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Distinguish conditional distributions from unconditional ones and outline what regime switching is.

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

The idea

A conditional distribution is the distribution of tomorrow's return given what is known today, such as the latest volatility estimate or the current market state. The unconditional distribution is the average across all such states, which is what a long-run histogram shows. Because volatility changes, the conditional standard deviation can be far above or below the unconditional one. VaR built from the unconditional figure is too low in turbulent periods and too high in calm ones. Regime switching is one way to model this. Returns are drawn from different distributions, for example a calm state with low volatility and a crisis state with high volatility. The state switches randomly with given transition probabilities, usually persistent. The unconditional distribution is then the probability-weighted mixture of the regime distributions. Its variance is the weighted average of regime variances, plus a term for any differences between regime means, and it has fat tails.

Formulas

Unconditional variance of a two-regime model

unconditional variance = p × σ₁² + (1 − p) × σ₂²

p
the long-run probability of the first regime
σ₁, σ₂
the standard deviation of returns in each regime

Holds when the regimes share the same mean; otherwise add a term for the differences between regime means. Average variances, never standard deviations.

Long-run share of time in the crisis regime

crisis share = p(calm to crisis) ÷ [p(calm to crisis) + p(crisis to calm)]

p(calm to crisis)
the probability of switching from calm to crisis in one period
p(crisis to calm)
the probability of switching from crisis back to calm in one period

The stationary probability of a two-state switching model. Quoting the switching probability itself as the long-run share is the trap.

A practice question

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

Which statement correctly distinguishes a conditional distribution of returns from an unconditional one?

  1. A conditional distribution describes the return averaged over all states; an unconditional one uses today's information.Why not: Swaps the two definitions.
  2. A conditional distribution describes only the worst 5% of returns; an unconditional one describes the rest.Why not: Confuses conditioning on information with conditioning on being in the loss tail, as in expected shortfall.
  3. A conditional distribution is the distribution after subtracting the mean, leaving only volatility.Why not: Mistakes conditioning on information for de-meaning the data.
  4. A conditional distribution describes the next return given current information such as recent volatility; an unconditional one averages over all possible information states. - correct

Conditioning means using information available now, for example the latest volatility estimate. The unconditional distribution is the long-run average across all such states and is what a full-sample histogram shows.

A harder one

A market is in a calm regime 75% of the time with annual volatility 10% and in a crisis regime 25% of the time with annual volatility 30%. Both regimes have zero mean. What is the unconditional annual volatility?

  1. 15.0%Why not: Takes the probability-weighted average of the volatilities instead of the variances.
  2. 20.0%Why not: Takes the simple average of 10% and 30%, ignoring the regime probabilities.
  3. 17.3% - correct
  4. 3.0%Why not: Computes the weighted variance (0.03) but quotes it as a percentage without taking the square root.

Unconditional variance = 0.75 x 0.01 + 0.25 x 0.09 = 0.03, so volatility = square root of 0.03 = 17.3%. Zero means in both regimes make this the full variance.

The hardest tier

A model has a calm state (daily variance 1 in %², standard deviation 1%) and a crisis state (daily variance 9, standard deviation 3%). The probability of staying in crisis from one day to the next is 0.80, and of moving from calm to crisis is 0.05. The market is in crisis today. What is the expected daily volatility two days from now (the state two steps ahead), measured as the square root of the expected variance?

  1. 2.72%Why not: Uses the one-step-ahead crisis probability of 0.80 instead of compounding the transition matrix twice.
  2. 2.49% - correct
  3. 1.61%Why not: Uses the long-run stationary crisis probability (0.20) and ignores that today's crisis state still matters.
  4. 3.00%Why not: Assumes the crisis state persists with certainty for two days.

P(crisis in two days) = 0.80 x 0.80 + 0.20 x 0.05 = 0.65. Expected variance = 0.65 x 9 + 0.35 x 1 = 6.2, so volatility = 2.49%. The forecast sits between today's 3% and the long-run level.

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