FRM一级模拟题(2)
FRM一级模考
FRM一级模拟题1 . Which of the following statements regarding the lease rate in commodity futures contracts is incorrect?I The lease rate is the return required by the lender in exchange for lending a commodity.II Assuming it is positive, as the lease rate increases, the futures price for a commodity increases.III In a cash-and-carry arbitrage, the lease rate is earned whether or not the underlying commodity is actually loaned.IV Lease rates are similar to dividends paid lo the lender of a share of common stock.V If the lease rate is less than the risk-free rate, the forward market is said to be in contango.A . II and IIIB . III and VC . I, III, and VD . II and IVAnswer: AThe lease rate is the amount that a lender requires as compensation for Jending a commodity. In determining the price of a commodity futures contract, the lease rate, 81, is subtracted from the risk-free rate, r, as follows:Assuming a positive lease rate, the lease rate effectively reduces the futures price, all else constant.This also assumes that there is an active market for lending the commodity underlying the futures contract. The lease rate can only be earned by actually lending the underlying commodity.2 . .Consider the factors that affect the price of futures contracts on various commodities. Which of the following statements does not accurately describe the relationship between a commodity's futures price and its underlying factors?A. Gold futures have an implicit lease rate which, because it is not actually paid by commodity borrowers, creates incentive to' hold physical rather than synthetic gold as ideal strategy to gain gold exposure.B. Natural gas is produced relatively consistently but has seasonal demand, causing the futures price to rise steadily in the fall months, since natural gas is too expensive to store.C. The cost of storing corn, which has relatively constant demand, causes the futures price to rise until the next harvest at which point the price falls.D. Relatively constant worldwide demand for oil and its ability to be cheaply transported keep oil prices relatively stable in the absence of short-run supply and demand.Answer: AGold futures have an implicit lease rate, because it is not actually paid by commodity borrowers, which creates incentive to hold physical rather than synthetic gold as ideal strategy to gain gold exposure.Gold can be loaned out to financial intermediaries and other investors willing to pay the lease rate (the price for borrowing the gold) to the lender. Thus, holding physical gold requires the investor to forgo earning the lease rate while also incurring storage costs. Therefore, the ideal gold exposure strategy is generally to hold synthetic gold.3 . The S&P 500 index is trading at l,025. The S&P 500 pays an expected dividend yield of l .2% and the current risk-free rate is 2.75%. The value of a 3-month futures contract on the S&P 500 index is closest to:A. $1,028.98B. $1,108.59C. $984.86D. $1,025.00Answer: A4 . Which of the following statements describing the role of a convenience yield in pricing commodity futures is true? The convenience yield:I will cause contango in the futures pricing relationship.II Effectively reduces the cost of carry in the futures pricing relationship.III Eliminates the potential for arbitrage between the futures and spot price.IV Accounts for additional costs for storing an asset in the futures pricing Relationship.A. I onlyB. II onlyC. II, III, an d IV onlyD. I and II onlyAnswer: BThe convenience yield suggests there is a benefit, or convenience, to owning the spot asset. This generally means the spot price of the underlying asset will be above the futures price (normal backwardation). The convenience yield serves to reduce the cost of carry in the futures pricing relationship. .5 . Consider a 6-month futures contract on the S&P 500, and suppose the current value of the index is1330. Suppose the dividend yield is l.5% annually for the stocks underlying the index, and that the continuously compounded risk-free interest rate is 5.5% annually. What is the cost of carry for this futures contract?A. 4.0%B. -4.0%C. 2.0%D. -2.0%。
FRM一级模考
