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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Model risk</span></span>
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</style><table class="sidebar nomobile nowraplinks" style="border:1px solid #B2BEB5"><tbody><tr><td class="sidebar-pretitle" style="background:#ACE1AF">Categories of</td></tr><tr><th class="sidebar-title-with-pretitle" style="background:#ACE1AF;padding-bottom:0.15em;"><a href="Financial_risk" title="Financial risk">Financial risk</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"></span></td></tr><tr><th class="sidebar-heading" style="background:#ACE1AF">
<a href="Credit_risk" title="Credit risk">Credit risk</a></th></tr><tr><td class="sidebar-content plainlist">
<ul><li><a href="Settlement_risk" title="Settlement risk">Settlement risk</a></li>
<li><a href="Concentration_risk" title="Concentration risk">Concentration risk</a></li>
<li><a href="Sovereign_credit_risk" title="Sovereign credit risk">Sovereign risk</a></li>
<li><a href="Default_(finance)" title="Default (finance)">Default risk</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#ACE1AF">
<a href="Market_risk" title="Market risk">Market risk</a></th></tr><tr><td class="sidebar-content plainlist">
<ul><li><a href="Interest_rate_risk" title="Interest rate risk">Interest rate risk</a></li>
<li><a href="Inflation_risk" class="mw-redirect" title="Inflation risk">Inflation risk</a></li>
<li><a href="Currency_risk" class="mw-redirect" title="Currency risk">Currency risk</a></li>
<li><a href="Equity_risk" title="Equity risk">Equity risk</a></li>
<li><a href="Commodity_risk" title="Commodity risk">Commodity risk</a></li>
<li><a href="Volatility_risk" title="Volatility risk">Volatility risk</a></li>
<li><a href="Systemic_risk" title="Systemic risk">Systemic risk</a></li>
<li><a href="Systematic_risk" title="Systematic risk">Systematic risk</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#ACE1AF">
<a href="Liquidity_risk" title="Liquidity risk">Liquidity risk</a></th></tr><tr><td class="sidebar-content plainlist">
<ul><li><a href="Refinancing_risk" title="Refinancing risk">Refinancing risk</a></li>
<li><a href="Deposit_risk" title="Deposit risk">Deposit risk</a></li>
<li><a href="Margining_risk" title="Margining risk">Margining risk</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#ACE1AF">
<a href="Investment_risk" class="mw-redirect" title="Investment risk">Investment risk</a></th></tr><tr><td class="sidebar-content plainlist">
<ul>
<li><a href="Execution_risk" class="mw-redirect" title="Execution risk">Execution risk</a></li>
<li><a href="Valuation_risk" title="Valuation risk">Valuation risk</a></li></ul></td>
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<a href="Business_risk" class="mw-redirect" title="Business risk">Business risk</a></th></tr><tr><td class="sidebar-content plainlist">
<ul><li><a href="Reputational_risk" class="mw-redirect" title="Reputational risk">Reputational risk</a></li>
<li><a href="Operational_risk" title="Operational risk">Operational risk</a></li>
<li><a href="Country_risk" title="Country risk">Country risk</a></li>
<li><a href="Political_risk" title="Political risk">Political risk</a></li>
<li><a href="Legal_risk" title="Legal risk">Legal risk</a></li>
<li><a href="Moral_hazard" title="Moral hazard">Moral hazard</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#ACE1AF">
<a href="Profit_risk" title="Profit risk">Profit risk</a></th></tr><tr><th class="sidebar-heading" style="background:#ACE1AF">
<a href="Non-financial_risk" title="Non-financial risk">Non-financial risk</a></th></tr><tr><td class="sidebar-content plainlist">
<ul><li><a href="Stranded_asset" title="Stranded asset">Stranded asset</a></li></ul></td>
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<p><b>Case studies:</b>
</p>
<ul><li><a href="Merrill_Lynch" class="mw-redirect" title="Merrill Lynch">Merrill Lynch</a> (1970s; $70m loss) - for <a href="Zero-coupon_bond#Strip_bonds" title="Zero-coupon bond">stripped bonds</a>, used a single <a href="Yield_to_maturity" title="Yield to maturity">par yield</a> instead of separate annuity yield-curves and zero-coupon curves for the resultant <a href="Interest_only" class="mw-redirect" title="Interest only">IO</a> and <a href="Principal_only" class="mw-redirect" title="Principal only">PO</a> securities.<sup id="cite_ref-Gibson_1-0" class="reference"><a href="#cite_note-Gibson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul>
