# Calculating probability that a stock gained value and lost value

The major stock market indexes had mixed results in 2011. The mean one-year return for stocks in the S&P 500, a group of 500 very large companies, was 0.00%. The mean one-year return for the NASDAQ, a group of 3,200 small and medium-sized companies, was -1.8% Historically, the one-year returns are approximately normally distributed, the standard deviation in the S&P 500 is approximately 20%, and the standard deviation in the NASDAQ is approximately 30%.

a. What is the probability that a stock in the S&P 500 gained value in 2011?

b. What is the probability that a stock in the S&P 500 gained 10% or more in 2011?

c. What is the probability that a stock in the S&P 500 lost 20% or more in 2011?

d. What is the probability that a stock in the S&P 500 lost 40% or more in 2011?

e. Repeat a - d for a stock in the NASDAQ.

f. Discuss the risks associated with a large standard deviation.

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#### Solution Preview

For S&P 500, mean=0 and sd=20. SInce one-year returns are approximately normally distributed, z=(x-mean)/[sd]=(x-0)/[20].

a. P(a stock in the S&P 500 gained value in 2011)=P(X>0)=P(Z>0/20)=P(Z>0)=0.5 from standard normal table

b. P(a stock in the S&P 500 gained 10% or more in 2011)=P(X>10)=P(Z>10/20)=P(Z>0.5)=0.3085 from standard normal table

c. P(a stock in the S&P 500 lost 20% or more in ...

#### Solution Summary

The solution gives detailed steps on calculating the probability that a stock gained value and that a stock lost value.