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Financial Data Analysis. Mathematical Formula Sheet. -distribution 1  2 = 2 1  1 2 2  2 where 2 1 and 2 2 are two independent 2 distribution with d.f. 1 and 2 , respectively. 1 = 2 1  1 where 2 1 is a 2 distribution with d.f. 1 .L e p t o k u r t i c Distributions. The kurtosis is de f ned as = ³ ( ) 4 4 ´ Skewness. = ³ ( ) 3 3 ´ . Suppose that we try to f nd the root of ( )=  The Newton algorithm is +1 = + ( ) 0 ( ) . In the mean variance analysis the optimal weights are given by = C 1 ( R 1 ) where = = 2 . For a sequence of i.i.d. random vectors X 1 , X 2 , ..., X , which has the unknown population mean vector is R 1 × and covariance is C × , the James-Stein estimator which shrinks toward the common sample mean ¯ is given by R =(1 ) ˆ R + ¯ 1 ,whe re ˆ R is the individual sample mean ˆ R = 1 P =1 X and ¯ is the overall average for all stocks, ¯ = 1 P =1 ˆ Here is given by =min(1 ( 2)  ( ˆ R ¯ 1 ) > ( 1 ˆ C ) 1 ( ˆ R ¯ 1 ) ) The estimator constructed by Jorion (1986) is obtained by shrinking the sample averages toward a common value given by the mean of the minimum variance portfolio: R =(1 ) ˆ R +  min 1 ,where min = 1 > C 1 R 1 > C 1 1  = +  = ( +2)( 1) ³ ˆ R min 1 ´ > ˆ C 1 ³ ˆ R min 1 ´ ( 2) ¸ Black-Litterman model inputs: (1) X × 1 | R × 1 ( R × 1 C × ) (2) The prior distribution R × 1 ( π × 1 Σ × ) The two parameters π 1 × and Σ × areg iven . W ecanchoose π 1 × as π = 1 C .W ecanchoo se Σ = C . (3) The subjective views are given in a linear form P × · R × 1 = q × 1 + ε × 1 where P × and q × 1 are given, and ε × 1 is a noise vector with ε ×
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This note was uploaded on 10/16/2010 for the course IEOR 4709 taught by Professor Stevenkou during the Fall '10 term at Columbia.

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form - Financial Data Analysis. Mathematical Formula Sheet....

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