专注国际财经教育FRM一级模拟题1 . An Asset/Liability Management analyst at a community bank notices that a right to pay fixed swaption was purchased with a notional amount of 200 million with a strike in three months. The bank has only floating rate funding sources. The transaction most likely was done becausethe: .A. bank is anticipating a new $200M fixed rate loan.B. bank is anticipating a new $200M floating rate loan.C. bank is anticipating that a prepayment of $200M will occur on a flxed loan.D. trading area is speculating.Answer: AThe bank purchased the option in order to be able to convert variable rate funding to fixed rate if it needs to. The possibility of making a fixed rate loan would make this a rational strategy.2 . Consider a 2 int0 3-year Bermudan swaption (i.e., an option to obtain a swap that starts in 2 years and matures in 5 years). Consider the following statements:I A lower bound on the Bermudan price is a 2 int0 3 year European swaption.II An upper bound on the Bermudan price is a cap that starts in 2 years and matures in 5 years.III A lower bound on the Bermudan price is a 2 int0 5 year European optionWhich of the following statements is (are) TRUE?A. I onlyB. II onlyC. I and IID. III onlyAnswer: CSince a Bermudan option can be exercised on a discrete set of dates, it is at least as valuable as a European option, so I is correct. A cap would be exercisable continuously during the period, so it would represent the upper bound of the swap (and hence the option on the swap), so II is correct. III confuses the 2 int0 3 year Burmudan swaption and a 2- int0 5-year European option.3 . Which of the following actions would be most profitable when a trader expects a sharp rise in interest rates?A. Sell a payer swaption.B. Buy a payer swaption.C. Sell a receiver swaption.D. Buy a receiver swaption.Answer: BA payer swaption gives the holder the right to pay fixed rate and receive floating rate. Selling a receiver swaption would also be a profitable strategy as it would mean receiving a premium but with a sharp rise in interest rates buying a payer swaption should be a better deal.。
FRM一级模考
专注国际财经教育FRM一级模拟题1 . Bank regulators are examining the loan portfolio of a large, diversified lender. The regulators' main concern is that the bank remains solvent during turbulent economic times. Which of the following is most likely the area on which the regulators will want to focus?A. Expected loss, since each asset can expect, on average, to decline in value from a positive probability of default,B. Expected loss, given the decrease in underwriting standards of new loans.C. Unexpected loss, since the bank will need to set aside additional capital for the unlikely event that recovery rates are smaller than expected. .D. Unexpected loss, since the bank will need to set aside additional capital for the unlikely event that usage given default is smaller than expected.Answer: CUnexpected loss is a measure of the variation in expected loss: As a precaution, thebank needs to set aside sufficient capital in the event that actual losses exceedexpected losses with a reasonable likelihood. For example, smaller recovery rateswould be indicative of larger actual losses.2 . Which of the following is (are) characteristic of self-insurance as a means to hedge against catastrophic and operational losses? 'I Using captive insurers has tax benefits associated with self-insurance.II Exercising risk prevention and control is a form of self-insurance.III Establishing a contingent line of credit that becomes available in the event of a large operational loss is one method of self-insurance.A. I onlyB. II onlyC. II and III onlyD. I, II, and IIIAnswer: DCaptive insurers are off-shore, wholly owned subsidiaries that may deduct for tax purposes the discounted value of all future expected losses stemming front a claim spanning several years. Essential this allows self-insurers to deduct losses before they have even occurred. Incurring costs to manage and control operational risk achieves the same result in principle as self-insurance. A contingent line of credit is a form of self-insurance that provides liquidity in the event of a Joss rather than building up cash reserves in anticipation of a loss.。
FRM一级模考
