1、1 1 金程教育FRM一级上海强化班讲义 数量分析 强化习题 讲师：吴轶 CFA ，FRM，CQF ， CISI 日期：2010 年4 月 地点：上海 北京 深圳2 专业来自百分百的投入 Copyright 2010 By GFEDU 吴轶 职称： 金程教育高级董事，金融研究院院 长，金程教育资深培训师，FRM 首席 培训师，英国证券分析师协会认证教师，CFA（注册金融分析师），FRM （金融 风险管理师），CQF （特许金融工程师），CISI （特许英国证券投资分析师）， 麻省理工学院（MIT）斯隆管理学院金融工程硕士，复旦大学金融工程硕士，复 旦大学财务管理学士，香港理工大学金融 数学学士
2、，中欧国际工商管理学院中国 金融研究中心研究员，国家外汇管理局储 备管理司风险管理处研究员。 授课： 目前为止，共讲授CFA Level I 40次，CFA Level II 20 次， CFA Level 10 次，中国工商银行总行CFA 授课2 次，中国银行总行CFA 授课2 次，FRM 交 通银行内部培训班3 次，公开班10 次，CFRM 交通银行内部培训班3 次，摩根士丹 利全英文CISI 授课2 次，GMAC 上汽金融内训授课1 次，华立集团内训授课4 次。 著作： 参与编写金融计量经济学教程；ISTP 收入英文论文（独立作者）两 篇；数量经济技术经济研究两篇论文 ；中国民营经济创新
3、国际论坛出版英文 会议论文一篇；第二届全国高等计 量经济学年会出版会议论文一篇。 工作： 国家外汇管理局，高盛集团（亚洲）有限 公司，北京高华证券有限公司。 客户： 国家外汇管理局，上海证券交易所，高盛 集团（亚洲）有限公司，北京高 华证券有限公司，花旗银行（中国）有限 公司，莱曼兄弟（香港），德意志银行 上海分行，中国工商银行总行，中 国银行总行，交通银行总行。 联系： 电话：86-13917952237 Email: 3 专业来自百分百的投入 Copyright 2010 By GFEDU Total Probability Formula A portfolio of bonds co
4、nsists 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 专业来自百分百的投入 Copyright 2010 By GFEDU （ 2007FRM 真题） The joint probab
5、ility distribution of random variables X and Y is given by f (x, y)=kxy for x=1,2,3, y=1,2,3, and k is a positive constant. What is the probability that X+Y will exceed 5? A. 1/9 B. 1/4 C. 1/36 D. Cannot be determined5 专业来自百分百的投入 Copyright 2010 By GFEDU Covariance Given that x and y are random varia
6、bles, and a, b, c and d are constant, which one of the following definitions is wrong? A. , if x and y are correlated B. , if x and y are correlated C. , if x and y are correlated D. , if x and y are uncorrelated ()( )( ) E ax by ca E xb E y c += + + ()() Var ax by c Var ax by c + += + (,)( )( ) ()(
7、 , ) Cov ax by cx dy acVar x bdVar y ad bc Cov x y += + + ()()( )( ) Var x y Var x y Var x Var y = += +6 专业来自百分百的投入 Copyright 2010 By GFEDU 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
8、(5X + 2Y)? A. 11.0 B. 29.0 C. 29.4 D. 33.07 专业来自百分百的投入 Copyright 2010 By GFEDU Correlation coefficient 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. 14.408 专业来自百分百的投入
9、Copyright 2010 By GFEDU 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
10、 can be calculated by scaling the covariance between two random variables9 专业来自百分百的投入 Copyright 2010 By GFEDU Kurtosis What is the kurtosis of a normal distribution? A. 0 B. Cannot be determined. It depends on the variance of the particular normal distribution considered C. 2 D. 3 0 Excess kurtosis
11、3 Kurtosis platykurtic mesokurtic leptokurtic10 专业来自百分百的投入 Copyright 2010 By GFEDU The standard normal distribution 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%11 专业来自百分百的投入 Copyri
12、ght 2010 By GFEDU The standard normal distribution （2006FRM 真题） 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 l
13、evel 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 24.4512 专业来自百分百的投入 Copyright 2010 By GFEDU （2006FRM 真题） Let be a uniformly distributed random variable between-1 and 1 so that the standard deviation of x is 0.577. What
14、 percentage of the distributions will be less than 1096 standard deviations above the mean? A. 100% B. 97.5% C. 95% D. Insufficient information provlded.13 专业来自百分百的投入 Copyright 2010 By GFEDU Statistics inference: hypothesis testing An analyst has constructed the following t-test for a portfolio of f
15、inancial securities whose returns are normally distributed: Number of securities = 40. :Mean return 18% Significance level = 0.1. What is the rejection point for this test? A. 1.304 B. 1.684 C. 2.021 D. 2.023 0 H 14 专业来自百分百的投入 Copyright 2010 By GFEDU hypothesis testing Suppose the standard deviation
16、 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
17、6.355 and 9.645 D. Between -8.45 and 24.4515 专业来自百分百的投入 Copyright 2010 By GFEDU hypothesis testing （ 2007FRM 真题） Which of the following statements regarding hypothesis testing is incorrect? A. Type II error refers to the failure to reject the null hypothesis when it is actually false. B. Hypothesis
