Question

In: Math

5. Arsalaan A., a well-known financial analyst, selected 50 consecutive years of U.S. financial markets data...

5. Arsalaan A., a well-known financial analyst, selected 50 consecutive years of U.S. financial markets data at random. For 11 of the years, the rate of return for the Dow Jones Industrial Average [DJIA] exceeded the rates of return for both the S&P 500 Index and the NASDAQ Composite Index. For 8 of the years, the rate of return for the DJIA trailed the rates of return for both the S&P 500 and the NASDAQ. For 21 of the years, the rate of return for the DJIA trailed the rate of return for the S&P 500. Over the 50 years,

a. determine the probability the rate of return for the DJIA trailed the rate of return for the NASDAQ.

b. determine the probability the rate of return for the DJIA trailed the rate of return for at least one of the other two Indexes.

c. determine the probability the rate of return for the DJIA trailed the rate of return for the S&P 500 given it trailed the rate of return for the NASDAQ.

d. determine the probability the rate of return for the DJIA exceeded the rate of return for the S&P 500 given it exceeded the rate of return for the NASDAQ.

Solutions

Expert Solution

Let us define the following events:

S : the rate of return for the DJIA exceeded the rate of return for S&P 500 Index.

=> S^c : the rate of return for the DJIA trailed the rate of return for S&P 500 Index.

N : the rate of return for the DJIA exceeded the rate of return for NASDAQ.

=> N^c : the rate of return for the DJIA trailed the rate of return for NASDAQ.

Now, we are given the following details in the question:

The financial analyst selected 50 consecutive years of U.S. financial markets data at random.

"For 11 of the years, the rate of return for the Dow Jones Industrial Average [DJIA] exceeded the rates of return for both the S&P 500 Index and the NASDAQ Composite Index", which implies that:

[Since, there are a total of 50 years out of which 11 years had DJIA's rate of returns exceeding both S&P 500's rate of return and NASDAQ rate of returns]

"For 8 of the years, the rate of return for the DJIA trailed the rates of return for both the S&P 500 and the NASDAQ", which implies that:

"For 21 of the years, the rate of return for the DJIA trailed the rate of return for the S&P 500", which implies that:

Now, we know the following identity:

The probability that the rate of return for the DJIA trailed the rate of return for the NASDAQ is given by:

The probability that the rate of return for the DJIA trailed the rate of return for at least one of the other two Indexes is given by:

The probability that the rate of return for the DJIA trailed the rate of return for the S&P 500 given it trailed the rate of return for the NASDAQ is given by:

The probability that the rate of return for the DJIA exceeded the rate of return for the S&P 500 given it exceeded the rate of return for the NASDAQ is given by:

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milcah answered 8 months ago

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