FRM一级模拟题1. Given the following 1-year transition matrix, what is the probability that a Baa rated firm will default over a 2-year period?a. 5.00%.b. 9.75%.c. 14.50%.d. 20.00%.解析：bAt the end of year 1 there is a 5% chance of default and an 80% chance that the firm will maintain a Baa rating. In year 2, there is a 5% chance of default if the firm was rated Baa after 1 year (80% x 5% = 4%). There is a 0% chance of default if the firm was rated Aaa after 1 year (10% x 0% = 0%). Also, there is a 15% chance of default if the firm was rated Caa after 1 year (5% x 15% =0.75%). The probability of default is 5% from year 1 plus 4.75% chance of default from year 2(i.e., 4% + 0% + 0.75%) for a total probability of default over a 2-year period of 9.75%.2 . Isabelle Burns, FRM, is an investment advisor for a firm whose client base is composed of high net worth individuals. In her personal portfolio, Burns has an investment in Torex, a company that has developed software to speed up internet browsing. Burns has thoroughly researched Torex and believes the company is financially strong yet currently significantly undervalued. According to the GARP Code of Conduct, Burns may:a. not recommend Torex as long as she has a personal investment in the stock.b. not recommend Torex to a client unless her employer gives written consent to do so.c. recommend Torex to a client, but she must disclose her investment in Torex to the client.d. recommend Torex to a client without disclosure as long as it is a suitable investment for the client.解析：cStandards 2.1 and 2.2-Conflicts of Interest. Members and candidates must act fairly in all situations and must fully disclose any actual or potential conflict to all affected parties. Sell-side members and candidates should disclose to their clients any ownership in a security that they are recommending.3. Donaldson Capital Management, a regional money management firm, manages nearly $400 million allocated among three investment managers. All portfolios have the same objective, which is to produce superior risk-adjusted returns (by beating the market) for their clients. You have beenfollowing information based on the last ten years of returns.During the same time period the annual rate of return on the marker portfolio was 13% with a standard deviation of 19%. In order to assess the portfolio performance of the above managers, you should use:a. the Treynor measure of performance.b. the Sharpe measure of performance.c. the Jensen measure of performance.d. none of the above.解析：bThe Trcynor measure is most appropriate for comparing well-diversified portfolios. That is, the Treynor measure is the best to compare the excess returns per unit of systematic risk earned by portfolio managers, provided all portfolios are well diversified.All three portfolios managed by Donaldson Capital Management are clearly less diversified than the market portfolio. Standard deviation of returns for each of the three portfolios is higher than the standard deviation of the market portfolio, reflecting a low level of diversification.Jensen's alpha is the most appropriate measure for comparing portfolios that have the same beta. The Sharpe measure can be applied to all portfolios because it uses total risk and it is more widely used than the other two measures. Also, the Sharpe ratio evaluates the portfolio performance based on realized returns and diversification. A less diversified portfolio will have higher total risk and vice versa.4. Which of the following statements about the central limit theorem is least likely correct?a. The variance of the distribution of sample means is s2 / n.b. The central limit theorem has limited usefulness for skewed distributions.c. The mean of the population and the mean of all possible sample means are equal.d. When the sample size n is large, the sampling distribution of the sample means is approximately normal.解析：bThe central limit theorem holds for any distribution as long as the sample size is large (i.e., n > 30).5. Bank regulators are examining the loan portfolio of a large, diversified lender. The regulatorsfollowing is most likely the area that the regulator will want to focus on?a. Expected loss, since each asset can expect, on average, to decline in value from a positive probability of default.b. Expected loss, given the increase in underwriting standards of new loans.c. Unexpected loss, since the bank will need to set aside additional capital for the unlikely event that recovery rates are larger than expected.d. Unexpected loss, since the bank will need to set aside additional capital for the unlikely event that default losses are larger than expected.解析：dUnexpected loss is a measure of the variation in expected loss. As a precaution, the bank needs to set aside sufficient capital in the event that actual losses exceed expected losses with a reasonable likelihood.。
FRM一级模考题(二)
