My research papers, policy analysis and random thoughts on topics such as crisis, regulations and financial risk. http://www.riskresearch.org Copyright 2011 Jon Danielsson, All rights reserved. http://blogs.law.harvard.edu/tech/rss en-us j.danielsson@lse.ac.uk (Jon Danielsson) RiskResearch.org Financial institutions rely heavily on Value-at-Risk (VaR) as a risk measure, even though it is not subadditive. First, we theoretically show that the VaR portfolio measure is subadditive in the relevant tail region if asset returns are multivariate regularly varying, thus allowing for dependent returns. Second, we note that VaR estimated from historical simulations may lead to violations of subadditivity. This upset of the theoretical VaR subadditivity in the tail arises because the coarseness of the empirical distribution can affect the apparent fatness of the tails. Finally, we document a dramatic reduction in the frequency of subadditivity violations, by using semi--parametric extreme value techniques for VaR estimation instead of historical simulations. http://www.riskresearch.org/?paperid=35&catid=1#papercontent http://www.riskresearch.org/?paperid=35&catid=1#papercontent Wed, 28 Mar 2012 00:00:00 +0100 Statistical systemic risk measures (SRMs) have been proposed by several authors. Those generally depend on established methods from market risk forecasting. The two most common SRMs, MES and CoVaR, along with VaR, are compared theoretically and then critically empirically analyzed. They are found to contain a high degree of model risk so that the signal they produce is highly unreliable. Finally, the papers discusses the main problems in systemic risk forecasting and proposed evaluation criteria fur such models. http://www.riskresearch.org/?paperid=41&catid=2#papercontent http://www.riskresearch.org/?paperid=41&catid=2#papercontent Wed, 9 Nov 2011 00:00:00 +0000 The risks impacting financial markets are attributable (at least in part) to the actions of market participants. In turn, market participants' actions depend on perceived risk. In equilibrium, risk is the fixed point of the mapping from perceived risk to actual risk. When market players believe trouble is ahead, they take actions that bring about realized volatility. This is endogenous risk. A model of endogenous risk enables the study of the propagation of financial booms and distress. Among other things, we can make precise the notion that market participants appear to become more risk-averse in response to deteriorating market outcomes. For economists, preferences and beliefs would normally be considered independent of one another. We discuss modeling of endogenous risk and some of its distinctive features, both theoretical and empirical. http://www.riskresearch.org/?paperid=37&catid=4#papercontent http://www.riskresearch.org/?paperid=37&catid=4#papercontent Wed, 31 Aug 2011 00:00:00 +0100 Banks operating under Value-at-Risk constraints give rise to a well- defined aggregate balance sheet capacity for the banking sector as a whole that depends on total bank capital. Equilibrium risk and market risk premiums can be solved in closed form as functions of aggregate bank capital. We explore the empirical properties of the model in light of recent experience in the financial crisis and highlight the importance of balance sheet capacity as the driver of the financial cycle and market risk premiums. http://www.riskresearch.org/?paperid=39&catid=4#papercontent http://www.riskresearch.org/?paperid=39&catid=4#papercontent Tue, 30 Aug 2011 00:00:00 +0100 Extreme events in financial markets are often generated by shocks that are generated within the system, rather than those that arrive from outside the system. The combination of risk-sensitive behav- ior rules and the coordinated actions implied by mark-to-market ac- counting can result in outcome distributions with fat tails, even if the fundamental shocks are Gaussian. We illustrate such "endogenous extreme events" through the pricing density resulting from dynamic hedging of options and the "flash crash" of May 2010. http://www.riskresearch.org/?paperid=40&catid=1#papercontent http://www.riskresearch.org/?paperid=40&catid=1#papercontent Sat, 20 Aug 2011 00:00:00 +0100 The dependence of foreign exchange rates on order flow is investigated for four ma jor exchange rate pairs, EUR/USD, EUR/GBP, GBP/USD and USD/JPY, across sampling frequencies ranging from 5 minutes to 1 week. Strong dependence and explanatory power is discovered across sampling frequencies. In a new result, inter–market effect of order flows is