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Study guide · Session 1

Quantitative Methods

11-14% of the exam · session 1 · 11 learning modules

Quantitative Methods gives you the tools the rest of the curriculum relies on: measuring returns, discounting cash flows, describing and modelling return data, testing hypotheses, building portfolios, simulating outcomes and fitting a simple regression. Level I questions are often short calculations on a financial calculator, mixed with items on choosing the right measure, test or model and interpreting the result.

A study feature written by ExamDragon, not part of the real exam. Each heading is one of this topic's 11 learning modules in the 2027 curriculum, named in our own words.

At a glance

Session 1
11-14%

of the exam, as CFA Institute weights it for 2027

Weight
11-14%
Modules
11
Key terms
12

What this guide covers, module by module

  1. QM.1What returns are and what investors require
  2. QM.2Measuring returns in different ways
  3. QM.3Money-weighted, time-weighted and index returns
  4. QM.4Discounting cash flows and no-arbitrage pricing
  5. QM.5Describing return data: centre, spread, shape
  6. QM.6Probability models for prices and returns
  7. QM.7Sampling, confidence intervals and hypothesis tests
  8. QM.8Portfolio risk, diversification and the CML
  9. QM.9Historical, bootstrap and Monte Carlo simulation
  10. QM.10Simple regression and the CAPM beta
  11. QM.11Big data and machine learning in investing

QM.1What returns are and what investors require

  • A return has two parts: income (dividends, coupons, rent) and the change in price.
  • The required return is the minimum return an investor accepts for an investment's risk; it is set before investing, whereas a realised return is measured afterwards.
  • The nominal risk-free rate combines the real risk-free rate and expected inflation: (1 + nominal) = (1 + real) x (1 + inflation), roughly real + inflation.
  • A risky asset's required return adds premiums to the nominal risk-free rate for default risk, liquidity risk and maturity risk.
  • A premium can only be isolated by comparing instruments that differ in one respect: same issuer and terms but less trading gives the liquidity premium.
  1. The situationThe real risk-free rate is 1.5%, expected inflation is 2.5%, and a corporate bond carries a 1.2% default premium and a 0.3% liquidity premium.
  2. Step 1Nominal risk-free rate = 1.015 x 1.025 - 1 = 4.04%.
  3. Step 2Add the premiums: 4.04% + 1.2% + 0.3% = 5.54%.
  4. Where questions turnA rise in an asset's required return with the risk-free rate and inflation unchanged must come from its risk premiums, not from the real rate.
  5. The takeawayThe bond must offer about 5.54% to compensate for time, inflation and its own risks.
Practise QM.1

QM.2Measuring returns in different ways

  • Holding period return (HPR) = (P1 - P0 + income) / P0, for any length of period.
  • Arithmetic mean return is the simple average of period returns; the geometric mean, (product of (1 + Ri))^(1/n) - 1, is the compound growth rate and is never above the arithmetic mean.
  • Annualised return = (1 + HPR)^(periods per year) - 1, which lets you compare investments held for different lengths of time.
  • Continuously compounded return = ln(P1 / P0) = ln(1 + HPR); these returns add across periods.
  • Gross return is after trading costs but before fees; net return is after fees; after-tax return also deducts taxes; real return removes inflation.
  • A leveraged return = (return on all assets - interest paid) / own equity; borrowing magnifies gains and losses.
  1. The situationA fund gains 50% in year 1 and loses 50% in year 2.
  2. Step 1Arithmetic mean = (50% - 50%) / 2 = 0%.
  3. Step 2Geometric mean = (1.50 x 0.50)^(1/2) - 1 = 0.75^(1/2) - 1 = -13.40%.
  4. Step 3An investment of 100 is worth 75 after two years, which matches the geometric mean.
  5. Where questions turnDividing a cumulative return by the number of years, or multiplying a quarterly return by four, ignores compounding.
  6. The takeawayVolatile returns open a gap between the two means; the geometric mean tells you what an investor actually earned.
Practise QM.2

