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Explain how correlation between assets changes the risk of a portfolio

CFA® Level I · Portfolio Construction · Portfolio Risk and Return: Part I · PM.2.a

The idea

Correlation governs how much diversification reduces risk. At a correlation of +1 the assets move together and portfolio standard deviation is just the weighted average - no benefit. As correlation falls, risk reduction grows, and at -1 a weighting exists that eliminates volatility entirely. The practical implication is that benefit exists whenever correlation is below +1, so adding a volatile asset can lower total portfolio risk if it is weakly correlated with what you already hold. Correlations are unstable and tend to rise in crises, exactly when diversification is most needed.

Formulas

Correlation from covariance

ρ = Cov ÷ (σ₁ × σ₂)

Cov
the covariance of the two returns
σ₁, σ₂
the standard deviation of each return
ρ
the correlation, always between −1 and 1

Covariance says which way two assets move together but its size depends on the units. Dividing by both standard deviations scales that out, which is why correlation can be compared across pairs and covariance cannot.

A practice question

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

Correlation can take values in the range:

  1. any real numberWhy not: That is covariance, whose size depends on the units. Correlation is covariance scaled by the two standard deviations, which bounds it.
  2. −1 to +1 - correct
  3. 0 to +1Why not: Omits the negative half, where assets move in opposite directions - which is where the strongest diversification is found.

Correlation is a score for how closely two assets move together. It is covariance scaled by the size of each asset's swings, which is what keeps it in a fixed range. Formula: Correlation = Covariance / (SD A x SD B). The score runs from -1 (perfectly opposite) through 0 (unrelated) to +1 (perfectly in step). It cannot be any real number, because the scaling stops it going beyond these limits. It also is not limited to 0 to +1, since assets can move in opposite directions.

A harder one

At a correlation of −1, it is possible to construct a two-asset portfolio with:

  1. volatility equal to the lower of the twoWhy not: With perfect negative correlation, the right weights make the movements cancel completely, so risk can fall below even the safer asset.
  2. volatility equal to the weighted averageWhy not: The weighted average applies at a correlation of +1. At −1 the offsetting movements reduce risk well below it.
  3. zero volatility - correct

At -1 the two assets move in exactly opposite ways, so with the right mix the swings cancel completely and the portfolio does not fluctuate at all. Formula: Portfolio SD = absolute value of (w1 x SD1 - w2 x SD2). It equals zero when w1 x SD1 = w2 x SD2. Example: with SDs of 10% and 20%, put 2/3 in the 10% asset and 1/3 in the 20% asset: (2/3 x 10%) - (1/3 x 20%) = 0. Expected return is not affected and remains the weighted average of the two returns.

The hardest tier

Correlations between asset classes tend to rise during market crises. The practical implication is that:

  1. diversification benefits shrink when they are most needed - correct
  2. diversification works better in downturns, as assets fall at different ratesWhy not: The evidence points the other way. In crises assets tend to fall together, so correlations rise and the benefit shrinks.
  3. diversification is unaffected, since correlations stay constant over timeWhy not: Correlations are not constant. They tend to rise in crises, which is exactly when the benefit matters most.

Diversification works because assets that normally move differently soften each other's falls. In a crisis, investors tend to sell almost everything at once, so assets that usually move apart start falling together. Their correlation rises towards +1, and the benefit shrinks. Formula: as correlation approaches +1, Portfolio SD approaches the weighted average of the SDs, so little risk is cancelled. So diversification weakens just when you need protection most.

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