2. In 41 of the 63 years from 1950 through 2013 (in 2011 there was virtually no

Question

2. In 41 of the 63 years from 1950 through 2013 (in 2011 there was virtually no change), the S&P 500 finished higher after the first five days of trading. In 36 out of 41 year. Is a good first week a good omen for the upcoming year? The following table gives the first week and annual performance over this 63-year period:

S&P 500’S ANNUAL PERFORMANCE

FIRST WEEK

HIGHER

LOWER

HIGHER

36

5

LOWER

11

11

a. If a year is selected at random, what is the probability that S&P 500 finished higher for the year?

b. Given that S&P 500 finished higher after the first five days of trading, what is the probability that it finished higher for the year?

c. Are two events “first week performance” and “annual performance” independent? Explain.

d. Look up the performance after the first five days of 2014 and the annual performance of the S&P at finance.yahoo.com. Comment of the results.

S&P 500’S ANNUAL PERFORMANCE

FIRST WEEK

HIGHER

LOWER

HIGHER

36

5

LOWER

11

11

Explanation / Answer

Following table shows row and column total:

a)

Out of  63 years, in 47 years S&P 500 finished higher for the year so

P(finished higher for the year) = 47 / 63 = 0.7460

b)

P(finished higher after the first five days and finished higher for the year) = 36/63

P(finished higher after the first five days) = 41/63

So the probability that it finished higher for the year, given that S&P 500 finished higher after the first five days of trading is

P( finished higher for the year|finished higher after the first five days) =P(finished higher after the first five days and finished higher for the year) / P(finished higher after the first five days) = 36 /41 = 0.8780

(c)

No because P( finished higher for the year|finished higher after the first five days) is not equal to P(finished higher for the year).

S&P 500's Annual Performance First week Higher Lower Total Higher 36 5 41 Lower 11 11 22 Total 47 16 63

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