Module 9

The EPS for a large group of firms are normally distributed and have M=6 and standard deviation= $2. Find the probability that randomly selected firms earnings are greater than $9.70.

(9.70-6)/2= 1.85 go to z table z<_ .9678 z>_.1-.9678

Multivariate Normal Distribution mean=? Standard deviation=? variance=?

0 1 1

In a normal distribution, the skewness = ? kutosis = ? Tails?

0 3 Go on forever and just get smaller and smaller

90% confidence interval will be within +/-

1.65 standard deviations of the mean

95% confidence interval will be within +/-

1.96 standard deviations of the mean

Assume an investor purchases a stock for $50. One year later, the stock is worth $60. After one more year, the stock price has fallen to the original price of $50. Calculate the continuously compounded return for year 1 and year 2.

18.23% and -18.23% Given a holding period return of R, the continuously compounded rate of return is: ln(1 + R) = ln(Price1/Price0). Here, if the stock price increases to $60, r = ln(60/50) = 0.18232, or 18.23%. The return for year 2 is ln(50/60), or ln(0.833) = -18.23%.

99% confidence interval will be within +/-

2.58 standard deviations of the mean

Confidence Interval for a normal distribution=?

68% of the outcomes are within 1 standard deviation of the mean. 95% are within +/- 2 standard deviations

A stock portfolio's returns are normally distributed. It has had a mean annual return of 25% with a standard deviation of 40%. The probability of a return between -41% and 91% is closest to: A)90%. B)95%. C)65%.

A 90% confidence level includes the range between plus and minus 1.65 standard deviations from the mean. (91 - 25) / 40 = 1.65 and (-41 - 25) / 40 = -1.65. (Study Session 3, Module 9.2, LOS 9.k)

Bernoulli's Random Variable

A binomial random variable for which the number of trials is 1. Each trial is a Bernoulli random variable. There are only two possible outcomes

Cumulative Distribution Function

A function giving the probability that a random variable is less than or equal to a specified value. -Represents the cumulative value (or sum) of the probabilities for the outcomes up to an including a specified outcome

Multivariate normal distribution

A probability distribution for a group of random variables that is completely defined by the means and variances of the variables plus all the correlations between pairs of the variables. Basically specifies the probabilities associated with groups of random variables

A multivariate normal distribution that includes three random variables can be completely described by the means and variances of each of the random variables and the: A)correlations between each pair of random variables. B)conditional probabilities among the three random variables. C)correlation coefficient of the three random variables.

A) A multivariate normal distribution that includes three random variables can be completely described by the means and variances of each of the random variables and the correlations between each pair of random variables. Correlation measures the strength of the linear relationship between two random variables (thus, "the correlation coefficient of the three random variables" is inaccurate).

The continuously compounded rate of return that will generate a one-year holding period return of -6.5% is closest to: A)-6.7%. B)-5.7%. C)-6.3%.

A) Continuously compounded rate of return = ln(1 - 0.065) = -6.72%. (Study Session 3, Module 9.3, LOS 9.o)

Which of the following statements describes a limitation of Monte Carlo simulation? A)Outcomes of a simulation can only be as accurate as the inputs to the model. B)Variables are assumed to be normally distributed but may actually have non-normal distributions. C)Simulations do not consider possible input values that lie outside historical experience.

A) Monte Carlo simulations can be set up with inputs that have any distribution and any desired range of possible values. However, a limitation of the technique is that its output can only be as accurate as the assumptions an analyst makes about the range and distribution of the inputs.

A key property of a normal distribution is that it: A) has zero skewness. B) is asymmetrical. C) has zero kurtosis.

A) Normal distributions are symmetrical (i.e., have zero skewness) and their kurtosis is equal to three. (LOS 9.i)

There is an 80% chance of rain on each of the next six days. What is the probability that it will rain on exactly two of those days? A)0.01536. B)0.15364. C)0.24327.

A) P(2) = 6! / [(6 - 2)! × 2!] × (0.82) × (0.24) = 0.01536 = 6 nCr 2 × (0.8)2 × (0.2)4 (Study Session 3, Module 9.1, LOS 9.f)

Which of the following statements about probability distributions is least accurate? A)A probability distribution is, by definition, normally distributed. B)In a binomial distribution each observation has only two possible outcomes that are mutually exclusive. C)A probability distribution includes a listing of all the possible outcomes of an experiment.

A) Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures. (Study Session 3, Module 9.1, LOS 9.a)

Which of the following would least likely be categorized as a multivariate distribution? A)The days a stock traded and the days it did not trade. B)The return of a stock and the return of the DJIA. C)The returns of the stocks in the DJIA.

