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Unformatted text preview: ACTL3004: Week 2 Financial Economics for Insurance and Superannuation: Week 2 Risk Measures, Financial Data and the Efficient Market Hypothesis 1 / 26 ACTL3004: Week 2 Investment Risk Measures Investment Risk Measures Let X denote the rate of return random variable for a particular investment and let X f X ( ) , the density of X , with mean = E ( X ) . The riskiness of the investment can be assessed based on the following: Variance: Variance is a measure of how variable the return is relative to the mean. Var ( X ) = E h ( X ) 2 i = E ( X 2 ) 2 = Z  ( x ) 2 f X ( x ) dx SemiVariance of Return: It is sometimes called the downside semivariance. It accounts only for the variation below the mean. semi var ( X ) = Z  ( x ) 2 f X ( x ) dx . 2 / 26 ACTL3004: Week 2 Investment Risk Measures Expected Shortfall: It gives the shortfall below a certain level, say L , called the benchmark. It is useful for monitoring a funds exposure to risk. ES ( L ) = Z L ( L x ) f X ( x ) dx Skewness: It measures the extent to which a probability distribution is asymmetric about its mean. S ( X ) = E h ( X ) 3 i = Z  ( x ) 3 f X ( x ) dx 3 / 26 ACTL3004: Week 2 Investment Risk Measures Kurtosis: It measures the peakedness or pointedness of a distribution. K ( X ) = E h ( X ) 4 i = Z  ( x ) 4 f X ( x ) dx Shortfall Measures: It provides a measure of the expected underperformance relative to a given benchmark. Z L g ( L x ) f X ( x ) dx where g ( ) is some function. 4 / 26 ACTL3004: Week 2 Investment Risk Measures Special cases for this function include: 1. If g ( L x ) = 1 , then we have the socalled shortfall probability p ( L ) = Z L f X ( x ) dx = Pr ( X L ) . 2. If g ( L x ) = ( L x ) , then we have the socalled expected shortfall as defined above. 3. If g ( L x ) = ( L x ) 2 , then we have the socalled shortfall variance Z L ( L x ) 2 f X ( x ) dx . 5 / 26 ACTL3004: Week 2 ValueatRisk ValueatRisk I Valueatrisk measures the worst expected loss over a given horizon under normal market conditions at a given confidence level. I It is a risk management tool. I It is the answer to the following question: What is the maximum loss that can be incurred from holding a security or a portfolio of securities over a given time period so that there is a low probability, say 1 % , that the actual loss will be larger? I It describes the quantile of the projected portfolio loss distribution. I It is a measure of market risk: the risk that changes in financial market prices and rates will reduce the value of a security or a portfolio. 6 / 26 ACTL3004: Week 2 ValueatRisk Calculating ValueatRisk Assume that in a given time horizon, P is the initial value of your portfolio (which is known) and P is the projected value of your portfolio. Then, if R is the random rate of return, then in a single period, P = P ( 1 + R ) . Your loss will therefore be....
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This note was uploaded on 08/07/2011 for the course ACTL 3004 at University of New South Wales.
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 BRIAN

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