FRM2012数量分析(习题)

QUANTITATIVE ANALYSIS

1.Assume you have a portfolio of 10 obligors that are not correlated. Given that each obligor's

probability of default is 5%, what is the probability that no defaults will occur over the next year?

A. 5.0%

B.50.0%

C.60.0%

D.95.0%

2.The annual marginal probability of default of a bond is 15% in year 1 and 20% in year 2.

What is the probability of the bond surviving (i.e. no default) to the end of two years?

A.68%

B.65%

C.80%

D.85%

3. A portfolio of bonds consists of five bonds whose default correlation is zero. The one-year

probabilities of default of the bonds are: 1%, 2%, 5%, 10% and 15%. What is the one-year probability of no default within the portfolio?

A.71%

B.67%

C.85%

D.99%

4.The characteristic function of the product of independent random variables is equal to the:

A.square root of the product of the individual characteristic functions

B.product of the individual characteristic functions

C.exponential root of the product of the individual characteristic functions

D.sum of the individual characteristic functions

5.An investor is choosing one of twenty securities. Ten of the securities are stocks and ten are

bonds. Four of the ten stocks were issued by utilities; the other six were issued by industrial firms. Two of the ten bonds were issued by utilities; the other eight were issued by industrial firms. If the investor chooses a security at random, the probability that it is a bond or a security issued by an industrial firm is:

A.0.80

B.0.70

C.0.60

D.0.50

6.Dependent random variables are defined as variables where their joint probability is:

A.equal to zero

B.not equal to the product of their individual probabilities

C.greater than the product of their individual probabilities

D.equal to the product of their individual probabilities

7.Given the following data for a market variable, what is the best estimate of its variance?

Probability Value

24% -12

40% 4

36% 14

A. 6

B.10

C.32

D.97

8.An analyst is studying a stock that is currently trading at $35. The analyst estimates that there

is 33% probability that the stock will trade at $50 after one year, a 20% probability that the stock will trade at $42, and a 47% probability that the stock will trade at $20. What is the volatility of this stock price?

A.13%

B.24%

C.31%

D.39%

https://www.360docs.net/doc/ec3304610.html,ing the following joint probability distribution to answer the questions below.

Y=1 Y=2 Y=3

X=1 0.05 0.05 0.10

X=2 0.05 0.10 0.15

X=3 0.15 0.15 0.20

The expected value of Y is closest to:

A.0.2

B. 2.2

C. 1.0

D. 2.3

________________________________________

If you know that Y is equal to 2, the probability that X is equal to 1 is closest to:

A.0.05

B.0.25

C.0.20

D.0.17

________________________________________

The variance of X is closest to:

A.0.54

B.0.67

C.0.74

D.0.61

10.Given two random variables X and Y, what is the Variance of X given Variance[Y] = 100,

Variance [4X - 3Y] = 2,700 and the correlation between X and Y is 0.5?

A.56.3

B.113.3

C.159.9

D.225.0

11.You are given that X and Y are random variables, and each of which follows a standard

normal distribution with Covariance (X, Y) = 0.4. What is the variance of (5X + 2Y)?

A.11.0

B.29.0

C.29.4

D.33.0

12.Suppose returns are uncorrelated over time. You are given that the volatility over two days is

1.20%. What is the volatility over 20 days?

A.0.38%

B. 1.20%

C. 3.79%

D.12.0%

13.What is the covariance between populations A and B?

A B

17 22

14 26

12 31

13 29

B. 6.50

C.-3.61

D. 3.61

14.The covariance between variable A and variable B is 5. The correlation between A and B is

0.50. If the variance of A is 12, what is the variance of B?

A.10.00

B. 2.89

C.8.33

D.14.40

15.Which one of the following statements about the correlation coefficient is FALSE?

