Module 1 — Foundations

Risk and Return — The Fundamental Tradeoff

Markets, returns, and the language of risk. Builds the mental model every quant relies on.

Module lessons

What Quantitative Finance Actually Is Risk and Return — The Fundamental Tradeoff Time Series — The Quant's Native Language

Learning objectives

▸Define expected return and realized return precisely.
▸Understand variance as the canonical risk measure and its limitations.
▸Reason about why risk premia exist.

TEXT

Return is compensation for bearing risk

A risk-free instrument (e.g. a 3-month US T-bill) earns the risk-free rate. Anything riskier must offer an additional expected return — the risk premium — to convince a rational investor to hold it. The size of that premium depends on how much undiversifiable risk the asset carries.

FORMULA

Simple period return

r_t = (P_t - P_{t-1}) / P_{t-1}

FORMULA

Expected vs realized

E[r] = expected return (the mean of the unknown future distribution)
r̄ = realized return (the sample mean we actually observed)
They are not the same thing. A strategy with E[r]=8% can realize -20% in a year.

TEXT

Variance as a (flawed) risk measure

Variance treats upside and downside symmetrically — a 10% gain hurts the variance just as much as a 10% loss helps it. Real investors care more about downside, which is why later we'll meet semi-variance, drawdown, and conditional VaR. But variance is mathematically clean and remains the workhorse of MPT and Sharpe.

EXAMPLE

Why the equity risk premium exists

Stocks have averaged ~6–8% real return per year over a century while T-bills earned ~1%. Why doesn't capital flood into stocks until the gap closes? Because stocks lose 30–50% in recessions, exactly when investors most need their wealth. The premium pays you to endure that pain.

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