discovered, where the GBP exchange rate is dominated by EUR/USD order flow. The Meese and Rogoff (1983a,b) framework is used to investigate the forecasting power of order flow and it is shown that the order flow specifications reduce RMSEs, relative to a random walk, for virtually all exchange rates and sampling frequencies. http://www.riskresearch.org/?paperid=28&catid=3#papercontent http://www.riskresearch.org/?paperid=28&catid=3#papercontent Sun, 10 Apr 2011 00:00:00 +0100 Large once-off events cause large changes in prices but may not affect volatility and correlation dynamics as much as smaller events. Standard volatility models may deliver biased covariance forecasts in this case. We propose a multivariate volatility forecasting model that is accurate in the presence of large once-off events. The model is an extension of the dynamic conditional correlation model (DCC) model. Compared to the DCC model, our method produces more precise out-of-sample covariance forecasts and, when used in portfolio allocation, it leads to portfolios with similar return characteristics but lower turnover and hence higher profits. http://www.riskresearch.org/?paperid=38&catid=2#papercontent http://www.riskresearch.org/?paperid=38&catid=2#papercontent Sat, 5 Feb 2011 00:00:00 +0000 We exploit full order level information from an electronic FX broking system to provide a comprehensive account of the determination of its liquidity. We not only look at bid-ask spreads and trading volumes, but also study the determination of order entry rates and depth measures derived from the en- tire limit order book. We find strong predictability in the arrival of liquidity supply/demand events. Further, in times of low (high) liquidity, liquidity sup- ply (demand) events are more common. In times of high trading activity and volatility, the ratio of limit to market order arrivals is high but order book spreads and depth deteriorate. These results are consistent with market order traders having better information than limit order traders. http://www.riskresearch.org/?paperid=27&catid=3#papercontent http://www.riskresearch.org/?paperid=27&catid=3#papercontent Sat, 3 Jul 2010 00:00:00 +0100 We develop a coordination game to model interactions between fundamentals and liquidity during unstable periods in financial markets. We then propose a flexible econometric framework for estimation of the model and analysis of its quantitative implications. The specific empirical application is carry trades in the yen--dollar market, including the turmoil of 1998. We find a generally very deep market, with low information disparities amongst agents. We observe occasionally episodes of market fragility, or turmoil with up by the escalator, down by the elevator patterns in prices. The key role of strategic behavior in the econometric model is also confirmed. http://www.riskresearch.org/?paperid=29&catid=4#papercontent http://www.riskresearch.org/?paperid=29&catid=4#papercontent Mon, 1 Mar 2010 00:00:00 +0000 Market volatility reflects traders' actions, while their actions depend on perceptions of risk. Equilibrium volatility is the fixed point of the mapping that takes perceived risk to actual risk. We solve for equilibrium stochastic volatility in a dynamic setting where risk-neutral traders operate under Value-at-Risk constraints. We derive a closed form solution for the stochastic volatility function in the benchmark model with a single risky asset. Even though the underlying fundamental risks remain constant, the resulting dynamics generate stochastic volatility through traders' reactions in equilibrium. Volatilities, expected returns and Sharpe ratios are shown to be countercyclical. If the purpose of financial regulation is to shield the financial system from collapse, then basing regulation on individually optimal risk management may not be enough. http://www.riskresearch.org/?paperid=34&catid=4#papercontent http://www.riskresearch.org/?paperid=34&catid=4#papercontent Fri, 24 Jul 2009 00:00:00 +0100 The quality of statistical risk models is much lower than often assumed. Such models are useful for measuring the risk of frequent small events, such as in internal risk management, but not for systematically important events. Unfortunately, it is common to see unrealistic demands placed on risk models. Having a number representing risk seems to be more important than having a number which is correct. Here, it is demonstrated that even in what may be the easiest and most reliable modeling exercise, Value-at-Risk forecasts from the most commonly used risk models