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Explain the links between correlation risk and, on one side, systemic risk and, on the other, concentration risk.

FRM® Part II · Market Risk Measurement and Management · Correlation Basics: Definitions, Applications, and Terminology · FRM2.MR.8.g

The idea

Systemic risk is the risk that distress at one institution or market spreads and brings down much of the financial system. Correlation is the channel. In a crisis the correlations among asset prices, defaults and institutions rise, so diversification that works in normal times fails and losses arrive together. Individually diversified firms can still be highly correlated with each other if they hold the same assets or lend to one another. Concentration risk is the loss from exposure that is heavily weighted to one name, sector or region. Such exposures are highly correlated, so adding positions gives little benefit. For n equal positions with volatility σ and average correlation ρ, the portfolio volatility is given by Volatility = σ × √(1/n + (1 − 1/n) × ρ). With positive correlation, adding positions never takes volatility below the limit given by Limit = σ × √ρ, however large n becomes. Exam point: correlation sets the limit on diversification.

Formulas

Volatility of an equal-weighted portfolio with a common correlation

σp = σ × √(1 ÷ n + (1 − 1 ÷ n) × ρ)

σ
the volatility of each position
n
the number of equally weighted positions
ρ
the average pairwise correlation

Assumes equal weights and equal volatilities. As n grows it tends to σ × √ρ, not σ × ρ and not zero.

A practice question

From the question bank, with the reason each wrong option is wrong.

How are concentration risk and correlation risk related?

  1. They are unrelated: concentration risk is about position size and correlation about timing.Why not: Concentrated exposures share the same risk factors and are therefore correlated.
  2. Concentrated exposures tend to be highly correlated, so diversification benefits are small. - correct
  3. Concentration risk lowers correlation, because a concentrated book has few distinct risk factors.Why not: Exposures to the same factor are highly correlated, not less correlated.
  4. Concentration risk is lowest when correlations among holdings are highest.Why not: Reverses the relationship; high correlation makes any given number of holdings behave like a concentrated book.

Concentration risk and high correlation reinforce each other: a book concentrated in related exposures has little diversification and clustered losses.

A harder one

An equal-weighted portfolio holds n stocks, each with volatility 25%, and every pair has a correlation of 0.4. As n grows very large, portfolio volatility approaches:

  1. 15.8%. - correct
  2. 10.0%.Why not: Multiplies volatility by the correlation (25% x 0.4) instead of by its square root.
  3. 0.0%.Why not: Assumes all risk is diversified away, forgetting that the correlation sets a floor.
  4. 25.0%.Why not: Assumes no diversification benefit, which holds only at a correlation of one.

sigma_p = 0.25 x sqrt(1/n + (1 - 1/n) x 0.4) tends to 0.25 x sqrt(0.4) = 15.8%. Correlation sets the limit of diversification.

The hardest tier

Two portfolios each hold 10 equally weighted positions with a stand-alone volatility of 25%. Portfolio A is spread across unrelated sectors with an average pairwise correlation of 0.1. Portfolio B is concentrated in a single sector with an average correlation of 0.6. The ratio of the volatility of B to the volatility of A is closest to:

  1. 6.0.Why not: Takes the ratio of the correlations (0.6 / 0.1) as the ratio of volatilities.
  2. 3.4.Why not: Takes the ratio of portfolio variances instead of volatilities.
  3. 1.8. - correct
  4. 1.0.Why not: Assumes equal volatility because the number of positions and the stand-alone volatilities are the same.

A: 25% x sqrt(0.1 + 0.9 x 0.1) = 10.9%. B: 25% x sqrt(0.1 + 0.9 x 0.6) = 20.0%. The ratio is 1.84. Concentration in correlated exposures removes most of the diversification.

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