A.It always ranges from –1 to +1

B. A correlation coefficient of zero means that two random variables are independent

C.It is a measure of linear relationship between two random variables

D.It can be calculated by scaling the covariance between two random variables

16. A portfolio contains two perfectly negatively correlated investments with volatilities of 5%

and 7%. The proportion of these two securities that would lead to the lowest risk are:

A.58% and 42%

B.42% and 58%

C.34% and 66%

D.66% and 34%

17.Which statement best describes correlations and variances in times of financial crisis?

A.There are only marginal changes in correlations and variances in times of crisis and

therefore they do not need to be factored into risk management.

B.The diversification benefits decrease as correlations increase and therefore your risk

level increases.

C.The diversification benefits increase as correlations decrease and therefore your risk

level decreases.

D.VaR estimates using the Riskmetrics approach provide for the effects of increased

correlations during periods of crisis and therefore the effects are factored into current

positions.

18.The probability that an observation lies within 3 standard deviations of the mean for any

probability distribution is at least:

A.75%

C.89%

D.54%

19.The probability of an outcome being between 10 and 30 for a random variable that follows a

continuous uniform distribution over the range of 0 to 40 is closest to:

A.0.2

B.0.5

C.0.4

D.0.8

20.Given that a stock price rises on two days out of five and falls on three days out of five, what

is the probability that it will rise on exactly seven out of the next eight days?

A.0.8%

B. 4.5%

C.9.0%

D.12.0%

21.An analyst has determined that the probability that the S&P 500 index will increase on any

given day is 0.60 and the probability that it will decrease is 0.40. The expected value and variance of the number of up days in a 5-day period are closest to:

A. 3.0 and 1.2

B. 3.0 and 1.1

C. 2.0 and 0.5

D. 2.0 and 2.1

22.Which of the following statements are TRUE?

I.The sum of two random normal variables is also a random normal variable.

II.The product of two random normal variables is also a random normal variable.

III.The sum of two random lognormal variables is also a random lognormal variable.

IV.The product of two random lognormal variables is also a random lognormal variable.

A.I and II only

B.II and III only

C.III and IV only

D.I and IV only

23.Which of the following statements best characterizes the relationship between the normal and

lognormal distributions?

A.The lognormal distribution is the logarithm of the normal distribution

B.If the natural log of the random variable X is lognormally distributed, then X is normally

distributed

C.If X is lognormally distributed, then the natural log of X is normally distributed

D.The two distributions have nothing to do with each other

24.If a distribution with the same variance as a normal distribution has kurtosis greater than 3,

which of the following is TRUE?

A.It has fatter tails than normal distribution

B.It has thinner tails than normal distribution

C.It has the same tail fatness as the normal distribution since, variances are the same

D.Cannot be determined from the information provided

25.For a standard normal distribution, what is the approximate area under the cumulative

distribution function between the values -1 and 1?

A.50%

B.66%

C.75%

D.95%

26.Consider a stock with an initial price of $100. Its price one year from now is given by S =

100*exp(r), where the rate of return r is normally distributed with a mean of 0.1 and a standard deviation of 0.2. With 95% confidence, after rounding, S will be between:

A.$67.57 and $147.99

B.$70.80 and $149.20

C.$74.68 and $163.56

D.$102.18 and $119.53

27.Which of the following are characteristics of a normal distribution?

I. Skewness equals to zero

II. Mean less than median

III. Kurtosis greater than zero

IV. Continuous and unbounded

A.I and IV

B.II and III

C.I, II and III

D.I, III and IV

28.Which of the following statements is the most accurate about the relationship between a

normal distribution and a Student's t-distribution that have the same mean and standard

deviation?

A.They have the same skewness and the same kurtosis

B.The Student's t-distribution has larger skewness and larger kurtosis

C.The kurtosis of a Student's t-distribution converges to that of the normal distribution as

the number of degrees of freedom increases

D.The normal distribution is a good approximation for the Student's t-distribution when the

number of degrees of freedom is small

29. A call center receives an average of two phone calls per hour. The probability that they will

receive 20 calls in an 8-hour day is closest to:

A. 5.59%

B.16.56%

C. 3.66%

D. 6.40%

30.It is often said that distributions of returns from financial instruments are leptokurtotic. For

such distributions, which of the following comparisons with a normal distribution of the same mean and variance MUST hold?

