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二、Investment Tools: Quantitative Methods 1.A.: Time Value of Money Question ID: 20976 A certain investment product promises to pay $25,458 at the end of 9 years. If an investor feels this investment should produce a rate of return of 14 percent, compounded annually, what’s the most he should be willing to pay for it? A. B. C. D. $7,618. $8,342. $9,426. $7,828. D =9; I/Y=14; FV=-25,458; PMT=0; CPT PV=$7,828. Question ID: 20973 An investor deposits $4,000 in an account that pays 7.5 percent, compounded annually. How much will this investment be worth after 12 years? A. B. C. D. $9,527. $7,600. $9,358. $5,850. A N=12; I/Y=7.5; PV=-4,000; PMT=0; CPT FV=$9,527. Question ID: 18489 1 If a person needs $20,000 in 5 years from now and interest rates are currently 6% how much do they need to invest today? A. B. C. D. $15,301 $14,683 $14,945 $14,284 C PV=FV/(1+r)n =20000/(1.06)5 = 20000/1.33823 = 14,945 n = 5, i/yr = 6%, PMT = 0, FV = $20,000, Compute PV = -$14,945.16 Question ID: 20978 How long will it take $20,000 to grow to $50,000, given an investor can earn 12 percent (compounded annually) on his money? A. B. C. D. 5.8 years. 8.1 years. 7.5 years. 6.0 years. B I/Y=12; PV=-20,000; FV=50,000; PMT=0; CPT N=8.08 Question ID: 19376 A leading brokerage firm has advertised money multiplier certificates that will triple your 2 money in 9 years (i.e., invest $10,000 today, and it'll be worth $30,000 in 9 years). Approximately what rate of return will investors be earning on these certificates? A. B. C. D. 16.0%. 8.8%. 13.0%. 12.5%. C Using your calculator: N=9; PV=-10,000; FV=30,000; PMT=0; CPT I/Y=12.98. Question ID: 19378 What is the present value of a 12-year annuity due that pays $5,000 per year, given a discount rate of 7.5 percent? A. B. C. D. $36,577. $41,577. $38,676. $23,857. B Using your calculator: N=11; I/Y=7.5; PMT=-5,000; FV=0; CPT PV=36,577+5,000=$41,577. Or set your calculator to BGN mode and N=12; I/Y=7.5; PMT=-5,000; FV=0; CPT PV=$41,577. Question ID: 18492 An annuity will pay eight annual payments of $100, with the first payment to be received one year from now. If the interest rate is 12% per year, what is the present value of this annuity? 3 A. B. C. D. $556.38 $1,377.57 $496.76 $1,229.97 C n = 8, i/yr = 12%, PMT = -$100, FV = 0, Compute PV = $496.76 Question ID: 19919 Consider a 10-year annuity that promises to pay out $10,000 per year; given this is an ordinary annuity and that an investor can earn 10 percent on her money, the future value of this annuity, at the end of 10 years, would be: A. B. C. D. $175,312. $152,500. $110.000. $159,374. D Answer: N=10; I/Y=10; PMT=-10,000; PV=0; CPT FV=$159,374. Question ID: 20981 A 5-year, $50,000 loan requires annual end-of-the-year payments of $13,528. What rate of interest is the borrower paying on this loan (assume annual compounding)? A. 11.0%. 4 B. C. D. 9.5%. 10.5%. 9.0%. A N=5; PV=-50,000; FV=0; PMT=13,528; CPT I/Y=11.0% Question ID: 20982 The HGM Co. is thinking about taking out a $100,000 loan that is to be paid off in annual (end-of-the-year) payments of $12,042. If the bank agrees to lend the money at 8.5 percent, compounded annually, how many years will it take HGM to pay off the loan? A. B. C. D. 15.0 years. 10.9 years. 8.3 years. 12.5 years. A I/Y=8.5; PV=-100,000; PMT=12,042; FV=0; CPT N=15.0. Question ID: 20985 Given investors require an annual return of 12.5 percent, a perpetual bond (i.e., a bond with no maturity/due date) that pays $87.50 a year in interest should be valued at: A. B. C. $1,093. $875. $700. 5 D. $70. C 87.50/.125=$700. Question ID: 19381 Nortal Industries has a preferred stock outstanding that pays (fixed) annual dividends of $3.75 a share. If an investor wants to earn a rate of return of 8.5 percent, how much should he be willing to pay for a share of Nortel preferred stock? A. B. C. D. $37.50. $31.88. $44.12. $42.10. C PV=3.75/.085 = $44.12. Question ID: 20986 The preferred shares of AWA Co. are currently trading in the market at $75.00 a share. What rate of return are investors earning on these preferred stock, given they pay (fixed) annual dividends of $6.00 a share? A. B. C. D. 8.0%. 12.5%. 7.5%. 6.0%. 6 A 6.00/75.00=0.08. Question ID: 20989 Given the following cash flow stream: End of Year 1 2 3 4 Annual Cash Flow $4,000 $2,000 -0-$1,000 Using a 10 percent discount rate, the present value of this cash flow stream is: A. B. C. D. $3,636. $2,225. $3,415. $4,606. D PV(1): N=1; I/Y=10; FV=-4,000; PMT=0; CPT PV=3,636 PV(2): N=2; I/Y=10; FV=-2,000, PMT=0; CPT PV=1,653 PV(3): 0 PV(4): N=4; I/Y=10; FV=1,000; PMT=0; CPT PV=-683 Total PV=3,636+1,653+0-683=4,606 Question ID: 18503 Suppose you are going to deposit $1,000 this year, $1,500 next year, and $2,000 the following year in an savings account. How much money will you have in three years if the 7 rate of interest is 10% each year? A. B. C. D. $4,000.00 $5,346.00 $5,750.00 $6,800.00 B Future value of? $1000 for 3 periods at 10% is 1331 Future value of $1500 for 2 periods at 10% is? 1815 Future value of $2000 for 1 period at 10% is??? 2200 Total = $5346 n = 3, PV = -$1000, i/yr = 10%, Compute FV = $1331 n = 2, PV = -$1500, i/yr = 10%, Compute FV = $1815 n = 1, PV = -$2000, i/yr = 10%, Compute FV = $2200 Question ID: 19925 The BBC Co. is evaluating an investment that promises to generate the following annual cash flows: End of Year Cash Flows 1 2 3 4 5 6 7 8 9 $5,000 $5,000 $5,000 $5,000 $5,000 -0-0$2,000 $2,000 8 Given BBC uses an 8 percent discount rate, this investment should be valued at: A. B. C. D. $19,963. $22,043. $27,027. $23,529. B Answer: PV(1-5): N=5; I/Y=8; PMT=-5,000; FV=0; CPT?¨PV=19,963 PV(6-7): 0 PV(8): N=8; I/Y=8; FV=-2,000; PMT=0; CPT?¨PV=1,080 PV(9): N=9; I/Y=8; FV=-2,000; PMT=0; CPT?¨PV=1,000 Total PV=19,963+0+1,080+1,000=22,043. Question ID: 19383 Given the following cash flow stream: End of Year 1 2 3 4 Annual Cash Flow $4,000 $2,000 0 -$1,000 Using a 10 percent discount the present value of this cash flow stream is: A. B. C. D. $2,225. $4,606. $3,415. $3,636. 9 B PV(1): N=1; I/Y=10; FV=-4,000; PMT=0; CPT PV=3,636 PV(2): N=2; I/Y=10; FV=-2,000, PMT=0; CPT PV=1,653 PV(3): 0 PV(4): N=4; I/Y=10; FV=1,000; PMT=0; CPT PV=-683 Total PV = 3,636 + 1,653 + 0 - 683 = 4,606 Question ID: 19385 What is the maximum price an investor should be willing to pay (today) for a 10 year annuity that will generate $500 per quarter (such payments to be made at the end of each quarter), given he wants to earn 12 percent, compounded quarterly? A. B. C. D. $6,440. $11,557. $5,862. $11,300. B Using a financial calculator: N= 10x4 = 40; I/Y = 12/4 = 3; PMT= -500; FV=0; CPT PV = 11,557. Question ID: 18508 Given: an 11% annual rate paid quarterly; PV = 8000; time is 2 years; compute FV. A. B. C. D. $8,446 $9,939 $8,962 $9,857 10 B Divide the interest rate by the number of compound periods and multiply the number of years by the number of compound periods. i=11/4=2.75; n=(2)(4)=8; PV =8000 Question ID: 20995 A major brokerage house is currently selling an investment product that offers an 8 percent rate of return, compounded monthly. Based on this information, it follows that this investment has: A. B. C. D. a nominal effective rate of 6.67% an effective annual rate of 8.00% a periodic interest rate of 0.667%. a stated rate of 0.830%. C Periodic rate = 8.0/12=0.667. Stated rate is 8.0% and effective rate is 8.30%. Question ID: 20996 Which of the following statements about stated and effective annual interest rates is/are TRUE? I. The stated rate adjusts for frequency of compounding. II. The periodic interest rate is used to find the effective annual rate. III. So long as interest is compounded more than once a year, the effective rate will always be more than the stated rate. A. B. III only. II and III only. 11 C. D. I only. I and II only. B Statements II and III are true; statement I is wrong since it’s the effective annual rate that adjusts for frequency of compounding. Question ID: 19387 A local bank advertises that it will pay interest at the rate of 4.5 percent, compounded monthly, on regular savings accounts. What is the effective rate of interest that the bank is paying on these accounts? A. B. C. D. 4.65%. 4.88%. 4.59%. 4.50%. C (1+.045/12) 12 - 1 = 1.0459 - 1 = 0.0459. Question ID: 19388 Lois Weaver wants to have $1.5 million in a retirement fund when she retires in 30 years. If Weaver can earn a 9 percent rate of return on her investments, approximately how much money must she invest at the end of each of the next 30 years in order to reach her goal? A. B. C. $11,005. $50,000. $7,350. 