FRM一级模考题（二）1. Based on a sample size of 100 and sample mean of $30, you estimate a 95%confidence interval for the mean weekly soft drink expenditures of students at alocal college. Your estimate of the confidence interval is $26.77 to $33.23. Sinceyou knew the standard deviation beforehand, your confidence interval was basedon a standard deviation closest to:A. 1.65.B. 6.59.C. 11.53.D. 16.48.Solution : DWith a known variance, the 95% confidence interval is constructed asSo you know that Solving for a provides 16.48.2. Consider a 1-year European call option with a strike price of $27.50 that iscurrently valued at $4.10 on a $25 stock. The 1-year risk-free rate is 6%. Whichof the following is closest to the value of the corresponding put option?A. $0.00B. $3.12.C. $5.00.D. $6.60.Solution : CUsing put-call parity: p = c + Xe-rT - So = 4.10 + 27.50e-0.06 - 25 = $5.00.3. A binomial interest-rate tree indicates a 1-year spot rate of 4%, and the price of the bond if rates decline is 95.25 and 93.75 if rates increase. The risk-neutral probability of an interest rate increase is 0.55. You hold a call option on the bond that expires in one year and has an exercise price of93.00. The option value is closest to:A. 1.17B. 0.97C. 1.44D. 1.37Solution : DThe call has payoff of 95.25 -93 = 2.25 if rates decline and payoffof 93.75- 93= 0.75 if rates increase. The expected discounted value of the payoffs is [0.55(0.75)+ 0.45(2.25)]/1.04 = 1.37.4. Cooper Industries is the pay-fixed counterparty in an interest rate swap. Theswap is based on a notional value of $2,000,000 and pays a floating rate basedon the 6-month Hong Kong Interbank Offered Rate (HIBOR). Cooper pays afixed rate of 7% semiannually. A swap payment has just been made. The swaphas a remaining life of 18 months, with pay dates at 6, 12, and 18 months. SpotHIBOR rates are shown in the table below.The value of the swap to Cooper Industries is closest to:A. $0.B. $6,346.C. $17,093.D. $72,486.Solution : CThe fixed payments made by Cooper are (0.07 / 2) x $2,000,000 = $70,000. Thepresent value of the fixed payments == $67,762 + $65,398 + $1,849,747 = $1,982,907The value of the floating rate payments received by Cooper at the payment date is thevalue of the notional principal, or $2,000,000.The value of the swap to Cooper Industries is ($2,000,000 - $1,982,907) =$17,093.5. A stack-and-roll hedge as described in the Metallgesellschaft case is bestdescribed as:A. buying futures contracts of different expirations and allowing them to expirein sequence.B. buying futures contracts of different expirations and closing out the positionshortly before expiration.C. using short-term futures to hedge a long-term risk exposure by replacingthem with longer-term contracts shortly before they expire.D. using short-term futures contracts with a larger notional value than thelong-term risk they are meant to hedge.Solution : CA stack is a bundle of futures contracts with the same expiration. Over time, a firmmay acquire stacks with various expiry dates. To hedge a long-term risk exposure, a firmwould close out each stack as it approaches expiry and enter into a contract with a moredistant delivery, known as a roll. This strategy is called a stack-and-roll hedge and isdesigned to hedge long-term risk exposures with short-term contracts. Using short-termfutures contracts with a larger notional value than the long-term risk they are meant tohedge could result in "over hedging" depending on the hedge ratio.。
FRM一级模考
FRM一级模拟题1 . What is the lower pricing bound for a European call option with a strike price of 80 and one year until expiration? The price of the underlying asset is 90, and the. 1-year interest rate is 5% per annum. Assume continuous compounding of interest.A. 14.61B. 13.90C. 10.00D. 5.90Answer: BThe lower bound is the difference between the stock price and the present value of the strike price2 . The price of an American call stock option is equal to an otherwise equivalent European call stock option at time t when:I The stock pays continuous dividends from t to option expiration T. .. II The interest rates follow a mean-reverting process between t and T. .III The stock pays no dividends from t to option expiration T.IV Interest rates are non-stochastic between t and T.A. II and IVB. III onlyC. I and III .D. None of the above; an American option is always worth more than a European option. Answer: BThe only economic reason to exercise early is if the stock pays a dividend, and the full effect of the dividend payment is not expressed in a change in 'the stock price. If the stock does not pay a dividend, the price