金融风险管理师(FRM)2012考试真题.doc

Question bankMonte Carlo MethodsLet N be an n x 1 vector of independent draws from a standard normal distribution, 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 zero and covariance matrix V to be used in a Monte Carlo simulation?NCCNNCE LECannot be determined from data givenConsider a stock that pays no dividends, has a volatility of 25% pa and an expected return of 13% pa. The current stock price is S0 = $30. This implies the model St+1 = St(1 + 0.13 At + 0.25yAt e), where e is a standard normal random variable. To implement this simulation, you generate a path of the stock price by starting at t = 0, generating a sample for e, 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 generating a sample for e will implement this simulation properly?Generate a sample for e by using the inverse of the standard normal cumulative distribution of a sample value drawn from a uniform distribution between 0 and 1.Generate a sample for e by sampling from a normal distribution with mean 0.13 and standard deviation 0.25.Generate a sample for e 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.Generate a sample for e by sampling from a normal distribution with mean 0.13 and standard deviation 0.25. Use Cholesky decomposition to correlate this sample with the sample from the previous time interval.Continuing with the previous question, you have implemented the simulation process discussed above using a time interval At = 0.001, and you are analyzing the following stock price path generated by your implementation.tSt1eA S030.000.09300.03130.030.84930.21230.230.96170.23330.470.24600.06430.530.47690.12530.650.71410.18Given this sample, which of the following simulation steps most likely contains an error.Calculation to update the stock priceGeneration of random sample value for eCalculation of the change in stock price during each periodNone of the aboveIn the geometric Brownian motion process for a variable S,I. S is normally distributed.II. dln(S) is normally distributed.III. dS/S is normally distributed.IV. S is lognormally distributed.a. I onlyb. II, III, and IVc. IV onlyd. III and IVConsider that a stock price S that follows a geometric Brownian motiondS = aSdt + bSdz, with b strictly positive. Which of the following statements is false?a. If the drift a is positive, the price one year from now will be above todays price.b. The instantaneous rate of return on the stock follows a normal distribution.c. The stock price S follows a lognormal distribution.d. This model does not impose mean reversion.The Vasicek model defines a risk-neutral process for r which is dr =a(b r )dt + dz, where a, b, and are constant, and r represents the rate of interest. From this equation we can conclude that the model is aa. Monte Carlo-type modelb. Single-factor term-structure modelc. Two-factor term-structure modeld. Decision tree modelThe term a(b r ) in the previous question represents which term?a. Gammab. Stochasticc. Reversiond. VegaWhich group of term-structure models do the Ho-Lee, Hull-White, and Heath, Jarrow, and Morton models belong to?a. No-arbitrage modelsb. Two-factor modelsc. Lognormal modelsd. Deterministic modelsA plausible stochastic process for the short-term rate is often considered to be one where the rate is pulled back to some long-run average level. Which one of the following term-structure models does not include this characteristic?a. The Vasicek modelb. The Ho-Lee modelc. The Hull-White modeld. The Cox-Ingersoll-Ross modelWhich of the following statements about Monte Carlo simulation is false?a. Monte Carlo simulation can be used with a lognormal distribution.b. Monte Carlo simulation can generate distributions for portfolios that contain only linear positions.c. One drawback of Monte Carlo simulation is that it is computationally very intensive.d. Assuming the underlying process is normal, the standard error resulting from Monte Carlo simulation is inversely related to the square root of the number of trials.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.The measurement error in VAR, due to sampling variation, should be greater witha. More observations and a high confidence level (e.g., 99%)b. Fewer observations and a high confidence levelc. More observations and a low confidence level (e.g., 95%)d. Fewer observations and a low confidence levellet the pdf f(x) be positive at x=-1,0,1 and zero else where.1. if f(0) = 0.25 , find E(x2)2. if f(0) = 0.25 and if E(X) =0.25 , determine f(-1) and f(1)One card is drawn from a standard 52 card deck. In describing the occurrence of two possible events, an Ace and a King, these two events are said to be: (a) independent (b) mutually exclusive (c) random variables (d) randomly independent.Which of the following is NOT a possible probability? a. 25/100 b. 1.25 c. 1 d. 0Among twenty-five articles, nine are defective, six having only minor defects and three having major defects. Determine the probability that an article selected at random has major defects given that it has defects. a. 1/3 b. .25 c. .24 d. .08Among twenty-five articles eight are defective, six having only minor defects and two having major defects. Determine the pro- bability that an article selected at random has major defects given that it has defects. (a) .08 (c) 1/3 (b) .25 (d) .24A dormitory on campus houses 200 students. 