12 D. $28,725. A Using a financial calculator: N=30; I/Y=9; FV=-1,500,000; PV=0; CPT PMT = 11,004.52. Question ID: 20999 The First State Bank is willing to lend $100,000 for 4 years at a 12 percent rate of interest, with the loan to be amortized in equal semi-annual payments. Given the payments are to be made at the end of each 6-month period, how big will each loan payment be? A. B. C. D. $18,500. $16,104. $32,925. $25,450. B N=4x2=8; I/Y=12/2=6; PV=-100,000; FV=0; CPT PMT=16,103.59. Question ID: 18509 Given: $1,000 investment, compounded monthly at 12% find the future value after one year. A. B. C. D. $1,126.83 $1,120.00 $1,121.35 $1,1233.57 A 13 Divide the interest rate by the number of compound periods and multiply the number of years by the number of compound periods. i=12/12=1; n=(1)(12)=12; PV=1000 Question ID: 19924 Given a preferred stock is trading in the market at $100.00 a share, if the rate of return on this investment is 15 percent, it follows that the stock must be paying (fixed) annual dividends of: A. B. C. D. $5.00 a share. $12.50 a share. $15.00 a share. $7.50 a share. C Answer: $100x.15=$15.00. Question ID: 18488 How much will $10,000 grow in 5 years if the annual interest rate is 8%, compounded monthly? A. B. C. D. $14,693.28 $14,000.00 $14,898.46 $14,802.44 C FV(t=5) = $10,000 x (1+.08/12)60 = $14,898.46 n = 60 (12*5), PV = -$10,000, i/yr = 8%, Compute FV = $14,898.46 14 Question ID: 18493 An annuity will pay eight annual payments of $100, with the first payment to be received three years from now. If the interest rate is 12% per year, what is the present value of this annuity? A. B. C. D. $496.76 $400.00 $800.53 $396.01 D First solve for present value in 3 years: n = 8, PMT = 100, i/yr = 12%, FV = 0, Compute PV = $496.76 Now discount back two years: n = 2, PMT = 0, i/yr = 12%, FV = $496.76, Compute PV = $396.01 Question ID: 20006 What’s the effective rate of return on an investment that generates a return of 12 percent, compounded quarterly? A. B. C. D. 13.33% 12.55% 12.00% 14.34% B Answer: (1+.12/4)4-1=1.1255-1=.1255. 15 1.B: Statistical Concepts and Market Returns Question ID: 19684 Which of the following statements regarding the terms population and sample is FALSE? A. B. C. A sample includes all members of a specified group. A descriptive measure of a sample is called a statistic. A sample seeks to describe a population as a whole. Observing all members of a population can be expensive or time consuming. D. A A population includes all members of a specified group. A sample is a portion, or subset of the population of interest. Question ID: 19685 Which of the following is an example of a parameter? A. B. C. D. Sample standard deviation. Sample mode. Sample mean. Population variance. D A parameter is any descriptive measure of a population. The population variance describes a population while the sample standard deviation, sample mean, and sample mode are each descriptive measures of samples. 16 Question ID: 19686 Which measure of scale has a true zero point as the origin? A. B. C. D. Ordinal scale. Nominal scale. Ratio scale. Interval scale. C Ratio scales are the strongest level of measurement; they quantify differences in the size of data and have a true zero point as the origin. Question ID: 19687 Fifty mutual funds are ranked according to performance. The five best performing funds are assigned the number 1, while the five worst performing funds are assigned the number 10. This is an example of the: A. B. C. D. ratio scale. nominal scale. interval scale. ordinal scale. D The ordinal scale of measurement categorizes and orders data with respect to some characteristic. In this example, the ordinal scale tells us that a fund ranked “1” performed better than an fund ranked “10,” but it does not tell us anything about the difference in performance. Question ID: 19688 17 Which of the following statements regarding frequency distributions is FALSE? Frequency distributions: A. B. C. D. organize data into overlapping groups. work with all types of measurement scales. help in the analysis of large amounts of statistical data. summarizes data into a small number of intervals. A Data in a frequency distribution must belong to only one group or interval. Intervals are mutually exclusive and non-overlapping. Question ID: 18880 Which of the following best describes a frequency distribution? A frequency distribution is a grouping of: selected data into classes so that the number of observations in each of the non-overlapping classes can be seen and tallied. raw data into classes so that the number of observations in each of the non-overlapping classes can be seen and tallied. independent classes so that they can be seen and tallied. dependent classes so that they can be seen and tallied. A. B. C. D. B By definition. Question ID: 18878 Use the results from the following survey of 500 firms to answer the question. 18 Number of Employees Frequency 300 up to 400 400 up to 500 500 up to 600 600 up to 700 700 up to 800 800 up to 900 40 62 78 101 131 88 The class frequency of the third class is: A. B. C. D. 500 78 180 156 B The third class in 500 - 600 with a frequency of 78 Question ID: 19689 A stock is currently worth $75. If the stock was purchased one year ago for $60, and the stock paid a $1.50 dividend over the course of the year, that is the holding period return? A. B. C. D. 24.0%. 22.0%. 2.5%. 27.5%. D 19 (75 – 60 + 1.50)/60 = 27.5% Question ID: 19371 A bond was purchased exactly one year ago for $910 and was sold today for $1,020. During the year, the bond made two semi-annual coupon payments of $30. What is the holding period return? A. B. C. D. 18.7%. 12.1%. 6.0%. 6.6%. A HPY = (1,020 + 30 + 30 –910)/910 = 0.1868 or 18.7% Question ID: 18937 If an investor started with $1,000 and four months later ended up with $1,100, what would be their annualized holding period return? A. B. C. D. 1.33 2.10 3.30 1.10 A HPR=1100/1000=1.1 which represents a 4 month total return of 10% that must be annualized. Annualized HPR = HPR =(1.1) 1/n 1/.333 =(1.1) =1.33 3 20 Question ID: 19690 Within a frequency distribution, an interval: A. B. contains all members of a specified group. is a set of return values within which an observation falls. is calculated by taking the frequency divided by the total number of frequencies. is the difference between the largest and smallest values in a data set. C. D. B An interval within a frequency distribution is defined as the set of return values within which an observation falls. A population contains all members of a specified group; the relative frequency is calculated by taking the frequency divided by the total number of frequencies; and the range is the difference between the largest and smallest values in a data set. Question ID: 19373 Given the following frequency distribution: Return -10% up to 0% 0% up to 10% 10% up to 20% 20% up to 30% 30% up to 40% Frequency 5 7 9 6 3 What is the relative frequency of the 0 percent to 10 percent interval? A. B. C. D. 23.3%. 33.3%. 40.0%. 10.0%. 21 A Total number of frequencies = 30. 7/30 = 23.3%. Question ID: 19692 Given the following frequency distribution: Return -10% up to 0% 0% up to 10% 10% up to 20% 20% up to 30% 30% up to 40% Frequency 5 7 9 6 3 What is the relative frequency of the 30 percent up to 40 percent return interval? A. B. C. D. 23.3%. 40.0%. 10.0%. 33.3%. C Total number of frequencies = 30. 3/30 = 10.0% Question ID: 19693 In a frequency distribution histogram, the frequency of an interval is given by the: A. B. width of the corresponding bar. sum of the width of all bars. 22 C. D. height multiplied by the width of the corresponding bar. height of the corresponding bar. D In a histogram, intervals are placed on the horizontal axis, and frequencies are placed on the vertical axis. The frequency of a particular interval is given by the value on the vertical axis, or the height of the corresponding bar. Question ID: 19695 Given the following annual returns, what are the median and mode returns, respectively? 1995: 15%, 1996: 2%, 1997: 5%, 1998: -7%, 1999: 0%. Median 0.00% Median Mode 2.75% Mode no exists Mode 3.00% Mode mode A. B. 2.00% C. Median 3.00% Median D. no exists median 3.00% B Median: Arrange the return values from largest to smallest and take the middle value: (7%), 0%, 2%, 5%, 15%. The middle value is 2.00%. Mode: The mode is defined as the value that most often shows up in a distribution. Because no return value shows up more than once, this distribution has no mode. Question ID: 19377 Given the following annual returns, what are the geometric and aritmetic mean returns, 23 respectively? 1995: 15%, 1996: 2%, 1997: 5%, 1998: -7%, 1999: 0%. A. B. C. D. 3.00%, 2.00%. 2.00%, 2.75%. 8.00%, 3.00%. 2.75%, 3.00%. D Geometric Mean: (1.15 x 1.02 x 1.05 x 0.93 x 1.0) 1/5 –1 = 1.1454 1/5 –1 = 2.75% Arithmetic Mean: (15% + 2% + 5% - 7% + 0%) / 5 = 3.00% Question ID: 19696 Which of the following statements regarding the geometric and arithmetic means is FALSE? If all return values in a distribution are equal, the geometric and arithmetic means will also be equal. If returns are variable by period, the geometric mean will be greater than the arithmetic mean. The difference between the arithmetic and geometric mean increases as variability between returns increases. The geometric mean is used to calculate compound rates of return. A. B. C. D. B The difference between the arithmetic mean and the geometric mean increases as variability between return values increase. If there is no variability (all values are equal), the geometric and arithmetic mean values will also be equal. If returns are variable, the geometric mean will always be less than the arithmetic mean. 24 Question ID: 18530 A stock had the following returns over the last five years: 98 percent, 50 percent, 13 percent, 62 percent, 34 percent. What is the geometric mean for this stock? A. B. C. D. 48.76%. 