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Michael Brenndoerfer

Capital Market Theory - CAPM and the Efficient Frontier › Systematic and Unsystematic Risk

Michael BrenndoerferDecember 14, 20257 min read

Quantitative FinanceAll chapters

Part 9 of 15Portfolio Theory and Investment Analysis

Chapter 47Capital Market Theory - CAPM and the Efficient Frontier › Systematic and Unsystematic Risk

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CAPM decomposes total risk into two components: systematic risk and unsystematic risk. This decomposition explains why diversification reduces portfolio risk and why the market only rewards one type of risk. Understanding this distinction is essential because it determines which risks you should care about and which risks you can safely ignore.

Systematic Risk

Systematic risk (also called market risk or non-diversifiable risk) affects all assets in the economy. It arises from macroeconomic factors like interest rate changes, recessions, inflation, and geopolitical events. Because all assets are exposed to these factors, systematic risk cannot be eliminated through diversification.

Unsystematic Risk

Unsystematic risk (also called idiosyncratic risk, specific risk, or diversifiable risk) affects only individual assets or small groups of assets. Examples include management decisions, product failures, lawsuits, and labor disputes. Unsystematic risk can be eliminated through diversification.

The intuition behind this decomposition is straightforward. When a recession hits, nearly all companies suffer to some degree, so holding more stocks doesn't protect you from recessions. That is systematic risk. But when a single company's CEO resigns unexpectedly, other companies are largely unaffected. If you hold many stocks, the bad surprises at some companies tend to be offset by good surprises at others. That is unsystematic risk, and it averages out in a diversified portfolio.

Consider a simple model where each asset's return depends on a market factor plus an asset-specific component. This single-factor structure captures the idea that all assets respond to common market conditions while also having their own unique sources of variation:

R_i = \alpha_i + \beta_i R_M + \epsilon_i

where:

$R_i$ : return on asset $i$
$\alpha_i$ : asset's return component independent of the market
$\beta_i$ : sensitivity to market movements
$R_M$ : market return
$\epsilon_i$ : idiosyncratic component (with $E[\epsilon_i] = 0$ and $Cov(\epsilon_i, R_M) = 0$ )

This model states that an asset's return can be decomposed into three parts: a constant component $\alpha_i$ that represents the asset's baseline expected return unrelated to the market, a systematic component $\beta_i R_M$ that moves with the market, and an idiosyncratic component $\epsilon_i$ that captures firm-specific events. The key assumption is that the idiosyncratic component is uncorrelated with market returns, meaning firm-specific shocks are independent of broader economic conditions.

Using the properties of variance and the assumption that $Cov(\epsilon_i, R_M) = 0$ , we decompose the total variance. This derivation applies the standard variance formula for a sum of random variables, recognizing that the constant term $\alpha_i$ contributes nothing to variance and that the cross-term vanishes due to the independence assumption:

\begin{aligned} \sigma_i^2 &= Var(R_i) \\ &= Var(\alpha_i + \beta_i R_M + \epsilon_i) && \text{(substitute return model)} \\ &= \beta_i^2 Var(R_M) + Var(\epsilon_i) + 2\beta_i Cov(R_M, \epsilon_i) && \text{(variance properties)} \\ &= \beta_i^2 \sigma_M^2 + \sigma_{\epsilon_i}^2 && \text{(uncorrelated residuals)} \end{aligned}

where:

$\sigma_i^2$ : total variance of asset $i$
$\beta_i^2 \sigma_M^2$ : systematic variance (market risk)
$\sigma_{\epsilon_i}^2$ : unsystematic variance (idiosyncratic risk)

The first term represents systematic variance (exposure to market risk), while the second term represents unsystematic variance. The decomposition shows that total variance splits cleanly into two additive components. The systematic component depends on the square of beta (measuring how strongly the asset responds to market movements) times the market variance. The unsystematic component is simply the variance of the firm-specific shocks.

We can visualize this decomposition for a set of hypothetical assets to see how the proportions of systematic and unsystematic risk vary across different types of securities.

Out[7]:

Visualization

Stacked bar chart showing systematic and unsystematic variance for different asset types. — Decomposition of total variance into systematic and unsystematic components for different asset types. The market index has only systematic risk, while individual stocks carry substantial idiosyncratic risk. Diversified portfolios reduce the unsystematic component.

In a well-diversified portfolio, the unsystematic risks of individual assets cancel out through diversification, leaving only systematic risk. To see why, consider an equally-weighted portfolio of $n$ stocks. The portfolio's idiosyncratic variance equals the average of the individual idiosyncratic variances divided by $n$ . As $n$ grows large, this term approaches zero. However, the systematic component does not diversify away because all stocks share exposure to the same market factor.

In[8]:

Code

import numpy as np

# Demonstrate diversification effect
np.random.seed(123)

# Simulate returns for many assets
n_assets_sim = 500
n_periods = 252  # One year of daily returns

# Market returns
market_returns = np.random.normal(
    0.0004, 0.01, n_periods
)  # ~10% annual, 16% vol

# Asset-specific parameters
betas = np.random.uniform(0.5, 1.5, n_assets_sim)
idio_vols = np.random.uniform(
    0.005, 0.02, n_assets_sim
)  # Daily idiosyncratic vol

# Generate asset returns
asset_returns = np.zeros((n_periods, n_assets_sim))
for i in range(n_assets_sim):
    idio_returns = np.random.normal(0, idio_vols[i], n_periods)
    asset_returns[:, i] = betas[i] * market_returns + idio_returns

# Calculate portfolio volatility for different portfolio sizes
portfolio_sizes = list(range(1, 101))
avg_volatilities = []

for n in portfolio_sizes:
    # Average over many random portfolios of size n
    vols = []
    for _ in range(100):
        selected = np.random.choice(n_assets_sim, n, replace=False)
        port_returns = asset_returns[:, selected].mean(axis=1)
        vols.append(port_returns.std() * np.sqrt(252))  # Annualized
    avg_volatilities.append(np.mean(vols))

# Theoretical systematic risk of equal-weighted portfolio
market_vol = market_returns.std() * np.sqrt(252)
systematic_risk = market_vol * betas.mean()  # Average beta times market vol

We plot the average portfolio volatility against the number of holdings to visualize the diversification benefit.

Out[9]:

Visualization

Line chart showing portfolio volatility decreasing and leveling off as number of holdings increases. — Portfolio volatility decreases as the number of holdings increases, but converges to the systematic risk level. Beyond approximately 30-40 stocks, additional diversification provides minimal risk reduction.

The figure demonstrates a fundamental principle: while adding more securities to a portfolio reduces total volatility, there's a floor below which risk cannot be reduced. This floor is the systematic risk component. The shaded area represents diversifiable risk, the portion of total volatility that disappears as you add more securities. Notice how quickly the curve drops initially; the first few securities you add provide substantial risk reduction. But the curve flattens out around 30 to 40 stocks, indicating diminishing returns to further diversification.

Since investors can eliminate unsystematic risk at no cost through diversification, the market provides no compensation for bearing it. Only systematic risk commands a risk premium. This insight is the economic foundation of CAPM. If you could earn extra return by bearing idiosyncratic risk, every investor would pile into high-idiosyncratic-risk stocks until prices rose and expected returns fell to the level justified by systematic risk alone.