READING 3: PROBABILITY CONCEPTS QB
A firm will select two of four vice presidents to be added to the investment committee. How many different groups of two are possible? A. 6 B. 12 C. 24
A
All else being equal, as the correlation between two assets approaches +1.0, the diversification benefits: A. decrease. B. stay the same. C. increase.
A
From an approved list of 25 funds, a portfolio manager wants to rank 4 mutual funds from most recommended to least recommended. Which formula is most appropriate to calculate the number of possible ways the funds could be ranked? A. Permutation formula B. Multinomial formula C. Combination formula
A
Gerd Sturm wants to sponsor a contest with a $1 million prize. The winner must pick the stocks that will be the top five performers next year among the 30 stocks in a well-known large-cap stock index. He asks you to estimate the chances that contestants can win the contest. What are the chances of winning if the contestants must pick the top five stocks without regard to order? If choosing five stocks randomly, a contestant's chance of winning is one out of: A. 142,506. B. 17,100,720. C. 24,300,000.
A
Himari Fukumoto has joined a new firm and is selecting mutual funds in the firm's pension plan. If 10 mutual funds are available, and she plans to select four, how many different sets of mutual funds can she choose? A. 210 B. 720 C. 5,040
A
If the probability for an event Z is 14% (i.e., P(Z) = 14%), the odds for Z are closest to: A. 0.163. B. 0.071. C. 0.123.
A
If the probability that Zolaf Company sales exceed last year's sales is 0.167, the odds for exceeding sales are closest to: A. 1 to 5. B. 1 to 6. C. 5 to 1.
A
If two events, A and B, are independent, and the probability of A does not equal the probability of B [i.e., P(A) ≠ P(B)], then the probability of event A given that event B has occurred [i.e., P(A | B)] is best described as: A. P(A). B. P(B). C. P(B | A).
A
The conditional expected value of a random variable is best described as the: A. expected value of a random variable given an event or scenario. B. probability-weighted average of the possible outcomes of the random variable. C. weighted average of the probabilities of an event given all possible scenarios.
A
The correlation of returns between Stocks A and B is 0.50. The covariance between these two securities is 0.0043, and the standard deviation of the return of Stock B is 26%. The variance of returns for Stock A is: A. 0.0011. B. 0.0331. C. 0.2656.
A
The probability that the DJIA will increase tomorrow is 2/3. The probability of an increase in the DJIA stated as odds is: A. two-to-one. B. one-to-three. C. two-to-three.
A
US and Spanish bonds have return standard deviations of 0.64 and 0.56, respectively. If the correlation between the two bonds is 0.24, the covariance of returns is closest to: A. 0.086. B. 0.335. C. 0.390.
A
Given a portfolio of five stocks, how many unique covariance terms, excluding variances, are required to calculate the portfolio return variance? A. 10 B. 20 C. 25
A A covariance matrix for five stocks has 5 × 5 = 25 entries. Subtracting the 5 diagonal variance terms results in 20 off-diagonal entries. Because a covariance matrix is symmetrical, only 10 entries are unique (20/2 = 10).
A group of fund analysts have to select the first, second, and third best fund manager of the year for 2012 based on their subjective judgment. If 10 fund managers are candidates for the three awards, the number of ways in which each analyst can make his ranking is closest to: A. 30. B. 720. C. 120.
B
A tree diagram is most likely used when dealing with investment problems that involve outcomes that are: A. independent at each node. B. mutually exclusive. C. unconditional at each node.
B
At a charity ball, 800 names are put into a hat. Four of the names are identical. On a random draw, what is the probability that one of these four names will be drawn? A. 0.004. B. 0.005. C. 0.010.
B
Consider a universe of 10 bonds from which an investor will ultimately purchase six bonds for his portfolio. If the order in which he buys these bonds is not important, how many potential 6-bond combinations are there? A. 7. B. 210. C. 5,040.
B
Gerd Sturm wants to sponsor a contest with a $1 million prize. The winner must pick the stocks that will be the top five performers next year among the 30 stocks in a well-known large-cap stock index. He asks you to estimate the chances that contestants can win the contest. What are the chances of winning if the contestants must pick the five stocks in the correct order of their total return? If choosing five stocks randomly, a contestant's chance of winning is one out of: A. 142,506. B. 17,100,720. C. 24,300,000.