A.The skewness of the leptokurtotic distribution is greater

B.The kurtosis of the leptokurtotic distribution is greater

C.The skewness of the leptokurtotic distribution is smaller

D.The kurtosis of the leptokurtotic distribution is smaller

31. A bank would like to estimate the number of operational risk events due to problems with

tellers (large mistakes, fraud, and so on). The bank decides to model teller operational risk events as a Poisson process with rate λ(number of events per year). With this model, teller operational risk events are assumed to occur independently of one another and the number of teller operational risk events in a year is Poisson distributed with mean λ. Other properties of a Poisson distribution with mean λ include:

Variance: λ

λ?

Skewness: 0.5

Excess Kurtosis: 1/λ

Based on historical data regarding the number of teller operational risk events that occurred in previous years, the bank determines that the average number of events has been 5 per year and decides to set λto 5. Which of the following is true regarding that model?

A.The corresponding exponential distribution that describes the time between two teller

operational risk events has a mean value of 0.25 years.

B.The variance of the number of teller operational events in a year is 25.

C.The number of teller operational risk events in a year cannot exceed 25.

D.The model is not appropriate if a teller is more likely to have an operational risk event

because his friend who is also a teller has been caught stealing.

32.Assume the true distribution of returns is leptokurtotic. If we assume normality when we

calculate the VaR, then which of the following statements is true:

A.The 95% VaR is overstated.

B.The 95% VaR is understated.

C.The 95% VaR is appropriate.

D.We cannot state the relationship between the true VaR and the calculated VaR.

33.The last three year-end returns for a stock are 5%, –2%, and 1%. Using an arithmetic mean,

the sample standard deviation is closest to:

A. 3.51%

B. 2.87%

C. 1.33%

D. 3.11%

34.The mean equity risk premium over a 40-year period is equal to 8.0%. The standard deviation

of the sample is 12%. The standard error of the sample mean is closest to:

A.0.30%

B. 1.90%

C. 1.26%

D.8.00%

35.The mean age of the 80 employees in a company is 35 and the standard deviation is 15.

Assuming that the ages are normally distributed and using 95% confidence level, we can say that the employees within the firm fall between:

A.20.0 and 50.0 years

B.31.7 and 38.3 years

C.33.8 and 36.2 years

D.34.6 and 35.4 years

36.Which of the following statements are NOT true?

I. Type I error occurs when the null hypothesis is not rejected when it is actually false.

II. Type II error occurs when the null hypothesis is rejected when it is actually true.

III. Type I error occurs when the alternate hypothesis is wrongly accepted.

IV. Minimizing the probability of Type II error maximizes the power of the test.

A.I and II

B.I and III

C.II and IV

D.I, II and IV

37.According to the Basel backtesting framework guidelines, penalties start to apply if there are

five or more exceptions during the previous year. The Type I error rate of this test is 11%. If the true coverage is 97% of exceptions instead of the required 99%, the power of the test is 87%. This implies that there is a (an):

A.89% probability regulators will reject the correct model

B.11% probability regulators will reject the incorrect model

C.87% probability regulators will not reject the correct model

D.13% probability regulators will not reject the incorrect model

38.An analyst wants to test whether the standard deviation of return from pharmaceutical stocks

is lower than 0.2. For this purpose, he obtains the following data from a sample of 30 pharmaceutical stocks. Mean return from pharmaceutical stocks = 8%. Standard deviation of return from pharmaceutical stocks = 12%. Mean return from the market = 12%. Standard deviation of return from the market = 16%. What is the appropriate test statistic for this test?