<ul><li><a href="NatWest" title="NatWest">NatWest</a> (1997; £90m loss) - incorrect model specification, "a naive volatility input in their systems",<sup id="cite_ref-Gibson_1-1" class="reference"><a href="#cite_note-Gibson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> for interest rate options and swaptions.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul>
<ul><li><a href="Bank_of_Tokyo-Mitsubishi" class="mw-redirect" title="Bank of Tokyo-Mitsubishi">Bank of Tokyo-Mitsubishi</a> (1997; $83m loss) - a "systematic pricing bias"<sup id="cite_ref-Gibson_1-2" class="reference"><a href="#cite_note-Gibson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> for <a href="Out-of-the-money" class="mw-redirect" title="Out-of-the-money">out-of-the-money</a> and <a href="Option_style#Bermudan_option" title="Option style">Bermuda</a> swaptions which had been calibrated to <a href="At-the-money" class="mw-redirect" title="At-the-money">at-the-money</a> <a href="Option_(finance)#Option_styles" title="Option (finance)">vanilla</a> swaptions.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul>
<ul><li><a href="Barclays_de_Zoete_Wedd" class="mw-redirect" title="Barclays de Zoete Wedd">Barclays de Zoete Wedd</a> (1997; £15m loss) - mispriced currency options.<sup id="cite_ref-New_England_Econ_Review_4-0" class="reference"><a href="#cite_note-New_England_Econ_Review-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul>
<ul><li><a href="LTCM" class="mw-redirect" title="LTCM">LTCM</a> (1998; required $3.65 billion in recapitalization) - <a href="Financial_risk_modeling" title="Financial risk modeling">risk models</a> had drastically under-estimated the risks of a profound <a href="Economic_crisis" class="mw-redirect" title="Economic crisis">economic crisis</a>, due to an <a href="Long-Term_Capital_Management#Analysis" title="Long-Term Capital Management">insufficient data-window</a> and a lack of <a href="Stress_test_(financial)" title="Stress test (financial)">stress testing</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>
<ul><li><a href="National_Australia_Bank" title="National Australia Bank">National Australia Bank</a> (2001; $2.2 Billion AUD loss) - <a href="National_Australia_Bank#HomeSide_write-downs" title="National Australia Bank">its Homeside</a> interest rate model made inconsistent use of rates.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7:30_report_7-0" class="reference"><a href="#cite_note-7:30_report-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul>
<ul><li><a href="2008_financial_crisis" title="2008 financial crisis">2008 financial crisis</a> – Over-reliance on <a href="David_X._Li" title="David X. Li">David X. Li</a>'s <a href="Gaussian_copula" class="mw-redirect" title="Gaussian copula">Gaussian copula</a> model misprices the risk of <a href="Collateralized_debt_obligation" title="Collateralized debt obligation">collateralized debt obligations</a>.<sup id="cite_ref-Wired.com_8-0" class="reference"><a href="#cite_note-Wired.com-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul>
</td></tr></tbody></table>
<p>In <a href="Finance" title="Finance">finance</a>, <b>model risk</b> is the risk of loss resulting from using insufficiently accurate <a href="Financial_model" class="mw-redirect" title="Financial model">models</a> to make decisions, originally and frequently in the context of valuing <a href="Financial_securities" class="mw-redirect" title="Financial securities">financial securities</a>.<sup id="cite_ref-derman-model-risk_9-0" class="reference"><a href="#cite_note-derman-model-risk-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>
</p><p>Here, Rebonato (2002) defines model risk as "the risk of occurrence of a significant difference between the <a href="Mark-to-model" class="mw-redirect" title="Mark-to-model">mark-to-model</a> value of a complex and/or <a href="Illiquid" class="mw-redirect" title="Illiquid">illiquid</a> instrument, and the price at which the same instrument is revealed to have <a href="Mark_to_market" class="mw-redirect" title="Mark to market">traded in the market</a>".
