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# Lecture 2 - ACTL3004 Week 2 Financial Economics for...

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ACTL3004: Week 2 Financial Economics for Insurance and Superannuation: Week 2 Risk Measures, Financial Data and the Efficient Market Hypothesis 1 / 26

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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 μ = 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 - μ ) 2 i = E ( X 2 ) - μ 2 = Z -∞ ( x - μ ) 2 f X ( x ) dx Semi-Variance of Return: It is sometimes called the downside semi-variance. It accounts only for the variation below the mean. semi - var ( X ) = Z μ -∞ ( x - μ ) 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 fund’s 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 - μ ) 3 i = Z -∞ ( x - μ ) 3 f X ( x ) dx 3 / 26

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ACTL3004: Week 2 Investment Risk Measures Kurtosis: It measures the ”peakedness” or ”pointedness” of a distribution. K ( X ) = E h ( X - μ ) 4 i = Z -∞ ( x - μ ) 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 so-called shortfall probability p ( L ) = Z L -∞ f X ( x ) dx = Pr ( X L ) . 2. If g ( L - x ) = ( L - x ) , then we have the so-called expected shortfall as defined above. 3. If g ( L - x ) = ( L - x ) 2 , then we have the so-called shortfall variance Z L -∞ ( L - x ) 2 f X ( x ) dx . 5 / 26

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ACTL3004: Week 2 Value-at-Risk Value-at-Risk I Value-at-risk 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 Value-at-Risk Calculating Value-at-Risk Assume that in a given time horizon, P 0 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 0 ( 1 + R ) . Your loss will therefore be L = P 0 - P 0 ( 1 + R ) = - P 0 R .

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Lecture 2 - ACTL3004 Week 2 Financial Economics for...

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