provide very inconsistent results. http://www.riskresearch.org/?paperid=33&catid=0#papercontent http://www.riskresearch.org/?paperid=33&catid=0#papercontent Sun, 26 Oct 2008 00:00:00 +0100 We characterize the partial orderings induced by the most common risk measures and compare them to the partial orderings induced by first and second order stochastic dominance, respectively. We show which risk measures are consistent in the sense that they induce the same partial orderings as stochastic dominance. We also demonstrate which risk measures exhibit the property that stochastic dominance among risky choices imply consistency, and whether the reverse is true. Finally, we find that tail conditional expectation does not meet these consistency criteria. http://www.riskresearch.org/?paperid=25&catid=2#papercontent http://www.riskresearch.org/?paperid=25&catid=2#papercontent Fri, 20 Jun 2008 00:00:00 +0100 In the microstructure literature, it has been demonstrated that order flow, one way buying or selling pressure, carries information to the market. When assessing \textit{how} informative order flow is, the VAR methodology is typically employed, using impulse response functions, following \citeasnoun{Hasbrouck91a}. However, in such analyses, the direction of causality runs explicitly from order flow to asset return. If data are sampled at anything other than at the highest frequencies then any feedback trading may well appear contemporaneous; trading in period $t$ depends on the asset return in that interval. The implications of contemporaneous feedback trading are examined in the spot USD/EUR currency market and we find that when data are sampled at the one and five minute frequency, such trading strategies cause the price impact of order flow to be significantly larger than when feedback trading is ruled out. http://www.riskresearch.org/?paperid=32&catid=3#papercontent http://www.riskresearch.org/?paperid=32&catid=3#papercontent Fri, 20 Jun 2008 00:00:00 +0100 We characterize the investor's optimal portfolio allocation subject to a budget constraint and a probabilistic VaR constraint in complete markets environments with a finite number of states. The set of feasible portfolios might no longer be connected or convex, while the number of local optima increases exponentially with the number of states, implying computational complexity. The optimal constrained portfolio allocation may therefore not be monotonic in the state--price density. We propose a type of financial innovation, which splits states of nature, that is shown to weakly enhance welfare, restore monotonicity of the optimal portfolio allocation in the state–price density, and reduce computational complexity http://www.riskresearch.org/?paperid=11&catid=2#papercontent http://www.riskresearch.org/?paperid=11&catid=2#papercontent Sun, 5 Aug 2007 00:00:00 +0100 Hedge funds have emerged as influential players within the financial system, simultaneously providing considerable public benefits and contributing to systemic risk. Any attempts at regulating \hfs should preserve their contributions to the efficiency of the financial system, whilst directly targeting the systemic risk, recognizing that standard financial regulations do not lend themselves to the hedge fund sector. Supervisors should leave the ongoing activities of \hfs unencumbered, and focus their attention on crisis events with potential systemic consequences, with prime brokers and client banks legally obliged to participate in resolving such crisis events. http://www.riskresearch.org/?paperid=30&catid=5#papercontent http://www.riskresearch.org/?paperid=30&catid=5#papercontent Sat, 5 May 2007 00:00:00 +0100 We provide an equilibrium multi-asset pricing model with micro-founded systemic risk and heterogeneous investors. Systemic risk arises due to excessive leverage and risk taking induced by free-riding externalities. Global risk-sensitive financial regulations are introduced with a view of tackling systemic risk, with Value-at-Risk a key component. The model suggests that risk-sensitive regulation can lower systemic risk in equilibrium, at the expense of poor risk-sharing, an increase in risk premia, higher and asymmetric asset volatility, lower liquidity, more comovement in prices, and the chance that markets may not clear. http://www.riskresearch.org/?paperid=26&catid=2#papercontent http://www.riskresearch.org/?paperid=26&catid=2#papercontent Thu, 15 Feb 2007 00:00:00 +0000 The conference presentation was based on Bruche et al (2006) who study contagion in foreign exchange markets in the