Chartered Financial Analyst Level I Quantitative Methods

1.Adrian paid 100 for a portfolio and later marked the same holdings at 80. No cash moved in or out. What holding-period return do those two values imply?

2.Bianca's account rose 100% in the first year and then fell 50% in the second. She wants the compound rate that links the start to the end. What is the geometric mean return?

3.Caleb's account rose 100% one year and fell 50% the next. He averages those two yearly rates with equal weight and does not compound them. What arithmetic mean does he get?

4.Dahlia's account stayed at 100 until she deposited 20, which lifted the balance to 120. It then finished at 130. What time-weighted return uses the value just before that deposit?

5.Elena added a large cash deposit the day before her fund lost 10%. The time-weighted result stays near -10%, while the money-weighted result is worse. Which return does that deposit hurt?

6.Felix is quoted a stated annual rate of 12% with monthly compounding. What effective annual rate does a full year of that monthly compounding produce?

7.Grace is offered a stated annual rate of 8% that compounds once each quarter. What effective annual rate does she earn if the money stays in for a full year?

8.Hector will receive 1,000 in three years and discounts that single payment at 10% a year. What present value belongs on today's books?

9.Iris sets aside 200 for two years at 5% a year, compounded annually, and adds nothing along the way. What future value does that deposit reach?

10.Jonah will receive four payments of 100, the first one year from now, and he discounts at 10%. What is the present value of that ordinary annuity, to the nearest unit?

11.Keiko is promised 6 a year forever, with the first cash arriving in one year. She discounts the stream at 8%. What is that perpetuity worth today?

12.Liam's project costs 100 today and pays 121 in one year. He discounts that payoff at 10%. What net present value does the project produce?

13.Maya spends 100 today on a project that returns 110 in one year. What internal rate of return equates that payoff with the cost?

14.Nadia's sample is the three values 8, 10, and 12. She divides by the sample count minus one, not by the population count. What is the sample standard deviation?

15.Omar's sample has a mean of 10 and a sample standard deviation of 2. What coefficient of variation do those two figures imply?

16.Petra measures a covariance of 0.01 between two returns whose standard deviations are 0.20 and 0.10. What correlation do those three numbers imply?

17.Quinn finds that an asset's covariance with the market is 0.03 and that the market variance is 0.04. What beta do those two inputs give the asset?

18.Rosa holds two assets at weights of 50/50. Their standard deviations are 10 and 20, and the correlation is +1. What is the portfolio standard deviation?

19.Samir models a return as normal, with a stated mean and standard deviation. About what share of the observations fall within one standard deviation of the mean?

20.Tessa plots monthly returns and sees a long right tail, with a few very large gains. Relative to the median, where does the mean of that distribution sit?

21.Ulrich's hypothesis test rejects a null that later turns out to be true. The data looked extreme, but the null was correct. What kind of error is that rejection?

22.Vera's test returns a p-value of 0.03, and she set the significance level at 0.05 before seeing the data. What should she do with the null?

23.Wesley built a 90% confidence interval from a sample, then rebuilt it at 99% with that same sample. What happens to the width of the interval?

24.Ximena invests 100 at a continuous rate of 12% and leaves it for one year. What is the ending value, using e^0.12?

25.Yara's note pays 3% each six months, so the bond-equivalent yield is 6%. What effective annual yield does that semiannual compounding produce?

26.Zane earned a holding-period return of 10% and wants the matching continuously compounded rate. Using ln(1.10), what rate does he record?

27.Amara's prior P(A) is 0.25. Signal B appears with probability 0.80 if A is true and 0.10 if A is false. What is P(A|B), to two decimals?

28.Bruno faces two equally likely outcomes, a gain of 6 and a loss of 4, each with probability 0.5. What is the expected value of one draw?

29.Celine's two outcomes are +6 and -4, each with probability 0.5, and the expected value is 1. What is the standard deviation of one draw?

30.Diego's sample standard deviation is 6, and the sample contains 36 observations. What is the standard error of the sample mean?

31.Esther computed a standard error from a sample she now plans to quadruple. By what factor does that larger sample multiply the standard error?

32.Farid studies a skewed population of fund flows and draws a sample of n = 200. What does the central limit theorem say about the mean of samples like his?

33.Gina computes a sample variance from 15 monthly returns. How many degrees of freedom does that sample variance use?

34.Hugo wants a share that does not assume a normal shape. Using the Chebyshev bound with k = 2, at least what fraction of observations lie within two standard deviations of the mean?

35.Ingrid's portfolio earned an excess return of 8. Its standard deviation is 16, and she refuses to put beta in the denominator. What Sharpe ratio results?

