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Economics 434

Economics 434. Financial Markets Professor Burton University of Virginia Fall 2014. Tomorrow. s 1. Today. s 2. s 3. And, we may not have any idea what the probabilities of s 1 , s 2 , s 3 may be!!. Another way of looking at tomorrow:.

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Economics 434

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  1. Economics 434 Financial Markets Professor Burton University of Virginia Fall 2014

  2. Tomorrow s1 Today s2 s3 And, we may not have any idea what the probabilities of s1, s2, s3 may be!!

  3. Another way of looking at tomorrow: As a probability distribution of returns (usually a normal distribution) Probability Density Function -- Returns -- Now, instead of three (or any finite number of) states, there are an infinity of states possible with various probabilities of returns assigned

  4. Need Some Statistics • Mean, Variance, Covariance • Standard Deviation • Mean --- roughly means ‘average’ • Variance – squared differences summed up • Covariance – two variable, take differences of each from their means, multiply the differences observation by observation • Standard Deviation = Square Root of Variance • We will typically use normal distribution – but that is not necessary. We could use other distributions

  5. Because of uncertainty, we need a different utility function • Why not just expected value? • Instead, we use expected utility • What’s the difference?

  6. Some Definitions  = √ 2 (Xi1- µi ) 1,2 1,2 12

  7. Modern Portfolio Theory • Randomness • Construction of efficient (best) portfolios Harry Markowitz

  8. The Markowitz Assumptions • Every asset is a probability distribution of returns • To keep things simple, let’s assume a normal distribution • Asset Xi has mean: μi • Asset Xi has variance: σ2i • Xi ~ (μi , σ2i )

  9. Every Asset Xiis a return distribution +σ2 -σ2 μi -- Returns --

  10. For ConvenienceFinances Uses Standard DeviationIn Place of Variance Standard deviation = σ The square root of σ2 So asset Xi: Xi ~ (μi , σi )

  11. How Do You Create A Portfolio?Try It with Two Assets Mean “X1” “X2” Where are the Portfolios That Can Be Created from Just These Two Assets ? Standard Deviation

  12. Back to the ½, ½ Portfolio Mean of X1 Mean of P Mean of X2 Where P = ½ [X1] + 1/2 [X2]

  13. Covariance Looks forbidding!!

  14. Covariance (Between X1 and X2) Need “deviations from the mean” Xi1- µi ith observation minus the mean (Xi1- µi ) times (Xi2- µi ) Plus if they move together Minus if they move in opposite directions Covariance adds them all up and divides by n

  15. Correlation Coefficient • Adjusting covariance, so that it is confined to be a fraction (or at most +1 or -1) • Simply divide covariance by the product of the standard deviations of the two variables • Then +1 is perfectly correlated, 0 is uncorrelated, -1 is exact opposites

  16. Back to the ½, ½ Portfolio If  = 1 Mean of X1 Mean of P Mean of X2 Where P = ½ [X1] + 1/2 [X2] 1,2 = 1/21 + 1/22

  17. Back to the ½, ½ Portfolio If   1 1,2  1/21 + 1/22 Mean of X1 Mean of P Mean of X2 Where P = ½ [X1] + ½ [X2]

  18. If   1 Then all the portfolios are here

  19. This Means the “boundary”of the possible portfolioslooks like this:

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