CFA Level 1 — Quantitative Methods

Holding Period Return (HPR) — The Foundation

The most basic return measure: what did you earn on one investment over one holding period? It captures both income (dividends/coupons) and price appreciation in a single number.

Where \(P_0\) = beginning price, \(P_1\) = ending price, \(CF_1\) = cash received (dividend, coupon).

Three Types of Mean Returns Critical

1. Arithmetic Mean Return

Simple average of periodic returns. Best use: estimate the expected return in a single future period, given a history of returns.

2. Geometric Mean Return

The compound annual growth rate (CAGR). Best use: describe the actual historical growth rate of a portfolio that was left to compound. Always ≤ arithmetic mean (equal only when all returns are identical).

3. Harmonic Mean Return

Used specifically for dollar-cost averaging — when you invest a fixed €amount each period (not a fixed number of shares). The average cost per share when investing equal amounts is the harmonic mean of prices.

Relationship: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean (equality holds only if all values are identical).

Money-Weighted vs. Time-Weighted Returns Critical

This is one of the most heavily tested concepts in the entire CFA curriculum. The key question: whose decision are we evaluating?

Interest Rate Conversions — EAR, APR, and Compounding

Effective Annual Rate (EAR)

EAR is the true annual return after accounting for intra-year compounding. It's the standardised way to compare investments with different compounding frequencies.

Where \(m\) = number of compounding periods per year.

Real vs. Nominal Returns

A nominal return includes inflation. A real return strips it out, showing purchasing power growth.

The Core Principle

Future Value and Present Value Formula

Where \(r\) = interest rate per period, \(N\) = number of periods.

Annuities — Repeated Equal Cash Flows Critical

An annuity is a series of equal payments at equal intervals. Mortgages, lease payments, pension distributions — all are annuities.

Ordinary Annuity (payments at END of period — most common)

Annuity Due (payments at BEGINNING of period)

Simply multiply the ordinary annuity by (1+r) — you receive each payment one period earlier, so each is worth more by one period of interest.

Perpetuities — Payments Forever

A perpetuity pays the same amount forever. While no real instrument literally pays forever, this formula is used to value preferred stocks and is the basis for the Gordon Growth Model.

Net Present Value (NPV) and Internal Rate of Return (IRR) Critical

NPV Decision Rule: Invest if NPV > 0 (project adds value). IRR is the discount rate that makes NPV = 0. IRR Decision Rule: Invest if IRR > required rate of return (hurdle rate).

Loan Amortisation

Each loan payment consists of an interest component and a principal repayment component. Over time, the proportion shifts: early payments are mostly interest; later payments are mostly principal. This is why you barely reduce your mortgage principal in the early years.

Measures of Central Tendency

Arithmetic Mean

Geometric Mean (for returns — covered in R1)

Use geometric mean for compounding problems; arithmetic mean for expected value of a single future period.

Weighted Mean

Used for portfolio returns: weight each asset's return by its portfolio weight.

Median and Mode

Median = middle value when sorted (50th percentile). Mode = most frequently occurring value. For skewed distributions, the median is a better measure of "typical" because it's less affected by outliers.

Measures of Dispersion

Variance and Standard Deviation Formula

Coefficient of Variation (CV) — Relative Risk Exam Favourite

CV answers: "How much risk am I taking per unit of return?" Lower CV = better risk-adjusted performance (all else equal). Unlike standard deviation, CV allows comparison across investments with different return scales.

Sharpe Ratio — Risk-Adjusted Performance

The Sharpe ratio measures excess return (above the risk-free rate) per unit of total risk. Unlike CV, it uses excess return, making it the standard for comparing risky portfolios.

Skewness — Asymmetry of the Distribution Critical

Kurtosis — Fat Tails Critical

Kurtosis measures the "tailedness" of a distribution — how much weight is in the extreme outcomes compared to a normal distribution.

Core Probability Rules

Conditional Probability

"The probability of A given B has occurred" — we've restricted our probability space to only the scenarios where B is true.

Bayes' Theorem — Updating Beliefs Exam Favourite

Bayes' theorem answers: "Given that an event occurred, what is the revised probability of its cause?" It's how rational people update beliefs when new information arrives.

