At the end of the first quarter of 2006, all the major stock market indexes had posted strong gains in the past 12 months. Mass Mutual Financial Group credited the increases to solid growth in corporate profits. The mean one-year return for stocks in the S&P 500, a group of 500 very large companies, was approximately 12%. The mean one-year return for companies in the Russell 2000, a group of 2000 small companies, was approximately 26%. Historically, the one-year returns are approximately normal, the standard deviation in the S&P 500 is approximately 20%, and the standard deviation in the Russell 2000 is approximately 35%.

a. What is the probability that a stock in the S&P 500 gained 25% or more in the last year? gained 50% or more?

b. What is the probability that a stock in the S&P 500 lost money in the last year? Lost 25% or more? lost 50% or more?

c. Repeat (a) and (b) for a stock in the Russell 2000.

I'm really having trouble with this one. I know it's lengthy, but any help would be greatly appreciated!

Aa

To solve this problem, we need to use the concept of standard deviation and the normal distribution. Below are the steps to calculate the probabilities:

a. Probability that a stock in the S&P 500 gained 25% or more in the last year:
1. Calculate the z-score for a 25% gain:
z-score = (X - mean) / standard deviation
z-score = (25 - 12) / 20 = 0.65

2. Look up the z-score in the standard normal table (also called the Z-table). The table provides the area under the curve to the left of the z-score.

3. The area under the curve to the left of a z-score of 0.65 is approximately 0.7422.

4. Subtract this value from 1 to get the probability of gaining 25% or more:
Probability = 1 - 0.7422 = 0.2578

So, there is approximately a 25.78% probability that a stock in the S&P 500 gained 25% or more in the last year.

b. Probability that a stock in the S&P 500 lost money in the last year:
1. To calculate the probability of losing money, we need to consider the negative returns. So, we can calculate the z-score for a 0% return and then find the area under the curve to the left of it.

2. Calculate the z-score for a 0% gain:
z-score = (0 - 12) / 20 = -0.6

3. Look up the z-score (-0.6) in the standard normal table.

4. The area under the curve to the left of a z-score of -0.6 is approximately 0.2743.

So, there is approximately a 27.43% probability that a stock in the S&P 500 lost money in the last year.

To calculate the probability of losing 25% or more or losing 50% or more, you can follow a similar process. Just replace the calculation for the z-score in step 2 with the desired percentage loss, and then find the area under the curve to the left of that z-score.

c. Repeat steps a and b for a stock in the Russell 2000, but use the mean, standard deviation, and desired returns specific to the Russell 2000 index.

Remember to look up the z-scores and corresponding areas under the standard normal curve in the Z-table to find the probabilities.

I hope this helps! Let me know if you have any further questions.