0.50%. 0.49%. 0.18%. A Geometric mean = (1.98)(1.5)(1.13)(1.62)(1.34)1/5 –1 = 7.28542.20-1 = 1.48762 – 1 = 0.48762 0.48762 x 100 = 48.76% Question ID: 18525 The annual rate of return for XYZ's common stock has been: 1995 1996 1997 1998 Return 14% 19% -10% 14% What is the geometric mean of the rate of return for XYZ's common stock over the four years? A. B. C. D. 8.62% 14.21% 14.00% 9.25% 25 A [(1.14)(1.19)(.90)(1.14)] 1/4 - 1 = 8.62% Question ID: 18527 A stock had the following returns over the last five years: 10 percent, -22 percent, 2 percent, 26 percent, -5 percent. What is the geometric mean for this stock? A. B. C. D. 0.01 0.93 0.39 0.13 B Geometric mean = (1.1)(0.78)(1.02)(1.26)(0.95)20 –1 = 1.0475720 – 1 = 1.00934 – 1 = 0.00934 0.00934 x 100 = 0.934 Question ID: 19382 Given the following annual returns, what are the population and standard deviation, respectively? 1995: 15%, 1996: 2%, 1997: 5%, 1998: -7%, 1999: 0%. A. B. C. D. 51.6, 7.2 64.5, 8.0 32.4, 5.7 22.0, 4.7 26 A The population variance is found by taking mean of all squared deviations from the mean. [ (15-3) + (2-3) + (5-3) + (-7-3) + (0-3) ] / 5 = 51.6 2 2 2 2 2 The population standard deviation is found by taking the square root of the population variance. 51.61/2 = 7.2 Question ID: 19700 Assume that the following returns are a sample of annual returns for firms in the clothing industry. Given the following sample of returns, what are the sample variance and standard deviation? Firm 1 15% Firm 2 2% Firm 3 5% Standard Deviation 7.2 Standard Deviation 4.7 Standard Deviation 5.7 Standard Deviation 8.0 Firm 4 (7%) Firm 5 0% A. Variance 51.6 B. Variance 22.0 C. Variance 32.4 D. Variance 64.5 D The sample variance is found by taking the sum of all squared deviations from the mean and dividing by (n-1). (15-3)2 + (2-3)2 + (5-3)2 + (-7-3)2 + (0-3)2/(5-1) = 64.5 27 The sample standard deviation is found by taking the square root of the sample variance. √ 64.5 = 8.03 Question ID: 19697 An investor has a portfolio with 10 percent cash, 30 percent bonds, and 60 percent stock. If last year’s return on cash was 2.0 percent, the return on bonds was 9.5 percent, and the return on stock was 25 percent, what was the return on the investor’s portfolio? A. B. C. D. 15.00%. 36.50%. 18.05%. 22.30%. C Find the weighted mean of the returns. (0.10 x 0.02) + (0.30 x 0.095) + (0.60 x 0.25) = 18.05% Question ID: 19380 An investor has a $15,000 portfolio consisting of $10,000 in stock A with an expected return of 20 percent and $5,000 in stock B with an expected return of 10 percent. What is the investor’s expected return on the portfolio? A. B. C. D. 15.8%. 7.9%. 12.2%. 16.7%. D Find the weighted mean where the weights equal the proportion of $15,000. [(10,000/15,000) *0.20] + [(5,000/15,000 * 0.10) = 16.7%. 28 Question ID: 19701 In a skewed distribution, approximately how many observations will fall between two standard deviations from the mean? A. B. C. D. 75%. 95%. 25%. 84%. A Because the distribution is skewed, we must use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1/k2). 1 – (1/22) = 0.75, or 75%. Note that for a normal distribution, 95% of observations will fall between +/- 2 standard deviations of the mean. Question ID: 19892 The mean monthly return on a sample of small stocks is 4.56 percent with a standard deviation of 3.56 percent. What is the coefficient of variation? A. B. C. D. 128%. 78%. 64%. 84%. B The coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and is found by CV = s/mean. 3.56/4.56 = 0.781, or 78%. 29 Question ID: 19384 The mean monthly return on (U.S. Treasury bills) T-bills is 0.42 percent with a standard deviation of 0.25 percent. What is the coefficient of variation? A. B. C. D. 840%. 84%. 168%. 60%. D The coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and is found by CV = s/mean, or 0.25/0.42 = 0.595, or 60%. Question ID: 18544 The mean and standard deviation of returns from Stock A and B are represented below. Arithmetic Mean Stock A Stock B 20% 15% Standard Deviation 8% 5% The coefficient of variation of the two stocks is given by: A. B. C. D. 2.5 and 3.0 0.16 and 1.67 0.4 and 0.33 0.10 and 0.125 C CV = Standard Deviation / Mean =(8/20)=0.4 and (5/15)=0.333 30 Question ID: 19893 Which of the following statements regarding the Sharpe ratio is TRUE? The Sharpe ratio measures: A. B. C. D. dispersion relative to the mean. peakedness of a return distrubtion. excess return per unit of risk. total return per unit of risk. C The Sharpe ratio measures excess return per unit of risk. Remember that the numerator of the Sharpe ratio is (portfolio return – risk free rate), hence the importance of excess return. Note that dispersion relative to the mean is the definition of the coefficient of variation, and the peakedness of a return distribution is measured by kurtosis. Question ID: 19896 A distribution with a mean that is less than its median: A. B. C. D. is positively skewed. is negatively skewed. has positive excess kurtosis. has negative excess kurtosis. D A distribution with a mean that is less than its median is a negatively skewed distribution. A negatively skewed distribution is characterized by many small gains and a few extreme losses. Note that kurtosis is a measure of the peakedness of a return distribution. 31 Question ID: 19895 If a distribution is positively skewed: A. B. C. D. the mode is greater than the median. the mean is greater than the median. the median, mean, and mode are equal. the mode is greater than the mean. B For a positively skewed distribution, the mode is less than the median, which is less than the mean (the mean is greatest). Remember that investors are attracted to positive skewness because the mean return is greater than the median return. Question ID: 19897 Which of the following statements regarding skewness is FALSE? In a skewed distribution, 95% of all values will lie within plus or minus two standard deviations of the mean. A positively skewed distribution is characterized by many small losses and a few extreme gains. Skewness refers to a distribution that is not symmetrical. A normal distribution will have a mean that is equal to its median. A. B. C. D. A For a normal distribution, the mean will be equal to its median and 95% of all observations will fall within plus or minus two standard deviations of the mean. For a skewed distribution, because it is not symmetrical, this may not be the case. Chebyshev’s inequality tells us that at least 75% of observations will lie within plus or minus two standard deviations from the mean. 32 Question ID: 19386 Which of the following statements about kurtosis is FALSE? Kurtosis: measures the peakedness of a distribution reflecting a greater or lesser concentration of returns around the mean. describes the degree to which a distribution is not symmetric about its mean. is used to reflect a departure from the normal distribution. is used to reflect the probability of extreme outcomes for a return distribution. A. B. C. D. B The degree to which a distribution is not symmetric about its mean is measured by skewness. Kurtosis is used to reflect a departure from the normal distribution, measures the peakedness of a distribution, and is used to reflect the probability of extreme outcomes. Question ID: 19899 A distribution of returns that has a greater percentage of small deviations from the mean and a greater percentage of extremely large deviations from the mean: A. B. C. D. has negative excess kurtosis. is a normal distribution. is positively skewed. has positive excess kurtosis. D A distribution that has a greater percentage of small deviations from the mean and a greater percentage of extremely large deviations from the mean will be leptokurtic and will exhibit positive excess kurtosis. The distribution will be taller with fatter tails than a normal distribution. 