The Capital Asset Pricing Model (CAPM)

A Masterclass Edition lesson on the Nobel Prize-winning model for risk and expected return.

In our last lesson, we focused on building the "perfect" portfolio from a universe of assets. We were architects, designing an optimal structure.

Now, we shift our perspective. We become analysts. We are given a single stock—say, Apple (AAPL)—and we need to answer one of the most fundamental questions in finance: What is a fair expected return for this stock?

The insight of CAPM, which won William Sharpe the Nobel Prize, is that an investor should not be rewarded for all risk, but only for the risk they *cannot* diversify away.

Part 1: The Two Flavors of Risk

The core idea of CAPM is to split a stock's total risk into two parts:

  1. Systematic Risk (Market Risk): This is the risk inherent to the entire market (e.g., recessions, interest rate changes). You cannot get rid of it by diversifying.
  2. Idiosyncratic Risk (Specific Risk): This is the risk specific to a single company (e.g., a drug trial failing, a factory fire). This risk is **diversifiable**.

The Central Premise of CAPM: The market will only compensate you (with higher expected returns) for taking on systematic risk—the risk you are forced to bear.

Part 2: Measuring Sensitivity with Beta (β)
Beta measures how much a stock tends to move when the overall market moves. It is the measure of an asset's systematic risk.
  • `β = 1`: The stock moves in line with the market.
  • `β &gt 1`: The stock is more volatile than the market.
  • `β &lt 1`: The stock is less volatile than the market.
Part 3: The Linear Model
CAPM proposes a simple linear relationship:
(Rs−Rf)=α+β(Rm−Rf)+ϵ(R_s - R_f) = \alpha + \beta (R_m - R_f) + \epsilon
  • y=(Rs−Rf)y = (R_s - R_f): The stock's excess return (dependent variable).
  • x=(Rm−Rf)x = (R_m - R_f): The market's excess return (independent variable).
  • `β` (Beta): The **slope** of the line, measuring systematic risk.
  • `α` (Alpha): The **y-intercept**, representing performance not explained by the market.
  • `ε` (Epsilon): The **error term**, representing idiosyncratic risk.
Part 4: The Linear Algebra Revelation
Finding the best-fit `α` and `β` from historical data is a classic **Least Squares Problem**.

For a set of historical returns, we have an overdetermined system `Ax = b` where `x = [α, β]ᵀ`. The solution is found by solving the **Normal Equations**:

ATAx^=ATbA^TA\hat{x} = A^Tb

The matrix `A` is constructed with a column of ones for the intercept `α` and a column of the market's excess returns for `β`.

Summary: The Triumph of Regression
  1. The Goal: Find a fair expected return for a stock based on its non-diversifiable market risk.
  2. The Concept: We measure this risk with **Beta (`β`)**.
  3. The Formulation: We model this as a simple **linear regression**.
  4. The Solution: We solve the **Normal Equations `AᵀAx̂ = Aᵀb`** to find the estimated `α` and `β`.

**Up Next:** We will use **Eigendecomposition / SVD** to perform a **Principal Component Analysis (PCA)** on asset returns, uncovering the hidden "factors" that drive the entire financial system.