cyber21

If a stock's return is normally distributed with a mean of 16% and a standard deviation of 50%, what is the probability of a negative return in a given year?

A) 0.3745.

B) 0.5000.

C) 0.0001.

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The selected random value is standardized (its z-value is calculated) by subtracting the mean from the selected value and dividing by the standard deviation. This results in a z-value of (0 − 16) / 50 = -0.32. Changing the sign and looking up +0.32 in the z-value table yields 0.6255 as the probability that a random variable is to the right of the standardized value (i.e. more than zero). Accordingly, the probability of a random variable being to the left of the standardized value (i.e. less than zero) is 1 − 0.6255 = 0.3745.

cyber21

The average amount of snow that falls during January in Frostbite Falls is normally distributed with a mean of 35 inches and a standard deviation of 5 inches. The probability that the snowfall amount in January of next year will be between 40 inches and 26.75 inches is closest to:

A) 79%.

B) 87%.

C) 68%.

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To calculate this answer, we will use the properties of the standard normal distribution. First, we will calculate the Z-value for the upper and lower points and then we will determine the approximate probability covering that range.  Note: This question is an example of why it is important to memorize the general properties of the normal distribution.
Z = (observation – population mean) / standard deviation

• Z26.75 = (26.75 – 35) / 5 = -1.65. (1.65 standard deviations to the left of the mean)
• Z40 = (40 – 35) / 5 = 1.0 (1 standard deviation to the right of the mean)

Using the general approximations of the normal distribution:

• 68% of the observations fall within ± one standard deviation of the mean. So, 34% of the area falls between 0 and +1 standard deviation from the mean.

• 90% of the observations fall within ± 1.65 standard deviations of the mean. So, 45% of the area falls between 0 and +1.65 standard deviations from the mean.

Here, we have 34% to the right of the mean and 45% to the left of the mean, for a total of 79%.

cyber21

Three portfolios with normally distributed returns are available to an investor who wants to minimize the probability that the portfolio return will be less than 5%. The risk and return characteristics of these portfolios are shown in the following table:

Portfolio	Expected return	Standard deviation
Epps	6%	4%
Flake	7%	9%
Grant	10%	15%

Based on Roy’s safety-first criterion, which portfolio should the investor select?

A) Epps.

B) Grant.

C) Flake.

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Roy’s safety-first ratios for the three portfolios:
Epps = (6 - 5) / 4 = 0.25
Flake = ( 7 - 5) / 9 = 0.222
Grant = (10 - 5) / 15 = 0.33
The portfolio with the largest safety-first ratio has the lowest probability of a return less than 5%. The investor should select the Grant portfolio.

cyber21

The mean and standard deviation of four portfolios are listed below in percentage terms. Using Roy's safety first criteria and a threshold of 3%, select the respective mean and standard deviation that corresponds to the optimal portfolio.

A) 5; 3.

B) 14; 20.

C) 19; 28.

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According to the safety-first criterion, the optimal portfolio is the one that has has the largest value for the SFRatio (mean − threshold) / Standard Deviation. A mean = 5 and Standard Deviation = 3 yields the largest SFRatio from the choices given: (5 − 3) / 3 = 0.67.

cyber21

The safety-first criterion rules focuses on:

A) SEC regulations.

B) margin requirements.

C) shortfall risk.

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The safety-first criterion focuses on shortfall risk which is the probability that a portfolio’s value or return will not fall below a given threshold level. The safety-first criterion usually dictate choosing a portfolio with the lowest probability of falling below the threshold level or return.

cyber21

If the threshold return is higher than the risk-free rate, what will be the relationship between Roy’s safety-first ratio (SF) and Sharpe’s ratio?

A) The SF ratio will be lower.

B) The SF ratio may be higher or lower depending on the standard deviation.

C) The SF ratio will be higher.

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Since each ratio has the standard deviation of returns in the denominator, the difference depends upon the effect on the numerator. Since both the risk-free rate (in the Sharpe ratio) and the threshold rate (in the SF ratio) are subtracted from the expected return, a larger threshold rate would result in a smaller SF ratio value.

cyber21

The mean and standard deviation of three portfolios are listed below in percentage terms. Using Roy's safety-first criteria and a threshold of 4%, select the respective mean and standard deviation that corresponds to the optimal portfolio.

A) 14; 20.

B) 19; 28.

C) 5; 3.

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According to the safety-first criterion, the optimal portfolio is the one that has the largest value for the SFRatio (mean − threshold) / Standard Deviation. A mean = 19 and Standard Deviation = 28 yields the largest SFRatio from the choices given: (19 − 4) / 28 = 0.5357.

cyber21

The farthest point on the left side of the lognormal distribution:

A) is skewed to the left.

B) can be any negative number.

C) is bounded by 0.

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The lognormal distribution is skewed to the right with a long right hand tail and is bounded on the left hand side of the curve by zero.

cyber21

Which of the following statements regarding the distribution of returns used for asset pricing models is most accurate?

A) Normal distribution returns are used for asset pricing models because they will only allow the asset price to fall to zero.

B) Lognormal distribution returns are used for asset pricing models because they will not result in an asset return of less than -100%.

C) Lognormal distribution returns are used because this will allow for negative returns on the assets.

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Lognormal distribution returns are used for asset pricing models because this will not result in asset returns of less than 100% because the lowest the asset price can decrease to is zero which is the lowest value on the lognormal distribution. The normal distribution allows for asset prices less than zero which could result in a return of less than -100% which is impossible.

cyber21

If a random variable x is lognormally distributed then ln x is:

A) abnormally distributed.

B) normally distributed.

C) defined as ex.

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For any random variable that is normally distributed its natural logarithm (ln) will be lognormally distributed. The opposite is also true: for any random variable that is lognormally distributed its natural logarithm (ln) will be normally distributed.