18、testing is used to make inferences about the parameters of a given population on the basis of statistics computed foe a sample that is drawn from that population. C. All else being equal, the decrease in the chance of making a Type I error comes at the cost of increasing the probability of making a
19、Type II error. D. The p-value decision rule is to reject the null hypothesis if the p-value is greater than the significance level.16 专业来自百分百的投入 Copyright 2010 By GFEDU Type and Type Errors According to the Basel backtesting framework guidelines, penalties start to apply if there are five or more ex
20、ceptions 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 regulat
21、ors will reject the incorrect model C. 87% probability regulators will not reject the correct model D. 13% probability regulators will not reject the incorrect model17 专业来自百分百的投入 Copyright 2010 By GFEDU Type and Type Errors （2007FRM 真题） What does a hypothesis test at the 5% significance level mean?
22、A. P (not reject H0 | H0 is true)=0.05 B. P (not reject H0 | H0 is false)=0.05 C. P (reject H0 | H0 is true)=0.05 D. P (reject H0 | H0 is false)=0.0518 专业来自百分百的投入 Copyright 2010 By GFEDU Regression The number of sample observations in the regression estimation and total sum of squares (TSS), respect
23、ively, are closest to: Observations TSS A. 30 100 B. 30 1950 C. 31 1950 D. 31 100 25 925 error 5 1025 Regression DF Sum of Squares(SS)19 专业来自百分百的投入 Copyright 2010 By GFEDU Regression （ 2007FRM 真题） Consider two stocks, A and B. Assume their annual returns are jointly normally distributed, the margina
24、l distribution of each stock has mean 2% and standard deviation 10%, and the correlation is 0.9. What is the expected annual return of stock A if the annual return of stock B is 3%? A. 2% B. 2.9% C. 4.7% D. 1.1%20 专业来自百分百的投入 Copyright 2010 By GFEDU （ 2006FRM 真题） Identify the incorrect statement rega
25、rding the ordinary least squares(OLS) regression assumption: A. Due to heteroscedasticity, estimated coefficients and their standard errors become biased. B. Lagged variables often cause auto-correlation. C. Principal component analysis can be used to remove the effects the multicollinearity as prin
26、cipal components are uncorrelated. D. All of the above.21 专业来自百分百的投入 Copyright 2010 By GFEDU （ 2006FRM 真题） Paul Graham, FRM is analyzing the sales growth of a baby product launched three years ago by a regional company. He determines that three factors have contributed heavily toward the growth and
27、comes up with the following results: Sum of squared regression SSR=869.76 Sum of squared errors SSE 22.12 Determine what proportion of sales growth is explained by the regression results. A. 0.36 B. 0.98 C. 0.64 D. 0.55 123 1.5 1.2 3 Yb X X X =+22 专业来自百分百的投入 Copyright 2010 By GFEDU EWMA （ 2007FRM 真题
28、） An investment bank uses the Exponentially Weighted Moving Average (EWMA) technique with of 0.9 to model the daily volatility of a security. The current estimate of the daily volatility is 1.5%. The closing price of the security is USD 20 yesterday and USD 18 today. Using continuously compounded re
29、turns, what is the updated estimate of the volatility? A. 3.62% B. 1.31% C. 2.96% D. 5.44%23 专业来自百分百的投入 Copyright 2010 By GFEDU （ 2006FRM 真题 ） Using a daily RiskMetrics EWMA MODEL WITH A DECAY FACTOR =0.95 to develop a forecast of the conditional variance. which weight will be applied to the return
30、that is 4 days old? A. 0.000 B. 0.043 C. 0.048 D. 0.95024 专业来自百分百的投入 Copyright 2010 By GFEDU GARCH The GARCH model is useful for simulating asset returns. Which of the following statements about this model is FALSE? A. The Exponentially Weighted Moving Average (EWMA) approach of RiskMetrics is a par
31、ticular case of a GARCH process. B. The GARCH allows for time-varying volatility. C. The GARCH can produce fat tails in the return distribution. D. The GARCH imposes a positive conditional mean return.25 专业来自百分百的投入 Copyright 2010 By GFEDU GARCH A risk manager estimates daily variance( )using a GARCH