QM.3Money-weighted, time-weighted and index returns

  • The money-weighted return (MWR) is the internal rate of return of the investor's own cash flows: deposits, withdrawals and the ending value.
  • The time-weighted return (TWR) links the returns of sub-periods between cash flows geometrically, so it is unaffected by the size or timing of deposits and withdrawals.
  • Use TWR to judge a manager who does not control client flows; MWR shows the investor's own experience, including the timing of their flows.
  • A price-weighted index weights each stock by its price; its return is the change in the sum of prices.
  • A market-capitalisation-weighted index weights by market value (often float-adjusted) and rebalances itself; an equal-weighted index tilts toward small companies and needs regular rebalancing.
  • A price return index ignores income; a total return index reinvests dividends and interest.
  1. The situationAn investor puts 100 into a fund that returns 10% in year 1, adds another 100 at the end of year 1, and the fund then returns -5% in year 2, ending at 199.5.
  2. Step 1TWR = (1.10 x 0.95)^(1/2) - 1 = 2.23% a year.
  3. Step 2MWR solves 100 + 100 / (1 + r) = 199.5 / (1 + r)^2, which gives r = -0.17%.
  4. Step 3The MWR is lower because more money was invested just before the weak year.
  5. Where questions turnMWR rewards or punishes the timing of cash flows, so it is the wrong measure for a manager who has no say over them.
  6. The takeawayThe same fund can show a positive TWR and a negative MWR when the investor adds money at the wrong time.
Practise QM.3

QM.4Discounting cash flows and no-arbitrage pricing

  • Present value: PV = FV / (1 + r)^n; future value: FV = PV x (1 + r)^n.
  • A bond's price is the present value of its coupons and face value; a discount rate above the coupon rate gives a price below par.
  • A perpetuity is worth A / r; the constant growth (Gordon) model values a share at D1 / (r - g), with D1 = D0 x (1 + g).
  • Rearranged, the same equations give implied returns, r = (FV / PV)^(1/n) - 1 for a zero-coupon bond, and implied growth, g = r - D1 / P0, for a share.
  • Cash flow additivity: the value of a set of cash flows is the sum of their present values, so two packages with identical cash flows must have the same price (no arbitrage).
  • No arbitrage gives implied forward rates, (1 + S2)^2 = (1 + S1) x (1 + f1,1), forward exchange rates and, by replication, option values.
  1. The situationA preferred share pays GBP 4.00 a year forever and investors require 6.4%. The 1-year spot rate is 3% and the 2-year spot rate is 4%.
  2. Step 1Preferred share value = 4.00 / 0.064 = GBP 62.50.
  3. Step 2Implied 1-year forward rate one year ahead = 1.04^2 / 1.03 - 1 = 5.01%.
  4. Where questions turnThe Gordon model needs next year's dividend; using the dividend just paid understates the value.
  5. The takeawayOne discounting idea prices a perpetuity, and the same no-arbitrage logic pulls a forward rate out of two spot rates.
Practise QM.4

QM.5Describing return data: centre, spread, shape

  • Central tendency: arithmetic mean, median (middle value, robust to outliers), mode; trimmed and winsorised means limit the effect of extremes.
  • Location: quartiles, quintiles, deciles and percentiles split ordered data; the interquartile range is Q3 - Q1.
  • Dispersion: range, mean absolute deviation, variance and standard deviation; the sample variance divides by n - 1.
  • Target semi-deviation measures only returns below a target, which matches how investors see downside risk; the coefficient of variation, CV = s / mean, gives risk per unit of return.
  • Skewness: positive skew means a long right tail and mean > median > mode. Excess kurtosis above zero means fatter tails than the normal distribution.
  • Covariance shows whether two variables move together; correlation = covariance / (s1 x s2), between -1 and +1, measures only linear association.
  1. The situationA fund's annual returns are 2%, 6% and 10%.
  2. Step 1Mean = 6%.
  3. Step 2Sample variance = (16 + 0 + 16) / (3 - 1) = 16, so s = 4%.
  4. Step 3CV = 4 / 6 = 0.67.
  5. Where questions turnZero correlation rules out only a linear relationship; it does not prove the returns are independent.
  6. The takeawayThe fund carries 0.67 units of standard deviation per unit of mean return, which you can compare with other funds.
Practise QM.5