33 Question ID: 19900 Which of the following statements about semi-logarithmic scales is FALSE? Semi-logarithmic scales provide a more realistic picture when graphing past performance. Arithmetic scales are based on straight numerical changes in an index. Semi-logarithmic scales use a logarithmic scale on the horizontal axis, but an arithmetic scale on the vertical axis. Semi-logarithmic scales are based on percentage changes of an index. A. B. C. D. C Semi-logarithmic scales use a logarithmic scale on the vertical axis, but an arithmetic scale on the horizontal axis. This allows changes to be reflected on a percentage basis rather than a straight numerical basis, resulting in a more realistic picture when graphing past performance. Question ID: 18531 A stock had the following returns over the last four years: 15 percent, 2 percent, 9 percent, 44 percent, 23 percent. What is the geometric mean for this stock? A. B. C. D. 0.09 0.43 0.18 17.76 D Geometric mean = (1.15)(1.02)(1.09)(1.44)(1.23) 2.2646020 – 1 = 1.17760 – 1 = 0.17760 20 –1 = 34 0.17760 x 100 = 17.76 Question ID: 18528 A stock had the following returns over the last four years: 45 percent, -12 percent, 15 percent, 75 percent, -9 percent. What is the geometric mean for this stock? A. B. C. D. 0.44 18.50 0.19 0.31 B Geometric mean = (1.45)(0.88)(1.15)(1.75)(0.91)20 –1 = 2.3368320 – 1 = 1.18502 – 1 = 0.18502 0.18502 x 100 = 18.50 Question ID: 19369 Given the following frequency distribution, the sample size and frequency of the second interval are, respectively: Return -10% up to 0% 0% up to 10% 10% up to 20% 20% up to 30% 30% up to 40% Frequency 3 7 3 2 3 A. B. 5, 10 18, 7 35 C. D. 10, 3 10, 7 B The sample size is found by totaling all of the frequencies in the frequency distribution. (3+7+3+2+3) = 18. The frequency of the second interval is found simply by looking at the table: 7. Question ID: 18523 A portfolio realized a 10% return in Year 1 and a -10% return in Year 2. The geometric mean return over the two year period is: A. B. C. D. 0.99% 0.00% 0.995% -0.50% D (1+.10)(1-.10)1/2 -1 = (1.1)(0.90)1/2 -1 = square root of 0.99 - 1 = -0.005 =-0.5% Question ID: 19898 If a distribution is skewed: A. B. C. each side of a return distribution is the mirror image of the other. the distribution can be described completely by its mean and its variance. the magnitude of positive deviations from the mean is different from the 36 magnitude of negative deviations from the mean. it will be more or less peaked reflecting a greater or lesser concentration of returns around the mean. D. C Skewness is caused by the magnitude of positive deviations from the mean being either larger or smaller than the magnitude of negative deviations from the mean. Note that each side of the distribution being a mirror image of the other, and the ability to describe the distribution completely by its mean and variance are characteristics of a normal distribution. Peakedness of a distribution is measured by kurtosis. Question ID: 18879 Use the results from the following survey of 500 firms to answer the question. Number of Employees Frequency 300 up to 400 400 up to 500 500 up to 600 600 up to 700 700 up to 800 800 up to 900 40 62 78 101 131 88 The number of classes in this frequency table is: A. B. C. D. 100 5 6 600 C 37 300 - 400 = 1, 400 - 500 = 2, 500 - 600 = 3, 600 - 700 = 4, 700 - 800 = 5, 800 - 900 = 6, Total = 6 Question ID: 19694 Which of the following statements is FALSE? Histograms and frequency polygons are graphical tools used for portraying frequency distributions. A frequency polygon is constructed by plotting the midpoint of each interval on the horizontal axis. A histogram connects points with a straight line. A histogram and a frequency polygon both plot the absolute frequency on the vertical axis. A. B. C. D. C In constructing a frequency polygon, the midpoint of each interval is plotted on the horizontal axis and the frequency of each interval is plotted on the vertical axis. Each point is then constructed with a straight line. A histogram is a bar chart of data that has been grouped into a frequency distribution – because it is a bar chart, there are no individual points to connect. Question ID: 19379 A portfolio realized a 15 percent return in year 1 and a –15 percent return in year 2. The geometric mean over the two-year period is: A. B. C. D. 1.13%. 0.00%. 0.95%. -1.13%. D 38 Geometric mean = [(1.15)(0.85)] 1/2 – 1 = 0.9775 1/2 – 1 = 0.9887 – 1 = -1.13% Note that this problem could be answered without making any difficult calculations. Because the returns are variable by year and the arithmetic mean return is 0%, we know that the geometric mean return must be less than 0%. –1.13% is the only possible answer. Question ID: 18938 Suppose an investor started with $10,000 and 3 years later they have $8,500. What is their annualized holding-period return? A. B. C. D. 0.947 1.056 0.674 0.283 A 8,500/10,000=0.85 1/3=(0.85).333=0.947 Question ID: 19702 In a skewed distribution, approximately how many observations will fall between 1.5 standard deviations from the mean? A. B. C. D. 44%. 95%./font> 25%. 56%. D Because the distribution is skewed, we must use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1/k ). 2 39 1 – (1/1.5 ) = .5555, or 56% 2 1.C: Probability Concepts Question ID: 19429 For a newly issued U.S. Treasury bond which does not have default risk or call risk, which of the following is a random variable? A. B. C. D. The coupon rate. The nominal value of coupons to be received. The realized yield to maturity. The maturity of the bond. C Of the choices, only the realized yield to maturity is a quantity that is uncertain. This is because it would be determined by the interest rates between now and the maturity of the bond. All the other choices are certain. Question ID: 19428 When randomly selecting a stock from the S&P 500, which of the following is NOT an applicable random variable? A. B. C. D. Its current ratio. Its most recent closing price. Its P/E ratio. Its stock symbol. D 40 A random variable must be a number. Sometimes there is an obvious method for assigning a number, such as when the random variable is a number itself, like a P/E ratio. A stock symbol of a randomly selected stock could have a number assigned to it like the number of letters in the symbol. The symbol itself cannot be a random variable. Question ID: 19430 Which of the following sets of numbers does NOT represent a set of probabilities? A. B. C. D. (0.10, 0.20, 0.30, 0.40). (0.50, 0.50). (0.25, 0.25, 0.25, 0.25). (0.10, 0.20, 0.30, 0.40, 0.50). D A set of probabilities must sum to one. Question ID: 19360 Which of the following is an a priori probability? On a random draw, the probability of choosing a stock of a particular industry from the S&P 500. A politician's prediction concerning a recession prior to an election. The probability the Fed will lower interest rates prior to the end of the year. For a stock, based on prior patterns of up and down days, the probability of the stock having a down day tomorrow. A. B. C. D. A There are three types of probabilities: a priori, empirical, and subjective. A priori probability is based on logical analysis. Given the number of stocks in the airline industry in the S&P500 for 41 example, the a priori probability of selecting an airline stock would be that number divided by 500. Question ID: 19432 Which of the following is an empirical probability? A. A politician’s prediction concerning a recession prior to an election. For a stock, based on prior patterns of up and down days, the probability of the stock having a down day tomorrow. The probability the Fed will lower interest rates prior to the end of the year. On a random draw, the probability of choosing a stock of a particular industry from the S&P 500. B. C. D. B There are three types of probabilities: a priori, empirical, and subjective. An empirical probability is based on what has happened in the past. Question ID: 19433 In investment analysis, the best example of a consequence of probabilities that are inconsistent is: A. that one asset is overpriced relative to another. applying binomial distribution when a normal distribution is more appropriate. recent price patterns are different from those seen years before. that the probabilities in a distribution do not sum to one. B. C. D. A 42 Inconsistent probabilities mean that asset prices incorporate different values for the probability of a particular event. This would lead to assets being under and overpriced relative to each other. Question ID: 19435 For an unconditional probability: A. B. C. D. there is only one random variable of concern. the addition rule is important. the joint probability rule is important. there are at least two events. A The unconditional probability gives the expected value of a random variable regardless of what other events occur. Question ID: 19361 For a given corporation, which of the following is an example of a conditional probability? The probability the corporation's: A. B. C. D. dividend increases given its earnings increase. earnings increase and dividend increases. inventory improves. financial condition improves. A A conditional probability involves two events. One of the events is a given, and the probability of the other event depends upon that given. 