10_ISE_5434_11_Sensitivity_Risk_Analysis_2 - ISE 5434 Economic Evaluation of(Industrial Projects and Processes Unit#11 Risk Analysis Part 2 Analysis of

10_ISE_5434_11_Sensitivity_Risk_Analysis_2 - ISE 5434...

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ISE 5434 Economic Evaluation of (Industrial) Projects and Processes
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Unit #11 Risk Analysis - Part 2 -
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Analysis of Risk Risk analysis provides behavior of the measure of economic effectiveness* due to random events potential for reversal in preferences for investment alternatives Risk analysis measures the “sensitivity” of a decision to random events with known distribution Risk analysis is generally performed via Monte Carlo simulation * aka measure of merit
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Monte Carlo Given Model independent variables(s): x i dependent variable(s): y = f(x) Generate variate for each x distributions known Determine Cholesky Factor for Covariance Matrix C = L T L; L ~ Cholesky Factor, C ~ covariance matrix x = Lg + m; g ~ variate of x w/ mean = 0 and std dev = 1 Compute y Repeat
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Random Event A random event is a phenomenon described by a random variable A random variable is a function that assigns real numbers to elements of a sample space, i.e., the set of all possible events coin flip sample space: S = {head, tail} ~ {h,t} random variable, X: S reals, e.g., X(h) = 1, X(t) = 0 A random event is a subset of a sample space, to which a probability is assigned (by a random variable)
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Probability Distribution Associated with every random variable, X, is a cumulative probability distribution , F = = x x i i ) x ( p } x X Pr{ ) x ( F - = = x dt ) t ( f } x X Pr{ ) x ( F discrete continuous probability density function ~ frequency probability mass function ~ frequency
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Probability Mass Function Sample space is countable: S= {x 1 , x 2 ,…} 1 ) x ( p S x = ] 1 , 0 [ S : p = = 0 ) x X Pr( ) x ( p x S x S 1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 x p(x) pmf for a fair die
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Probability Density Function Sample space is uncountable (continuous) ] 1 , 0 [ S : f 1 dt ) t ( f = - f(t) t F(x) t = x
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Useful Excel Functions RAND() U(a,b) ~ RAND()*(b-a)+a 0 RAND() 1 NORMSDIST(z) NORMDIST(x,mean,standard_dev,cumulative) cumulative = 1: f(x) cumulative = 0: F(x) NORMSINV(probability) NORMINV(probability,mean,standard_dev) F -1
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Expectation Values The expected value of a random variable X, E[X], is the sum of X’s values weighted by X’s distribution discrete continuous - = = = μ dx ) x ( xf X ] X [ E = = = = μ n 1 i i i ) x ( p x X ] X [ E
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Expectation Values ] y [ E ] x [ E ] y x [ E + = + b ] x [ aE ] b ax [ E + = + P1 P2
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Proofs: P1 = = 0 ) x X Pr( ) x ( p p(ax) = p(x) p(x+b) = p(x) multiplying X by a constant does not change Pr(X=x) adding to X a constant does not change Pr(X=x) b ] x [ aE ) x ( p ) b ax ( ) b ax ( p ) b ax ( ] b ax [ E N 1 i i i N 1 i i i + = + = + + = + = =
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Proofs: P2 ) y Y , x X Pr( ) y x ( ] y x [ E j n 1 i n 1 j i j i = = + = + ∑∑ = = ) y Y , x X Pr( y ) y Y , x X Pr( x j n 1 i n 1 j i j j n 1 i n 1 j i i = = + = = = ∑∑ ∑∑ = = = =
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Proofs: P2 ) x X Pr( ) y Y , x X Pr( i n 1 j j i = = = = = ) y Y Pr( ) y Y , x X Pr( j n 1 i j i = = = = = = = = + = = + n 1 j j j n 1 i i i ) y Y Pr( y ) x X Pr( x ] y x [ E ] y [ E ] x [ E ] y x [ E + = +
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Variance ] ]) x [ E x [( E ) x ( VAR 2 - = 2 2 2 ] x [ E ] x [ E σ = - ] ] x [ E ] x [ xE 2 x [ E ] ]) x [ E x [( E 2 2 2 + - = - 2 2 2 ] x [ E ] x [ E ] ]) x [ E x [( E - = -
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Variance ) x ( VAR a ) b ax ( VAR 2 = + vanishes when independent correlation = 0 ) y , x ( COV 2 ) y ( VAR ) x ( VAR ) y x ( VAR + + = + P3 P4
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Proof: P3 ] ) b ] x [ aE b ax [( E ) b
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  • Normal Distribution, Probability theory, probability density function, Cumulative distribution function

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