QM.6Probability models for prices and returns

  • Expected value = sum of probability x outcome; variance = sum of probability x (outcome - expected value)^2.
  • The total probability rule gives an unconditional expectation from conditional ones: E(R) = sum of P(scenario) x E(R | scenario).
  • Bayes' formula updates a prior with new information: P(A | B) = P(B | A) x P(A) / P(B).
  • The normal distribution is described by its mean and variance; about 68% of outcomes lie within 1 standard deviation, 95% within 1.96 and 99% within 2.58.
  • A standard normal z = (x - mean) / standard deviation; the lognormal distribution, bounded below by zero and skewed to the right, suits asset prices.
  • The t-distribution has fatter tails than the normal and approaches it as degrees of freedom rise; chi-square and F distributions are used for variances.
  1. The situationAn analyst gives a 25% probability to a recession, with an expected equity return of -4% in a recession and 10% otherwise.
  2. Step 1E(R) = 0.25 x (-4%) + 0.75 x 10%.
  3. Step 2E(R) = -1.0% + 7.5% = 6.5%.
  4. Where questions turnIn Bayes' formula the denominator is the total probability of the new information across all states, not the prior.
  5. The takeawayThe unconditional expected return of 6.5% weights each scenario's conditional expectation by its probability.
Practise QM.6

QM.7Sampling, confidence intervals and hypothesis tests

  • The central limit theorem: for large samples (often n of 30 or more) the sample mean is approximately normal with standard error s / sqrt(n), whatever the population's shape.
  • Confidence interval = point estimate plus or minus reliability factor x standard error; use z with a known population variance or a large sample, t otherwise.
  • Sampling can be simple random, stratified (by subgroup), cluster or convenience; biases include data snooping, survivorship and look-ahead bias.
  • Hypothesis testing steps: state H0 and Ha, choose the test statistic and significance level, set the decision rule, calculate, decide.
  • Type I error: reject a true H0 (probability alpha). Type II error: fail to reject a false H0 (probability beta). Power = 1 - beta.
  • Non-parametric tests, such as the Spearman rank correlation test, suit ranked data, small samples or data that breach parametric assumptions.
  1. The situationA strategy's 100 monthly returns have a mean of 0.8% and a standard deviation of 4.0%. Test H0: mean = 0 at the 5% level, two-tailed.
  2. Step 1Standard error = 4.0 / sqrt(100) = 0.4%.
  3. Step 2Test statistic = 0.8 / 0.4 = 2.0.
  4. Step 32.0 exceeds the critical value of about 1.98 (t, 99 degrees of freedom), so reject H0.
  5. Where questions turnUse the standard error, s / sqrt(n), in intervals and tests about the mean, never the standard deviation of the data itself.
  6. The takeawayThe mean return is significantly different from zero at the 5% level, though the margin is small.
Practise QM.7

QM.8Portfolio risk, diversification and the CML

  • Portfolio expected return is the weighted average of asset expected returns.
  • Two-asset variance = w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 rho s1 s2; any correlation below +1 reduces portfolio risk below the weighted average.
  • The minimum-variance frontier holds the lowest-risk portfolio for each expected return; its upper part, above the global minimum-variance portfolio, is the efficient frontier.
  • Combining the risk-free asset with the tangency portfolio gives the capital allocation line with the highest Sharpe ratio, (E(R) - Rf) / s.
  • Each investor picks the point where an indifference curve touches the line: more risk-averse investors lend, less risk-averse investors borrow.
  • With homogeneous expectations the tangency portfolio is the market portfolio and the line becomes the capital market line (CML).
  1. The situationTwo assets each have a standard deviation of 20%, a correlation of zero, and are held 50/50.
  2. Step 1Variance = 0.25 x 0.04 + 0.25 x 0.04 + 0 = 0.02.
  3. Step 2Standard deviation = sqrt(0.02) = 14.14%.
  4. Where questions turnRisk aversion decides how much to put in the risk-free asset, not which risky portfolio to hold.
  5. The takeawayUncorrelated assets cut risk from 20% to about 14% with no loss of expected return.
Practise QM.8

QM.9Historical, bootstrap and Monte Carlo simulation

  • Simulation estimates the distribution of outcomes, such as a portfolio's value at risk, when a formula is impractical.
  • Historical simulation replays observed past returns; it makes no distributional assumption but cannot produce events that never occurred.
  • Bootstrap resampling draws repeatedly, with replacement, from the observed sample to build many simulated samples and the sampling distribution of a statistic.
  • Monte Carlo simulation draws from an assumed distribution with chosen parameters; it can model scenarios outside history and complex payoffs, at the cost of model risk.
  • Steps of a Monte Carlo study: specify the variables and their distributions, draw many trials, value the portfolio in each, then summarise the results.
  1. The situationA risk team runs 10,000 Monte Carlo trials of a portfolio's one-month value and wants the 5% value at risk.
  2. Step 1Sort the 10,000 simulated outcomes from worst to best.
  3. Step 2The 500th worst outcome marks the 5th percentile.
  4. Step 3The loss at that point is the 5% one-month value at risk.
  5. Where questions turnDrawing from an assumed distribution is Monte Carlo; resampling the data itself with replacement is the bootstrap.
  6. The takeawaySimulation turns assumptions into a full distribution from which any percentile can be read.
Practise QM.9