43 Question ID: 19436 Which of the following is a joint probability? The probability that a: A. B. C. D. stock increases in value after an increase in interest rates has occurred. stock pays a dividend and splits next year. company merges with another firm next year. stock either splits or joins the NYSE. B A joint probability applies to two events that both must occur, but neither is certain. Question ID: 18428 Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of suffering from allergies and not suffering from allergies? Suffer from Allergies Don't Suffer from Allergies Total Smoker Nonsmoker Total 35 55 90 25 185 210 60 240 300 A. B. C. D. .50 .24 1.00 .00 D 44 These are mutually exclusive, so there is no joint probability. Question ID: 19363 In a given portfolio, half of the stocks have a beta greater than one. Of those with a beta greater than one, a third are in a computer-related business. What is the probability of a randomly drawn stock from the portfolio having both a beta greater than one and being in a computer-related business? A. B. C. D. 0.500. 0.167. 0.667. 0.333. B This is a joint probability. From the information: P(beta > 1)=0.500 and P(comp.stock|beta > 1)=0.333. Thus, the joint probability is the product of these two probabilities. Question ID: 18426 100 tourists visit New York City, 75 go to the Empire State Building, 50 go to a Broadway show, and 40 go to Central Park. Which is closest to the probability of a tourist visiting two of the three sites. A. B. C. D. 0.90 0.40 1.15 90% B Possible outcomes for this joint probability are: (75/100)(50/100) = (.75)(.50) = 0.375 or about 0.40, 45 (75/100)(40/100) = (.75)(.40) = 0.3, and (50/100)(40/100) = (.50)(.40) = 0.2 Question ID: 19438 Three out of four times in the past, an inverted yield curve has been followed by a Bear Market. The probability of an inverted yield curve next year is 20 percent. What is the probability of an inverted yield curve next year followed by a Bear market? A. B. C. D. 0.150. 0.330. 0.500. 0.375. A This is a joint probability. From the information: P(Bear Market / inverted yield curve)=0.75 and P(inverted yield curve)=0.20. The joint probability is the product of these two probabilities. Question ID: 18424 Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of being either a nonsmoker or not suffering from allergies? Suffer from Allergies Don't Suffer from Allergies Total Smoker Nonsmoker Total 35 55 90 25 185 210 60 240 300 A. B. C. .383 .500 .883 46 D. 1.500 C Probability of being a nonsmoker = 240/300 = 0.8 Probability of not suffering from allergies = 210/300 = 0.7 Probability of being a nonsmoker and not suffering from allergies = 185/300 = 0.617 Since the question asks for the probability of being either a nonsmoker or not suffering from allergies we have to add the probability of being a nonsmoker plus the probability of not suffering from allergies and subtract the probability of being both. 0.8 + 0.7 - 0.617 = 0.883 Question ID: 18423 Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of being either a smoker or someone who suffers from allergies? Suffer from Allergies Don't Suffer from Allergies Total Smoker Nonsmoker Total 35 55 90 25 185 210 60 240 300 A. B. C. D. .383 .883 .117 .20 A 47 .2+.3-.117 = .383 Question ID: 19439 There is a 50 percent chance that the Fed will cut interest rates tomorrow. On any given day, there is a 67 percent chance the DJIA will increase the next day. On days the Fed cuts interest rates, the probability the DJIA will go up is 90 percent. What is the probability of the Fed cutting interest rates or the DJIA going up? A. B. C. D. 1.00. 0.95. 0.72. 0.33. C This requires the addition formula. From the information: P(cut interest rates)=0.50 and P(DJIA increase)=0.67, P(DJIA increase / cut interest rates)=0.90. The joint probability is 0.50 x 0.90=0.45. Thus P(inverted yield curve or cut interest rates)=0.50 + 0.67 – 0.45 = 0.72. Question ID: 19366 A company says that whether it increases its dividends or not is dependent on whether its earnings increase. From this we know: P(dividend increase | earnings increase) is not equal to P(earnings increase). P(earnings increase | dividend increase) is not equal to P(earnings increase). P(both dividend increase and earnings increase) = P(dividend increase). P(dividend increase or earnings increase) = P(both dividend and earnings increase). A. B. C. D. B 48 If the two events are dependent, then the conditional probabilities of each will not equal its unconditional probability. The other choices may or may not occur, e.g., P(Y | X) = P(X) is possible but not necessary. Question ID: 19441 If X and Y are independent events, which of the following must be TRUE? A. B. C. D. P(X or Y) = P(X) x P(Y). P(X or Y) = P(X) + P(Y). X and Y cannot occur together. P(X / Y) = P(X). D By the definition of independent events: P(X / Y) = P(X) Question ID: 19368 A bond portfolio consists of four BB-rated bonds. Each has a probability of default of 24 percent and that probabilities are independent. Which of the following is the probability of all the bonds defaulting and the probability of all the bonds not defaulting respectively? A. B. C. D. 0.04000, 0.9600. 0.96000, 0.0400. 0.06000, 0.1900. 0.00332, 0.3336. D For the four independent events where the probability is the same for each, the probability of all defaulting is (0.24)4. The probability of all not defaulting is (1-0.24)4. 49 Question ID: 19444 The events Y and Z are mutually exclusive and exhaustive: P(Y)=0.4 and P(Z)=0.6. If the probability of X given Y is 0.9, and the probability of X given Z is 0.1, what is the unconditional probability of X? A. B. C. D. 0.40. 0.66. 0.42. 0.33. C The formula is P(X)= P(X/Y) x P(Y)+ P(X/Z) x P(Z)= 0.4 x 0.9 + 0.6 x 0.1 = 0.42 Question ID: 19443 Firm A can fall short, meet, or exceed its earnings forecast. Each of these events is equally likely. Whether firm A increases its dividend will depend upon these outcomes. Respectively, the probabilities of a dividend increase conditional on the firm falling short, meeting or exceeding the forecast are 20 percent, 30 percent, and 50 percent. The unconditional probability of a dividend increase is: A. B. C. D. 0.333. 0.500. 0.167. 1.000. A The unconditional probability is the weighted average of the conditional probabilities where the weights are the probabilities of the conditions. In this problem the three conditions fall short, meet, or exceed its earnings forecast are all equally likely. Therefore, the unconditional probability is the simple average of the three conditional probabilities:(0.2+0.3+0.5)/3. 50 Question ID: 19445 A two-sided but very thick coin is expected to land on its edge twice out of every 100 flips. And the probability of face up (heads) and the probability of face down (tails) are equal. When the coin is flipped, the prize is $1 for heads, $2 for tails, and $50 when the coin lands on its edge. What is the expected value of the prize on a single coin toss? A. B. C. D. $2.47. $1.50. $17.67. $26.50. A Since the probability of the coin landing on its edge is 0.02, the probability of each of the other two events is 0.49. The expected payoff is: (.02 x $50)+(0.49 x $1)+(0.49 x 2) Question ID: 19370 Given P(X=2)=0.3, P(X=3)=0.4, P(X=4)=0.3. What is the variance? A. B. C. D. 3.0. 0.6. 0.3. 0.5 B The variance is the expected value of the squared deviations around the expected value. The expected value is E(X)=0.3*2+0.4*3+0.3*4=3. The variance is 0.3*(2-3) +0.4*(3-3) +0.3*(4-3) =0.6. 2 2 2 51 Question ID: 19447 When creating intervals around the mean to indicate the dispersion of outcomes, which of the following measures is the most useful? A. B. C. D. The variance. The median. The standard deviation. The mode. C The standard deviation is more useful than the variance because the standard deviation is in the same units as the mean. The median and mode do not help in creating intervals around the mean. Question ID: 19448 An analyst announces that an increase in the discount rate next quarter will double her earnings forecast for a firm. This is an example of a: A. B. C. D. use of counting rules. conditional expectation. use of Bayes’ formula joint probability. B This is a conditional expectation. The analyst indicates how an expected value will change given another event. Question ID: 19450 52 Outcomes P(X)=0.3 and P(Y)=0.7. The expected value of Z given X is 100. If Y occurs, then the expected value of Z is 200. The unconditional expected value of Z is: A. B. C. D. 130. 170. 180. 150. B The problem gives two conditional expectations: E(Z/X)= 100 and E(Z/Y)= 200. The unconditional expected value is E(Z)=100 x 0.3 + 200 x 0.7 = 170. Question ID: 19451 A gambler flips a coin. If the outcome is heads, she rolls one die and lets X equal the number on the one die. If the outcome is tails, she rolls two dice and lets X equal the total number on the two dice. Assuming the coin and dice are fair, the expected value of X is: A. B. C. D. 5.25. 4.75. 4.50. 