A.t-statistic

B.z-statistic

C.F-statistic

D.Chi-squared statistic

39.If the sample size is greater than 30 and population variance is unknown, the appropriate test

for the sample mean is the:

A.t-test

B.z-test

C.p-test or F-test

D.t-test or z-test

https://www.360docs.net/doc/ec3304610.html,ing a sample size of 61 observations, an analyst determines that the standard deviation of

the returns from a stock is 21%. Using a 0.05 significance level, the analyst:

A.can conclude that the standard deviation of returns is higher than 14%

B.cannot conclude that the standard deviation of returns is higher than 14%

C.can conclude that the standard deviation of returns is not higher than 14%

D.none of the above

41.An analyst is given the task of determining whether a group of 16 active portfolio managers

have achieved a significantly higher performance (using a significance level of 0.05) than the average for all portfolio managers over a certain period. Over the period of the study, the active portfolio managers achieved a mean return of 15%. Over the same period the mean return for all portfolio managers was 12% and their standard deviation was 8%. The correct conclusion from this study is that:

A.the performance of active portfolio managers is significantly higher than the average for

all portfolio managers

B.the performance of active portfolio managers is not significantly higher than the average

for all portfolio managers

C.the performance of active portfolio managers is significantly lower than the average for

all portfolio managers

D.the performance of active portfolio managers is the same as the average for all portfolio

managers

42.Suppose the standard deviation of a normal population is known to be 10 and the mean is

hypothesized to be 8. Suppose a sample size of 100 is considered. What is the range of sample means that allows the hypothesis to be accepted at a level of significance of 0.05?

A.Between -11.60 and 27.60

B.Between 6.04 and 9.96

C.Between 6.355 and 9.645

D.Between -8.45 and 24.45

43.Under what circumstances could the explanatory power of regression analysis be overstated?

A.The explanatory variables are not correlated with one another.

B.The variance of the error term decreases as the value of the dependent variable

increases.

C.The error term is normally distributed.

D.An important explanatory variable is omitted that influences the explanatory variables

included, and the dependent variable.

44.Many statistical problems arise when estimating relationships using regression analysis. Some of

these problems are due to the assumptions behind the regression model. Which one of the following is NOT one of these problems?

A.Stratification

B.Multicollinearity

C.Heteroscedasticity

D.Autocorrelation

45. An analyst is seeking to generate a simple linear regression equation that will allow him to

estimate the returns of a beverage company’s stock against the returns of the S&P 500. The analyst has compiled the following information:

S&P 500σ= 16.4%；beverage company σ= 22.0% ; ρ = 0.82

mean annual return S&P 500 = 8.2%; risk-free interest rate = 4.0%

What is the slope coefficient of the regression equation?

A. 1.85

B. 1.10

C. 2.34

D. 0.80

46. A simple linear regression of a stock’s returns on an industry index provides the following

results:

Coefficient Standard Error

Intercept 4.2 2.66

Industry Index 2.4 0.65

Sum of Squared Regression 998.56

Sum of Squared Errors 180.37

Sum of Squared Total 1,178.93

Assume that the sample has 8 years of quarterly observations.

Which of the following statements regarding the interpretation of the regression is (are) CORRECT?

I. The coefficient of determination is 84.7%.

II. The industry index coefficient is significant at the 99% confidence level.

III. The correlation coefficient between the stock’s returns and the return on the industry

index is 0.42.

IV . The correlation coefficient between the stock’s returns and the return on the industry

index is statistically different from zero at the 99% confidence level.

A. II and IV only

B. I and II only

C. I, II, and IV only

D. I only

47. You need to update a daily volatility forecast using the RiskMetrics TM exponential method

with a decay factor of 0.97. Yesterday's forecast of standard deviation was 1%. Given that you just observed a return of 2%, what will be the new forecast of standard deviation?