B
P(A|B) = 40% and P(B) = 30% and P(A) = 40%. It is most likely that: A. A and B are dependent. B. A and B are independent. C. A and B are mutually exclusive.
B
There are 10 sprinters in the finals of a race. How many different ways can the gold, silver, and bronze medals be awarded? A. 120. B. 720. C. 1,440.
B
Two mutually exclusive events: A. will both occur. B. cannot both occur. C. may both occur.
B
Which of the following correlation coefficients indicates the weakest linear relationship between two variables? A. -0.67 B. -0.24 C. 0.33
B
Which of the following is a property of two dependent events? A. The two events must occur simultaneously. B. The probability of one event influences the probability of the other event. C. The probability of the two events occurring is the product of each event's probability.
B
After estimating the probability that an investment manager will exceed his benchmark return in each of the next two quarters, an analyst wants to forecast the probability that the investment manager will exceed his benchmark return over the two-quarter period in total. Assuming that each quarter's performance is independent of the other, which probability rule should the analyst select? A. Addition rule B. Multiplication rule C. Total probability rule
B Because the events are independent, the multiplication rule is most appropriate for forecasting their joint probability. The multiplication rule for independent events states that the joint probability of both A and B occurring is P(AB) = P(A)P(B).
A discrete uniform distribution (each event has an equal probability of occurrence) has the following possible outcomes for : [1, 2, 3, 4]. The variance of this distribution is to: A. 1.00. B. 1.25. C. 2.00.
B Expected value=(1/4)(1+2+3+4)=2.5Variance = (1/4)[(1 - 2.5)2 + (2 - 2.5)2 + (3 - 2.5)2 + (4 - 2.5)2] = 1.25 Note that since each observation is equally likely, each has 25% (1/4) chance of occurrence. (LOS 3.k)
An analyst estimates that a share price has an 80% probability of increasing if economic growth exceeds 3%, a 40% probability of increasing if economic growth is between zero and 3%, and a 10% probability of increasing if economic growth is negative. If economic growth has a 25% probability of exceeding 3% and a 25% probability of being negative, what is the probability that the share price increases? A. 22.5%. B. 42.5%. C. 62.5%.
B The three outcomes given for economic growth are mutually exclusive and exhaustive. The probability that economic growth is positive but less than 3% is 100% - 25% - 25% = 50%. Using the total probability rule, the probability that the share price increases is (80%)(25%) + (40%)(50%) + (10%)(25%) = 42.5%. (LOS 3.g)
A manager will select 20 bonds out of his universe of 100 bonds to construct a portfolio. Which formula provides the number of possible portfolios? A. Permutation formula B. Multinomial formula C. Combination formula
C
A portfolio manager estimates the probabilities of the following events for a mutual fund: Event A: the fund will earn a return of 5%. Event B: the fund will earn a return below 5%. The least appropriate description of the events is that they are: A. dependent. B. mutually exclusive. C. exhaustive.
C
An event that includes all of the possible outcomes is said to be: A. random. B. exclusive. C. exhaustive.
C
Assuming no short selling, a diversification benefit is most likely to occur when the correlations among the securities contained in the portfolio are: A. greater than +1. B. equal to +1. C. less than +1.
C
By definition, the probability of any Event E is a number between: A. zero and positive infinity. B. minus one and positive one. C. zero and positive one.
C
Event X and Event Y are independent events. The probability of X is 0.2 [P(X) = 0.2] and the probability of Y is 0.5 [P(Y) = 0.5]. The joint probability of X and Y, P(XY), is closest to: A. 0.7. B. 0.3. C. 0.1.
C
If events A and B are mutually exclusive, then: A. P(A | B) = P(A). B. P(AB) = P(A) × P(B). C. P(A or B) = P(A) + P(B).
C
In probability theory, exhaustive events are best described as the set of events that: A. have a probability of zero. B. are mutually exclusive. C. include all potential outcomes.
