Estimating Risk with Copulas

Estimating Risk with Copulas

Iva Mihaylova (University of St. Gallen, Switzerland)
Copyright: © 2014 |Pages: 14
DOI: 10.4018/978-1-4666-5202-6.ch079
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Introduction

It is hardly possible to imagine the elaboration of any rational decision-making strategy in financial, capital, commodities markets, real sectors of the economy and international trade, without the proper consideration of the key concept of ‘risk’. However, there is a lack of consensus in the literature with regard to a uniform and consistent definition of the concept ‘risk’, which successfully to capture all of its elements. For example, Machina & Rothschild (2008) discuss the fundamental difference between ‘risk’ and ‘uncertainty’ as follows: ‘A situation is said to involve risk if the randomness facing an economic agent presents itself in the form of exogenously specified or scientifically calculable objective probabilities, as with gambles based on a roulette wheel or a pair of dice. A situation is said to involve uncertainty if the randomness presents itself in the form of alternative possible events, as with bets on a horse race, or decisions involving whether or not to buy earthquake insurance.’ Similarly, according to ISO 31000:2009, a globally-accepted standard for risk management, authored by the International Standards Organization (2009), risk can be described as the ‘effect of uncertainty on objectives’. In these two definitions, ‘risk’ is represented as a symmetric concept: with a potential for gain or loss. In contrast, McNeil, Frey, & Embrechts (2005) define an asymmetric version of the same concept as: ‘the quantifiable likelihood of loss or less-than expected returns’. Borghesi & Gaudenzi (2013) formulate the concept in an analogous way as ‘an unfavorable event capable of generating a negative sign deviation from a given expected situation, such as a smaller gain or a greater loss’ and as a ‘set of hindrances that threaten the pursuit of the business’s objectives’. These definitions with emphasis only on adverse results are adhered to in the regulatory risk management. Its central goal is to quantify the downside of risk, for example, in common tasks such as assessment of the decrease of the value of a portfolio, due to its exposure to various risk classes. A key risk that might adversely affect a portfolio is the market risk, defined as the overall uncertainty about the future asset price. For example, a press announcement for a merger between two key market players is supposed to affect their shares’ prices. When the goal of the market participant is to assess his/her extra loss when a position must be quickly closed or changed due to transaction uncertainty, the situation involves assessment of liquidity risk. In contrast, credit risk involves inability of the second contractual party, for example a borrower or an issuer of corporate bonds, to meet his/her pre-established obligations. Counterparty risk is a subset of the credit risk category, as it covers cases where a second party of a specific transaction is unable to complete it at expiration. The emphasis of this chapter is exclusively on market risk, to which are exposed the returns of financial assets, and on copula-based estimation methods of one of the most popular market risk measures in quantitative risk management, Value at Risk (VaR). A second contribution is the presentation of the current experts’ debate whether VaR can be successfully substituted by alternative risk measures.

Key Terms in this Chapter

Backtesting: In order to validate (assess for presence of evidence of model misspecification) risk measures, for example Value-at-Risk (VaR), a systematic comparison is performed between the actual portfolio profit and loss (P&L) observations and the predicted VaR at the respective confidence level. If the actual P&L observations considerably exceed the VaR forecast, this means that the risk measure model underestimates the (market) risk. The opposite situation signals for risk overestimation.

Copula: A dependence model, defined as a multivariate distribution function over a hypercube with uniform marginal distributions.

Value-at-Risk (VaR): A a risk measure, defined as the maximum loss in the value of a portfolio for a given confidence level alpha (usually between 90 and 100 per cent) over a pre-defined holding period, and based on a price changes distribution (Profit and Loss distribution).

Risk Management: The concept involves actions aimed at maintaining an adequate profitability (return on investments), solvency margin (assets in excess of liabilities), increase of the company’s shareholder value, as well as risk reduction (for example by hedging and diversification). The conceptual term ‘risk management’ is to be distinguished from the technical term ‘risk measurement’. For example, in the Value-at-Risk setting, the latter term describes the construction of a portfolio profit and loss distribution and the calculation of certain percentile.

Risk: According to ISO 31000:2009, a global risk management standard, authored by the International Standards Organization, risk can be described as the ‘effect of uncertainty on objectives’. In this definition, ‘risk’ is considered as a symmetric concept: with a potential for a gain and a loss.

Expected Shortfall (ES): A risk measure, defined as the expected value of the portfolio loss, provided that a Value-at-Risk exceedance has been registered.

Coherent Risk Measures: A risk measure is coherent, if it possesses the following four desirable properties, with economic interpretations in brackets according to McNeil et al. (2005) , p. 240: (1) translation invariance (the consequence of an increase/ a decrease with a constant of a position, to which corresponds a loss quantity L, leads to a modification of the risk requirements with exactly the same amount); (2) positive homogeneity (no diversifying or netting of losses in a portfolio); (3) monotonicity (more risk capital must be allocated to positions, associated with higher losses); (4) subadditivity (diversification decreases risk).

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