Question

TABLE 6-4

According to Investment Digest, the arithmetic mean of the annual return for common stocks from 1926-2010 was 9.5% but the value of the variance was not mentioned. Also 25% of the annual returns were below 8% while 65% of the annual returns were between 8% and 11.5%. The article claimed that the distribution of annual return for common stocks was bell-shaped and approximately symmetric. Assume that this distribution is normal with the mean given above. Answer the following questions without the help of a calculator, statistical software, or statistical table.

15) Referring to Table 6-4, find the probability that the annual return of a random year will be less than 11.5%.____?

16) Referring to Table 6-4, find the probability that the annual return of a random year will be more than 11.5%_____?

17) Referring to Table 6-4, find the probability that the annual return of a random year will be between 7.5% and 11%.________?

18) Referring to Table 6-4, what is the value above which will account for the highest 25% of the possible annual returns?_________

19) Referring to Table 6-4, 75% of the annual returns will be lower than what value?___________

Answer 1

Arithmetic mean of the annual return = 9.5%

The annual return below 8% is 25%

And we know that the distribution is symmertric, so the value at above 11% will also be 25%. As the difference between 8 to 9.5 and 9.5 to 11 is same.

15) Percent on the either side of the mean is 50%.

Annual return between 8% and 11% = 25+25% = 50%

And Annual returns between 8% and 11.5% is 65%

So Annual return between 11 to 11.5% = 65- 50 = 15%

Probability that the annual return of a random year will be less than 11.5%= 50+ 25+15

=  90% = 0.90

16) Probability that the annual return of a random year will be more than 11.5% = 1 - P(less than 11.5%)

= 1- 0.90 = 0.10

17) Annual return between 7.5 to 8 and 11to 11.5 will be same i.e 15% as the distribution is normal. And are equal distance away from the mean.

Annual return between 8% and 11% = 25+25% = 50%

robability of annual return of a random year between 7.5% and 11% =0.15+ 0.50 = 0.65

18) Value which will account for the highest 25% of the possible annual returns = 11% (see graph)

19) 75% of the annual returns will be lower than what value = 11%

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