A. 1.030%

B. 1.044%

C. 1.970%

D. 1.977%

48. Which of the following GARCH models will take the shortest time to revert to its mean?

A. t h = 0.05 + 0.0321t r

?+ 0.961t h ? B. t h = 0.03 + 0.0221t r

?+ 0.951t h ? C. t h = 0.02 + 0.0121t r

?+ 0.971t h ? D. t h = 0.01 + 0.0121t r

?+ 0.981t h ?

49. A risk manager estimates daily variance(t h )using a GARCH model on daily

returns(t r ):112

t=01t t r h ααγβ??++Assume the model parameter values are

0α=0.005, 1α=0.04, β=0.94. The long-run annualized volatility is approximately

A.

13.54% B.

7.94% C.

72.72% D. 25.00%

50. Consider the following GARCH model estimated using daily return data:

11296.003.005.0??++=t t t h r h

Which of the following statements are correct?

i. The persistence factor of the model is 0.99.

ii. The long-run average daily variance is 5.

iii. The model does not have a long-run average variance.

iv. Given ht, is 6, the forecast of conditional variance for day t+1 is 5.99.

A. Only I and II

B. Only l and Ill

C. I and lV

D. I, II and IV

51. You have implemented the simulation process using a time interval Δt = 0.001, and you are

analyzing the following stock price path generated by your implementation.

Given this sample, which of the following simulation steps most likely contains an error?

A.Calculation to update the stock price

B.Generation of random sample value for ε

C.Calculation of the change in stock price during each period

D.None of the above

52.Which of the following statements about Extreme Value Theory (EVT) and its application to

value at risk are true?

I.EVT extends the Central Limit Theorem to the distribution of the tails of independent,

identically distributed random variables drawn from an unknown distribution.

II.For empirical stock market data, the shape parameter in EVT is negative implying tails that disappear more rapidly than a normal distribution.

III.EVT can help avoid a shortcoming of the historical simulation method which may have difficulty calculating VaR reliably due to a lack of data in the tails.

IV.For empirical stock market data, standard value at risk estimates at the 95% confidence level are exceeded more often than 5% of the time and would therefore benefit from the

use of extreme value theory.

A.I and III

B.II and IV

C.I, III and IV

D.III and IV

53.Which of the following statement is FALSE regarding the implementation of MC simulation

for V AR?

I MC simulation involves the creation of a distribution of pricing paths created

via a random generation procedure

II After samples are made, a pricing model is used to determine financial values

III MC simulation uses actual market movements of past price data.

A.I only

B.II only

C.I and III

D. I, II and III

54. A risk manager has been requested to provide some indication of accuracy of a Monte Carlo

simulation. Using 1,000 replications of a normally distributed variable S, the relative error in the one-day 99% VaR is 5%. Under these conditions,

A. Using 1,000 replications of a long option position on S should create a larger relative

error.

B. Using 10,000 replications should create a larger relative error.

C. Using another set of 1,000 replications will create an exact measure of 5.0% for relative

error.

D. Using 1,000 replications of a short option position on S should create a larger relative

error.

55. In pricing a derivative using the Monte Carlo method, we need to stimulate a reasonable

number of paths for the price of the underlying asset. Suppose we use a simple model for the return of the underlying asset:

()()t y t drift vol e t =?Δ+，and e (t) is distributed ~ N (0,1)

Where drift and vol are known parameters and ?t is a step size. The generation of each path

requires a number of steps. Which of the following describes the correct procedure?

A. Generate a random number from a normal distribution N (0,1), use the cumulative

normal function to get e (t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.

B. Generate a random number from a normal distribution N (0,1), use the inverse normal

function to get e (t), which will be fed into the model to get y (t). Repeat the same procedure until you get the full desired path.

C. Generate a random number from a uniform distribution defined in [0,1], use the

cumulative normal function to get e (t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.

D. Generate a random number from a uniform distribution defined in [0,1], use the inverse

cumulative normal function to get e (t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.