Expected Value and Variance of a Random Variable

Properties of Expected Value

• \(E(aX + b) = a \cdot E(X) + b\)
• \(E(X + Y) = E(X) + E(Y)\) (always, regardless of dependence)
• \(Var(aX + b) = a^2 \cdot Var(X)\) (adding a constant doesn't change variance)
• \(Var(X + Y) = Var(X) + Var(Y) + 2 \cdot Cov(X,Y)\) (the portfolio variance formula!)

Portfolio Expected Return

Portfolio expected return IS the weighted average of individual expected returns — simple and linear.

Portfolio Variance — The Non-Trivial Part Critical

Two-Asset Portfolio

So the formula can also be written:

The Role of Correlation in Diversification Critical

Covariance Matrix Formula

For a portfolio with n assets, variance requires n variances and n(n−1)/2 unique covariances. With 100 assets, you need 100 variances + 4,950 covariances = 5,050 pieces of information. This is why covariance estimation is the hardest part of portfolio construction in practice.

Note: when i = j, Cov(R_i, R_i) = Var(R_i) = σ_i² — so the variance terms are just special cases of the covariance matrix diagonal.

Minimum Variance Portfolio (Two Assets)

The weight in Asset 1 that minimises portfolio variance:

Historical Simulation

Use actual past data (historical returns, historical factor moves) directly — resample from the empirical distribution, preserving the real-world distribution including skewness, kurtosis, and any correlations present in the data.

Monte Carlo Simulation Critical

Specify a probability model (e.g., normal distribution with given mean and variance, or a correlated multivariate distribution), then use a computer to draw thousands of random samples from that model and simulate outcomes.

Bootstrap Simulation

A hybrid approach: resample WITH REPLACEMENT from the actual historical data. Unlike historical simulation which uses each observation once, bootstrap can draw the same observation multiple times, creating new synthetic histories.

Sampling and the Central Limit Theorem

Point Estimates

A single number used to estimate a population parameter. The sample mean \(\bar{x}\) estimates population mean \(\mu\). The sample variance \(s^2\) estimates population variance \(\sigma^2\).

Standard Error of the Mean Formula

The standard deviation of the sampling distribution of the mean — how much would \(\bar{x}\) vary across different random samples?

Central Limit Theorem (CLT) Critical

Regardless of the shape of the underlying population distribution, the sampling distribution of the sample mean approaches a normal distribution as n increases (roughly n ≥ 30). This is the foundational justification for using normal distribution tools in hypothesis testing.

Confidence Intervals Critical

A confidence interval gives a range of values that is expected to contain the true population parameter with a specified probability (confidence level).

When to Use z vs. t Distribution Exam Favourite

The Hypothesis Testing Framework

Step 1: State the Hypotheses

H₀ (null hypothesis): The baseline claim to be tested. Usually the "no effect" or "no difference" statement. Always includes equality (=, ≤, ≥).

Hₐ (alternative hypothesis): What we're trying to find evidence for. Sets the direction of the test.

Step 2: Choose Significance Level (α)

α = the maximum probability of making a Type I error (rejecting a true H₀) you're willing to accept. Common choices: 1%, 5%, 10%.

Step 3: Compute the Test Statistic

Step 4: Compare to Critical Value and Make Decision

If |test statistic| > critical value (or equivalently, p-value < α): Reject H₀. Otherwise: Fail to reject H₀.

Type I and Type II Errors Critical

The key trade-off: Lowering α (being more strict) reduces Type I errors but increases Type II errors. You can only reduce BOTH by increasing sample size (n).

p-Value — The Modern Standard Exam Favourite

The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming H₀ is true.

Tests Beyond the Mean

Chi-Square Test (χ²) — Testing Variance

Tests whether a population variance equals a hypothesised value. The chi-square distribution is always positive and right-skewed, so critical values are asymmetric for one-tailed tests.

F-Test — Comparing Two Variances

Tests whether two population variances are equal. Convention: put the larger variance in the numerator (F ≥ 1). Used heavily in regression to test overall model fit.

