4 - Probability Distributions Flashcards
A stock increased in value last year. Which will be greater, its continuously compounded or its holding period return?
When a stock increases in value, the holding period return is always greater than the continuously compounded return that would be required to generate that holding period return. For example, if a stock increases from $1 to $1.10 in a year, the holding period return is 10%. The continuously compounded rate needed to increase a stock’s value by 10% is Ln(1.10) = 9.53%.
An investment has a mean return of 15% and a standard deviation of returns equal to 10%. If returns are normally distributed, which of the following statements is least accurate? The probability of obtaining a return:
Sixty-eight percent of all observations fall within +/- one standard deviation of the mean of a normal distribution. Given a mean of 15 and a standard deviation of 10, the probability of having an actual observation fall within one standard deviation, between 5 and 25, is 68%. The probability of an observation greater than 25 is half of the remaining 32%, or 16%. This is the same probability as an observation less than 5. Because 95% of all observations will fall within 20 of the mean, the probability of an actual observation being greater than 35 is half of the remaining 5%, or 2.5%.
A stock portfolio has had a historical average annual return of 12% and a standard deviation of 20%. The returns are normally distributed. The range –27.2 to 51.2% describes a:
The upper limit of the range, 51.2%, is (51.2 – 12) = 39.2 / 20 = 1.96 standard deviations above the mean of 12. The lower limit of the range is (12 – (-27.2)) = 39.2 / 20 = 1.96 standard deviations below the mean of 12. A 95% confidence level is defined by a range 1.96 standard deviations above and below the mean.
A group of investors wants to be sure to always earn at least a 5% rate of return on their investments. They are looking at an investment that has a normally distributed probability distribution with an expected rate of return of 10% and a standard deviation of 5%. The probability of meeting or exceeding the investors’ desired return in any given year is closest to:
The mean is 10% and the standard deviation is 5%. You want to know the probability of a return 5% or better. 10% - 5% = 5% , so 5% is one standard deviation less than the mean. Thirty-four percent of the observations are between the mean and one standard deviation on the down side. Fifty percent of the observations are greater than the mean. So the probability of a return 5% or higher is 34% + 50% = 84%.
(Module 4.2, LOS 4.h)
Standardizing a normally distributed random variable requires the:
All that is necessary is to know the mean and the variance. Subtracting the mean from the random variable and dividing the difference by the standard deviation standardizes the variable.
(Module 4.2, LOS 4.i)