of an American option should be equal to the price of a European option. If the position pays a continuous dividend, the stock price will be adjusted for the value of the dividend3 . The current price for shares of ABC Co. is $100 and they pay no dividends. Interest rates are 6.00% per (annual 30/360), i.e. a completely flat yield curve. What is the approximate price of a perpetual call option with a strike price of 100 0n I share of ABC, if the call will be automatically exercised only when ABC reaches $150?A. $18B. $25C. $33D. $50Answer: CWith a flat yield curve and a rate of 6%, it will take 7 years for the FV of $100 to teach $150. The present value of $50 received in 7 years at 6 percent is $33.3. You can relate this without the use of the interest rate:4 . Which statement is false about the value of an option?A. At expiration, its premium equals intrinsic value.B. Before expiration, its premium is the sum of time and intrinsic value.C. It is determined by an option-pricing model.D. Intrinsic value is the difference between the market price and the strike price. Answer: CA case can be made for answer C based upon semantics; it can be argued that models don't determine values. The value of an option is the market price at which it trades.5 . A six-month call option sells for $30, with a strike price of $120. lf the stock price is $100 per share and the risk-free interest rate is 5%, what is the price of a 6-month put option with a strike price of $120?A. $39.20B. $44.53 .C. $46.28D. $47.04Answer: D。
FRM一级模考
FRM一级模拟题1 . A hedger calculates the covariance between the spot and the futures prices to be 0.05, the spot standard deviation to be 0.3, and the futures standard deviation to be 0.2. What is the optimal hedge ratio for this position?a. 0.075.b. 0.033.c. 0.250.d. 1.250.2. For an option-free bond, what are the effects of the convexity adjustment on the magnitude (absolute value) of the approximate bond price change in response to an increase in yield and in response to a decrease in yield, respectively?Decrease in yield Increase in yielda. Increase in magnitude Decrease in magnitudeb. Increase in magnitude Increase in magnitudec. Decrease in magnitude Decrease in magnituded. Decrease in magnitude Increase in magnitude3. Which of the following statements about the univariate, multivariate, and standard normal distributions is least likely correct?a. A univariate distribution describes a single random variable.b. A multivariate distribution specifies the probabilities for a group of related random variables.c. The standard normal random variable, denoted Z, has mean equal to 1 and variance equal to 1.d. The need to specify correlations is a distinguishing feature of the multivariate normal distribution in contrast to the univariate normal distribution.4. Which of the following statements regarding hypothesis testing are false?I. A Type I error is rejecting the null hypothesis when it is true.II. Reject the null hypothesis if p-value < significance level.III. The critical z-value for a one-tailed test of significance at the 0.05 level will be either +1.96 or -1.96.IV. The test statistic for hypotheses concerning equality of variances is computed asa. I and II.b. I and III.c. II and III.d. III and IV.5. Which of the following statements incorrectly describe(s) the Taylor Series approximation?I. The first and second derivatives of a function for the relationship between a financial derivative and its underlying asset estimate the delta and rate of change of delta, respectively.II. The Taylor Series provides good approximation estimates of price changes in callable bonds or mortgage-backed securities.III. The Taylor Series provides a good approximation for changes in a well-behaved quadratic function.a. I and II.b. II only.c. III only.d. II and III.。
FRM一级模考