32、 model on daily returns( ): Assume the model parameter values are =0.005, =0.04, =0.94. The long-run annualized volatility is approximately A. 13.54% B. 7.94% C. 72.72% D. 25.00% 1 2 =0 1 1 tt t hrh + 0 1 t r t h t r26 专业来自百分百的投入 Copyright 2010 By GFEDU Monte Carlo simulation （2007FRM 真题） A risk man
33、ager 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
34、 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.27 专业来自百
35、分百的投入 Copyright 2010 By GFEDU Monte Carlo simulation （2007FRM 真题） 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: ， and e (t) is distri
36、buted 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)
37、, 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
38、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 unif
39、orm 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. () () tt ytd r i ftv o le t = + 28 专业来自百分百的投入 Copyright 2010 By GFEDU Monte Carlo simulation （ 2006FR
40、M真题 ） To implement this simulation, you generate a path of the stock price by starting at t =0, generating a ample for , updating the stock , price according to the model, incrementing t by 1 and repeating this process until the end of the horizon is reached. Which of the following strategies for ge
41、nerating a sample for will implement the simulation properly. A. Generate a sample for , by suing the inverse of the standard normal cumulative distribution between 0 and 1 B. Generate a sample for , by sampling from a normal distribution with the mean0.13and standard deviation 0.25 C. Generate a sa
42、mple for , by using the inverse of the standard normal cumulative distribution of a sample value drawn from a uniform distribution between 0 and 1. use cholesky decomposition to correlate this sample with the sample from the previous time interval D. Generate a sample for , by sampling from a normal
43、 distribution with mean 0.13and standard deviation 0.25. Use cholesky decomposition to correlate this sample with the sample from the previous tome interval. 29 专业来自百分百的投入 Copyright 2010 By GFEDU Cholesky factorization （2007FRM 真题） Let N be an n vector of independent draws from a standard normal dis
44、tribution, and let V be a covariance matrix of market time-series data. Then, if L is a diagonal matrix of the eigenvalues of V, E is a matrix of the eigenvectors of V, and CC is the Cholesky factorization of V, which of the following would generate a normally distributed random vector with mean zer
45、o and covariance matrix V to be used in a Monte Carlo simulation? A. NCCN B. NC C. ELE D. Cannot be determined from data given30 专业来自百分百的投入 Copyright 2010 By GFEDU EVT Which of the following statements regarding extreme value theory (EVT) is incorrect? A. In contrast to conventional approaches for e
46、stimating VaR, EVT only considers the tail behavior of the distribution. B. Conventional approaches for estimating VaR that assume that the distribution of returns follows a unique distribution for the entire range of values may fail to properly account for the fat tails of the distribution of retur
47、ns. C. EVT attempts to find the optimal point beyond which all values belong to the tail and then models the distribution of the tail separately. D. By smoothing the tail of the distribution, EVT effectively ignores extreme events and losses that can generally be labeled outliers.31 专业来自百分百的投入 Copyr
48、ight 2010 By GFEDU Pareto distribution （ 2007FRM真题） Company A uses a Pareto distribution to model the loss severity of its low-frequency, high-severity operational risk events. A Pareto distribution has the following properties, given parameter and k: Mean: for k 1 Variance: , for k 2 Cumulative dis
49、tribution function: After fitting the distribution to historical loss data, the parameters are estimated as k = 2.4, = 10,000. What is the unexpected loss of a low- frequency, high severity operational risk event at 99% confidence level? A. 40,703 B. 23,560 C. 50,986 D. 68,129 1 m kX k m X 2 2 (1 ) (2 ) m kX kk 1( ) m k X X m X
