CFA Level I — Quantitative Methods
2025 Program Curriculum · Volume 1 · Modules 1–11 · Complete Exam Reference
Returns
Statistics
Probability
Hypothesis Tests
Regression
Big Data
Statistics
Probability
Hypothesis Tests
Regression
Big Data
Module 1 — Interest Rates & Return Measures
Interest Rate ComponentsBuilding the Rate
r = Real risk-free rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium
Nominal risk-free (approx.)Real rate + Inflation premium
Exact nominal relation(1+nominal) = (1+real)(1+inflation)
Liquidity premiumCompensation for illiquid markets
Maturity premiumSensitivity to rate changes at longer maturities
T-bills proxy for the nominal risk-free rate in their country
Return FormulasKey Equations
Holding Period Return
R = (P₁ − P₀ + I₁) / P₀
Arithmetic Mean
R̄ = (1/T) Σ Rᵢₜ
Geometric Mean
R̄G = [∏(1+Rᵢₜ)]^(1/T) − 1
Harmonic Mean
X̄H = n / Σ(1/Xᵢ)
Always: Arithmetic ≥ Geometric ≥ Harmonic (unless all equal)
When to Use Which MeanMean Selection
ArithmeticSingle-period avg, no compounding
GeometricMulti-period compound growth
HarmonicRates, ratios, P/E, cost averaging
Trimmed/WinsorizedPresence of extreme outliers
Geometric is always ≤ arithmetic; gap grows with variance
Money-Weighted ReturnIRR Method
Σ CFₜ/(1+IRR)ᵗ = 0
- Accounts for timing & size of cash flows
- Investor inflows = outflows for fund (sign flip)
- Two investors, same fund → different MWR if different cash flow timing
- Excel: =IRR(values)
MWR ≠ appropriate for comparing portfolio managers — use TWR
Time-Weighted ReturnSub-period Link
- Breaks portfolio into sub-periods at each cash flow date
- Calculate HPR for each sub-period
- Chain-link (compound) all sub-period returns
- Take geometric mean of linked returns
- Not sensitive to cash flow timing
TWR is preferred for evaluating portfolio managers
Annualizing ReturnsCompounding & Non-Annual
EAR = (1 + periodic rate)ᵐ − 1
Continuous compounding: EAR = eʳ − 1
Non-annual compoundingAdjust rate & periods (r/m, n×m)
Continuously compounded rln(1 + HPR)
Return TypesGross / Net / Real / Leveraged
Gross returnBefore mgmt & admin expenses
Net returnAfter all expenses — what investor earns
After-tax nominalTotal return minus taxes on div/int/gains
Real returnAdjusts for inflation (useful cross-period)
Leveraged returnAmplifies gains AND losses
Module 2 — Time Value of Money in Finance
Fixed Income TVMBond Pricing
PV = Σ PMTₜ/(1+r)ᵗ + FV/(1+r)ᴺ
Discount bondFV only at maturity
YTMIRR of all promised cash flows
Implied return (discount)r = (FV/PV)^(1/t) − 1
Perpetuity: PV = PMT / r
Equity TVMDividend Models
Constant div: PV = D / r
Constant growth (Gordon): PV = D(1+g)/(r−g)
Implied return: r = D₁/PV + g
Implied growth: g = r − D₁/PV
Two-stage modelHigh growth (gs) then stable (gl)
Higher g reduces denominator → higher PV; r must exceed g
Cash Flow AdditivityNo-Arbitrage
- Identical future cash flows must have same PV today
- Forward rates derived by equating PV of cash flows
- FX forward: no-arbitrage rate from interest rate parity
- Option pricing: replicating portfolio + no-arbitrage
Principle: Two instruments with identical cash flows must have identical prices
Module 3 — Statistical Measures of Asset Returns
Central TendencyMeasures of Location
MeanΣxᵢ/n — sensitive to outliers
MedianMiddle value — robust to outliers
ModeMost frequent value
Percentile / QuartileDivide sorted data into portions
IQRQ3 − Q1 (middle 50%)
Outlier fenceQ3 + 1.5×IQR (upper) / Q1 − 1.5×IQR (lower)
Box & whisker plot: box = IQR, line inside = median
Dispersion MeasuresRisk Quantification
MAD = Σ|Xᵢ − X̄| / n
s² = Σ(Xᵢ − X̄)² / (n−1)
s = √s² (same units as data)
Semideviation = √[Σ(Xᵢ−B)²/(n−1)] for Xᵢ ≤ B
CV = s / X̄ (risk per unit of return)
Use n−1 (degrees of freedom) for sample variance — ensures unbiased estimate
Distribution ShapeSkewness & Kurtosis
Positive skew (right)Mean > Median > Mode; right fat tail
Negative skew (left)Mean < Median < Mode; left fat tail
Excess kurtosis > 0Leptokurtic — fat tails (more extreme events)
Excess kurtosis < 0Platykurtic — thin tails
Normal distributionSkew = 0, excess kurtosis = 0
Most financial return series are negatively skewed with positive excess kurtosis (fat tails)
CorrelationLinear Relationship
Cov(X,Y) = Σ(Xᵢ−X̄)(Yᵢ−Ȳ) / (n−1)
r_XY = Cov(X,Y) / (sₓ · sᵧ)
Range−1 ≤ r ≤ +1
r = 0No linear relationship
Spurious correlationChance, third-variable, or calculation artifact
Correlation ≠ causation. Cannot measure nonlinear relationships.