post- Asian crisis era. They develop a factor model specific to foreign exchange markets, where a key explanatory variable is foreign exchange trading volume. They find that their model identifies crisis episodes where volume has significant explanatory power. This suggests that trading volume would be an important variable policy makers should monitor in their analysis of currency crisis http://www.riskresearch.org/?paperid=31&catid=4#papercontent http://www.riskresearch.org/?paperid=31&catid=4#papercontent Thu, 1 Feb 2007 00:00:00 +0000 This paper explores the potential for violations of VaR subadditivity both theoretically and by simulations, and finds that for most practical applications VaR is subadditive. Hence, there is no reason to choose a more complicated risk measure than VaR, solely for reasons of coherence. http://www.riskresearch.org/?paperid=23&catid=1#papercontent http://www.riskresearch.org/?paperid=23&catid=1#papercontent Sat, 19 Nov 2005 00:00:00 +0000 Using regular variation to define heavy tailed distributions, we show that prominent downside risk measures produce similar and consistent ranking of heavy tailed risk. Expected shortfall, though, may not always distinguish between the differing risk levels of assets. http://www.riskresearch.org/?paperid=24&catid=1#papercontent http://www.riskresearch.org/?paperid=24&catid=1#papercontent Wed, 2 Nov 2005 00:00:00 +0000 There are increasing calls for the regulation of hedge funds, both for consumer protection and systemic reasons. We argue that the consumer protection arguments for direct regulation are not convincing, but find that the systemic concerns are sufficiently serious to warrant some forms of regulation. Existing regulatory methods, disclosure and activity restrictions, are unsuitable for hedge funds. Any future regulation must reduce the likelihood and potential costs of the failure of systemically important hedge funds whilst at the same time preserving the wider market benefits of hedge funds' ongoing activities. http://www.riskresearch.org/?paperid=19&catid=5#papercontent http://www.riskresearch.org/?paperid=19&catid=5#papercontent Sun, 4 Sep 2005 00:00:00 +0100 Many financial applications, such as risk analysis and derivatives pricing, depend on time scaling of risk. A common method for this purpose, though only correct when returns are iid normal, is the square-root-of-time rule where an estimated quantile of a return distribution is scaled to a lower frequency by the square-root of the time horizon. The aim of this paper is to examine time scaling of risk when returns follow a jump diffusion process. It is argued that a jump diffusion is well-suited for the modeling of systemic risk, which is the raison d'etre of the Basel capital adequacy proposals. We demonstrate that the square-root-of-time rule leads to a systematic underestimation of risk, whereby the degree of underestimation worsens with the time horizon, the jump intensity and the confidence level. As a result, even if the square-root-of-time rule has widespread applications in the Basel Accords, it fails to address the objective of the Accords. http://www.riskresearch.org/?paperid=21&catid=4#papercontent http://www.riskresearch.org/?paperid=21&catid=4#papercontent Sat, 6 Aug 2005 00:00:00 +0100 The introduction of risk sensitive bank capital charges into currency dependent economies exasperates the inherent procyclicality of banking regulations and frustrates the conduct of monetary policy. By requiring capital charges resulting from foreign currency lending to be denominated in the same foreign currency, the capital charge becomes countercyclical. http://www.riskresearch.org/?paperid=22&catid=4#papercontent http://www.riskresearch.org/?paperid=22&catid=4#papercontent Sat, 6 Aug 2005 00:00:00 +0100 An order flow model, where the coded identity of the counterparties of every trade is known, hence providing institution level order flow, is applied to both stable and crisis periods in a large and liquid overnight repo market in an emerging market economy. Institution level order flow is much more informative than cross sectionally aggregated order flow. The informativeness of institution level order flow increases with financial instability, with considerable heterogeneity in the yield impact across institutions. http://www.riskresearch.org/?paperid=18&catid=4#papercontent http://www.riskresearch.org/?paperid=18&catid=4#papercontent Tue, 26 Oct 2004 00:00:00 +0100 Implicit government guarantees induce moral hazard. The potential for moral hazard under the new Basel Capital