QM.10Simple regression and the CAPM beta

  • Simple linear regression fits Y = b0 + b1 X + e by ordinary least squares, minimising the sum of squared residuals; b1 = cov(X, Y) / var(X) and b0 = mean of Y - b1 x mean of X.
  • Assumptions: linearity, homoskedasticity (constant residual variance), independence of residuals and normally distributed residuals; residual plots reveal violations.
  • Goodness of fit: R^2 = explained variation / total variation = (SST - SSE) / SST; in simple regression R^2 equals the squared correlation.
  • ANOVA: F = MSR / MSE; with one independent variable, F equals the square of the slope's t-statistic. The standard error of the estimate is sqrt(SSE / (n - 2)).
  • A prediction interval for Y is wider than the standard error of the estimate alone because it also reflects uncertainty in the coefficients.
  • Functional forms: log-lin (ln Y on X), lin-log (Y on ln X), log-log (both logged, slope is an elasticity). In the CAPM regression of excess returns, the slope is beta and the intercept is alpha.
  1. The situationA regression of a stock's excess return on the market's excess return gives an intercept of 0.2% and a slope of 1.1. Next month the market's excess return is forecast at 4%.
  2. Step 1Predicted excess return = 0.2 + 1.1 x 4.
  3. Step 2Predicted excess return = 4.6%.
  4. Where questions turnDivide SSE by n - 2, not n or n - 1, for the standard error of the estimate in simple regression.
  5. The takeawayA beta of 1.1 makes the stock move slightly more than the market, and the 0.2% intercept is its estimated alpha.
Practise QM.10

QM.11Big data and machine learning in investing

  • Big data is described by volume (quantity), velocity (speed), variety (types and sources) and veracity (reliability); value is often added.
  • Sources include traditional data (filings, prices) and alternative data from individuals (social media), business processes (transactions) and sensors (satellites, devices).
  • Supervised learning trains on labelled data to predict a target (regression, classification); unsupervised learning finds structure in unlabelled data (clustering, dimension reduction); deep learning uses layered neural networks.
  • Overfitting means a model learns the noise of its training data and fails on new data; it is checked by testing on data held out from training.
  • Investment uses include text analysis with natural language processing, fraud detection, algorithmic trading, robo-advice and risk analysis.
  1. The situationA lender has five years of loan records, each marked as repaid or defaulted, and wants to score new applicants.
  2. Step 1The records carry the outcome, so this is a supervised classification problem.
  3. Step 2Train the model on part of the data and test it on the rest to check for overfitting.
  4. Where questions turnNatural language processing names the kind of data handled (text), not whether learning is supervised.
  5. The takeawayWhether the target is labelled decides between supervised and unsupervised learning.
Practise QM.11

Key terms

Holding period return
Price change plus income, divided by the beginning price, for the period held.
Geometric mean return
The constant compound rate that links the beginning and ending values.
Money-weighted return
The internal rate of return of an investor's own cash flows.
Time-weighted return
Sub-period returns linked geometrically, so the timing of client flows has no effect.
Cash flow additivity
The value of a set of cash flows equals the sum of their present values.
Coefficient of variation
Standard deviation divided by the mean: risk per unit of return.
Excess kurtosis
Kurtosis minus 3; positive values mean fatter tails than the normal distribution.
Standard error
The standard deviation of a sample statistic, for the mean s / sqrt(n).
Power of a test
The probability of correctly rejecting a false null hypothesis, 1 - beta.
Capital market line
Combinations of the risk-free asset and the market portfolio.
Bootstrap
Resampling the observed data with replacement to simulate many samples.
Coefficient of determination
R^2, the share of the variation in Y that the regression explains.

After this guide: You should be able to calculate and compare the main return measures, value cash flows, describe and test return data, compute two-asset portfolio risk, choose a simulation method and read a simple regression.

Practise Quantitative Methods

Built to the topics and weights CFA Institute publishes for the 2027 Level I curriculum. The module names, explanations, worked examples and traps are ExamDragon's own. Every source this path is built on.