Quantitative Methods for Valuation
Reading 9: Common Probability Distributions
1. Which of the following statements regarding probability distributions is least likely?
A. The probability of an outcome cannot be less than zero.
B. The probability of an outcome must lie between −1 and +1.
C. The sum of the probabilities of all possible distinct outcomes of an event must equal 1.
2. Which of the following statements is most likely?
A. For a discrete distribution, p(x) equals 0 when x cannot occur, while for a continuous distribution, p(x) equals zero even though x can occur.
B. For a continuous distribution, p(x) equals 0 when x cannot occur, while for a discrete distribution, p(x) equals zero even though x can occur.
C. For a continuous distribution, p(x) > 0 if x is a possible outcome.
Use the following information to answer Questions 3 to 7:
The probability function for a random variable is given as p(x) = x/100.
The set of possible values that the random variable, X, can take is given by X = (10, 20, 30, 40). For other values of x, p(x) = 0.
3. p(10) equals:
A. 0.1 B. 0.2 C. 0
4. p(30) equals:
A. 0.1 B. 0.2 C. 0.3
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5. p(35) equals A. 0.1
B. 0.35 C. 0
6. F(35) equals:
A. 0.35 B. 0.6 C. 0
7. F(40) equals:
A. 0 B. 0.40 C. 1
Use the following information to answer Questions 8 to 11:
The probability that a baseball player will swing his bat on any given pitch equals 0.35. He faces 20 pitches in each batting practice session.
8. How many different combinations are there of swinging the bat 8 times in 20 pitches?
A. 125,970
B. 5,079,110,400 C. 160
9. The probability of swinging on 8 pitches from 20 or P(x = 8) is closest to:
A. 0.16 B. 0.014 C. 0.23
10. How many times is the player expected to swing the bat at any given batting practice?
A. 6
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B. 7 C. 8
11. What is the standard deviation of the number of times he will swing the bat at any given practice session?
A. 4.55 B. 2.13 C. 20.7
12. The average annual return on a stock over its entire history is 10% and the standard deviation of returns equals 7.5%. Given that the stock’s returns are distributed normally, calculate the 95% confidence interval for the return in any given year.
A. 4.7% to 24.7%
B. −4.7% to 24.7%
C. −2.375% to 22.375%
13. What does an observation’s z-score essentially represent?
A. The number of standard deviations away from the variance the given observation actually lies.
B. The number of standard deviations away from the mean the given observation’s expected value lies.
C. The number of standard deviations away from the mean the given observation actually lies.
Use the following information to answer Questions 14 to 17:
The points, X, scored by students in a class on their final exam are normally distributed with a mean of 65 and standard deviation of 20.
14. What is the probability that the points scored by a given student will be less than 80 points?
A. 80.23%
B. 77.34%
C. 74.22%
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15. What is the probability that the points scored by a given student will be more than 75?
A. 69.15%
B. 65.54%
C. 30.85%
16. What is the probability that the points scored by a given student will be less than 35?
A. 97.72%
B. 47.72%
C. 6.68%
17. What is the probability that the points scored by a given student will be between 75 and 90?
A. 20.29%
B. 58.59%
C. 10.25%
18. Portfolio A has an expected return of 9% and a standard deviation of 12%, while Portfolio B has an expected return of 12% and a standard deviation of 15%. Using Roy’s Safety-first criterion and assuming that the target return equals 5%, which portfolio should be preferred?
A. Portfolio B B. Portfolio A
C. The investor would be indifferent between the two portfolios.
19. Which of the following statements is most likely?
A. A portfolio with a higher SF Ratio also has a higher probability of attaining returns higher than the threshold level.
B. A portfolio with a lower SF Ratio also has a lower probability of attaining returns lower than the threshold level.
C. A portfolio with a higher SF Ratio also has a lower probability of attaining returns higher than the threshold level.
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20. If the effective annual rate is given as 7.34%, the continuously compounded stated annual interest rate equals?
A. 7.08%
B. 55.04%
C. 7.62%
21. An investment of $1,000 appreciates to a value of $1,450 in 3.5 years.
What is the continuously compounded annual return on this investment?