9.00. A The conditional expectations are E(X/heads)= 3.5 and E(X/tails)= 7. Since P(heads)=P(tails)=0.5, then E(X) = 0.5 x 3.5 + 0.5 x 7 = 5.25. Question ID: 19452 Given Cov(X,Y)=1,000,000. What does this indicate about the relationship between X and Y? 53 A. B. C. D. Only that it is positive. It is weak and positive. It is strong and positive. It is curvilinear. A A positive covariance indicates a positive linear relationship but nothing else. The magnitude of the covariance by itself is not informative with respect to the strength of the relationship. Question ID: 19611 With respect to the units each is measured in, which of the following is the most easily directly applicable measure of dispersion? A. B. C. D. The covariance. The standard deviation. The variance. The kurtosis. B The standard deviation is in the units of the random variable itself and not squared units like the variance. The covariance would be measured in the product of two units of measure. Kurtosis is not a measure of dispersion. Question ID: 19610 If given the standard deviations of the returns of two assets and the correlation between the two assets, which of the following would an analyst NOT necessarily be able to derive from these? A. The variance of each return. 54 B. C. D. The expected returns. The covariance between the returns. The strength of the linear relationship between the two. B The correlations and standard deviations cannot give a measure of central tendency like the expected value. Question ID: 19372 For asset A and B we know the following: E(RA)=0.10, E(RB)=0.20, V(RA)=0.25, V(RB)=0.36 and the correlation of the returns is 0.6. What is the variance of the return of a portfolio that is equally invested in the two assets? A. B. C. D. 0.3050. 0.2275. 0.2425. 0.1500. C You are not given the covariance in this problem but instead you are given the correlation coefficient and the variances of assets A and B from which you can determine the covariance by Cov = (correlation of A,B)(Std A)(Std B). Since it is an equally weighted portfolio, the solution is: [( 0.5 )* 0.25 ] + [(0.5 ) * 0.36 ] + [ 2 * 0.5 * 0.5 * 0.6 * ( 0.25 0.0625 + 0.09 + 0.09 = 0.2425. 2 2 0.5 ) * ( 0.36 0.5 )] = 55 Question ID: 19612 For assets A and B we know the following: E(RA)=0.10, E(RB)=0.20, V(RA)=0.25, V(RB)=0.36 and the correlation of the returns is 0.6. What is the expected return of a portfolio that is equally invested in the two assets? A. B. C. D. 0.3050. 0.2500. 0.2275. 0.1500. D Since it is an equally weighted portfolio, the solution is: 0.5 x 0.1 + 0.5 x 0.2 = 0.15 Question ID: 19614 Given P(X=2, Y=10)=0.3, P(X=6, Y=2.5)=0.4, and P(X=10 , Y=0)=0.3, then COV(XY)is: A. B. C. D. -12.0. -4.9. 6.0. 24.0. A The expected values are: E(X)=0.3 x 2 + 0.4 x 6 + 0.3 x 10=6 and E(Y) = 0.3 x 10.0 + 0.4 x 2.5+0.3 x 0.0 = 4 COV(XY) =0.3 x (2-6)x(10-4) + 0.4 x (6-6)x(2.5-4) + 0.3 x (10-6)x(0-4) = -12 Question ID: 19613 56 Given P(X=20, Y=0)=0.4, and P(X=30 , Y=50)=0.6, then COV(XY)is: A. B. C. D. 125.00. 11.18. 25.00. 120.00. D The expected values are: E(X)=0.4 x 20+0.6 x 30=26 and E(Y)= 0.4 x 0+0.6 x 50=30 COV(XY) =0.4 x (20-26)x(0-30)+0.6 x (30-26)x(50-30)=120 Question ID: 18442 An analyst expects that 20 percent of all publicly traded companies will experience a decline in earnings next year. The analyst has developed a ratio to help forecast this decline. If the company is headed for a decline, there is a 90% chance that this ratio will be negative. If the company is not headed for a decline, there is only a 10% chance that the ratio will be negative. The analyst randomly selects a company with a negative ratio. Based on Bayes' theorem, the posterior probability that the company will experience a decline is: A. B. C. D. 44% 69% 18% 26% B (.20)(.90)/(.20)(.90) + (.80)(.10) = .18/.18 + .08 = .18/.26 = .69 or 69% Question ID: 19374 John purchased 60 percent of the stocks in a portfolio, while Andrew purchased the other 40 57 percent. Half of John’s stock-picks are considered good, while a fourth of Andrew’s are considered to be good. If a randomly chosen stock is a good one, what is the probability John selected it? A. B. C. D. 0.25. 0.75. 0.40. 0.30. B Using the information of the stock being good, the probability is updated to a conditional probability: P(John | good) = P(good and John) / P(good). P(good and John) = P(good | John) * P(John) = 0.5 * 0.6 = 0.3. P(good and Andrew) = 0.25 * 0.40 = 0.10. P(good)=P(good and John) + P (good and Andrew) = 0.40. P(John | good) = P(good and John) / P(good) = 0.3 / 0.4 = 0.75. Question ID: 19616 James Bernard, CFA, has a list of five stocks and three bonds that he is recommending to his clients. A client wants one stock and one bond. How many ways can Bernard combine one stock and one bond from his lists? A. B. C. D. 225. 4. 15. 8. 58 C This uses the multiplication rule: 5 x 3 = 15 Question ID: 19617 A firm produces three types of shoes and ten types of socks. As a promotion, it plans to offer two pairs of socks with each pair of shoes. For each pair of shoes, the two pair of socks can be the same or different. How many ways can the three items be packaged? A. B. C. D. 33. 30. 270. 300. D This uses the multiplication rule: 3 x 10 x 10 = 300. The fact that the shoe can be combined with two socks of the same type means using this formula. If the socks had to be different, then the answer would have been: 3 x 10 x 9. Question ID: 19618 A firm wants to select a team of five from a group of ten employees. How many ways can the firm compose the team of five? A. B. C. D. 120. 50. 25. 252. D 59 This is a labeling problem where there are only two labels: chosen and not chosen. Thus, the combination formula applies: 10!/(5! x 5!) Question ID: 19620 If a firm is going to create three teams of four from twelve employees. Which approach is the most appropriate for determining how the twelve employees can be selected for the three teams? A. B. C. D. Multiplication rule of counting. Binomial formula. Permutation formula. Multinomial formula. D This problem is a labeling problem where the 12 employees will be assigned one of three labels. It requires the multinomial formula. Question ID: 19621 For the task of arranging a given number of items without any sub-groups, this would require: A. B. C. D. the multiplication rule of counting. the binomial formula. only the factorial function. the multinomial formula. C The factorial function, denoted n!, tells how n items can be arranged where all the items are in the arrangement. 60 Question ID: 19622 The task is to choose six objects out of nine, how many ways can this be done if order is important? A. B. C. D. 60,480. 84. 10,080. 504. A This is a choose six from nine problem where order is important. Thus, it requires the permutation formula: 9!/3!=60,480. Question ID: 18443 Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies. The prior probability that a person in general suffers from allergies is (90/300)=.30. If you were told that a specific person was a smoker, what would be your revised probability that that person suffers from allergies? Suffer from Allergies Don't Suffer from Allergies Total Smoker Nonsmoker Total 35 55 90 25 185 210 60 240 300 A. B. C. D. .30 .73 .58 .25 61 C Bayes' P(A given that S has occured)=(90/300)(35/90)/(90/300)(35/90)+(210/300)(25/210)=35/60=.58 Question ID: 19449 A conditional expectation involves: A. B. C. D. estimating the skewness. calculating the conditional variance. determining the expected joint probability. refining a forecast because of new information. D This is the definition of a conditional expectation. The expectation changes as new information is revealed. Question ID: 19437 A joint probability of A and B must always be: A. B. C. D. less than or equal to the conditional probability of A given B. greater than or equal to than the probability of A or B. greater than or equal to the conditional probability of A given B. less than or equal to product of the probability of A and the probability of B. A By the formula for joint probability: P(AB)=P(A/B) x P(B), since P(B) ≤ 1, then P(AB) ≤ P(A|B). None of the other choices must hold. 62 Question ID: 19442 On the flip of two fair coins and the roll of two fair six-sided dice, what is the probability of getting two heads on the coins and two sixes on the dice? A. B. C. D. 0.0039. 0.0069. 0.4167. 0.8333. B For the four independent events defined here, the probability of the specified outcome is 0.5000 x 0.5000 x 0.1667 x 0.1667=0.0069. Question ID: 18425 The following table summarizes the availability of trucks with air bags and bucket seats at a dealership. Bucket seats No Bucket Seats Total Air Bags 75 50 60 110 125 95 220 No Air Bags 35 Total 110 What is the probability of selecting a truck at random that has either air bags or bucket seats? A. B. C. D. 50% 107% 73% 34% 63 C General rule of addition. P(air bags) + P(bucket seats) - P(air bags and bucket seats) = .57 + .50 - .34 = .73 or 73% Question ID: 19431 Which of the following would qualify as a probability? The number of days IBM increased in a given period divided by the number of days it decreased. The forecast of IBM’s earnings for the next year. The number of shares of IBM in a portfolio of stocks divided by the total number of all shares of stock in that portfolio. The return on IBM yesterday divided by the total return of the previous week. A. B. C. D. C A probability must be between zero and one. The correct answer here is the only choice that we know must be between zero and one all the time. Question ID: 18427 An investor has all her money invested in either of two mutual funds (A and B). She knows that there is a 40% probability that fund A will rise in price and a 60% chance that fund B will rise in price if fund A rises in price. What is the probability that both fund A and fund B will rise in price? A. B. C. D. 0.24 0.40 1.00 0.70 64 A (0.60)(0.40)=0.24 Question ID: 19619 A portfolio manager wants to eliminate four stocks from a portfolio that consists of six stocks. How many ways can the four stocks be sold when the order of the sales is important? A. B. C. D. 720. 