A) The number of days a stock traded and did not trade describes only one random variable. Both of the other cases involve two or more random variables. (Study Session 3, Module 9.2, LOS 9.j)

If a smooth curve is to represent a probability density function, what two requirements must be satisfied? The area under the curve must be: A)one and the curve must not fall below the horizontal axis. B)zero and the curve must not fall below the horizontal axis. C)one and the curve must not rise above the horizontal axis.

A) If a smooth curve is to represent a probability density function, the total area under the curve must be one (probability of all outcomes equals 1) and the curve must not fall below the horizontal axis (no outcome can have a negative chance of occurring).

A multivariate distribution: A)specifies the probabilities associated with groups of random variables. B)applies only to binomial distributions. C)gives multiple probabilities for the same outcome.

A) This is the definition of a multivariate distribution. (Study Session 3, Module 9.2, LOS 9.j)

A random variable that has a countable number of possible values is called a: A)discrete random variable. B)probability distribution. C)continuous random variable.

A) A discrete random variable is one for which the number of possible outcomes are countable, and for each possible outcome, there is a measurable and positive probability. A continuous random variable is one for which the number of outcomes is not countable. (Study Session 3, Module 9.1, LOS 9.b)

For a given stated annual rate of return, compared to the effective rate of return with discrete compounding, the effective rate of return with continuous compounding will be: A)higher. B)lower. C)the same.

A) A higher frequency of compounding leads to a higher effective rate of return. The effective rate of return with continuous compounding will, therefore, be greater than any effective rate of return with discrete compounding. (Study Session 3, Module 9.3, LOS 9.o)

A recent study indicated that 60% of all businesses have a fax machine. From the binomial probability distribution table, the probability that exactly four businesses will have a fax machine in a random selection of six businesses is: A) 0.138. B) 0.276. C) 0.311.

A) C Success = having a fax machine. [6! / 4!(6 − 4)!](0.6)4(0.4)6-4 = 15(0.1296)(0.16) = 0.311. (LOS 9.f)

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.

A) 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. (Study Session 3, Module 9.3, LOS 9.m)

Ten percent of all college graduates hired stay with the same company for more than five years. In a random sample of six recently hired college graduates, the probability that exactly two will stay with the same company for more than five years is closest to: A) 0.098. B) 0.114. C) 0.185.

A) Success = staying for five years. [6! / 2!(6 − 2)!](0.10)2(0.90)6-2 = 15(0.01)(0.656) = 0.0984. (LOS 9.f)

Which of the following is least likely a condition of a binomial experiment? A) There are only two trials. B) The trials are independent. C)If p is the probability of success, and q is the probability of failure, then p + q = 1.

A) There may be any number of independent trials, each with only two possible outcomes. (LOS 9.e)

For a lognormal distribution: A) the mean equals the median. B) the probability of a negative outcome is zero. C) the probability of a positive outcome is 50%.

B) A lognormally distributed variable is never negative. (LOS 9.n)

Using hypothesized parameter values and a random number generator to study the behavior of certain asset returns is part of: A) historical analysis. B) Monte Carlo simulation. C) standardizing a random variable.

B) Monte Carlo simulation involves modeling asset prices or returns by generating random values for the risk factors that affect the price of a security. (LOS 9.p, 9.q)

A stated interest rate of 9% compounded continuously results in an effective annual rate closest to: A)9.20%. B)9.42%. C)9.67%.

B) The effective annual rate with continuous compounding = er - 1 = e0.09 - 1 = 0.09417, or 9.42%. (Study Session 3, Module 9.3, LOS 9.o)

A dealer in a casino has rolled a five on a single die three times in a row. What is the probability of her rolling another five on the next roll, assuming it is a fair die? A)0.200. B)0.167. C)0.001.

B) The probability of a value being rolled is 1/6 regardless of the previous value rolled. (Study Session 3, Module 9.1, LOS 9.a)

If a stock decreases from $90 to $80, the continuously compounded rate of return for the period is: A) -0.1000. B) -0.1178. C) -0.1250.

B) This is given by the natural logarithm of the new price divided by the old price; ln(80 / 90) = -0.1178. (Study Session 3, Module 9.3, LOS 9.o)

For the standard normal distribution, the z-value gives the distance between the mean and a point in terms of: A) the variance. B) the standard deviation. C) the center of the curve.

B) This is true by the formula for z. (LOS 9.l)

Which of the following parameters is necessary to describe a multivariate normal distribution? A) Beta. B) Correlation. C) Degrees of freedom.

B) To describe a multivariate normal distribution, we must consider the correlations among the variables as well as the means and variances of the variables. (

For a continuous random variable X, the probability of any single value of X is: A) one. B) zero. C) determined by the cdf.

B) Explanation For a continuous distribution p(x) = 0 for all X; only ranges of value of X have positive probabilities. (LOS 9.a)

In a normal distribution, the: A)mean is less than the mode. B)median equals the mode. C)mean is greater than the median.