Accord is explored with three different incomplete markets models. First, where investment decisions are affected by direct risk regulation causing more risky investments to be selected. Second, how risk regulation restricts banks' alternative (off--equilibrium) project selection. Third, principal--agent relationships between a bank's board and its risk manager. In all three cases the government intervention has the potential to increase unintended risk levels due to market incompleteness. http://www.riskresearch.org/?paperid=17&catid=0#papercontent http://www.riskresearch.org/?paperid=17&catid=0#papercontent Tue, 20 May 2003 00:00:00 +0100 Global financial regulations are increasingly becoming risk sensitive. The economic implications of such a system are analyzed in a general equilibrium model with agents that are heterogeneous in risk preferences, wealth, and degree of supervision. Excessive risk taking arises due to externalities, giving rise to systemic risk. The model suggests that risk sensitive regulation can lower systemic risk, at the expense of an increase in risk premia, higher asset volatility, lower liquidity, more comovement in prices, and the chance that markets may not clear. In some situations, however, systemic risk can be worsened by risk sensitive regulation. http://www.riskresearch.org/?paperid=10&catid=2#papercontent http://www.riskresearch.org/?paperid=10&catid=2#papercontent Thu, 1 May 2003 00:00:00 +0100 Many financial applications, such as risk analysis and derivatives pricing, depend on time scaling of risk. A common method for this purpose, though only correct when returns are iid normal, is the square-root-of-time rule where an estimated quantile of a return distribution is scaled to a lower frequency by the square-root of the time horizon. The aim of this paper is to examine time scaling of risk when returns follow a jump diffusion process. It is argued that a jump diffusion is well-suited for the modeling of systemic risk, which is the raison d'être of the Basel capital adequacy proposals. We demonstrate that the square-root-of-time rule leads to a systematic underestimation of risk, whereby the degree of underestimation worsens with the time horizon, the jump intensity and the confidence level. As a result, even if the square-root-of-time rule has widespread applications in the Basel Accords, it fails to address the objective of the Accords. http://www.riskresearch.org/?paperid=16&catid=-1#papercontent http://www.riskresearch.org/?paperid=16&catid=-1#papercontent Thu, 27 Feb 2003 00:00:00 +0000 Endogenous Risk http://www.riskresearch.org/?paperid=15&catid=0#papercontent http://www.riskresearch.org/?paperid=15&catid=0#papercontent Sun, 22 Sep 2002 00:00:00 +0100 Risk based regulation has emerged as the primary ingredient in the Basel-II proposals, where a bank capital is to become a direct function of a bank's riskiness. While the notion that bank capital be risk sensitive is intuitively appealing, the actual implementation, in the form of Basel-II, carries with it a host of potentially perverse side effects. Basel-II may increase financial risk, both for individual institutions and the entire banking system, and hence promote financial instability. This can happen, e.g., due to the endogenous nature of risk. http://www.riskresearch.org/?paperid=13&catid=2#papercontent http://www.riskresearch.org/?paperid=13&catid=2#papercontent Fri, 13 Sep 2002 00:00:00 +0100 Most financial risk regulations assume that asset returns are exogenous, where risk is estimated by historical data. This assumption fails to take into account the feedback effect of trading decisions on prices. We investigate this by means of simulations of a general equilibrium model and compare the result to the case when risks regulations are not present. Prices and liquidity are lower in the presence of risk regulations, while volatility is higher. These effects are especially pronounced during crisis. Far from promoting stability, adoption of risk regulations may have the perverse effect of exacerbating financial instability. http://www.riskresearch.org/?paperid=9&catid=2#papercontent http://www.riskresearch.org/?paperid=9&catid=2#papercontent Tue, 20 Aug 2002 00:00:00 +0100 Extreme value theory has been applied to many areas of economics where the data is heavy tailed, e.g. in the analysis of market structure and risk forecasting. Accurate inference has, however, been hindered by the lack of consistent procedures for determining the start of the tail. A double subsample bootstrap procedure is proposed to solve