Module 4 — Probability Trees & Conditional Expectations
Expected Value & VarianceProbability-Weighted
E(X) = Σ P(Xᵢ) · Xᵢ
σ²(X) = Σ P(Xᵢ) · [Xᵢ − E(X)]²
σ(X) = √σ²(X)
Conditional Expected Value
E(X|S) = Σ P(Xᵢ|S) · Xᵢ
Total Probability Rule
E(X) = E(X|S₁)P(S₁) + E(X|S₂)P(S₂) + …
Bayes’ FormulaUpdating Probabilities
P(Event|New Info) = [P(New Info|Event) / P(New Info)] × P(Event)
- Rational method to update prior probabilities with new evidence
- P(new info) = unconditional probability — calculated using total probability rule
- Used in credit analysis, earnings revision, scenario updating
Probability tree: visualize scenarios → conditional expectations → weighted sum = E(X)
1Prior P
→
2New Evidence
→
3Posterior P
Module 5 — Portfolio Mathematics
Portfolio Return & Variance2-Asset Formulas
E(Rₚ) = w₁E(R₁) + w₂E(R₂)
σ²(Rₚ) = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)
Cov(R₁,R₂) = ρ₁₂ · σ₁ · σ₂
3-Asset Extension
σ²(Rₚ) = Σᵢ Σⱼ wᵢwⱼCov(Rᵢ,Rⱼ)
Add 2wᵢwⱼCov terms for each unique pair i≠j
Safety-First & Normal DistributionRoy’s Criterion
SFRatio = [E(Rₚ) − Rₗ] / σₚ
- Choose portfolio with highest SFRatio
- Rₗ = minimum acceptable (threshold) return
- Shortfall risk = P(Rₚ < Rₗ) — minimize this
- Higher E(R) and lower σ both reduce shortfall risk
- Assumes normally distributed returns
SFRatio is the z-score distance from the threshold — higher is safer
Module 6 — Simulation Methods
Lognormal DistributionAsset Price Model
- Asset prices are lognormally distributed (bounded at 0)
- If ln(X) is normal → X is lognormal
- Positively skewed, right-tailed
Continuously compounded r = ln(P₁/P₀)
Lognormal is appropriate for prices; normal for returns
Monte Carlo SimulationSimulation-Based
- Generate thousands of random scenarios using specified distributions
- Produces full distribution of outcomes (not just point estimate)
- Used for option pricing, risk assessment, retirement planning
- Requires specifying all input distributions and correlations
Garbage in → garbage out: model quality depends on distributional assumptions
BootstrappingResampling Method
- Repeatedly draw samples with replacement from observed data
- No distributional assumptions required
- Empirical sampling distribution estimated from actual data
- Useful when distributions are unknown or non-normal
Monte Carlo uses assumed distribution; bootstrapping uses actual data
Module 7 — Estimation & Inference
Sampling MethodsProbability vs Non-Probability
Simple randomEach member equally likely; can include luck
Stratified randomPopulation divided into strata; sample from each
ClusterClusters selected; all members within sampled
ConvenienceNon-prob; whatever is easiest (biased)
JudgementalNon-prob; expert selects sample
Non-probability samples may not represent population — selection bias risk
Central Limit TheoremSampling Distribution
For large n (≥30), sample means are approximately normally distributed, regardless of population distribution
Standard error of mean = s / √n
As n increasesStandard error decreases
Population σ knownUse z-distribution
Population σ unknownUse t-distribution (df = n−1)
CLT allows inference about means even when the parent distribution is non-normal
Module 8 — Hypothesis Testing
Hypothesis Test Framework6-Step Process
1State H₀ and Hₐ
2Select the appropriate test statistic (t, z, χ², F)
3Specify significance level α
4State the decision rule (critical values)
5Calculate the test statistic from sample data
6Make the decision (reject / fail to reject H₀)
Fail to reject H₀ ≠ H₀ is true. We never “accept” the null.