B) Explanation In a normal distribution, the mean, median, and mode are all equal. (Study Session 3, Module 9.2, LOS 9.i)

A probability distribution is least likely to: A)have only non-negative probabilities. B)give the probability that the distribution is realistic. C)contain all the possible outcomes.

B) Explanation The probability distribution may or may not reflect reality. But the probability distribution must list all possible outcomes, and probabilities can only have non-negative values.

A stock doubled in value last year. Its continuously compounded return over the period was closest to: A) 18.2%. B) 69.3%. C) 100.0%.

B) Explanation ln(2) = 0.6931 (LOS 9.o)

A stock priced at $100 has a 70% probability of moving up and a 30% probability of moving down. If it moves up, it increases by a factor of 1.02. If it moves down, it decreases by a factor of 1/1.02. What is the probability that the stock will be $100 after two successive periods? A)21%. B)42%. C)9%.

B) For the stock to be $100 after two periods, it must move up once and move down once: $100 × 1.02 × (1/1.02) = $100. This can happen in one of two ways: 1) the stock moves up during period one and down during period two; or 2) the stock moves down during period one and up during period two. The probability of either event is 0.70 × 0.30 = 0.21. The combined probability of either event is 2(0.21) = 0.42 or 42%.

Which of the following is least likely an example of a discrete random variable? A) The number of stocks a person owns. B) The time spent by a portfolio manager with a client. C) The number of days it rains in a month in Iowa City.

B) Time is usually a continuous random variable; the others are discrete. (LOS 9.a)

Which of the following statements least accurately describes the binomial distribution? A) It is a discrete distribution. B) The probability of an outcome of zero is zero. C) The combination formula is used in computing probabilities.

B) With only two possible outcomes, there must be some positive probability for each. If this were not the case, the variable in question would not be a random variable, and a probability distribution would be meaningless. It does not matter if one of the possible outcomes happens to be zero. (LOS 9.e)

Many analysts prefer to use Monte Carlo simulation rather than historical simulation because: A) it is much easier to generate the required variables. B) computers can manipulate theoretical data much more quickly than historical data. C) past distributions cannot address changes in correlations or events that have not happened before.

C While the past is often a good predictor of the future, simulations based on past distributions are limited to reflecting changes and events that actually occurred. Monte Carlo simulation can be used to model based on parameters that are not limited to past experience. (Study Session 3, Module 9.3, LOS 9.q)

A study of hedge fund investors found that their annual household incomes are normally distributed with a mean of $175,000 and a standard deviation of $25,000. The percentage of hedge fund investors that have incomes greater than $150,000 is closest to: A) 34.13%. B) 68.26%. C) 84.13%.

C) (150,000-175,000)/25000 (look at z table) 1 − F(-1) = F(1) = 0.8413. There is an 84.13% probability that a randomly chosen income is not more than one standard deviation below the mean. (LOS 9.k)

A multivariate distribution is best defined as describing the behavior of: A)two or more independent random variables. B)a random variable with more than two possible outcomes. C)two or more dependent random variables.

C) A multivariate distribution describes the relationships between two or more random variables, when the behavior of each random variable is dependent on the others in some way. (Study Session 3, Module 9.2, LOS 9.j)

For a standard normal distribution, F(0) is: A) 0.0. B) 0.1. C) 0.5.

C) By the symmetry of the z-distribution and F(0) = 0.5. Half the distribution lies on each side of the mean. (LOS 9.l)

Which of the following statements about probability distributions is least accurate? A)The skewness of a normal distribution is zero. B)A discrete random variable is a variable that can assume only certain clearly separated values resulting from a count of some set of items. C)A binomial probability distribution is an example of a continuous probability distribution.

C) The binomial probability distribution is an example of a discrete probability distribution. There are only two possible outcomes of each trial and the outcomes are mutually exclusive. For example, in a coin toss the outcome is either heads or tails. The other responses are both correct definitions.

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: A)99% confidence interval. B)68% confidence interval. C)95% confidence interval.

C) 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.

In a multivariate normal distribution, a correlation tells the: A)overall relationship between all the variables. B)relationship between the means and variances of the variables. C)strength of the linear relationship between two of the variables.

C) This is true by definition. The correlation only applies to two variables at a time. (Study Session 3, Module 9.2, LOS 9.j)

Joan Biggs, CFA, acquires a large database of past returns on a variety of assets. Biggs then draws random samples of sets of returns from the database and analyzes the resulting distributions. Biggs is engaging in: A)Monte Carlo simulation. B)discrete analysis. C)historical simulation.

C) Explanation This is a typical example of historical simulation. (Study Session 3, Module 9.3, LOS 9.q)

Consider a random variable X that follows a continuous uniform distribution: 7 ≤ X ≤ 20. Which of the following statements is least accurate? A)F(12 ≤ X ≤ 16) = 0.307. B)F(10) = 0.23. C)F(21) = 0.00.