36.Jules will receive four payments of 100, the first of them today, and the discount rate is 10%. The ordinary annuity of those payments is about 317. What is the annuity due worth, about?

37.Kira expects the next cash flow to be 3, then growth of 4% a year forever. She discounts at 10%. What is that growing perpetuity worth today?

38.Leon values a conventional project at 8% and again at 12%. The promised inflows do not change. What happens to net present value when the discount rate rises?

39.Mira can finance both of two independent projects. One has a net present value of +4 and the other has a net present value of -1. Which should she accept?

40.Nico must pick one of two mutually exclusive projects. At his cost of capital, the first has the higher internal rate of return and the lower net present value. Which does he take?

41.Olga ranks two projects by a two-year payback. North pays nothing for two years, then a large sum. South returns cash sooner and then stops, so payback picks South. What does that ranking miss?

42.Paolo's five observations are 4, 4, 5, 9, and 40. The median is 5 and the mean is 12.4. Which measure does the outlier 40 move?

43.Rhea treats two research signals as independent, with probabilities 0.40 and 0.25. What is the joint probability that both signals occur?

44.Soren's desk will hire one analyst or the other, never both. Those two hiring outcomes cannot occur together. What is their joint probability?

45.Tara combines two uncorrelated positions with variances 16 and 9 at equal weights of 50/50. The correlation is 0. What is the portfolio standard deviation?

46.Uma mixes two assets with standard deviations 10 and 20 and correlation -1. What weight on the lower-risk asset, the one with standard deviation 10, drives portfolio risk to zero?

47.Victor models a price as lognormal, so the curve stays above zero, while the return can still be negative. What cannot happen to the modeled price?

48.Willa's client will leave the money invested across the whole multi-year stretch, with no withdrawals. Which average should summarize that compound growth?

49.Xavier prices a Treasury bill with face value 10,000 at a price of 9,800, with 90 days left. What is the bank discount yield?

50.Yvette holds a Treasury bill with face 10,000, price 9,800, and 90 days to maturity. What money-market yield does that quote imply?

51.Zachary is quoted a bond-equivalent yield of 8%, which means 4% each six months. What effective annual yield does compounding those two periods produce?

52.Aisha deposits 100 at the end of each year for three years at 10%. What future value is in the account just after the third payment?

53.Blake puts a 60% weight on an asset with expected return 8% and a 40% weight on an asset with expected return 14%. What is the portfolio expected return?

54.Camille's two assets have a correlation of 0.40 and standard deviations of 10% and 20%. What covariance do those inputs imply?

55.Darius averages only the funds that are still open. Several funds on the original list closed after losses and were dropped. What does that average do to someone who could have picked a fund that later closed?

56.Elise forms this year's portfolio from market caps measured at the end of that same year. In January, that year-end size was not known. What is wrong with using it for the January holdings?

57.Farah tried twenty valuation ratios and published the one that worked inside her sample. What does she still have to do before treating that winner as a rule?

58.Gideon finds a return link that shows up in only one decade, then writes it down as a rule for every later period. What is off about that claim?

59.Hana wants every client account on a numbered list to have the same chance of being picked. Which draw is that simple random sample?

60.Ivan groups the bonds by maturity, then makes a random draw inside each maturity group. Which sampling procedure did he follow?

61.Juliet's 95% interval for a fund's mean return runs from 4% to 9%. A colleague says the mean wanders inside that interval with probability 95%. What does the interval claim instead?

62.Kenji has 12 draws from a normal population, and the variance is unknown. Which distribution should he use for a test of the mean?

63.Lila tests whether a tracking difference differs from zero, and she will reject for a large positive gap or a large negative gap. What kind of test is that?

64.Marcus moves his significance level from 5% to 1% and leaves the rest of the test alone. What happens to Type I and Type II errors?

65.Naomi's test fails to detect a real 2% mispricing 20% of the time. What is the power of that test?

66.Oscar's sample mean has an expected value equal to the population mean he is trying to estimate. Which property does that equality describe?

67.Rafael keeps adding years of returns, and his estimate tightens on the true mean as the record grows. Which property is that tightening?

68.Selene has two unbiased estimators of the same mean return. One of them has a smaller variance. Which description fits the smaller-variance estimator?

69.Tomas records three results, 1, 2, and 6. What is the mean absolute deviation of that trio?

70.Una spends the same dollar amount on a share at a price of 2 and again at a price of 8. What harmonic mean price did those two purchases produce?

71.Vincent sees a nominal rate of 8% and inflation of 3%. Using the exact relation, not the plain gap between those rates, what real rate does he get?