120 are male, 50 are upper division students, and 40 are upper division male students. A student is selected at random. The probability of selecting a lower division student, given the student is a female, is: (a) 7/8 (d) 7/20 (b) 7/15 (e) 1/4 (c) 2/5Which of the following statements regarding linear regression is false?A. Heteroskedasticity occurs when the variance of residuals is not the same across all observations in the sample.B. Unconditional heteroskedasticity leads to inefficient estimates, whereas conditional heteroskedasticity can lead to problems with both inference and estimation.C. Serial correlation occurs when the residual terms are correlated with each other.D. Multicollinearity occurs when a high correlation exists between or among two or more of the independent variables in a multiple regression.Consider two stocks, A and B. Assume their annual returns are jointly normallydistributed, the marginal distribution of each stock has mean 2% andstandard deviation 10%, and the correlation is 0.9. What is the expectedannual return of stock A if the annual return of stock B is 3%?A. 2%B. 2.9%C. 4.7%D. 1.1%Consider the following linear regression model: Y = a +b X+ e. Supposea = 0.05, b = 1.2, SD(Y) = 0.26, SD(e) = 0.1, what is the correlation betweenX and Y?A. 0.923B. 0.852C. 0.701D. 0.462Under what circumstances could the explanatory power of regression analysis be overstated?The explanatory variables are not correlated with one another.The variance of the error term decreases as the value of the dependent variable increases.The error term is normally distributed. An important explanatory variable is omitted that influences the explanatory variables included, and the dependent variable.What does a hypothesis test at the 5% significance level mean?P(not reject H0 | H0 is true) = 0.05P(not reject H0 | H0 is false) = 0.05P(reject H0 | H0 is true) = 0.05P(reject H0 | H0 is false) = 0.05Assume we calculate a one-week VAR for a natural gas position by rescal- ing the daily VAR using the square-root rule. Let us now assume that we determine the true gas price process to be mean-reverting and recalculate the VAR.Which of the following statements is true?The recalculated VAR will be less than the original VAR.The recalculated VAR will be equal to the original VAR.The recalculated VAR will be greater than the original VAR.There is no necessary relation between the recalculated VAR and the original VAR.On a multiple-choice exam with four choices for each of six questions, what is the probability that a student gets fewer than two questions correct simply by guessing?0.46%23.73%35.60%53.39%1.The current yield on a bond is equal to _. A)annual interest divided by the current market price B)the yield to maturity C)annual interest divided by the par value D)the internal rate of return Answer: A Difficulty: Easy Rationale: A is current yield and is quoted as such in the financial press.2.If a 7% coupon bond is trading for $975.00, it has a current yield of _ percent. A)7.00 B)7.24 C)8.53 D)7.18 Answer: D Difficulty: Easy Rationale: 70/975 = 7.18.3.If a 6% coupon bond is trading for $950.00, it has a current yield of _ percent. A)6.5 B)6.3 C)6.1 D)6.0 Answer: B Difficulty: Easy Rationale: 60/950 = 6.3.4.If an 8% coupon bond is trading for $1025.00, it has a current yield of _ percent. A)7.8 B)8.7 C)7.6 D)7.9 Answer: A Difficulty: Easy Rationale: 80/1025 = 7.8.5.If a 7.5% coupon bond is trading for $1050.00, it has a current yield of _ percent. A)7.0 B)7.4 C)7.1 D)6.9 Answer: C Difficulty: Easy Rationale: 75/1050 = 7.1.6.A coupon bond pays annual interest, has a par value of $1,000, matures in 4 years, has a coupon rate of 10%, and has a yield to maturity of 12%. The current yield on this bond is _. A)10.65% B)10.45% C)10.95% D)10.52% Answer: A Difficulty: Moderate Rationale: FV = 1000, n = 4, PMT = 100, i = 12, PV= 939.25; $100 / $939.25 = 10.65%.7.A coupon bond pays annual interest, has a par value of $1,000, matures in 12 years, has a coupon rate of 11%, and has a yield to maturity of 12%. The current yield on this bond is _. A)10.39% B)10.43% C)10.58% D)10.66% Answer: D Difficulty: Moderate Rationale: FV = 1000, n = 12, PMT = 110, i = 12, PV= 938.06; $100 / $938.06 = 10.66%.8.Of the following four investments, _ is considered the safest. A)corporate bonds B)U. S. Agency issues C)Treasury bonds D)Treasury bills Answer: D Difficulty: Easy Rationale: Only Treasury issues are insured by the U. S. government; the shorter-term the instrument, the safer the instrument.9.To earn a high rating from the bond rating agencies, a firm should have A)a low times interest earned ratio B)a low debt to equity ratio C)a high quick ratio D)B and C Answer: D Difficulty: Easy Rationale: High values for the times interest and quick ratios and a low debt to equity ratio are desirable indicators of safety.10.At issue, coupon bonds typically sell _. A)above par value B)below par