C) F(21) = 1.00. For a cumulative distribution function, the expression F(x) refers to the probability of an outcome less than or equal to x. In this distribution all the possible outcomes are between 7 and 20. Therefore the probability of an outcome less than or equal to 21 is 100%. The other choices are true. F(10) = (10 - 7) / (20 - 7) = 3 / 13 = 0.23 F(12 ≤ X ≤ 16) = F(16) - F(12) = [(16 - 7) / (20 - 7)] - [(12 - 7) / (20 - 7)] = 0.692 - 0.385 = 0.307

A continuous uniform distribution has the parameters a = 4 and b = 10. The F(20) is: A) 0.25. B) 0.50. C) 1.00.

C) F(x) is the cumulative probability, P(x < 20) here. Because all the observations in this distribution are between 4 and 10, the probability of an outcome less than 20 is 100%. (LOS 9.h)

Continuous Compounding

Compounding interest literally all the time. Equivalent to compounding interest an infinite number of times per year.

Probability Distribution

Describes the probabilities of all of the possible outcomes for a random variable' The outcomes must sum to 1

Roys Safety First Ratio

Measure that has been used to measure shortfall risk. Measures the number of standard deviations of some target returns that are below the expected return/value

Are binomial distribution trials interdependent?

No- success of one doesn't impact the other

Continuous Random Variable

One for which the number of possible outcomes is infinite, even if lower and upper bounds exist.

Roys Safety First Ratio Formula

SFRatio = (expected return − threshold return)/(standard deviation of return) the larger the better

Lognormal Distribution

The function e^x where x is normally distributed; Positively skewed; Bound to the left by 0 ;Price relative is the ending price divided by the starting price ln(new price / old price)

Binomial Distribution

The number of success in a given number of trials, whereby the outcome of either can be "Success" or "failure" Probability of success stays constant- trials change.

Discrete Random Variable

Variable where the number of outcomes can be counted and each outcome has a measurable and positive probability

Probability Function

a function, denoted by f(x), that specifies the probability that a random variable is equal to a specific value 0 <_ p(x)<_ 1

Confidence Interval

a range of values so defined that there is a specified probability that the value of a parameter lies within it.

Monte Carlo Simulation

a risk analysis technique in which probable future events are simulated on a computer, generating estimated rates of return and risk indexes specifies distributions of random variables such as interest rates and underlying stock prices

A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account? A)0.750. B)0.950. C)0.977.

c) Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ - 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 - 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.

A stock price decreases in one period and then increases by an equal amount in the next period. The investor calculates a holding period return for each period and calculates their arithmetic mean. The investor also calculates the continuously compounded rate of return for each period and calculates the arithmetic mean of these. Which of the arithmetic means will be greater? A)The mean of the continuously compounded returns. B)Neither, because both will equal zero. C)The mean of the holding period returns.

c) The holding period returns will have a positive arithmetic mean. For example, a fall from 100 to 90 is a decrease of 10%, but a rise from 90 to 100 is an increase of 11.1%. The continuously compounded returns will have an arithmetic mean of zero. Using the same example values, ln(90/100) = -10.54% and ln(100/90) = 10.54%. (Study Session 3, Module 9.3, LOS 9.o)

Given a holding period return of R, the continuously compounded rate of return is: A)eR - 1. B)ln(1 - R) - 1. C)ln(1 + R).

c) This is the formula for the continuously compounded rate of return.

discrete probability distribution

consists of the values a random variable can assume and the corresponding probabilities of the values (binomial tree)

Effective Annual Rate (EAR) for continuous compounding

e^(continuously compounded rate) -1

standard normal distribution

has a mean = 0 and a standard deviation that is =1 look at each random variable and determine how many standard deviations it is from the mean(z)

Skewness definition

how symetric it is.

Continuously Compounded Rate

ln(1+ holding period return)

If a stock's initial price is $20 and its year end price is $23, then it's continuously compounded rate of return is

ln(23/20)

Continuous Compounding Forumula

ln(e^x)

Normal Distribution is described by...

mean and variance mean=0

Discrete Distribution

p(x) = 0 when x cannot occur, or p(x) > 0 if it can

Continuous Distribution

p(x)= 0 even though x can occur. We can only consider P(X1<_ X<_X2) where X1 and X2 are real numbers

A probability distribution must be

positive and have a value between 0 and 1

Shortfall Risk

probability that a portfolio value will fall below a particular value over a given time period

The reward/variability ratio is also called

sharpe ration

Historical Simulation

uses randomly selected past changes in risk factors to generate a distribution of possible security values, in contrast to Monte Carlo simulation, which uses randomly generated values. A limitation of historical simulation is that it cannot consider the effects of significant events that did not occur in the sample period.

the number of standard deviation from the mean (z) formula

z= x-m (mean)/ standard deviation (need z table)