23.single_index_ex

# 23.single_index_ex - 370.61670 0.004105009 1-0.0043 0.94...

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University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Single index model - example For three stocks you are given the following data based on the single index model: Stock α β σ 2 ± A -0.0043 0.94 0.0033 B 0.0059 0.61 0.0038 C 0.0048 1.12 0.0046 Assume ¯ Rm = 0 . 01 and σ 2 m = 0 . 0018. Below you are given the solution to the problem (point of tangency) when short sales are allowed and R f = 0 . 005 using two methods: A. Using the formula Z = Σ - 1 R : Z = Σ - 1 R = 0 . 00489048 0 . 00103212 0 . 00189504 0 . 00103212 0 . 00446978 0 . 00122976 0 . 00189504 0 . 00122976 0 . 00685792 ! - 1 0 . 0051 - 0 . 005 0 . 0120 - 0 . 005 0 . 0160 - 0 . 005 ! = - 0 . 883563202 1 . 327096101 1 . 610164293 ! . The sum of the z i ’s is 3 i =1 z i = 2 . 053697192 and therefore the x i ’s are: x 1 = - 0 . 4302 ,x 2 = 0 . 6462 ,x 3 = 0 . 7840. B. Using the single index model: Ranking the stocks based on the excess return to beta ratio. Stock i α i ˆ β i ¯ R i ˆ σ 2 ±i R i - R f ˆ β i ( ¯ R i - R f ) ˆ β i ˆ σ 2 ±i i j =1 ( ¯ R j - R f ) ˆ β j ˆ σ 2 ±j ˆ β 2 i ˆ σ 2 ±i i j =1 ˆ β 2 j ˆ σ 2 ±j C i 2 0.0059 0.61 0.0120 0.0038 0.0114754098 1.12368421 1.123684 97.92105 97.92105 0.001719548 3 0.0048 1.12 0.0160 0.0046 0.0098214286 2.67826087 3.801945 272.69565
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Unformatted text preview: 370.61670 0.004105009 1-0.0043 0.94 0.0051 0.0033 0.0001063830 0.02848485 3.830430 267.75758 638.37428 0.003208254 From the table above we get C * = 0 . 003208254 (short sales are allowed therefore C * is the last C i . Using, z i = β i σ 2 ±i ± ¯ R i-R f β i-C * ² we compute the z i s . We get: z 1 = . 94 . 0033 [0 . 0001063830-. 003208254] =-. 8835632 z 2 = . 61 . 0038 [0 . 0114754098-. 003208254] = 1 . 3270961 z 3 = . 1 . 12 . 0046 [0 . 0098214286-. 003208254] = 1 . 610164 The sum of the z i s is: 3 X i =1 z i = 2 . 053697 , and therefore using x i = z i ∑ n i =1 z i we compute the x i s : x 1 =-. 8835632 2 . 053697 =-. 4302305. x 2 = 1 . 3270961 2 . 053697 = 0 . 6461985. x 3 = 1 . 610164 2 . 053697 = 0 . 7840320. The two methods give exactly the same answer....
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## This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.

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