What is the approximate probability that the S&P 500 Index value will increase b

Question

What is the approximate probability that the S&P 500 Index value will increase by more than 1 standard deviation over the next year.

cut v E Wrap Text Calibri A A General BE BE Co og Conditional Format as Cell Paste rt Delete Fo mat B I U erge & Cent Format Painter Formatting Table Clipboard Fon Alignment Number M1 1 Date open High Low close Volume Adj Close Daily Return Daily Annual 1/20/2017 2269.96 2276.96 2265.01 22 31 3524970000 2221. 1006 0.34% Average 0.05% 11.77% 3 1/19/2017 2271.9 2274.33 2258.41 226 3.69 3 165970000 2 26 3.68994 -0.36% Standard Deviation 0.81% 12.82% 4 1/18/2017 2269.14 2272.01 226 3.35 2271.89 3315 250000 2271.88989 a 18% 5 1/17/2017 2269.14 2272.08 2262.81 2267.89 3584 990000 2267.88989 -0.30% 1/13/2017 2272.74 2278.68 2271.51 2274.64 3081270000 2274.63989 7 1/12/2017 2271.14 2271.78 2254.25 2270.44 34621 30000 2270.43994 -0.21% 8 1/11/2017 2268.6 2275.32 2260.83 2275.32 3620410000 2275. 2007 a 28% 1/10/2017 2269.72 2279.27 2265.27 2268.9 3638790000 2268-8999 a 0096 10 1/9/2017 2273.59 2275.49 2268.9 2268.9 3217610000 2268-8999 -0.35% 11 1/6/2017 2271.14 2282.1 2264.06 2276.98 3339890000 2276.97998 a 35% 12 1/5/2017 2268.18 2271.5 2260.45 2269 3761820000 2269 -0.08% 13 1/4/2017 2261.6 2272.82 2261.6 2270.75 3764890000 2270.75 0.57% 14 1/3/2017 2251.57 2263.88 2245.13 2257.83 37705 30000 225 3008 85% 15 12/30/2016 2251.61 2253.58 2233.62 2238.83 2670900000 2238.83008 -0.46% 16 12/29/2016 2249.5 2254.51 2244.56 2249.26 2336370000 2 249. 26001 -0.03% 12/28/2016 2270.23 2271.31 2249.11 2249.92 2392360000 2249.91992 18 12/27/2016 2266.23 2273.82 2266.15 2268.88 1987080000 2268.87988 0.22% 19 12/23/2016 2260.25 2263.79 2258.84 2263.79 2020550000 2 26 3.79004 0.13% 20 12/22/2016 2262.93 2263.18 2256.08 2260.96 2876 20000 2260 95996 -0.19% 21 12/21/2016 2270.54 2271.23 2265.15 2265.18 2852230000 2265.17993 22 12/20/2016 2266.5 2272.56 2266.14 2270.76 3 298780000 2270 76001 0.36% 23 12/19/2016 2259.24 2267.47 2258.21 2262.53 3 248 370000 2262.5 300 0.20% READY Autosu Sort & Find Clear ilter Select Editing NTT

Explanation / Answer

Here, X = mean - sigma

= 11.77 - 12.82

= - 1.05

mean = 11.77 , sigma = 12.82

By using central limit theorem,

z = (X - mean) / sigma

= (- 1.05 - 11.77) / 12.82

= -1

we need to find P(Z > X) by using z standard right tail,

P(Z > - 1) = 0.1587

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