Quantitative Methods — Study Notes

CFA® Level I topic weight 6–9% · the toolkit every other topic borrows

Time value of money & returns

Core TVM relation: FV = PV × (1 + r)ⁿ, and PV = FV ÷ (1 + r)ⁿ. Everything else — annuities, perpetuities, loan payments — is layered on this. A perpetuity’s value is PV = PMT ÷ r.
Ordinary annuity vs annuity due: an annuity due pays at the beginning of each period, so its value equals the ordinary annuity value × (1 + r). Getting the timing setting wrong on the calculator is the single most common quant error.
Effective annual rate: EAR = (1 + periodic rate)^m − 1, where m is compounding periods per year. More frequent compounding raises the EAR above the stated annual rate; continuous compounding gives EAR = e^r − 1.
Money-weighted vs time-weighted return: the money-weighted return is an IRR that reflects the size and timing of the investor’s cash flows; the time-weighted return chains sub-period returns geometrically and is the standard for judging managers, because they do not control client contributions and withdrawals.

Describing data

Arithmetic vs geometric mean: the geometric mean compounds — use it for multi-period growth rates. It is always ≤ the arithmetic mean, with the gap widening as volatility rises. The harmonic mean suits averaging ratios like price paid per share under equal-dollar purchases (cost averaging).
Dispersion: variance is the average squared deviation from the mean (sample variance divides by n − 1); standard deviation is its square root. The coefficient of variation, CV = s ÷ mean, measures risk per unit of return and lets you compare dispersion across series with different scales.
Target downside deviation measures dispersion only of observations below a target return — the intuition behind “bad” volatility.
Skewness: a positively skewed distribution has a long right tail, with mode < median < mean; negative skew reverses the order. Kurtosis: a leptokurtic distribution (excess kurtosis > 0) has fatter tails than the normal — more frequent extreme outcomes, which matters for risk.

Probability & distributions

Conditional probability: P(A|B) = P(AB) ÷ P(B). The multiplication rule, total probability rule, and Bayes’ formula (updating a prior with new information) are tested as short scenario calculations.
Expected value and variance of a random variable weight each outcome by its probability. Covariance and correlation extend this to pairs — correlation is covariance scaled to [−1, +1].
Normal distribution: symmetric, fully described by mean and variance. Standardize with z = (x − μ) ÷ σ. Roughly 68% of outcomes fall within ±1σ, 90% within ±1.65σ, 95% within ±1.96σ, 99% within ±2.58σ.
Lognormal distribution: bounded below by zero and right-skewed — used for asset prices, while continuously compounded returns are modeled as normal. The binomial distribution handles fixed numbers of success/failure trials.

Sampling, testing & regression

Central limit theorem: the sampling distribution of the mean approaches normal as n grows (n ≥ 30 as the working rule) regardless of the population’s shape. The standard error of the mean is σ ÷ √n.
Hypothesis testing: state H₀ and H₁, choose significance level α, compute the test statistic, compare with the critical value or use the p-value (reject when p < α). A Type I error rejects a true null (probability = α); a Type II error fails to reject a false null. Power = 1 − P(Type II).
Simple linear regression: Y = b₀ + b₁X + ε. The slope is the change in Y per unit of X; R² is the proportion of Y’s variation explained — in simple regression it equals the squared correlation. Assumptions include linearity, independent errors, and constant error variance (homoskedasticity).
Correlation is not causation: spurious correlation can arise from chance, outliers, or a third variable driving both series — a favorite conceptual question.

Common traps

Leaving the calculator in end-of-period mode for an annuity-due problem (or vice versa) — always check the timing assumption first.
Using the arithmetic mean for multi-year compound growth — compounding questions want the geometric mean.
Confusing Type I and Type II errors — α is the probability of rejecting a true null, and lowering α raises the chance of a Type II error.
Mixing up money-weighted and time-weighted returns — large deposits before strong periods inflate the money-weighted return, not the time-weighted one.
Applying the normal distribution to prices — prices are modeled lognormally; continuously compounded returns are the normal variable.

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