72.Wendy sees a 2-year yield of 4 and a 1-year yield of 3. Lenders want extra pay for tying money up. In that liquidity-preference description, what can the extra 1 include?

73.Yasmin's project cash flows are -100, then +230, then -132. The sign changes more than once. What should she not do with the internal rate of return?

74.Andre mixes a risk-free asset with one risky fund and plots expected return against risk. What shape is that capital allocation line?

75.Bridget's mean estimate is 20 and the standard error is 2. Using 2 standard errors for about 95% of a normal sampling distribution, what interval does she report?

76.Cole's test statistic is 2.4, and the critical value is 1.96 for a two-sided 5% test. What should he conclude about the null?

77.Delia's wealth relatives over three years are 1.10, 0.90, and 1.20. What three-year holding-period return do those relatives produce?

78.Emilio owns a common stock and cannot be asked to pay in more than he invested. What is the worst holding-period return that limited liability allows?

79.Fiona searched many trading rules on one history and found a rule that looked strong. What is the honest next step after that search?

80.Graham's bill has a face value of 1,000, a price of 990, and 180 days to maturity. What is the money-market yield?

81.Hannah's portfolio return is 11, the risk-free rate is 3, beta is 1.25, and the market excess return is 8. What is Jensen's alpha?

82.Isaac finds positive autocorrelation in a stock's weekly returns. How does the annual variance compare with 52 times the weekly variance?

83.Jade runs twenty independent tests, each at a 5% significance level, and every null is actually true. How many false rejections does she expect?

84.Karim invests 1 at 8% a year for 3 years, compounded annually. What future-value factor, 1.08^3, should he use?

85.Leah will receive 126 in three years and discounts at 8% a year. Using 1.08^3 = 1.259712, what present value does she record?

86.Malik has a sample that contains one return and nothing else. What is the sample standard deviation of that single observation?

87.Nora models returns with a normal curve and asks for the probability that the return is exactly 4.137%. What probability does that continuous model assign to the single point?

88.Otto holds two assets at weights of 50/50. The variances are 25 and 16, and the covariance is -4. What is the portfolio variance?

89.Pia buys 1,000 dollars of shares at 20 and another 1,000 dollars of the same shares at 30. What average price per share did she pay?

90.Quentin already has a confidence interval for the mean return. He now wants a range for one future return from the same process. How does that prediction interval compare?

91.Renata must assign 3 different jobs to 3 people, and which person gets which job matters. How many ordered assignments are there?

92.Stefan must choose 2 funds out of 5, and the order inside each pair does not matter. How many different pairs can he form?

93.Thea assigns probability 0.2 to a default. What are the odds against that default?

94.Usman reviews 200 bonds. Sixty of them are callable, and 30 of those 60 callable bonds are high-yield. What is the probability a bond is high-yield given that it is callable?

95.Violet's desk holds 200 bonds, and 40 of them are high-yield. What is the unconditional probability that a bond drawn from the desk is high-yield?

96.Walter's active returns are 1, -1, 1, and -1, so the mean active return is 0. What is the sample standard deviation of those four active returns, about?

97.Xia's portfolio beat its benchmark by an active return of 2%, and the tracking error of that active return is 4%. What information ratio do those two figures give?

98.Yusuf sets a shortfall level of 4%. His portfolio has an expected return of 10% and a standard deviation of 12%. A rival portfolio has a standardized distance of 0.25. What is this portfolio's ratio?

99.Zara subtracts a minimum acceptable return of 4, not the Treasury bill yield of 1, and then divides by the standard deviation. Which ratio did she form?

100.Ansel compounds 100 continuously at 6% for two years. What ending value does e^(0.12) give him?

101.Beatrice will receive 100 in two years. The annual effective rate is 6%, so the two-year growth factor is 1.1236. What present value does she record?

102.Cyrus has three paired deviations, (1, 2), (0, 0), and (-1, -2), and he uses the sample divisor. What sample covariance do those pairs give?

103.Dana's return history has more large negative surprises than large positive ones. Where does the mean sit relative to the median?

104.Edgar, who watches for crashes, finds a return distribution whose kurtosis sits above the normal value of 3. What does that kurtosis say about the tails?

105.Freya's spreadsheet asks for the covariance of a return series with that same series. What should the cell return?

106.Grant's returns are 2, 4, and 6, and the sample standard deviation is 2. He adds 10 to every return, getting 12, 14, and 16. What is the new sample standard deviation?

107.Helena's sample is 2, 4, and 6, with sample standard deviation 2. She multiplies every return by 2. What is the sample standard deviation of the doubled sample?

108.Idris holds three assets at equal weights of one-third. Each variance is 9, and each correlation is 0. What is the portfolio variance?