429week04f07 - Week 4: Sep 17 Equity Risk Risk Defined...

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Week 4: Sep 17 Equity Risk Risk Defined Random Walk Stat Review Portfolio Theory Portfolios Diversification (2 Securities) Examples/Progressive
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Equity Risk Risk Defined What is Risk? Risk involves taking a chance => Probabilities Payouts must be represented by Expected Values. Risk is defined by variability in the possible future payouts For Bonds, we know exactly what the payouts will be. Is there risk? Variability is quantified by Variance 1. The existence of the possibility of large losses vs gains 2. The relative possibility of loss vs gains
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Equity Risk Random Walk To describe Returns with just the Expected Values and the Variance we need a model Normality of Returns We assume that returns follow a normal distribution Allows us to use standard tables to describe probability of returns Assume returns follow a random walk The next price will be any other price with a probability described by the Normal
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Equity Risk Stock prices are NOT determined fundamentally, but by random motion => roots in Physics Empirical Evidence We do NOT find that stock prices follow a 0 10 20 30 40 50 60 3 2.5 2 1.5 1 0.5 0 - norm Value-Weighted
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Equity Risk Based on monthly data from CRSP (62- Present) Too much probability near the mean Fat tails (esp. on negative side) Positively Skewed
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Equity Risk Stat Review Expected Value “Average” = what you would expect to receive from a game of chance (random outcomes) if you could play the game many times Notation: E(a) = Expected Value of a Properties Expected Value of Sum is Sum of Expected Values s x of x x x prob Value Expected ' # * ) ( = = ) ( ) ( ) ( y E x E y x E + = +
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Equity Risk Expectation of a Constant times x is the Constant times Expected Value of x For a linear combination of random variables We are interested in Expected Values because we know that if we have returns, we know the price ) ( * ) ( x E K Kx E = ) ( * ) ( * ) ( ) ( ) ( y E L x E K Ly E Kx E Ly Kx E + = + = + Return) ( 1 P(0) E(P(t)) Return 1 ) 0 ( ) ( E P t P + = + =
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Equity Risk Variance Farther x’s are from E(x), larger the variance “The more outcomes there are that are very hi or very low, the more risk there is.” More likely x E(x), larger the variance Notation: Variance(x) = V(x) = σ x 2 Standard Deviation = Variance .5 = σ x s x' of # (x)) - (x Prob(x) * )) ( ( ) ( 2 2 = - = E x E x x V
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Equity Risk Properties Variance of constant times variable is Constant squared times Variance of the variable Variance of Sum of weighted variables is sum of weights squared times variances of each plus twice the covariance » Covariance is how much on variable moves with the other (direction and magnitude). » Notation: Covariance of x and y = Cov(x,y) ) ( )) ( ( )) ( ( )) ( ( ) ( 2 2 2 2 2 x V a x E x a x aE ax ax E ax ax V = - = - = - =
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Equity Risk Note: Cov(x,y)= ρ xy σ x σ y » -1 ρ xy 1 is the correlation coefficient ) , ( 2 ) ( ) ( ) , ( 2 ) ( ) ( 2 2 ) ( ) ( ), ( )) ( ) ( ( ) ( 2 2 2 2 2 2 2 y x abCov y V b x V a by ax Cov by V ax V gh h g gh h g by ax V by E by h ax E ax g Let by E by ax E ax by ax V + + = + + = + + = + + = + - = - = - + - = +
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429week04f07 - Week 4: Sep 17 Equity Risk Risk Defined...

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