 Allen N Berger – Iftekhar Hasan – Mingming Zhou Bank ownership and efficiency in China: what lies ahead in the world’s largest nation? 2007. 47 p. ISBN 978-952-462-380-3, print; ISBN 978-952-462-381-0, online. 17/2007 Jozsef Molnar Pre-emptive horizontal mergers: theory and evidence. 2007. 37 p. ISBN 978-952-462-382-7, print; ISBN 978-952-462-383-4, online. 18/2007 Federico Ravenna – Juha Seppälä Monetary policy, expected inflation and inflation risk premia. 2007. 33 p. ISBN 978-952-462-384-1, print; ISBN 978-952-462-385-8, online. 19/2007 Mikael Bask Long swings and chaos in the exchange rate in a DSGE model with a Taylor rule. 2007. 28 p. ISBN 978-952-462-386-5, print; ISBN 978-952-462-387-2, online. 20/2007 Mikael Bask Measuring potential market risk. 2007. 18 p. ISBN 978-952-462-388-9, print; ISBN 978-952-462-389-6, online.
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