Error Types & PowerDecision Errors
Type I error (α)Reject true H₀ (false positive)
Type II error (β)Fail to reject false H₀ (false negative)
Power of test1 − β = P(reject false H₀)
p-valueSmallest α that rejects H₀ with calculated stat
Decreasing α (stricter) reduces Type I but increases Type II errors — trade-off
Reject H₀ if p-value < α, or if |test stat| > critical value
Test of Mean (t-test)Single & Difference
t = (X̄ − μ₀) / (s/√n), df = n−1
Two independent meansPooled variance t-test
Paired means (dependent)t on mean of differences d̄
Paired test more powerful — eliminates between-group variation
Test of VarianceChi-Square & F-test
χ² = (n−1)s² / σ₀², df = n−1
F = s₁² / s₂², df = n₁−1, n₂−1
χ² testSingle population variance
F-testEquality of two variances
Nonparametric TestsDistribution-Free
- Used when data are non-normal or ordinal
- Spearman rank correlation: nonparametric correlation test
- Contingency table: chi-square test of independence
- Less powerful than parametric tests when assumptions hold
Spearman rₛ = 1 − 6Σdᵢ²/[n(n²−1)]
Module 10 — Simple Linear Regression
Regression BasicsOLS Estimation
Ŷᵢ = b̂₀ + b̂₁Xᵢ + εᵢ
b̂₁ = Cov(X,Y) / Var(X)
b̂₀ = Ȳ − b̂₁X̄
b̂₁ interpretationChange in Ŷ per unit change in X
b̂₀ interpretationExpected Y when X = 0
R²Proportion of variation in Y explained by X
R² = r²_XY (simple regression)
CLRM AssumptionsRequired for Valid OLS
1Linearity: True relationship is linear in parameters
2Homoskedasticity: Constant variance of errors
3Independence: Errors are uncorrelated across observations
4Normality: Errors are normally distributed
Violations (heteroskedasticity, autocorrelation) distort standard errors and test statistics
ANOVA in RegressionPartitioning Variance
TSSTotal sum of squares (total variation)
RSSRegression SS (explained by model)
SSEError SS (unexplained)
TSS = RSS + SSE
R² = RSS / TSS
F = (RSS/1)/(SSE/(n−2))
Hypothesis Test — Coefficientst-test for Slope
t = b̂₁ / se(b̂₁), df = n−2
CI: b̂₁ ± t* × se(b̂₁)
H₀: β₁ = 0X has no linear effect on Y
F-test (overall) and t-test (slope) equivalent in simple regression
Functional FormsLog Transformations
Lin-LinY = b₀ + b₁X (standard)
Log-Linln(Y) = b₀ + b₁X (Y grows exponentially)
Lin-LogY = b₀ + b₁ln(X) (diminishing returns)
Log-Logln(Y) = b₀ + b₁ln(X) (elasticity = b₁)
Choose form based on economic rationale and residual analysis
Module 11 — Big Data Techniques & Fintech
Big Data CharacteristicsThe 3 Vs
VolumeEnormous size of datasets
VarietyStructured + unstructured (text, image, audio)
VelocitySpeed at which data is generated/processed
SourcesWeb, IoT, social media, financial transactions
Machine Learning TypesAI in Finance
SupervisedLabeled data → predict outcomes
UnsupervisedFind hidden patterns, clusters, no labels
Deep learningNeural networks with many layers
ReinforcementAgent learns via rewards/penalties
Overfitting: model learns noise, not signal — use cross-validation
Text Analytics & NLPUnstructured Data
- Sentiment analysis of earnings calls, news
- Information extraction from filings (10-K, 10-Q)
- Bag-of-words: word frequency features
- Named entity recognition (NER)
- Applied to earnings surprises, risk signal detection
High-Frequency Exam Traps & Quick Reference
Common Exam TrapsWatch Out For These
Arithmetic vs GeometricMulti-period → always geometric
MWR vs TWRManager evaluation → always TWR
P-value decisionReject if p-value < α (not >)
Two-tailed critical valueSplit α/2 on each tail
Degrees of freedomn−1 for variance, n−2 for regression
“Fail to reject”Never “accept” H₀
Correlation vs CausationHigh r ≠ causal relationship
Positive skewMean > Median — right tail is longer
Quick Formula ReferenceMust-Know Numbers & Formulas
Normal: 68/95/99.7±1σ / ±2σ / ±3σ coverage
z critical (5%, two-tail)±1.96
z critical (1%, two-tail)±2.576
Arith × Harmonic= (Geometric)²
Roy’s SFR optimalMax [E(Rₚ) − Rₗ] / σₚ
CLT min sample sizen ≥ 30 (rule of thumb)
Spearman rₛ1 − 6Σdᵢ²/[n(n²−1)]
EAR from continuouseʳ − 1