Question

The Standard and Poor 500 (S\&P 500 ) is a weighted average of the stocks for 500 large companies in the United States. It is commonly used as a measure of the overall performance of the US stock market. Between January 1,2009 and January \(1,2012,\) the S\&P 500 increased for 423 of the 756 days that the stock market was open. We will investigate whether changes to the S\&P 500 are independent from day to day. This is important, because if changes are not independent, we should be able to use the performance on the current day to help predict performance on the next day. (a) What is the probability that the S\&P 500 increased on a randomly selected market day between January 1,2009 and January \(1,2012 ?\) (b) If we assume that daily changes to the \(S \& P\) 500 are independent, what is the probability that the S\&P 500 increases for two consecutive days? What is the probability that the S\&P 500 increases on a day, given that it increased the day before? (c) Between January 1, 2009 and January 1,2012 the S\&P 500 increased on two consecutive market days 234 times out of a possible \(755 .\) Based on this information, what is the probability that the S\&P 500 increases for two consecutive days? What is the probability that the S\&P 500 increases on a day, given that it increased the day before? d) Compare your answers to part (b) and part (c). Do you think that this analysis proves that daily changes to the S\&P 500 are not independent?

Short answer

\(a)P(I) = 423/756\) \n\(b)P(I, I) = (423/756) * (423/756), P(I_2 | I_1) = 423/756\) \n\(c) P(I_2 , I_1) = 234/755, P(I_2 | I_1) = 234/755\) \n\(d) If P(I_2 , I_1) and P(I, I) are significantly different, and if P(I_2 | I_1) and P(I) are significantly different, this suggests that the changes to the S&P 500 are not independent. Compare these probabilities and draw conclusions.

Step-by-step solution

Calculate Probability of Increase in S&P 500 on a Given Day

To calculate the probability that the S&P 500 increased on a randomly selected market day, we simply divide the number of days the index increased by the total number of market days. Thus, the probability \(P(I)\) is \(423/756\)

Calculate Probability of Increase for Two Consecutive Days assuming Independence

Assuming the changes in S&P 500 are independent, the probability that the S&P 500 increases for two consecutive days is the product of the probability that it increases on a single day. So, \(P(I, I) = P(I) * P(I) = (423/756) * (423/756)\). For independent events A and B, \(P(A | B) = P(A)\), i.e., the probability of event A given event B is just the probability of event A. So, \(P(I_2 | I_1) = P(I) = 423/756\) where \(I_1\) and \(I_2\) are increases on two consecutive days.

Calculate Probability of Increase for Two Consecutive Days based on Given Data

From the data, we know that the S&P 500 increased on two consecutive market days 234 times out of a possible 755. So, \[P(I_2 , I_1) = 234/755\]. And the probability that the S&P 500 increases on a day, given that it increased the day before is also \[P(I_2 | I_1) = 234/755\].

Comparing and Concluding

Compare the results from step 2 and 3. If the results in the two steps are significantly different, this suggest that changes to the S&P 500 are not independent. Discuss this conclusion with reference to the results.