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Chapter 9: Measuring risk

Financial risk is the chance that an investment, such as a portfolio of stocksand options, will lose value. Conventionally, in finance, risk is classified asmarket risk, credit risk and operational risk. There are other types of risks,such as liquidity and reputational risk, which are more difficult to characterizeand model. They are the object of current research in academia and industry.

Market risk is the risk of loss associated with modifications in the conditionsof the market. For instance, the loss of value of an investment portfolio due tochanges in asset prices. In this course we will focus on this type of risk.

Unit 9.1describes how to measure the risk of an investment portfolio. Thegoal is to model the distribution of losses at a specified time horizon (e.g. 5days). This distribution quantifies the probability of changes in the value of theportfolio that result from variations in the risk factors (e.g. the asset prices inshort-term market risk). To define the risk measures we need to set a probabilitylevel,p. The most relevant events for the measurement of risk are extreme eventsin the right tail of the loss distribution. Therefore large values ofp (e.g. 95% or99%) are commonly used to compute the risk measures. The standard measureof risk is the Value at Risk. It is defined as the p-quantile of the loss distribution.VaR, or value at risk, is not a coherent risk measure. This means that the sumof the values at risk of two different portfolios can be smaller than the value atrisk for the single portfolio that results from merging these two portfolios. Thisis contrary to the intuition that diversification should lead to a reduction of risk.A coherent measure of risk that properly reflects the benefits of diversificationis the expected shortfall. The expected shortfall is defined as the average of thelosses that are larger than the Value at Risk. This coherent risk measure is also

known as tail or conditional VaR.The value at risk and the expected shortfall can be computed using simulatedscenarios for changes in the risk factors on which the value of the portfoliodepends [Unit 9.2]. Such scenarios can be obtained from the recent history ofchanges in the relevant risk factors. This method, which is illustrated in Unit9.3, is simple and has the advantage of naturally incorporating the dependenciesamong different risk factors. Instead of relying on recent history, we can generateforward-looking scenarios using Monte Carlo simulation. To fulfill this task weneed to learn how to model the distribution of changes of the risk factors andthe dependency structure. This is what well do in the following units.

InUnit 9.4we illustrate how to model an individual risk factor. The modeldistribution can be calibrated to data using either the method of moments orby maximum likelihood.

Our portfolio includes investments that involve several assets. To measureits risk we need to make models of the joint distribution of the future pricesof these assets. A simple model is to assume that the joint distribution of theasset log returns is a multivariate Gaussian [Unit 9.5].

The multivariate Gaussian distribution can model only linear correlationsamong different variables. For more general dependencies, we can use copulas

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