180. 30. 24. C This is a choose four from six problem where order is important. Thus, it requires the permutation formula: 6!/4!=30 1.D: Common Probability Distributions Question ID: 19623 Which of the following is NOT a probability distribution? A. B. C. D. DJIA: P(increase)=0.67, P(not increase)=0.33. Roll an irregular die: p(1)=p(2)=p(3)=p(4)=0.2 and p(5)=p(6)=0.1. Zeta Corp.: P(dividend increases)=0.60, P(divided decreases)=0.30. Flip a coin: P(H)=P(T)=0.5. 65 C All the probabilities must be listed. In the case of Zeta Corp. the probabilities do not sum to one. Question ID: 19624 A probability distribution does NOT do which of the following? A. B. C. D. List all the possible outcomes. Give the probability that the distribution is realistic. Have only non-negative probabilities. Apply to continuous random variables. B The probability distribution may or may not reflect reality. But the probability distribution must list all possible outcomes, and it can apply to continuous random variables. Probabilities can only have non-negative values. Question ID: 19625 Which of the following is a discrete random variable? A. B. C. D. The realized return on a corporate bond. The amount of time between trades of a stock in a day. The weight of debris swept off the floor of the NYSE in a day. The number of advancing stocks in the DJIA in a day. D Since the DJIA consists of only 30 stocks, the answer associated with it would be the most discrete random variable of the choices. Random variables measuring time, rates of return and weight will be continuous. 66 Question ID: 19627 Which of the following could be the set of all possible outcomes for a random variable that follows a binomial distribution? A. B. C. D. (0, 0.5, 1, 1.5, 2, 2.5, 3). (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11). (1, 2). (-1, 0, 1). B This reflects a basic property of binomial outcomes. They take on whole number values that must start at zero up to the upper limit n. The upper limit in this case is 11. Question ID: 19628 A continuous uniform distribution has a lower bound of zero and f(X)=2. What is P(X≤ 0.25)? A. B. C. D. 0.00. 0.50. 0.25. 1.00. B The height of a pdf for a continuous distribution is f(X). The area under the pdf must equal one; therefore, the length of the pdf is 0.5. Since the lower bound is zero, the upper bound is 0.5, P(X≤0.25)=0.5. Question ID: 19629 A probability function: 67 A. B. C. D. only applies to continuous distributions. specifies the probability that the random variable take on a specific value. is completely determined by its median and mode. is often referred to as the "cdf." B This is true by definition. Question ID: 19634 Which of the following is a key property of a probability function? A. B. C. D. Sum of all p(x)=1. P(A and B) = P(A) + P(B). F(x) ≤p(x). p(x) ≤F(x). A The probabilities must sum to one. This is one of the two key properties of a probability function. Question ID: 19633 Which of the following is a key property of a probability function? A. B. C. D. 0≤p(x) ≤1. P(A or B) = P(A/B) x P(A). 0≤F(X). E(ex) = eµ. 68 A The probabilities must be between zero and one. This is one of the two key properties of a probability function. Question ID: 19362 Which of the following qualifies as a cumulative distribution function? A. B. C. D. F(1)=0.5, F(2)=0.25, F(3)=0.25. F(1)=0, F(2)=0.5, F(3)=0.5, F(4)=0. F(1)=0.3, F(2)=0.6, F(3)=0.3. F(1)=0, F(2)=0.25, F(3)=0.50, F(4)=1. D For successfully larger numbers, the cumulative probability values must stay the same or increase. Question ID: 19636 The number of days a particular stock increases in a given five-day period is uniformly distributed between zero and five inclusive. In a given five-day trading week, what is the probability that the stock will increase exactly three days? A. B. C. D. 0.333. 0.200. 0.600. 0.167. D If the possible outcomes are X:(0,1,2,3,4,5), then the probability of each of the six outcomes is 1/6=0.167. 69 Question ID: 19364 Which of the following random variables would be most likely to follow a discrete uniform distribution? A. The number of heads on the flip of two coins. The outcome of a roll of a standard, six-sided die where X equals the number facing up on the die. The number of red coats to yellow coats on the floor of the NYSE. The outcome of the roll of two standard, six-sided dice where X is the sum of the numbers facing up. B. C. D. B The discrete uniform distribution is characterized by an equal probability for each outcome. A single die roll is an often-used example of a uniform distribution. In combining two random variables, such as coin flip or die roll outcomes, the sum will not be uniformly distributed. The number of red coats to yellow coats is just a ratio and is not a possible answer. Question ID: 19637 Which of the following is NOT an assumption of the binomial distribution? A. B. C. D. Each trial can only have one of two possible outcomes. The trials are independent. Random variable X is discrete. The expected value is a whole number. D The expected value is n x p. A simple example shows us that the expected value does not have to be a whole number: n=5, p=0.5, n x p=2.5. All the other conditions are necessary for the binomial distribution. 70 Question ID: 19646 For a certain class of junk bonds, the probability of default in a given year is 0.2. Whether one bond defaults is independent of whether another bond defaults. For a portfolio of five of these junk bonds, what is the expected value of the number of bonds to default in the upcoming year? A. B. C. D. 0.20. 0.25. 0.40. 1.00. D The expected value is E(X)=n x p=5 x 0.2=1. Question ID: 19365 The DJIA goes up two out of every three days, but whether it goes up on one day is independent of whether it goes up on another day. In a year of 250 trading days, what is the variance of the distribution describing the number of days the DJIA will go up? A. B. C. D. 83.33. 166.67. 0.22. 55.56. D The variance of this binomial random variable is V(X)=n*p*(1-p)=250*0.67*0.34=55.56. Question ID: 19367 71 A stock can either increase in value one percent or decrease in value one percent on a given day. The probability of the stock increasing is P(U)=0.7 and the probability of a decrease is P(D)=0.3. Its current price is $200, what is the expected value two days from now? A. B. C. D. $204.02. $199.98. $196.02. $201.60. D After two steps, the outcomes, terminal values and associated probabilities are listed here in [ ]: [(UU), 204.2, 0.49], [(UD), 0.42, 199.98], [(DD), 0.09, 196.02]. 204.2*0.49+0.42*199.98+0.09*196.02=$201.60. Question ID: 19647 A binomial tree used to describe the movement of a stock price has five steps. How many different terminal values for the stock price will the tree have? A. B. C. D. 2. 120. 6. 32. C This is tricky because the binomial tree will have 32 branches after five steps, but the number of terminal values equals the number of steps plus one. The five terminal values will be associated with the outcomes: (UUUUU), (UUUUD), (UUUDD), (UUDDD), (UDDDD), (DDDDD). The terminal value associated with (UUUUD) can occur five different ways, the terminal value associated with (UUUDD) can occur 10 different ways, etc. 72 Question ID: 19653 A discount brokerage firm states that the time between a customer order for a trade and the execution of the order is uniformly distributed between three minutes and fifteen minutes. If a customer orders a trade at 11:54 A.M., what is the probability that the order is executed after noon? A. B. C. D. 0.250. 0.500. 0.125. 0.750. D The upper and lower limits of the uniform distribution are three and 15. Since the problem concerns time, it is continuous. Noon is six minutes after 11:54 A.M., thus the problem wants to find P(6 Question ID: 19648 Probability density function of a continuous uniform distribution is best described by a: A. B. C. D. horizontal line segment. line segment with a 45-degree slope. line segment with a curvilinear slope. vertical line segment. A By definition, for a continuous uniform distribution, the probability density function is a horizontal line segment over a range of values such that the area under the segment equals one. If the upper and lower limits of the distribution are zero and one, then f(x)=1. If the upper and lower limits of the distribution are two and seven, then f(x)=0.2. 