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Apply methods for estimating volatility or VaR over long horizons, and describe how mean reversion works under a GARCH (1,1) model.

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

The idea

Long-horizon risk is often obtained by scaling the daily figure over T days. The T-day volatility is: Volatility = daily volatility × √T. VaR scales the same way, under normality with a zero mean. This holds only if daily returns are independent with constant variance. Positive autocorrelation makes it understate, and mean reversion in volatility makes it misstate. Under GARCH(1,1), the expected variance k days ahead is: Forecast = V + (α + β)^k × (current variance − V). So the forecast decays geometrically toward the long-run variance V. When current volatility is above the long-run level, square-root scaling of today's volatility overstates long-horizon risk, and when it is below, it understates. The T-day variance is the sum of the expected daily variances, and the T-day volatility is the square root of that sum. The speed of this decay is summarised by the half-life, the time taken to close half of the gap to V: Half-life = ln(0.5) ÷ ln(α + β). It is measured in days. Distinguish the variance horizon sum from simply scaling today's figure.

Formulas

GARCH(1,1) variance forecast k days ahead

expected variance in k days = V + (α + β)^(k) × (current variance − V)

V
the long-run variance
α
the GARCH weight on the latest squared return
β
the GARCH weight on the previous variance
k
the number of days ahead
current variance
today's variance estimate

The forecast decays geometrically to V. For a T-day horizon, add the daily forecasts and take the square root, rather than scaling today's volatility by √T.

Half-life of a GARCH volatility shock

half-life = ln(0.5) ÷ ln(α + β)

α
the GARCH weight on the latest squared return
β
the GARCH weight on the previous variance

The days it takes for the gap between forecast and long-run variance to halve. Persistence close to 1 means shocks fade slowly.

A practice question

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

Which assumption justifies scaling a one-day VaR to a T-day VaR by multiplying by the square root of T?

  1. Daily returns are independent with identical, constant variance (no serial correlation or volatility change). - correct
  2. Daily returns are perfectly positively correlated from one day to the next.Why not: Perfect serial correlation would scale volatility by T, not by the square root of T.
  3. Volatility mean-reverts quickly toward its long-run level over the horizon.Why not: Mean reversion breaks the rule; the square root rule assumes a constant variance.
  4. Returns are normally distributed, whatever their dependence over time.Why not: Normality alone does not give the rule; a normal series can still be autocorrelated or have changing variance.

Variance adds across independent days: T-day variance = T x daily variance, so volatility scales with the square root of T. Autocorrelation or time-varying volatility invalidates the rule.

A harder one

A GARCH(1,1) model has a long-run daily variance of 0.0001 and alpha + beta = 0.97. The current daily variance is 0.00025. The forecast of expected daily variance k days ahead is V + (alpha + beta)^k x (current variance - V). What is the expected daily volatility 10 days ahead?

  1. 1.57%Why not: Uses a single factor of 0.97 instead of raising it to the 10th power.
  2. 1.45% - correct
  3. 1.18%Why not: Applies the decay weight to the long-run variance instead of to the gap, reversing the roles.
  4. 1.00%Why not: Assumes the forecast has already fully reverted to the long-run level after 10 days.

0.97^10 = 0.7374. Variance = 0.0001 + 0.7374 x (0.00025 - 0.0001) = 0.000211, so volatility is 1.45%. Only about a quarter of the gap has closed.

The hardest tier

A GARCH(1,1) model has a long-run daily variance of 0.0001 and alpha + beta = 0.98. Tomorrow's variance forecast (day 1) is 0.0004. The expected variance for day k is V + 0.98^(k-1) x (0.0004 - V). Assuming zero mean and normal returns, what is the 10-day 99% VaR on USD 60 million, using the sum of the daily forecasts over days 1 to 10?

  1. USD 8.83 millionWhy not: Holds variance at tomorrow's 0.0004 for ten days, ignoring mean reversion toward the long-run level.
  2. USD 4.41 millionWhy not: Assumes variance reverts to the long-run level immediately and ignores the decaying excess.
  3. USD 8.54 million - correct
  4. USD 2.79 millionWhy not: Reports a one-day VaR rather than the 10-day VaR.

Sum of variances = 10 x 0.0001 + 0.0003 x (1 - 0.98^10) / 0.02 = 0.003744. Ten-day volatility = 6.119%. VaR = 2.326 x 6.119% x 60 = 8.54 million, below the square-root-of-time figure of 8.83 million.

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