Parametric Tests — When Distribution Assumptions Hold

t-Test for Correlation Formula

Tests whether the population correlation coefficient ρ is significantly different from zero (i.e., whether two variables are actually linearly related).

Where \(r\) = sample correlation coefficient, \(n\) = number of observations.

Paired Comparison t-Test

Used when observations come in natural pairs (e.g., the same company before and after a policy change, or matched pairs of funds). You test whether the mean difference is zero.

Where \(\bar{d}\) = mean of paired differences, \(s_d\) = std dev of differences, \(\mu_{d_0}\) = hypothesised mean difference (usually 0).

Non-Parametric Tests — When Assumptions Break Down

Spearman Rank Correlation Exam Favourite

The non-parametric analogue of Pearson correlation. Rank both variables, then apply the standard Pearson correlation formula to the ranks.

Where \(d_i\) = difference in ranks for the i-th pair. This formula assumes no tied ranks.

Runs Test — Testing for Independence/Randomness

Tests whether a sequence of observations is random (independent). Counts "runs" — consecutive sequences of the same sign (e.g., consecutive positive or negative returns).

Too few runs: positive serial correlation (trending). Too many runs: negative serial correlation (mean-reverting). Exact random walk: number of runs consistent with random probability.

The Regression Model

Where \(Y_i\) = dependent variable (what we're predicting), \(X_i\) = independent variable (the predictor), \(b_0\) = intercept, \(b_1\) = slope coefficient, \(\varepsilon_i\) = error term (residual).

Ordinary Least Squares (OLS) Estimation Formula

OLS minimises the sum of squared residuals (SSE). This gives the "best fit" line through the data.

Measuring Fit: R², SEE, and ANOVA Critical

The total variation in Y (SST) can be decomposed into explained variation (SSR) and unexplained variation (SSE):

R² = the proportion of variation in Y explained by X. R² = 1 means perfect fit; R² = 0 means X explains nothing about Y.

SEE = the typical size of a prediction error. Lower SEE = better model fit.

Testing the Slope Coefficient Critical

H₀: b₁ = 0 (the independent variable has no linear relationship with Y). The t-test:

Where \(s_{\hat{b}_1}\) = standard error of the slope estimate.

F-Test — Overall Regression Significance

In simple regression, the F-test and the t-test for the slope coefficient test the same hypothesis. F = t². Always one-tailed (reject if F > critical value).

Assumptions of OLS Regression

Types of Data

Machine Learning Categories Critical

Overfitting — The Most Dangerous Problem Critical

Overfitting occurs when a model learns the training data too well — including the noise — and fails to generalise to new data. In investment research, an overfit model shows spectacular backtested returns but performs poorly (or disastrously) in live trading.

Bias-Variance Tradeoff

Preventing Overfitting: Regularisation and Cross-Validation

Regularisation — Penalise Complexity

Add a penalty to the loss function that discourages large coefficients. Two main types:

• LASSO (L1 regularisation): Penalty = λ × Σ|βᵢ|. Drives some coefficients exactly to zero — performs automatic variable selection. Use when you want a sparse model.
• Ridge (L2 regularisation): Penalty = λ × Σβᵢ². Shrinks all coefficients toward zero but rarely to exactly zero. Use when you believe all variables are relevant.

Higher λ = stronger regularisation = simpler model. The optimal λ is chosen by cross-validation.

k-Fold Cross-Validation

Divide the training data into k equal "folds." Train on k−1 folds, test on the remaining fold. Repeat k times, rotating the test fold. Average the test errors. This gives an unbiased estimate of out-of-sample performance — the true test of a model's value.

Natural Language Processing (NLP)

NLP converts unstructured text into structured, analysable data. Applications in investment analysis:

• Sentiment analysis: Classify text as positive/negative/neutral. Mine earnings call transcripts, analyst reports, news articles for market-moving tone shifts.
• Named entity recognition: Automatically identify companies, people, dates, and amounts in text.
• Word embeddings (Word2Vec, BERT): Represent words as numeric vectors that capture semantic meaning — "excellent" and "outstanding" will have similar vectors.

Neural Networks

Inspired by the human brain: layers of interconnected nodes (neurons). Each connection has a weight; the network learns by adjusting weights to minimise prediction error (backpropagation).