Diversification: Systematic vs. Idiosyncratic Risk

Two portfolios, ten firms each. Same firm-level risk. Wildly different portfolio behavior.

Setup. Each firm has two future states: good (return +50%) or bad (return −30%), with P(good) = 0.5. Portfolio S holds 10 firms whose only risk is one common (systematic) shock — they all win or all lose together. Portfolio I holds 10 firms whose only risk is independent (idiosyncratic) — each firm flips its own coin. S has just 2 possible portfolio outcomes; I has 2¹⁰ = 1024 outcomes — but they cluster tightly around the mean.

Number of firms (N)

11030

Good return (%)

Bad return (%)

P(good) — probability of good state

0.050.500.95

Portfolio S

Systematic only

All firms share one common shock.

This realization's portfolio return —

Portfolio I

Idiosyncratic only

Each firm flips its own independent coin.

This realization's portfolio return —

Distribution of portfolio returns

Bars = empirical (samples). Orange dots = theoretical PMF. Dashed line = mean.

Portfolio S — only 2 possible outcomes

Mass sits entirely on the two extremes. Diversification cannot move it.

Portfolio I — N+1 possible outcomes

Independent shocks cancel. Mass collapses toward the mean as N grows.

Risk comparison

Single firm

Mean—

Std dev (σ)—

Portfolio S (systematic)

Theoretical mean—

Theoretical σ—

Empirical mean—

Empirical σ—

Samples drawn0

Portfolio I (idiosyncratic)

Theoretical mean—

Theoretical σ—

Empirical mean—

Empirical σ—

Samples drawn0

Key relationship: σ_S = σ_firm (no diversification — risk is shared) | σ_I = σ_firm / √N (perfect diversification — independent shocks cancel)

With N = 10, the idiosyncratic portfolio's std dev is reduced to about 31.6% of a single firm's. Push N to 30 and watch the I distribution collapse onto the mean.