C)at or near par value D)at a value unrelated to par Answer: C Difficulty: Easy Rationale: If the investment banker has appraised the market and the quality of the bond correctly, the bond will sell at or near par (unless interest rates have changed very dramatically and very quickly around the time of issuance).11.Accrued interest A)is quoted in the bond price in the financial press. B)must be paid by the buyer of the bond and remitted to the seller of the bond. C)must be paid to the broker for the inconvenience of selling bonds between maturity dates. D)A and B. Answer: B Difficulty: Moderate Rationale: Accrued interest must be paid by the buyer, but is not included in the quotations page price.12.The invoice price of a bond that a buyer would pay is equal to A)the asked price plus accrued interest. B)the asked price less accrued interest. C)the bid price plus accrued interest. D)the bid price less accrued interest. Answer: A Difficulty: Easy Rationale: The buyer of a bond will buy at the asked price and will also be invoiced for any accrued interest due to the seller.1.Which of the following statements regarding risk-averse investors is true? A)They only care about the rate of return. B)They accept investments that are fair games. C)They only accept risky investments that offer risk premiums over the risk-free rate. D)They are willing to accept lower returns and high risk. Answer: C Difficulty: Moderate 2.Which of the following statements is (are) true?Risk-averse investors reject investments that are fair games.Risk-neutral investors judge risky investments only by the expected returns.Risk-averse investors judge investments only by their riskiness.Risk-loving investors will not engage in fair games. A)I only B)II only C)I and II only D)II and III only Answer: C Difficulty: Moderate Rationale: Risk-averse investors consider a risky investment only if the investment offers a risk premium. Risk-neutral investors look only at expected returns when making an investment decision.3.In the mean-standard deviation graph an indifference curve has a _ slope. A)negative B)zero C)positive D)northeast Answer: C Difficulty: Easy Rationale: The risk-return trade-off is one in which greater risk is taken if greater returns can be expected, resulting in a positive slope.4.In the mean-standard deviation graph, which one of the following statements is true regarding the indifference curve of a risk-averse investor? A)It is the locus of portfolios that have the same expected rates of return and different standard deviations. B)It is the locus of portfolios that have the same standard deviations and different rates of return. C)It is the locus of portfolios that offer the same utility according to returns and standard deviations. D)It connects portfolios that offer increasing utilities according to returns and standard deviations. Answer: C Difficulty: Moderate Rationale: Indifference curves plot trade-off alternatives that provide equal utility to the individual (in this case, the trade-offs are the risk-return characteristics of the portfolios).5.In a return-standard deviation space, which of the following statements is (are) true for risk-averse investors? (The vertical and horizontal lines are referred to as the expected return-axis and the standard deviation-axis, respectively.)An investors own indifference curves might intersect.Indifference curves have negative slopes.In a set of indifference curves, the highest offers the greatest utility.Indifference curves of two investors might intersect. A)I and II only B)II and III only C)I and IV only D)III and IV only E)none of the above Answer: D Difficulty: Moderate Rationale: An investors indifference curves are parallel, and thus cannot intersect and have positive slopes. The highest indifference curve (the one in the most northwestern position) offers the greatest utility. Indifference curves of investors with similar risk-return trade-offs might intersect.6.Elias is a risk-averse investor. David is a less risk-averse investor than Elias. Therefore, A)for the same risk, David requires a higher rate of return than Elias. B)for the same return, Elias tolerates higher risk than David. C)for the same risk, Elias requires a lower rate of return than David. D)for the same return, David tolerates higher risk than Elias. Answer: D Difficulty: Moderate Rationale: The more risk averse the investor, the less risk that is tolerated, given a rate of return.7.When an investment advisor attempts to determine an investors risk tolerance, which factor would they be least likely to assess? A)the investors prior investing experience B)the investors degree of financial security C)the investors tendency to make risky or conservative choices D)the level of return the investor prefers E)the investors feeling about loss Answer: D Difficulty: Moderate Use the following to answer questions 8-9:Assume an investor with the following utility function: U = E(r) - 3/2(s2). 8.To maximize her expected utility, she would choose the asset with an expected rate of return of _ and a standard deviation of _, respectively. A)12%; 20% B)10%; 15% C)10%; 10% D)8