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Work through an example of how a correlation estimate is updated.

FRM® Part I · Valuation and Risk Models · Measuring and Monitoring Volatility · FRM1.VR.3.i

The idea

Correlations are updated the same way as variances. With EWMA, New covariance = λ × old covariance + (1 − λ) × x × y. Here x and y are the latest returns on the two assets. The two variances are updated with the same λ. The updated correlation is the new covariance divided by the product of the two new volatilities. To start, turn the old correlation and volatilities into an old covariance. Covariance = correlation × volatility X × volatility Y. A GARCH version is Covariance = ω + α × x × y + β × old covariance. It uses parameters consistent with the variance models, and ω equals (1 − α − β) × long-run covariance. The usual errors are dividing by the old volatilities, using different λ values for the three quantities, and ignoring the sign of the return product. Returns of opposite sign pull the correlation down.

Formulas

EWMA covariance update

new covariance = λ × old covariance + (1 − λ) × x × y

λ
the decay factor, the same as in the variance updates
old covariance
the previous covariance estimate
x
the latest return on the first asset
y
the latest return on the second asset

Turn a starting correlation into a covariance first. Keep the sign of x × y: returns of opposite sign pull the covariance down.

Updated correlation

new correlation = new covariance ÷ (new σX × new σY)

new covariance
the updated covariance
new σX, new σY
the updated volatilities of the two assets

All three inputs must be updated with the same scheme and parameters. Dividing by the old volatilities is the usual error.

GARCH(1,1) covariance update

new covariance = ω + α × x × y + β × old covariance

ω
the constant, (1 − α − β) × long-run covariance
α
the weight on the latest return product
x
the latest return on the first asset
y
the latest return on the second asset
β
the weight on the previous covariance
old covariance
the previous covariance estimate

Use parameters consistent with the two variance models, and set ω from the long-run covariance, not the long-run variance.

A practice question

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

Which expression updates the covariance between returns x and y under EWMA with decay factor lambda?

  1. New covariance = (1 - lambda) x old covariance + lambda x x(n-1) x y(n-1)Why not: Swaps lambda and 1 - lambda, giving the latest product the dominant weight.
  2. New covariance = lambda x old covariance + (1 - lambda) x x(n-1) x y(n-1) - correct
  3. New covariance = lambda x old covariance + (1 - lambda) x (x(n-1) + y(n-1))Why not: Uses the sum of the returns instead of their product.
  4. New covariance = lambda x old covariance + (1 - lambda) x old correlationWhy not: Mixes a correlation into a covariance recursion, so the units are inconsistent.

The EWMA covariance update has the same form as the variance update, with the product of the two latest returns replacing the squared return.

A harder one

Daily returns on X and Y have volatilities of 1.0% and 1.5% and a correlation of 0.50. Using EWMA with lambda = 0.94, the latest returns were X = -2.0% and Y = -1.5%. What is the updated correlation, closest to?

  1. 0.59Why not: Divides the new covariance by the old volatilities instead of the updated ones.
  2. 0.50Why not: Assumes the correlation is unchanged by a single new observation.
  3. 0.54 - correct
  4. 0.98Why not: Swaps lambda and 1 - lambda throughout, letting one observation dominate.

Old covariance = 0.5 x 1.0 x 1.5 = 0.75. New covariance = 0.94 x 0.75 + 0.06 x 3.0 = 0.885. New variances: X = 0.94 x 1 + 0.06 x 4 = 1.18; Y = 2.25. Correlation = 0.885 / (sqrt(1.18) x sqrt(2.25)) = 0.543.

The hardest tier

A portfolio holds USD 12 million of asset X and USD 8 million of asset Y. Yesterday's EWMA (lambda = 0.94) estimates were daily volatilities of 1.0% for X and 2.0% for Y with a correlation of 0.30. Today's returns were -1.5% for X and +2.5% for Y. After updating both variances and the covariance, what is the one-day 99% VaR of the portfolio (zero mean, normal returns)?

  1. USD 0.528 millionWhy not: Uses the old volatilities and correlation without updating for today's returns.
  2. USD 0.512 million - correct
  3. USD 0.538 millionWhy not: Updates the variances but keeps the old covariance, ignoring the opposite-signed returns that reduce it.
  4. USD 0.668 millionWhy not: Adds the stand-alone VaRs, which assumes perfect correlation and ignores diversification.

New variances: X = 0.94 x 1 + 0.06 x 2.25 = 1.0750; Y = 0.94 x 4 + 0.06 x 6.25 = 4.1350. Old covariance = 0.3 x 1 x 2 = 0.6; new covariance = 0.94 x 0.6 + 0.06 x (-3.75) = 0.3390. Portfolio variance (USD million, %²) = 144 x 1.0750 + 64 x 4.1350 + 192 x 0.3390; volatility x 2.326 / 100 = 0.512 million.

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