A. 10.62%
B. 37.16%
C. 4.26%
22. Which of the following is least likely regarding the lognormal distribution?
A. It is bounded by zero at the lower end.
B. It is skewed to the left.
C. Its upper range is unbounded.
23. Do the Monte Carlo and historical simulation models facilitate “What if?” analysis?
Monte Carlo Simulation Historical Simulation
A. Yes Yes
B. Yes No
C. No Yes
24. Which of the following is least likely regarding the continuous uniform distribution?
A. P(X = x) = 0 for all values of x that lie between the parameters of the distribution.
B. The cumulative distribution function for the variable rises in steps.
C. P(X = x) = 0 for all values of x that lie outside the parameters of the distribution.
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Reading 9: Common Probability Distributions
1. Which of the following statements regarding probability distributions is least likely?
A. The probability of an outcome cannot be less than zero.
B. The probability of an outcome must lie between −1 and +1.
C. The sum of the probabilities of all possible distinct outcomes of an event must equal 1.
Answer: B
The probability of a given outcome must lie between zero and +1.
2. Which of the following statements is most likely?
A. For a discrete distribution, p(x) equals 0 when x cannot occur, while for a continuous distribution, p(x) equals zero even though x can occur.
B. For a continuous distribution, p(x) equals 0 when x cannot occur, while for a discrete distribution, p(x) equals zero even though x can occur.
C. For a continuous distribution, p(x) > 0 if x is a possible outcome.
Answer: A
For a discrete distribution, p(x) equals 0 when x cannot occur, while for a continuous distribution, p(x) equals zero even though x can occur. For a continuous distribution p(x) = 0 even if x is a possible outcome.
Use the following information to answer Questions 3 to 7:
The probability function for a random variable is given as p(x) = x/100.
The set of possible values that the random variable, X, can take is given by X = (10, 20, 30, 40). For other values of x, p(x) = 0.
3. p(10) equals:
A. 0.1
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B. 0.2 C. 0 Answer: A
p(10) = 10/100 = 0.1 4. p(30) equals:
A. 0.1 B. 0.2 C. 0.3 Answer: C
p(30) = 30/100 = 0.3 5. p(35) equals
A. 0.1 B. 0.35 C. 0 Answer: C
p(35) = 0 because 35 is not a possible outcome for the random variable.
6. F(35) equals:
A. 0.35 B. 0.6 C. 0 Answer: B
F(35) = p(10) + p(20) + p(30) = 0.1 + 0.2 + 0.3 = 0.6 7. F(40) equals:
A. 0 B. 0.40 C. 1
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Answer: C
F(40) = p(10) + p(20) + p(30) + p(40) = 0.1 + 0.2 + 0.3 + 0.4 = 1.0 Use the following information to answer Questions 8 to 11:
The probability that a baseball player will swing his bat on any given pitch equals 0.35. He faces 20 pitches in each batting practice session.
8. How many different combinations are there of swinging the bat 8 times in 20 pitches?
A. 125,970
B. 5,079,110,400 C. 160
Answer: A
The number of possible combinations of swinging the bat 8 times when facing 20 pitches equals 20C8 = 125,970
9. The probability of swinging on 8 pitches from 20 or P(x = 8) is closest to:
A. 0.16 B. 0.014 C. 0.23 Answer: A
Using the binomial distribution:
p(x = 8) = 20C8 (0.35ˆ8) (0.65ˆ12) = 16.14% or 0.16
10. How many times is the player expected to swing the bat at any given batting practice?
A. 6 B. 7 C. 8 Answer: B
E(X) = (n)(p) = (20)(0.35) = 7
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11. What is the standard deviation of the number of times he will swing the bat at any given practice session?
A. 4.55 B. 2.13 C. 20.7 Answer: B
Var(X) = (n)(p)(1−p) = (20)(0.35)(0.650) = 4.55 Std. Dev = 4.55 ˆ 0.5 = 2.13
12. The average annual return on a stock over its entire history is 10% and the standard deviation of returns equals 7.5%. Given that the stock’s returns are distributed normally, calculate the 95% confidence interval for the return in any given year.
A. 4.7% to 24.7%
B. −4.7% to 24.7%
C. −2.375% to 22.375%
Answer: B
The reliability factor for a 95% confidence interval equals 1.96.