</p><p>However, model risk is increasingly relevant in contexts other than financial securities valuation, including assigning consumer <a href="Credit_score" title="Credit score">credit scores</a>, real-time <a href="Fraud_detection" class="mw-redirect" title="Fraud detection">prediction of fraudulent</a> credit card transactions, and computing the probability of an air flight passenger being a terrorist.
In fact, Burke regards failure to use a model (instead over-relying on expert judgment) as a type of model risk.<sup id="cite_ref-Burke-risk-professional-interest-forum_10-0" class="reference"><a href="#cite_note-Burke-risk-professional-interest-forum-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Types">Types</h2></div>
<p>Derman describes various types of model risk that arise from using a model:<sup id="cite_ref-derman-model-risk_9-1" class="reference"><a href="#cite_note-derman-model-risk-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Wrong_model">Wrong model</h3></div>
<ul><li>Inapplicability of model.</li>
<li>Incorrect model specification.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Model_implementation">Model implementation</h3></div>
<ul><li>Programming errors.</li>
<li>Technical errors.</li>
<li>Use of inaccurate numerical approximations.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Model_usage">Model usage</h3></div>
<ul><li>Implementation risk.</li>
<li>Data issues.</li>
<li>Calibration errors.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Uncertainty_on_volatility">Uncertainty on volatility</h3></div>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Volatility_risk" title="Volatility risk">Volatility risk</a></div>
<p>Volatility is the most important input in risk management models and pricing models. Uncertainty on volatility leads to model risk. Derman believes that products whose value depends on a <a href="Volatility_smile" title="Volatility smile">volatility smile</a> are most likely to suffer from model risk. He writes "I would think it's safe to say that there is no area where model risk is more of an issue than in the modeling of the volatility smile."<sup id="cite_ref-derman-euronext-smile_11-0" class="reference"><a href="#cite_note-derman-euronext-smile-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>
Avellaneda & Paras (1995) proposed a systematic way of studying and mitigating model risk resulting from volatility uncertainty.<sup id="cite_ref-AvellanedaParas_12-0" class="reference"><a href="#cite_note-AvellanedaParas-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Time_inconsistency">Time inconsistency</h3></div>
<p>Buraschi and Corielli formalise the concept of 'time inconsistency' with regards to <a href="Arbitrage" title="Arbitrage">no-arbitrage</a> models that allow for a perfect fit of the term structure of the interest rates. In these models the current <a href="Yield_curve" title="Yield curve">yield curve</a> is an input so that new observations on the yield curve can be used to update the model at regular frequencies. They explore the issue of time-consistent and self-financing strategies in this class of models. Model risk affects all the three main steps of <a href="Financial_risk_management" title="Financial risk management">risk management</a>: specification, estimation and implementation.<sup id="cite_ref-Buraschi2_13-0" class="reference"><a href="#cite_note-Buraschi2-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Correlation_uncertainty">Correlation uncertainty</h3></div>
<p>Uncertainty on correlation parameters is another important source of model risk. Cont and Deguest propose a method for computing model risk exposures in multi-asset equity derivatives and show that options which depend on the worst or best performances in a basket (so called <a href="Rainbow_option" title="Rainbow option">rainbow option</a>) are more exposed to model uncertainty than index options.<sup id="cite_ref-ContDeguest2012_14-0" class="reference"><a href="#cite_note-ContDeguest2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>
</p><p>Gennheimer investigates the model risk present in pricing basket default derivatives. He prices these derivatives with various copulas and concludes that "... unless one is very sure about the dependence structure governing the credit basket, any investors willing to trade basket default products should imperatively compute prices under alternative copula specifications and verify the estimation errors of their simulation to know at least the model risks they run".<sup id="cite_ref-Gennheimer-copula_15-0" class="reference"><a href="#cite_note-Gennheimer-copula-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Complexity">Complexity</h3></div>
<p>Complexity of a model or a financial contract may be a source of model risk, leading to incorrect identification of its risk factors. This factor was cited as a major source of model risk for mortgage backed securities portfolios during the 2007 crisis.