C
The covariance of returns is positive when the returns on two assets tend to: A. have the same expected values. B. be above their expected value at different times. C. be on the same side of their expected value at the same time.
C
The multiplication rule of probability determines the joint probability of two events as the product of: A. two conditional probabilities. B. two unconditional probabilities. C. a conditional probability and an unconditional probability.
C
The number of permutations that are possible when choosing 4 objects from a total of 10 objects is closest to: A. 30. B. 210. C. 5,040.
C
The probability of Event A is 40%. The probability of Event B is 60%. The joint probability of AB is 40%. The probability (P) that A or B occurs, or both occur, is closest to: A. 40%. B. 84%. C. 60%.
C
The probability of an event given that another event has occurred is a: A. joint probability. B. marginal probability. C. conditional probability.
C
Two events are said to be independent if the occurrence of one event: A. means that the second event cannot occur. B. means that the second event is certain to occur. C. does not affect the probability of the occurrence of the other event.
C
When working backward from the nodes on a binomial tree diagram, the analyst is attempting to calculate: A. the number of potential outcomes. B. the probability of a given scenario. C. an expected value as of today.
C
Which of the following statements is most accurate? If the covariance of returns between two assets is 0.0023, then: A. the assets' risk is near zero. B. the asset returns are unrelated. C. the asset returns have a positive relationship.
C
Which of the following values cannot be the probability of an event? A. 0.00. B. 1.00. C. 1.25.
C
Which probability estimate most likely varies greatly between people? A. An a priori probability B. An empirical probability C. A subjective probability
C
Florence Hixon is screening a set of 100 stocks based on two criteria (Criterion 1 and Criterion 2). She set the passing level such that 50% of the stocks passed each screen. For these stocks, the values for Criterion 1 and Criterion 2 are not independent but are positively related. How many stocks should pass Hixon's two screens? A. Less than 25 B. 25 C. More than 25
C Let event A be a stock passing the first screen (Criterion 1) and event B be a stock passing the second screen (Criterion 2). The probability of passing each screen is P(A) = 0.50 and P(B) = 0.50. If the two criteria are independent, the joint probability of passing both screens is P(AB) = P(A)P(B) = 0.50 × 0.50 = 0.25, so 25 out of 100 stocks would pass both screens. However, the two criteria are positively related, and P(AB) ≠ 0.25. Using the multiplication rule for probabilities, the joint probability of A and B is P(AB) = P(A | B)P(B). If the two criteria are not independent, and if P(B) = 0.50, then the contingent probability of P(A | B) is greater than 0.50. So the joint probability of P(AB) = P(A | B)P(B) is greater than 0.25. More than 25 stocks should pass the two screens.
After six months, the growth portfolio that Rayan Khan manages has outperformed its benchmark. Khan states that his odds of beating the benchmark for the year are 3 to 1. If these odds are correct, what is the probability that Khan's portfolio will beat the benchmark for the year? A. 0.33 B. 0.67 C. 0.75
C The odds for beating the benchmark = P(beating benchmark) / [1 - P(beating benchmark)]. Let P(A) = P(beating benchmark). Odds for beating the benchmark = P(A) / [1 - P(A)].3 = P(A) / [1 - P(A)] Solving for P(A), the probability of beating the benchmark is 0.75
Which of the following best describes how an analyst would estimate the expected value of a firm using the scenarios of bankruptcy and non-bankruptcy? The analyst would use: A. the addition rule. B. conditional expected values. C. the total probability rule for expected value.
C The total probability rule for expected value is used to estimate an expected value based on mutually exclusive and exhaustive scenarios.
An analyst believes Davies Company has a 40% probability of earning more than $2 per share. She estimates that the probability that Davies Company's credit rating will be upgraded is 70% if its earnings per share are greater than $2 and 20% if its earnings per share are $2 or less. Given the information that Davies Company's credit rating has been upgraded, what is the updated probability that its earnings per share are greater than $2? A. 50%. B. 60%. C. 70%.
C This is an application of Bayes' formula. As the tree diagram below shows, the updated probability that earnings per share are greater than $2 is 28% / 28% + 12% = 70%