Therefore, the confidence interval is calculated as:
10 ± (1.96) × (7.5)
13. What does an observation’s z-score essentially represent?
A. The number of standard deviations away from the variance the given observation actually lies.
B. The number of standard deviations away from the mean the given observation’s expected value lies.
C. The number of standard deviations away from the mean the given observation actually lies.
Answer: C
An observation’s z-score basically represents the number of standard deviations away from the mean the given observation actually lies.
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Use the following information to answer Questions 14 to 17:
The points, X, scored by students in a class on their final exam are normally distributed with a mean of 65 and standard deviation of 20.
14. What is the probability that the points scored by a given student will be less than 80 points?
A. 80.23%
B. 77.34%
C. 74.22%
Answer: B
P(Z ≤ 0.75) = 0.7734
15. What is the probability that the points scored by a given student will be more than 75?
A. 69.15%
B. 65.54%
C. 30.85%
Answer: C.
P(Z ≥ 0.5) = 1 − P (Z ≤ 0.5) = 1 − 0.6915 = 0.3085
16. What is the probability that the points scored by a given student will be less than 35?
A. 97.72%
B. 47.72%
C. 6.68%
Answer: C
P (Z ≤ −1.5) = P (Z ≥ 1.5) = 1 − P (Z ≤ 1.5) = 1 − 0.9332 = 0.0668
17. What is the probability that the points scored by a given student will be between 75 and 90?
A. 20.29%
B. 58.59%
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C. 10.25%
Answer: A
P (Z ≤ 1.25) − P (Z ≤ 0.5) = 0.8944 − 0.6915 = 0.2029
18. Portfolio A has an expected return of 9% and a standard deviation of 12%, while Portfolio B has an expected return of 12% and a standard deviation of 15%. Using Roy’s Safety-first criterion and assuming that the target return equals 5%, which portfolio should be preferred?
A. Portfolio B B. Portfolio A
C. The investor would be indifferent between the two portfolios.
Answer: A
Portfolio B has a higher SF Ratio so it is preferred. (7/15 = 0.47 versus 4/12 = 0.33).
19. Which of the following statements is most likely?
A. A portfolio with a higher SF Ratio also has a higher probability of attaining returns higher than the threshold level.
B. A portfolio with a lower SF Ratio also has a lower probability of attaining returns lower than the threshold level.
C. A portfolio with a higher SF Ratio also has a lower probability of attaining returns higher than the threshold level.
Answer: A
A portfolio with a higher SF Ratio also has a higher probability of attaining returns higher than the threshold level. Alternatively, it has a lower probability of attaining returns lower than the threshold level.
20. If the effective annual rate is given as 7.34%, the continuously compounded stated annual interest rate equals?
A. 7.08%
B. 55.04%
C. 7.62%
Answer: A
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rcc = ln (EAR + 1). Therefore, rcc = ln (1.0734) = 7.08%
21. An investment of $1,000 appreciates to a value of $1,450 in 3.5 years.
What is the continuously compounded annual return on this investment?
A. 10.62%
B. 37.16%
C. 4.26%
Answer: A
rcc = [ln (HPR + 1)]/ t. Therefore, rcc = [ln (0.45 + 1)]/ 3.5 = 10.62%
22. Which of the following is least likely regarding the lognormal distribution?
A. It is bounded by zero at the lower end.
B. It is skewed to the left.
C. Its upper range is unbounded.
Answer: B
The lognormal distribution is skewed to the right (positively skewed).
23. Do the Monte Carlo and historical simulation models facilitate “What if?” analysis?
Monte Carlo Simulation Historical Simulation
A. Yes Yes
B. Yes No
C. No Yes
Answer: B
Monte Carlo simulation does lend itself to “What if?” analysis.
However, historical simulation does not.
24. Which of the following is least likely regarding the continuous uniform distribution?
A. P(X = x) = 0 for all values of x that lie between the parameters of
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the distribution.
B. The cumulative distribution function for the variable rises in steps.
C. P(X = x) = 0 for all values of x that lie outside the parameters of the distribution.
Answer: B
The cumulative distribution function (cdf) for the continuous uniform distribution is an upward sloping straight line. The cdf for the discrete uniform distribution rises in steps.
The probability of any outcomes outside a and b, the parameters of the distribution equals zero.
Since it is a continuous distribution, the probability of any individual outcome that lies within the parameters of the distribution is also zero.
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