Quantitative Methods

    Exam weight 6–9%
    166 questions in the bank
    9 subtopics

    Quant carries a modest weight but an outsized influence, because the rest of the curriculum runs on it. Discounting shows up in equity and fixed income; distributions and hypothesis testing show up wherever a result has to be defended. Time spent here pays interest elsewhere.

    What you need to be able to do

    • Move confidently between present value, future value, and annuities without re-deriving each time
    • Choose the right measure of return for the question asked: holding period, time-weighted, or money-weighted
    • State a null hypothesis correctly, then read a test statistic against it
    • Interpret a regression output rather than compute one

    Where candidates lose marks

    Calculator settings. Payments per year, begin versus end mode, and a stale register have failed more questions here than any concept.

    10 free Quant practice questions

    Real questions from the CFAQuiz bank, one or two per subtopic, with the full explanation. No account needed.

    1
    Hypothesis Testing
    easy
    QUANT-HYPO-005

    A quantitative researcher extracts a random sample of 20 observations from a normally distributed asset return population whose true variance is completely unknown. To execute a valid hypothesis test concerning the exact value of the population mean, the most appropriate test statistic to apply is the:

    1. A.z-statistic.
    2. B.t-statistic.
    3. C.Chi-square statistic.
    Show answer and explanation

    Correct answer: B

    When testing a hypothesis concerning a single population mean from a normally distributed population where the population variance is unknown, the t-statistic is the theoretically correct choice regardless of sample size. Option A is a named mistake (incorrectly choosing a z-test when population variance is unknown) because a z-test requires either a known population variance or a very large sample size where the sample standard deviation can serve as a robust asymptotic proxy via the Central Limit Theorem. Option C is a named mistake (confusing a test of the mean with a test of variance) because the Chi-square statistic is utilized to test hypotheses regarding a single population variance, not a population mean.

    2
    Linear Regression
    medium
    QUANT-REGR-009

    A quantitative researcher fits a simple linear regression model using a random sample of 27 daily observations. The estimated slope coefficient (b1b_1) is 2.10, and its standard error (sb1s_{b_1}) is 0.45. The critical t-value for a two-tailed 5% significance level with 25 degrees of freedom is 2.060, and for 26 degrees of freedom is 2.056. The lower bound of the 95% confidence interval for the true population slope coefficient is closest to:

    1. A.1.22
    2. B.1.65
    3. C.1.17
    Show answer and explanation

    Correct answer: C

    Step 1: Identify the correct degrees of freedom for a simple linear regression coefficient test, which is n−2n - 2. Given n=27n = 27, df=27−2=25\text{df} = 27 - 2 = 25. Therefore, the appropriate critical t-value is tc=2.060t_c = 2.060. Step 2: Calculate the margin of error by multiplying the critical value by the standard error of the slope:

    Margin of Error=tc×sb1=2.060×0.45=0.9270\text{Margin of Error} = t_c \times s_{b_1} = 2.060 \times 0.45 = 0.9270

    Step 3: Calculate the lower bound of the confidence interval by subtracting the margin of error from the estimated slope:

    Lower Bound=b1−Margin of Error=2.10−0.9270=1.1730≈1.17\text{Lower Bound} = b_1 - \text{Margin of Error} = 2.10 - 0.9270 = 1.1730 \approx 1.17

    Option A is the result of a named mistake (incorrect critical distribution choice) where the candidate uses the standard normal z-distribution critical value of 1.96 instead of the t-distribution value: 2.10−(1.96×0.45)=1.218≈1.222.10 - (1.96 \times 0.45) = 1.218 \approx 1.22. Option B is the result of a named mistake (omitting the critical value scaling factor) where the candidate simply subtracts the unscaled standard error from the slope coefficient: 2.10−0.45=1.652.10 - 0.45 = 1.65.