this problem. The accuracy of the procedure is accessed with Monte Carlo experiments. Subsequently it is applied to Gibrat's and Zipf's laws, as well as the estimation of financial risk. http://www.riskresearch.org/?paperid=4&catid=1#papercontent http://www.riskresearch.org/?paperid=4&catid=1#papercontent Sat, 2 Feb 2002 00:00:00 +0000 Under the new Capital Accord banks can choose between different type of risk management systems. Using a stylized model of risk management systems which differ in quality and by modelling the relationship between the bank board and the risk manager, we consider the incentives for the adoption of a particular system. We show that in some cases banks may adversely adopt an unsophisticated risk management system in order to evade regulation. http://www.riskresearch.org/?paperid=12&catid=2#papercontent http://www.riskresearch.org/?paperid=12&catid=2#papercontent Sun, 7 Oct 2001 00:00:00 +0100 Submitted in Response to the Basel Committee's Request for Comments http://www.riskresearch.org/?paperid=8&catid=0#papercontent http://www.riskresearch.org/?paperid=8&catid=0#papercontent Tue, 26 Jun 2001 00:00:00 +0100 This paper considers the properties of risk measures, primarily Value--at--Risk (VaR), from both internal and external (regulatory) points of view. It is argued that since market data is endogenous to market behavior, statistical analysis made in times of stability does not provide much guidance in times of crisis. In an extensive survey across data classes and risk models, the empirical properties of current risk forecasting models are found to be lacking in robustness while being excessively volatile. For regulatory use, the VaR measure may give misleading information about risk, and in some cases may actually increase both idiosyncratic and systemic risk. http://www.riskresearch.org/?paperid=1&catid=0#papercontent http://www.riskresearch.org/?paperid=1&catid=0#papercontent Wed, 6 Jun 2001 00:00:00 +0100 In this article we point to several important facts, which we feel have been neglected in the discussion about VaR models in general and the Basle internal models approach in particular. We argue that the current set of Basle requirements still provides disincentives for the development of more reliable VaR models, and show that considerable improvement of current VaR models is possible by means of techniques that explicitly focus on the properties of extreme return fluctuations. We then briefly discuss how a change in the determination of the Basle 'multiplication factor' may encourage the industry to adopt improved VaR models, such as those proposed here. http://www.riskresearch.org/?paperid=3&catid=0#papercontent http://www.riskresearch.org/?paperid=3&catid=0#papercontent Sat, 10 Jun 2000 00:00:00 +0100 Tail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e. the number of extreme order statistics on which the estimation is based. A complete solution to the sample fraction selection is given by means of a two step subsample bootstrap method. This method adaptively determines the sample fraction that minimizes the asymptotic mean squared error. Unlike previous methods, prior knowledge of the second order parameter is not required. In addition, we are able to dispense with the need for a prior estimate of the tail index which already converges roughly at the optimal rate. The only arbitrary choice of parameters is the number of Monte Carlo replications. http://www.riskresearch.org/?paperid=5&catid=1#papercontent http://www.riskresearch.org/?paperid=5&catid=1#papercontent Sat, 10 Jun 2000 00:00:00 +0100 Accurate prediction of the frequency of extreme events is of primary importance in many financial applications such as Value--at--Risk (VaR) analysis. We propose a semi--parametric method for VaR evaluation. The largest risks are modelled parametrically, while smaller risks are captured by the non--parametric empirical distribution function. The semi--parametric method is compared with historical simulation and the J. P. Morgan RiskMetrics technique on a portfolio of stock returns. For predictions of low probability worst outcomes, RiskMetrics analysis underpredicts the VaR while historical simulation overpredicts the VaR. However, the estimates obtained from applying the semi--parametric method are more accurate in the VaR prediction. In addition we study the role of an option that lowers the downside risk of the portfolio. http://www.riskresearch.org/?paperid=2&catid=1#papercontent http://www.riskresearch.org/?paperid=2&catid=1#papercontent Sun, 28 May 2000 00:00:00 +0100