</p>
<div class="mw-heading mw-heading3"><h3 id="Illiquidity_and_model_risk">Illiquidity and model risk</h3></div>
<p>Model risk does not only exist for complex financial contracts. Frey (2000) presents a study of how market illiquidity is a source of model risk. He writes "Understanding the robustness of models used for hedging and risk-management purposes with respect to the assumption of perfectly liquid markets is therefore an important issue in the analysis of model risk in general."<sup id="cite_ref-Frey_16-0" class="reference"><a href="#cite_note-Frey-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>
<a href="Convertible_bond" title="Convertible bond">Convertible bonds</a>, <a href="Mortgage-backed_security" title="Mortgage-backed security">mortgage-backed securities</a>, and <a href="High-yield_bond" class="mw-redirect" title="High-yield bond">high-yield bonds</a> can often be illiquid and difficult to value. Hedge funds that trade these securities can be exposed to model risk when calculating monthly NAV for its investors.<sup id="cite_ref-Keith_H_Black_17-0" class="reference"><a href="#cite_note-Keith_H_Black-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Spreadsheet_Errors">Spreadsheet Errors</h3></div>
<p>Many models are built using <a href="Spreadsheet" title="Spreadsheet">spreadsheet</a> technology, which can be particularly prone to implementation errors.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>
Mitigation strategies include adding consistency checks, validating inputs, and using specialized tools.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>
See <a href="Spreadsheet#Spreadsheet_risk" title="Spreadsheet">Spreadsheet risk</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Quantitative_approaches">Quantitative approaches</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Model_averaging_vs_worst-case_approach">Model averaging vs worst-case approach</h3></div>
<p>Rantala (2006) mentions that "In the face of model risk, rather than to base decisions on a single selected 'best' model, the modeller can base his inference on an entire set of models by using model averaging."<sup id="cite_ref-Rantala_20-0" class="reference"><a href="#cite_note-Rantala-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>
This approach avoids the "flaw of averages".<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>
</p><p>Another approach to model risk is the worst-case, or minmax approach, advocated in decision theory by Gilboa and Schmeidler.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>
In this approach one considers a range of models and minimizes the loss encountered in the worst-case scenario. This approach to model risk has been developed by Cont (2006).<sup id="cite_ref-Cont2006_23-0" class="reference"><a href="#cite_note-Cont2006-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>
</p><p>Jokhadze and Schmidt (2018) propose several model risk measures using Bayesian methodology. They introduce superposed risk measures that incorporate model risk and enables consistent market and model risk management. Further, they provide axioms of model risk measures and define several practical examples of superposed model risk measures in the context of financial risk management and contingent claim pricing.
</p>
<div class="mw-heading mw-heading3"><h3 id="Quantifying_model_risk_exposure">Quantifying model risk exposure</h3></div>
<p>To measure the risk induced by a model, it has to be compared to an alternative model, or a set of alternative benchmark models. The problem is how to choose these benchmark models.<sup id="cite_ref-ModelRiskTalanxAG_24-0" class="reference"><a href="#cite_note-ModelRiskTalanxAG-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In the context of derivative pricing Cont (2006) proposes a quantitative approach to measurement of model risk exposures in derivatives portfolios: first, a set of benchmark models is specified and calibrated to market prices of liquid instruments, then the target portfolio is priced under all benchmark models. A measure of exposure to model risk is then given by the difference between the current portfolio valuation and the worst-case valuation under the benchmark models. Such a measure may be used as a way of determining a reserve for model risk for derivatives portfolios.<sup id="cite_ref-Cont2006_23-1" class="reference"><a href="#cite_note-Cont2006-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Position_limits_and_valuation_reserves">Position limits and valuation reserves</h3></div>
<p>Jokhadze and Schmidt (2018) introduce monetary market risk measures that covers model risk losses. Their methodology enables to harmonize market and model risk management and define limits and required capitals for risk positions.