FRM一级模拟题1 . Research and model projections indicate that a specific event is likely to move the CHF against the USD. While the direction of the move is highly uncertain, it is highly likely that magnitude of the move will be significant. Based on this information, which of the following strategies would provide the largest economic benefit?A. Long a call option on USD/CHF and long a put option on USD/CHF with the same strike price and expiration date.B. Long a call option on USD/CHF and short a put option on USD/CHF with the same strike price and expiration date.C. Short a call option on USD/CHF and short a put option on USD/CHF with the same strike price and expiration date.D. Short a call option on USD/CHF and long a put option on USD/CHF with the same strike price and expiration date.Answer: AWhile the direction of the move is highly uncertain, it is highly likely that magnitude of the move will be significant; we can see that we should take a trend trading. So, we buy a call option and buy a put-option, this strategy calls straggle.2 . Which of the following strategies creates a calendar spread?A. Buy a call option with a certain strike price and buy a longer maturity call option with the same strike price.B . Sell a call option with a certain strike price and buy a longer maturity call option with the same strike price.C . Buy a call option with a certain strike price and sell a longer maturity call option with the same strike price.D . Sell a call option with a certain strike price and sell a longer maturity call option with the same strike price.Answer: BDefinition of calendar spread3 . Your bank is an active player in the commodity market. The view of the economist of the bank is that inflation is expected to rise moderately in the near term and market volatility is expected to remain low. The traders are advised to undertake deals on the metals exchange to align your book to conform with the expectations of the economist of the bank. As risk manager, you are asked to monitor the positions of the traders to make sure that they have the exposures to inflation and market volatility sought by the bank. Which trader has taken an appropriate position among the traders you are monitoring?A. Trader A bought a call and a put, both with 90 days to expiration and with strike price equal to the existing spot level.B. Trader B bought a put option with a down-and-in knock in feature.C. Trader C bought a call option at the existing spot levels and sold a call at a higher strike price, both with 90 days to expiration.' D. Trader D sold a call option and bought a put at the existing levels, both with 90 days to expiration.Answer: CInflation is expected to rise moderately in the near term and market volatility is expected to remain low, stock price will increase moderately, so, choose bull spread.4 . Your bank is using the Black-Scholes model for valuation and pricing of exchange rate options with implied volatility of the at-the money options imputed from the market quotes. However, the research staff is now suggesting that exchange rate markets are exhibiting a volatility smile and theexisting valuations are incorrect. As a risk manager, you will be more concerned in which of the following situations?A. When the portfolio mainly consists of at-the-money long calls and puts.B. When the portfolio mainly consists of short put positions in deep out-of money options.C. When the portfolio mainly consists of short positions in at-the-money calls and puts.D. When the portfolio mainly consists of long call positions in deep out-of-money options. Answer: BWhen exchange rate markets exhibit a volatility smile, option at the money has smallest volatility. So A and C is incorrect. Generally speaking, long call option suffer limited loss when underlying asset drops, but short put suffer a huge loss. So, risk manager will be more concerned with short put position. D is correct.。
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FRM一级模拟题（2）1、Which one of the foll owing four trading strategies coul d limit the investor's upside potential while reducing her d ownsid e risk compared to a naked long position in the stock?A. A long position in a put combined with a long position in a stockB. A short position in a put combined with a short position in a stockC.Buying a call option on a stock with a certain strike price and selling a call option on the same stock with ahigher strike price and the same expiration dateD.Buying a call and a put with the same strike price and expiration date2、Which one of the foll owing four statements is correct about the early exercise of American options? (1). It is never optimal to exercise an American call option on a non-divid end-paying stock before the expiration date.(2). It can be optimal to exercise an American put option on a non-dividend-paying stock early.(3). It can be optimal to exercise an American call option on a non-dividend-paying stock early.