Diversification and its limits

Owning twenty shares instead of one cuts risk sharply. Then a crisis arrives, almost everything falls at once, and the portfolio behaves like one big share. Diversification is strongest in calm markets and weakest in a crisis, just when it is needed.

L2.5.03: Correlation and diversification showed that two assets that do not move exactly together make a portfolio less volatile than either on its own. Here is why that keeps working as you add more, and why it stops.

Two kinds of news. A share moves on news about its own company: a product launch, a lawsuit, a new chief executive. That is specific riskRisk that comes from news about one asset only, such as a product failure. It cancels out across many assets., and across many companies the good and bad surprises cancel. A share also moves on news that hits nearly every company: a recession, a jump in interest rates, a pandemic. That is common riskRisk that comes from news that moves most assets together, such as a recession or a change in interest rates. Also called systematic or market risk., and adding more shares cannot cancel it, because they all feel it. DiversificationSpreading money across assets that do not move exactly together, so their ups and downs partly cancel. removes the first kind and leaves the second.

A simple model. Hold n assets in equal amounts. Give each the same volatility σ and every pair the same correlation ρ, in the notation of M2.5: First risk measures. Then

As n grows, the second term shrinks to nothing, and the volatility falls to a floorThe volatility a portfolio cannot go below however many similar assets it holds: σ√ρ. of $\sigma\sqrt{\rho}$. The correlation sets the floor: with ρ = 0.25 a portfolio can never be less than 50% as volatile as one asset; with ρ = 0.60, never less than about 77%.

Each asset has 30% volatility. Most of the benefit comes from the first fifteen or twenty assets; after that the curve is nearly flat.

Why correlations rise in a crisis. In a crisis one shock dominates every price. Investors who face margin calls or redemptions must sell what they can, not what they would like to, so even unrelated assets fall together. Volatility rises at the same time. Both the σ and the ρ in the formula go up, and the floor rises with them. This is crisis correlationThe tendency of correlations between risky assets to rise in a crisis, when most of them fall together., and it has happened again and again:

What risk managers do about it. They check portfolios with correlations and volatilities from stressed periods, not only calm ones (L2.5.09: Stress testing). They set concentrationRisk from having too much exposure to one name, sector, country or risk factor. limits by name, sector and country, so no single shock can do too much damage. And they look for protection that pays when correlations jump, such as the puts of L2.6.03: Hedging with options. A single correlation number is also a simplification: real portfolios need a full correlation matrix, and the way assets fall together in the tails goes beyond correlation. Both come later in the course (M5.1: Statistical foundations for risk models).

The formula comes from the variance of a sum of random variables (see the Math Atom on combining random variables), the same rule that gives the two-asset formula. With weights $w_i = 1/n$:

There are n terms in the first sum and n(n − 1) pairs in the second:

The common part stays whatever n is; the specific part shrinks like 1/n. The share of variance that is common is $\rho \,/\, \big(\rho + (1-\rho)/n\big)$. At n = 2 the formula is the two-asset formula with equal weights and equal volatilities.

Start in the calm market and slide n from 1 to 50: watch how quickly the curve flattens. Then press “Crisis”: the whole curve jumps, and the floor with it. Finally, set ρ to 0 and see what diversification could do if correlations really were zero.

An equity fund holds 20 shares in equal amounts. Each has an annual volatility of 30%. In calm years the average correlation between them is 0.25. The risk team asks what happens in a crisis, when correlations rise to 0.60 and each share's volatility rises to 45%. All figures are illustrative.

Rules limit concentration because diversification cannot be relied on to do it. The Basel large exposures framework caps a bank's exposure to a single counterparty, or a group of connected ones, at 25% of its Tier 1 capital (M4.4: Counterparty, operational and large exposures). EU UCITS funds may hold at most 10% of their assets in one issuer, and holdings above 5% may not add up to more than 40%. Solvency II's standard formula gives insurers credit for diversification between risks, but through fixed correlation matrices set with stressed markets in mind (M4.6: Solvency II and insurance regulation).

• Diversification cancels specific risk; it cannot cancel common risk.
• For n equal assets with common correlation ρ: σ_p = σ √(ρ + (1 − ρ)/n).
• The floor is σ√ρ. Most of the benefit comes from the first fifteen to twenty assets.
• In a crisis correlations and volatilities rise together, so the floor rises just when losses come.
• Risk managers test with stressed correlations, limit concentration and look for protection that pays in a crisis.

Diversification

Spreading money across assets that do not move exactly together, so their ups and downs partly cancel.

Specific risk

Risk that comes from news about one asset only, such as a product failure. It cancels out across many assets.

Common risk

Risk that comes from news that moves most assets together, such as a recession or a change in interest rates. Also called systematic or market risk.

Diversification floor

The volatility a portfolio cannot go below however many similar assets it holds: σ√ρ.

Crisis correlation

The tendency of correlations between risky assets to rise in a crisis, when most of them fall together.

Concentration risk

Risk from having too much exposure to one name, sector, country or risk factor.