</p><p>Kato and Yoshiba discuss qualitative and quantitative ways of controlling model risk. They write "From a quantitative perspective, in the case of pricing models, we can set up a reserve to allow for the difference in estimations using alternative models. In the case of risk measurement models, scenario analysis can be undertaken for various fluctuation patterns of risk factors, or position limits can be established based on information obtained from scenario analysis."<sup id="cite_ref-Kato-and-Yoshiba-2000_25-0" class="reference"><a href="#cite_note-Kato-and-Yoshiba-2000-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Cont (2006) advocates the use of model risk exposure for computing such reserves.
</p>
<div class="mw-heading mw-heading2"><h2 id="Mitigation">Mitigation</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Theoretical_basis">Theoretical basis</h3></div>
<ul><li>Considering key assumptions.</li>
<li>Considering simple cases and their solutions (model boundaries).</li>
<li><a href="Occam's_razor" title="Occam's razor">Parsimony</a>.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Implementation">Implementation</h3></div>
<ul><li>Pride of ownership.</li>
<li>Disseminating the model outwards in an orderly manner.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Testing">Testing</h3></div>
<ul><li><a href="Stress_testing_(software)" title="Stress testing (software)">Stress testing</a> and <a href="Backtesting" title="Backtesting">backtesting</a>.</li>
<li>Avoid letting small issues snowball into large issues later on.</li>
<li>Independent validation</li>
<li>Ongoing monitoring and against market</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Examples_of_model_risk_mitigation">Examples of model risk mitigation</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Parsimony">Parsimony</h3></div>
<p>Taleb wrote when describing why most new models that attempted to correct the inadequacies of the <a href="Black%E2%80%93Scholes" class="mw-redirect" title="Black–Scholes">Black–Scholes</a> model failed to become accepted:
</p>
<dl><dd>Traders are not fooled by the Black–Scholes–Merton model. The existence of a '<a href="Volatility_surface" class="mw-redirect" title="Volatility surface">volatility surface</a>' is one such adaptation. But they find it preferable to fudge one parameter, namely volatility, and make it a function of time to expiry and strike price, rather than have to precisely estimate another.<sup id="cite_ref-Taleb-Dynamic-Hedging_26-0" class="reference"><a href="#cite_note-Taleb-Dynamic-Hedging-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></dd></dl>
<p>However, Cherubini and Della Lunga describe the disadvantages of parsimony in the context of volatility and correlation modelling. Using an excessive number of parameters may induce <a href="Overfitting" title="Overfitting">overfitting</a> while choosing a severely specified model may easily induce model misspecification and a systematic failure to represent the future distribution.<sup id="cite_ref-Structured_Finance_27-0" class="reference"><a href="#cite_note-Structured_Finance-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Model_risk_premium">Model risk premium</h3></div>
<p>Fender and Kiff (2004) note that holding complex financial instruments, such as <a href="Collateralized_debt_obligation" title="Collateralized debt obligation">CDOs</a>, "translates into heightened dependence on these assumptions and, thus, higher model risk. As this risk should be expected to be priced by the market, part of the yield pick-up obtained relative to equally rated single obligor instruments is likely to be a direct reflection of model risk."<sup id="cite_ref-BIS_No163_28-0" class="reference"><a href="#cite_note-BIS_No163-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="2008_financial_crisis#Incorrect_pricing_of_risk" title="2008 financial crisis">2008 financial crisis § Incorrect pricing of risk</a></li>
<li><a href="Financial_risk_management#Banking" title="Financial risk management">Financial risk management § Banking</a></li>
<li><a href="Risk_management" title="Risk management">Risk management</a></li>
<li><a href="Statistical_model_specification" title="Statistical model specification">Statistical model specification</a></li>
<li><a href="Value_at_risk" title="Value at risk">Value at risk</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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