(4). It is never optimal to exercise an American put option on a non-dividend-paying stock before the expiration date.A. 1 and 2B. 1 and 4C. 2 and 3D. 3 and 43、Mr. Black has been asked by a client to write a large option on the S&P 500 ind ex. The option has an exercise price and maturities that are not availabl e for options traded on exchanges. He therefore has to hedge the position dynamically. Which of the foll owing statements about the risk of his position is not correct?A.By selling short index futures short, he can make his portfolio delta neutral.B.There is a short position in an S&P 500 futures contract that will make his portfolio insensitive to bothsmall and large moves in the S&P 500.C. A long position in a traded option on the S&P 500 will help hedge the volatility risk of the option he haswritten.D.To make his hedged portfolio gamma neutral, he needs to take positions in options as well as futures.4、The current price of stock ABC is $42 and the call option with a strike at $44 is trading at $3. Expiration is inone year. The corresponding put is priced at $2. Which of the foll owing trading strategies will result in arbitrage profits? Assume that the annual risk-free rate is 10%, and that there is a risk-free bond paying the risk-free rate that can be shorted costl essly. There are no transaction costs.A.Long position in both the call option and the stock, and short position in the put option and risk-free bondB.Long position in both the call option and the put option, and short position in the stock and risk-free bondC.Long position in both the call option and risk-free bond, and short position in the stock and the put optionD.Long position in both the put option and the risk-free bond, and short position in the stock and the putoption5、The foll owing table gives the prices of two out of three U.S. Treasury notes for settlement on August 30, 2008. All three notes will mature exactly one year later on August 30, 2009.Coupon Price2$98.404?6$101.30Approximately, what woul d the price of the 41/2 U.S. Treasury note?A.$99.20B.$99.40C.$99.80D.$100.206、The observed zero yiel d curve is given by the foll owing data:1-year spot rate = 3.65%2-year spot rate = 3.99%3-year spot rate = 4.11%Using the data for the spot curve, the forward rate on a one-year contract maturing in two years is closest to:A. 3.20%B. 3.79%C. 4.33%D. 4.15%7、The spot price of gold is US200/oz and the price of a one-year gold futures contract is US205/oz. Assuming that the annual risk-free rate remains 5% and there are no arbitrage opportunities, which of the foll owing situations would cause backwardation1. Future value of the net cost for carrying physical gol d per oz increases.2. Future value of the net cost for carrying physical gol d per oz decreases.3. There is a net positive benefit from carrying physical gold.4. There is a net negative cost from carrying physical gold.A. 1 and 4 onlyB. 3 onlyC. 2 and 4D. 1 and 38、The foll owing table gives the cl osing prices and yiel ds of a particular liquid bond over the past few days. Day Price YieldMonday $106.3 4.25%Tuesday $105.8 4.20%Wednesday $106.1 4.23%What is the approximate duration of the bond?A.18.8B.9.4C. 4.7D. 1.99、John Flag, the manager of a USD 150 million distressed bond portfolio, conducts stress tests on the portfolio. The portfolio's annualized return is 12%, with an annualized return volatility of 25%. In the past two years, the portfolio encountered several days when the daily value change of the portfolio was more than 3 standard deviations. If the portfolio woul d suffer a 4-sigma daily event, estimate the change in the value of this portfolio.A.$9.48 millionB.$23.70 millionC.$37.50 millionD.$150 million10、A single stock has a price of $10 and a current daily volatility of 2%. Using the delta-normal approximation, the VaR on a l ong at-the-money call on this stock over a one-day holding period is:A.$0.1645B.$0.329C.$1.645D.$16.45Answer and Explanation:1. Long position in a put combined with long position in a stock could limit only the d ownside risk; (A) is incorrect.Short position in a put combined with short position in a stock coul d limit only the upside risk; (B) is incorrect. Buying a call option on a stock with a certain strike price and selling a call option on the same stock with a higher strike price and the same expiration date could limit both the upside and d ownside risk; (C) is correct. Buying a call and a put with the same strike price and expiration date could limit only the d ownsid e risk; (D) is incorrect.2. There are no advantages to exercising early if the investor plans to keep the stock for the remaining life of