73 Question ID: 19659 A normal distribution is completely described by its: A. B. C. D. mean, mode, and skewness. median and mode. variance and mean. skewness and kurtosis. C By definition, a normal distribution is completely described by its mean and variance. Question ID: 19661 A stock portfolio has had a historical average annual return of 12 percent and a standard deviation of 20 percent. The returns are normally distributed. The range –27.2 to 51.2 percent describes a: A. B. C. D. 99% confidence interval. 68% confidence interval. 50% confidence interval. 95% confidence interval. D For a normal distribution, the 95% confidence interval for X is mean –1.96*s to mean +1.96*s. Question ID: 19662 A stock portfolio's returns are normally distributed. It has had an average annual return of 25 percent. The 90 percent confidence interval for the returns is –41 to 91 percent. What is the 99 percent confidence interval? A. –78.2 to 128.2%. 74 B. C. D. -20 to 20%. -66 to 116%. –58.4 to 98.4%. D First we have to determine the standard deviation. This found from s=(91-25)/1.65=40. The 99 percent confidence interval for X is mean –2.58*s to mean +2.58*s: 25-2.58 x 40=-78.2 and 25+2.58 x 40=128.2. Question ID: 19663 Which of the following represents the mean, standard deviation, and variance of a standard normal distribution? A. B. C. D. 1, 2, 4. 0, 2, 4. 1, 1, 1. 0, 1, 1. D By definition, for the standard normal distribution, the mean, standard deviation, and variance are 0, 1, 1. Question ID: 19389 John Cupp, CFA, has several hundred clients. The values of the portfolio Cupp manages are normally distributed with a mean of $800,000 and a standard deviation of $250,000. The probability of a randomly selected portfolio being in excess of $1,000,000 is: A. B. C. 0.2119. 0.6227. 0.3773. 75 D. 0.1057. A Although the number of clients is discrete, since there are several hundred of them, we can treat them as continuous. First we standardize the value $1,000,000: Z = ( 1000000 - 800000 ) / 250000 = 0.80. Then we recognize that P(1,000,000 < X )=P( 0.8 < Z )= 1 - F(0.8)= 1 - 0.7881 = 0.2119. Question ID: 19665 If a stock's return is normally distributed with a mean of 16 percent and a standard deviation of 50 percent, what is the probability of a negative return in a given year? A. B. C. D. 0.5000. 0.3745. 0.0001 0.4226. B Given the parameters, P(X<0)=P(Z<(0-16)/50)=P(Z<-0.32)=1-F(0.32)=1-0.6255. Question ID: 19666 A multivariate distribution: A. B. C. D. applies only to normal distributions. specifies the probabilities of groups of random variables. applies only to binomial distributions. gives multiple probabilities for the same outcome. B This is the definition of a multivariate distribution. 76 Question ID: 19668 In a multivariate normal distribution, a correlation tells the: A. B. C. D. relationship between the means and variances of the variables. relationship between the means and standard deviations of the variables. overall relationship between all the variables. strength of the linear relationship between two of the variables. D This is true by definition. The correlation only applies to two variables at a time. Question ID: 19669 In addition to the usual parameters that describe a normal distribution, to completely describe 10 random variables, a multivariate normal distribution requires knowing the: A. B. C. D. overall correlation. 10 correlations. 20 correlations. 45 correlations. D Since we can draw 10!/(8!2!)=45 pairs out of 10 objects, that is the number of correlations there will be in a multivariate normal distribution describing 10 variables. Question ID: 19670 Shortfall risk is the: A. B. risk that an earnings announcement falls below its forecast. probability that the next forecast falls below the previous forecast. 77 C. D. risk a portfolio will fall below a certain value. probability that an index portfolio will fall short of its benchmark. C Question ID: 19670 Shortfall risk is the: A. B. C. D. risk that an earnings announcement falls below its forecast. probability that the next forecast falls below the previous forecast. risk a portfolio will fall below a certain value. probability that an index portfolio will fall short of its benchmark. B The safety-first criterion says choose the portfolio with the largest value for SFR= (mean-four percent)/variance. The mean=19 and variance=28 yields the highest SFR. Question ID: 19673 The mean and variance of four portfolios are listed below in percentage terms. The returns are normally distributed. Using Roy's safety first criteria and a threshold of 3 percent, select the mean and variance that corresponds to the optimal portfolio. Mean 8 Mean 14 Mean 5 Mean 19 Variance 10 Variance 20 Variance 3 Variance 28 A. B. C. D. 78 C The safety-first criterion says choose the portfolio with the largest value for SFR= (mean-three percent)/variance. The mean=5 and variance=3 yields the highest SFR. Question ID: 19674 X is a normally distributed random variable, and Y= e . If the mean of X increases then we know that: X A. B. C. D. only the mean of Y increases. there is no effect on either the mean of Y or the variance of Y. both the mean and variance of Y increases. only the variance of Y increases. C Y follows a lognormal distribution where E(Y) = eE(X)+V(X)/2 and V(Y)= e2 x E(X)+V(X) – (eV(X) – 1). Increasing the mean of X will increase both the mean and variance of Y. This is because of the skewness of the lognormal distribution. Question ID: 19676 A stock increased in value last year. Which will be greater, its continuously compounded or its discrete return? A. B. C. D. Its continuously compounded return. Not enough information to answer. Neither, they will be equal. Its discrete return. D When a stock increases, the discrete return is always greater than the continuously compounded return. 79 Question ID: 19677 If a stock decreases in one period and then increases by an equal dollar amount in the next period, will the arithmetic average of the discretely and continuously compounded rates of return be positive, negative, or zero? Continuously Discretely Compounded positive Discretely Compounded positive Discretely Compounded zero Discretely Compounded zero A. Compounded positive Continuously B. Compounded zero Continuously C. Compounded positive Continuously D. Compounded zero B The discrete return will have an upward bias that will give a positive average. The continuously compounded return will have an arithmetic average of zero. Question ID: 19678 Given a holding period return of R, the continuously compounded rate of return is: A. B. C. D. 1/(1-R). eR</SUP.< font> R 0.5 ln(1+R). D This is the formula for the continuously compounded rate of return. 80 Question ID: 19680 In which of the following cases would Monte Carlo simulation NOT be needed? Payoff of a: A. B. C. D. European option. roulette wheel. GNME. convertible bond with a call feature. B The probability distribution of a roulette wheel would be easy to estimate using empirical or a priori methodology. Question ID: 19681 Monte Carlo simulation is necessary to: A. B. C. D. approximate solutions to complex problems. reduce sampling error. determine a threshold return. compute continuously compounded returns. A This is the purpose of this type of simulation. The point is to construct distributions using complex combinations of hypothesized parameters. Question ID: 19682 Joan Biggs, CFA, acquires a large database of past returns on a variety of assets. Biggs then draws random samples of sets of returns from the database and analyzes the resulting distributions. Biggs is engaging in: A. historical simulation. 81 B. C. D. continuous analysis. discrete analysis. Monte Carlo simulation. A This is a typical example of historical simulation. Question ID: 19679 A stock decreased from $90 to $80, the continuously compounded rate of return is: A. B. C. D. -0.2310. -0.1178. -0.1000. -0.1250. B This is given by the natural logarithm of the new price divided by the old price; ln(80/90)=-0.1178. Question ID: 19626 If P(0≤X≤100) and p(50)=0, with respect to whether X is a continuous or discrete random variable, a statistician can say: A. B. C. D. nothing. X is discrete. X is continuous. X is bimodal. 82 A A few simple examples illustrate this. A discrete example is P(X=25)=P(X=75)=0.5. For any continuous distribution, the probability of X equaling any single number is zero. Question ID: 19635 Random variable X is continuous and bounded between zero and five, X:(0≤X≤5). The cdf for X is F(x)=x/5. Find P(2≤X≤4). A. B. C. D. 0.25. 1.00. 0.50. 0.40. D For a continuous distribution, P(a≤X≤b) = F(b)-F(a). Here, F(4)=0.8 and F(2)=0.4. Question ID: 19664 Standardizing a normally distributed random variable requires the: A. B. C. D. mean, variance and skewness. mean and the standard deviation. natural logarithm of X. variance and kurtosis. B The formula is the value of the realization minus the mean and that difference divided by the standard deviation. No other information is necessary. 83 84