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Unformatted text preview: Risk Measures Lecture Notes for Actsc 445/845 Department of Statistics and Actuarial Science University of Waterloo Notes by Prof. K.S. Tan/Actsc 445/845 Risk Measures – p. 1/42 Introduction So far we have been examining risks associated exclusively with interest rates e.g. price risk, reinvestment risk These risks are just two of the market risks that are specified to fixedincome securities. for example, credit risks, currency risk Market risk is the exposure to the uncertain market value of a portfolio (i.e. risk relating to movements in market variables). Question: Suppose we have established the loss distribution of a portfolio, how do we utilize the characteristics of the distribution for pricing, reserving, and risk management? similar risk management concept applies to an enterprise such as a bank or an insurance company Notes by Prof. K.S. Tan/Actsc 445/845 Risk Measures – p. 2/42 Risk Measure A risk measure is a functional mapping a loss (or profit) distribution to the real numbers. If we represent the distribution by the appropriate random variable X , and let H represent the risk measure functional, then H : X → ℜ The risk measure is assumed in some ways to encapsulate the risk associated with a loss distribution. Risk measure is an attempt to provide a single number summarizing the total market risk in a portfolio of financial assets Risk measure attempts to quantify and thereby control market risk! Notes by Prof. K.S. Tan/Actsc 445/845 Risk Measures – p. 3/42 Examples of Loss Distributions Risk Example I: A loss which is normally distributed with mean 33 and standard deviation 109.0 Risk Example II: A loss with a Pareto distribution with mean 33 and standard deviation 109.0 Risk Example III: A loss of 1000 max(1 − S 10 , 0) , where S 10 is the price at time T = 10 of some underlying equity investment, with initial value S = 1 . We assume the equity investment price process, S t follows a lognormal process with parameters μ = 0 . 08 and σ = 0 . 22 . This means that S t ∼ lognormal ( μt, σ 2 t ) . This loss distribution has mean value 33.0, and standard deviation 109.0. This risk is a simplified version of the put option embedded in the popular ‘variable annuity’ contracts. Notes by Prof. K.S. Tan/Actsc 445/845 Risk Measures – p. 4/42 Probability Density Functions: Examples 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 .000 0 .001 0 .002 0 .003 0 .004 0 .005 L o s s P robab ility Dens ity Func tion L o g n o r m a l P u t O p t i o n N o r m a l P a r e t o 4 0 0 6 0 0 8 0 0 1 0 0 0 e+00 2 e04 4 e04 6 e04 L o g n o r m a l P u t O p t i o n N o r m a l P a r e t o L o s s P robab ility Dens ity Func tion Notes by Prof. K.S. Tan/Actsc 445/845 Risk Measures – p. 5/42 Value At Risk (VaR)– the Quantile Risk Measure Definition: VaR is a statistical measure of the risk that estimates the maximum loss that may be experienced on a portfolio over a given period of time, and with a given level of confidence....
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 Spring '09
 ChristianeLemieux
 Normal Distribution, probability density function, Prof. K.S. Tan/Actsc, Prof. K.S. Tan/Actsc 445/845

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