the call option, because the early exercise woul d sacrifice the interest that woul d be earned. If the strike price is paid out later on expiration date after the early exercise, the investor may suffer the risk that the stock price will fall bel ow the strike price. As the stock pays no dividend, the early exercise will earn no income from the stock. So it is never optimal to exercise an American call option on a non-divid end-paying stock before the expiration date.At any given time during its life, a put option shoul d always be exercised early if it is sufficiently deep in-the-money. So it can be optimal to exercise an American put option on a non-dividend-paying stock early. As a result, answer (A) is correct.3. The short ind ex futures contract makes the portfolio delta neutral. It does not help with large moves.4. (A) is incorrect as this would not yield arbitrage profit.(B) is incorrect as this woul d not yield arbitrage profit.(C) is correct.The put-call parity relation is: stock + put = pv(strike) + callTherefore, for no arbitrage opportunity the foll owing relation should hold:42 + 2 = (44/1.10) + 3But 44 > 43Therefore, there is an arbitrage opportunity. The arbitrage profit is 49 – 42 = 7 by taking a long position in a call and buying the risk-free bond and going short on the stock and the put.(D) is incorrect as this woul d not yield arbitrage profit.5. 2.875% * X +6.25% * (1 – X) = 4.5%X = 52%The portfolio that has cash fl ows identical to the 41/2 bond consists of 52% of the 27/8 and 48% of the 61/4 bonds. As this portfolio has cash fl ows identical to the 41/2 bond, precluding arbitrage, the price of the portfolio should equal to 52% * 97.4 + 48% * 101.30, or $99.806. (A) is incorrect because the rate cannot be bel ow the spot rates as the zero curve has an upward sl ope.(B) is incorrect because the zero curve is upward-sloping and the rate must be higher than 3.99%.(C) is the correct answer:Rf = [(1 + r2) ^ t2/(1 + r1) ^ t1] – 1Rf = (r2t2 – r1t1)/(t2 – t1)[((1 + 3.99%) ^ 2)/(1 + 3.65%)] – 1 = 4.33%(D) is impossibl e because if we invest for a year at 3.65% and the next year at 4.15%, it is not the same as investing during two years at 3.99%.7. The expected spot price 1 year later is US210/oz. So, if no arbitrage opportunity exists, the future value of the net cost for carrying the physical gol d is US5/oz.If the future value of the net cost for carrying the gold per oz increases and exceeds US10/oz, the spot price one year later will be less than the current spot price. Thus backwardation occurs.Also, if the net benefit for carrying the gol d exists and the future value of such benefit exceeds US5/oz (with holding future value of net cost being constant), backwardation occurs too.Also, if the net negative from carrying the gol d exists and the future value of such benefit exceeds US5/oz (with holding future value of net cost being constant), backwardation d oes not occur.Therefore, both statement 1 and statement 3 are correct.(A) is incorrect because statement 3 is correct too.(B) is incorrect because statement 1 is correct too.(C) is incorrect because statement 2 is incorrect.(D) is correct because both statement 1 and statement 3 are correct.Remark: The tricky part of the question is that the candidate may feel confused since the no-arbitrage future price should be US210/oz (if net cost of storage d oes not exist), rather than the price US205/oz given in the question (where the net cost of storage exists).8. The duration can be approximated from the price changes.(106.3 – 105.8)/106.3/.0005 = 9.4(106.3 – 106.1)/106.3/.0002 = 9.4etc.9. Daily volatility is equal to 0.25 * sqrt (1/250) = 0.0158. A 4-sigma event therefore implies a l oss equal to 4*0.0158*150 = 9,486,832The correct answer is (A).(B) calculates the daily volatility and multiplies the volatility by the value of the portfolio.(C) multiplies the portfolio value by its annual volatility, or divides the portfolio value by 4.(D) attempts the shortcut of reducing the portfolio value by 4 times 25%, which is 100% (i.e., the value of the portfolio).10. This question requires candidates to know the formula for the delta-normal VaR approximation, and also to know that the delta of an at-the-money call is 0.5.The correct answer is (A).(B) uses a delta of 1.(C) confuses the decimal point.(D) uses 2 instead of 2% for the volatility.参与FRM的考生可按照复习计划有效进行，另外高顿网校官网考试辅导高清课程已经开通，还可索取FRM 考试通关宝典，针对性地讲解、训练、答疑、模考，对学习过程进行全程跟踪、分析、指导，可以帮助考生全面提升备考效果。