    3
    Probability Concepts
    medium
    QUANT-PROB-008

    A portfolio manager wants to create an equally weighted satellite portfolio by picking exactly 3 unique stocks out of a pool of 8 eligible small-cap candidate stocks. The total number of unique stock combinations that can be selected is equal to:

    1. A.336
    2. B.56
    3. C.24
    Show answer and explanation

    Correct answer: B

    Step 1: Determine whether order matters. Because the portfolio is equally weighted, the order of stock selection is irrelevant, requiring the combinations counting rule (nCr_nC_r). Step 2: Apply the combinations mathematical formula:

    nCr=n!r!(n−r)!  ⟹  8C3=8!3!(8−3)!=8!3!×5!_nC_r = \frac{n!}{r!(n-r)!} \implies _8C_3 = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \times 5!}

    Step 3: Expand and simplify the factorials to find the numerical outcome:

    8C3=8×7×63×2×1=3366=56_8C_3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56

    Option A is the result of a named mistake (permutation formula substitution error) where the candidate uses the permutation formula (nPr_nP_r), incorrectly assuming that the order of stock selection changes the portfolio identity: $8! /5!=8×7×6=336/ 5! = 8 × 7 × 6 = 336. Option C is the result of a named mistake (linear cross-multiplication error) where the candidate simply multiplies the pool size by the sub-portfolio size: 8×3=248 \times 3 = 24.

    4
    Probability Distributions
    easy
    QUANT-DIST-010

    To completely define a multivariate normal distribution for a portfolio containing 5 unique assets, an analyst must specify a distinct number of correlation coefficients. The number of unique correlation coefficients required is equal to:

    1. A.10
    2. B.5
    3. C.25
    Show answer and explanation

    Correct answer: A

    Step 1: Recall the formula for the number of unique correlation coefficients in a multivariate normal distribution of nn variables:

    Number of correlations=n(n−1)2\text{Number of correlations} = \frac{n(n - 1)}{2}

    Step 2: Substitute n=5n = 5 into the formula:

    Number of correlations=5×(5−1)2=5×42=10\text{Number of correlations} = \frac{5 \times (5 - 1)}{2} = \frac{5 \times 4}{2} = 10

    Option B is the result of a named mistake (parameter dimension confusion) where the candidate equates the number of correlations to the number of assets, which actually represents the number of means or variances needed. Option C is the result of a named mistake (matrix sizing error) where the candidate squares the number of assets, confusing the unique pairs with the total number of entries in a full matrix grid: 52=255^2 = 25.

    5
    Rates of Return
    medium
    QUANT-RETS-010

    A private wealth portfolio consists of two asset classes:

    • 60% invested in taxable corporate bonds yielding a 5.00% pre-tax return.
    • 40% invested in tax-exempt municipal bonds yielding a 4.00% pre-tax return.

    Assuming the investor faces a marginal income tax rate of 30.00% on all interest from corporate bonds, the weighted after-tax return of the total portfolio is equal to:

    1. A.3.22%
    2. B.3.70%
    3. C.4.60%
    Show answer and explanation

    Correct answer: B

    Step 1: Calculate the after-tax return for each asset class individually. Taxable Corporate Bonds: Interest is fully taxed, so the after-tax yield is:

    Rafter-tax, corporate=5.00%×(1−0.30)=3.50%R_{\text{after-tax, corporate}} = 5.00\% \times (1 - 0.30) = 3.50\%

    Tax-exempt Municipal Bonds: No tax is levied, so the return remains:

    Rafter-tax, municipal=4.00%R_{\text{after-tax, municipal}} = 4.00\%

    Step 2: Use the asset allocation weights to find the total weighted portfolio after-tax return:

    Rportfolio=(0.60×3.50%)+(0.40×4.00%)R_{\text{portfolio}} = (0.60 \times 3.50\%) + (0.40 \times 4.00\%)

    Step 3: Sum the individual weighted products together:

    Rportfolio=2.10%+1.60%=3.70%R_{\text{portfolio}} = 2.10\% + 1.60\% = 3.70\%

    Option C is the result of a named mistake (pre-tax portfolio calculation) where the candidate calculates the standard pre-tax return of the portfolio, omitting taxes completely: (0.60×5%)+(0.40×4%)=4.60%(0.60 \times 5\%) + (0.40 \times 4\%) = 4.60\%. Option A is the result of a named mistake (blanket taxation error) where the candidate computes the pre-tax portfolio return (4.60%) and incorrectly subjects the entire amount to the 30% tax rate, ignoring the tax-exempt status of the municipal allocation: 4.60%×(1−0.30)=3.22%4.60\% \times (1 - 0.30) = 3.22\%.

    6
    Sampling and Estimation
    easy
    QUANT-SAMP-010

    According to the mathematical framework of the Central Limit Theorem, the mean and the variance of the sampling distribution of the sample mean (Xˉ\bar{X}) for a sample size nn drawn from an infinite population with mean μ\mu and variance σ2\sigma^2 are respectively equal to:

    1. A.μ\mu and σ2/n\sigma^2 / n
    2. B.μ\mu and σ/n\sigma / \sqrt{n}
    3. C.μ/n\mu / n and σ2/n\sigma^2 / n
    Show answer and explanation

    Correct answer: A

    The Central Limit Theorem defines the mean of the sampling distribution of the sample mean to be exactly equal to the population mean μ\mu. The variance of this sampling distribution is equal to the population variance divided by the sample size, σ2/n\sigma^2 / n. Option B is a named mistake (standard error confusion) because σ/n\sigma / \sqrt{n} represents the standard deviation of the sample mean (the standard error), not its variance. Option C is a named mistake that incorrectly downscales the mean parameter by the sample size denominator.

    7
    Statistical Measures
    easy
    QUANT-STAT-006

    An analyst compiles a small random sample of asset returns from an emerging market index: 4%, 7%, 8%, 9%, and 12%. The sample variance for these returns is closest to:

    1. A.6.80
    2. B.8.50
    3. C.2.92
    Show answer and explanation

    Correct answer: B

    Step 1: Calculate the sample mean: (4+7+8+9+12)/5=40/5=8%(4 + 7 + 8 + 9 + 12) / 5 = 40 / 5 = 8\%. Step 2: Compute the squared deviations from the mean:

    (4−8)2=16;(7−8)2=1;(8−8)2=0;(9−8)2=1;(12−8)2=16(4-8)^2 = 16; \quad (7-8)^2 = 1; \quad (8-8)^2 = 0; \quad (9-8)^2 = 1; \quad (12-8)^2 = 16

    Step 3: Sum the squared deviations: 16+1+0+1+16=3416 + 1 + 0 + 1 + 16 = 34. Step 4: Divide by n−1n - 1 because it is a sample variance:

    s2=345−1=344=8.50s^2 = \frac{34}{5 - 1} = \frac{34}{4} = 8.50

    Option A is the result of a named mistake where the candidate treats the sample as a population, incorrectly dividing the sum of squared deviations by nn instead of n−1n-1: 34/5=6.8034 / 5 = 6.80. Option C is the result of a named mistake where the candidate calculates the sample standard deviation instead of the sample variance: 8.50=2.92\sqrt{8.50} = 2.92.

    8
    Time Series and Big Data
    medium
    QUANT-TSBD-003

    An analyst estimates an autoregressive model of order 1, AR(1), for corporate sales growth:

    xt=b0+b1xt−1+ϵtx_t = b_0 + b_1 x_{t-1} + \epsilon_t

    The estimated intercept is b0=0.04b_0 = 0.04 and the slope coefficient is b1=0.60b_1 = 0.60. The current period sales growth is xT=0.05x_T = 0.05. The forecasted sales growth for two periods ahead, xT+2x_{T+2}, is:

    1. A.0.0820.082
    2. B.0.0700.070
    3. C.0.0580.058
    Show answer and explanation

    Correct answer: A

    Step 1: Calculate the forecast for one period ahead (T+1T+1):

    xT+1=b0+b1xT=0.04+0.60×0.05=0.04+0.03=0.07x_{T+1} = b_0 + b_1 x_T = 0.04 + 0.60 \times 0.05 = 0.04 + 0.03 = 0.07

    Step 2: Calculate the forecast for two periods ahead (T+2T+2) using the T+1T+1 forecast:

    xT+2=b0+b1xT+1=0.04+0.60×0.07=0.04+0.042=0.082x_{T+2} = b_0 + b_1 x_{T+1} = 0.04 + 0.60 \times 0.07 = 0.04 + 0.042 = 0.082

    Option B represents a forecasting horizon error, where the analyst stops after calculating the one-period-ahead forecast. Option C represents a compounding error, where the analyst incorrectly multiplies the slope by the one-period addition without adding the intercept first: 0.04+0.60×0.03=0.0580.04 + 0.60 \times 0.03 = 0.058.

    9
    Time Value of Money
    easy
    QUANT-TVM-001

    An institutional investor evaluates three different fixed-income certificates with the same credit risk and maturity. The certificates offer alternative compounding structures on a stated nominal annual interest rate of 6.00%. Which of these configurations provides the highest effective annual rate (EAR)?

    1. A.Continuous compounding.
    2. B.Daily compounding.
    3. C.Quarterly compounding.
    Show answer and explanation

    Correct answer: A

    By definition, for any given positive stated nominal interest rate, the effective annual rate (EAR) increases as the frequency of compounding increases. Continuous compounding represents the mathematical limit where interest is compounded infinitely many times per period, thus yielding the highest EAR. Option B represents daily compounding, which provides a lower EAR than continuous compounding because the compounding frequency is discrete and finite. Option C represents quarterly compounding, which has the lowest compounding frequency among the choices and therefore provides the lowest EAR. Mathematically, for a 6.00% nominal rate, quarterly compounding yields an EAR of (1+0.06/4)4−1=6.1364%(1 + 0.06/4)^4 - 1 = 6.1364\%, daily compounding yields (1+0.06/365)365−1=6.1831%(1 + 0.06/365)^{365} - 1 = 6.1831\%, and continuous compounding yields e0.06−1=6.1837%e^{0.06} - 1 = 6.1837\%.

    10
    Hypothesis Testing
    easy
    QUANT-HYPO-006

    An analyst performs a hypothesis test on a mutual fund manager's alpha and obtains a calculated test statistic that corresponds to a p-value of 0.035. If the test is evaluated against a standard 5% significance level, the analyst should most accurately conclude that the null hypothesis is:

    1. A.rejected because the p-value is less than or equal to the significance level.
    2. B.not rejected because the p-value is less than or equal to the significance level.
    3. C.not rejected because the calculated p-value indicates that there is a 3.5% probability that the null hypothesis is true.
    Show answer and explanation

    Correct answer: A

    The standard decision rule states that if the p-value is less than or equal to the significance level (α\alpha), the null hypothesis is rejected. Because 0.035≤0.050.035 \le 0.05, the result is statistically significant, and the null hypothesis must be rejected. Option B is a named mistake (inverted p-value decision rule) where the candidate flips the logical criteria for decision-making. Option C is a named mistake (misinterpretation of the p-value definition) because a p-value does not represent the probability that the null hypothesis is true; rather, it is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is completely true.

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    Other Level I topics

    Ethical and Professional Standards
    Exam weight 15–20%
    Economics
    Exam weight 6–9%
    Financial Statement Analysis
    Exam weight 11–14%
    Corporate Issuers
    Exam weight 6–9%
    Equity Investments
    Exam weight 11–14%
    Fixed Income
    Exam weight 11–14%
    Derivatives
    Exam weight 5–8%
    Alternative Investments
    Exam weight 7–10%
    Portfolio Management
    Exam weight 8–12%

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