ETF5952 · Quantitative Methods for Risk Analysis

Multi-Period Risk and the Square-Root-of-Time Rule

This topic from ETF5952 Quantitative Methods for Risk Analysis at Monash University answers the practical question every risk figure eventually faces: a volatility, Value-at-Risk or Expected Shortfall estimated over one day has to be reported over a longer horizon h — a 10-day capital figure, a monthly forecast. You learn to aggregate log returns over h periods, to show the h-period mean scales with h while the h-period variance is the sum of the one-step variances, to collapse those into the square-root-of-time rule (σ over h periods = √h × the one-period σ) under flat volatility, and then to replace that shortcut with the GARCH(1,1) multi-step forecast that mean-reverts to the long-run variance when volatility clusters. It is the horizon-scaling layer that turns single-period risk measures into the multi-period numbers used for capital and reporting.

What this chapter covers

• 011. Return aggregation - because log returns add across time, the h-period return is the sum of the one-period log returns, R(t+h,h) = Σ R(t+j)
• 022. h-period mean - under a constant conditional mean the expectation of the sum is μ(h) = hμ, so the mean scales linearly with the horizon
• 033. h-period variance - for conditionally uncorrelated returns the variance of the sum is the sum of the one-step variances (covariance terms vanish)
• 044. Square-root-of-time rule - if every one-step variance equals a constant σ², the h-period variance is hσ² and the volatility is √h × σ
• 055. Variance scales with h, volatility with √h - the single most common slip is scaling volatility (or VaR) linearly in h
• 066. Scaling a VaR - scale μ by h and σ by √h, then recompute VaR; scaling VaR directly by √h is exact only when the drift is zero
• 077. When the rule breaks - √h assumes a flat volatility term structure; under volatility clustering it over- or under-states multi-period risk
• 088. GARCH(1,1) long-run variance - if α+β < 1 the process is stationary with long-run variance σ-bar² = ω/(1-α-β)
• 099. GARCH multi-step forecast - the h-step variance is a weighted average of σ-bar² and the next-period variance, reverting at rate α+β
• 1010. EWMA as a special case - EWMA is a GARCH(1,1) with α+β = 1, so it has no mean reversion and a flat forecast term structure

A 10-day Value-at-Risk by square-root-of-time scaling

Key terms

Return aggregation (h-period return)

The return over a horizon of h periods. Because log returns add across time, the h-period log return is the sum of the one-period log returns, R(t+h,h) = Σ R(t+j) for j = 1 to h. This is exact for log returns and only approximate for simple returns, which compound (multiply the gross returns) rather than add.

h-period mean

The conditional expectation of the aggregated return. Under a constant conditional mean μ, the expectation of the sum of h returns is μ(h) = hμ, so the mean scales linearly with the horizon.

h-period variance

The conditional variance of the aggregated return. When returns are conditionally uncorrelated the covariance terms vanish and the variance of the sum is the sum of the one-step variances, σ²(h) = Σ Var_t(R(t+j)). If every one-step variance equals a constant σ² this reduces to hσ².

Square-root-of-time rule

Under a flat volatility term structure (i.i.d. returns, or an EWMA whose future one-step forecasts are all equal), the h-period variance is hσ² and the h-period volatility is √h × σ. Variance scales with h; volatility scales with the square root of h. Annualising uses σ(ann) = √N × σ with N periods per year (e.g. 252 trading days).

Flat volatility term structure

The assumption that the one-step-ahead conditional variance is the same at every future step. Only under this assumption is the square-root-of-time rule exact. Real returns show volatility clustering, so the term structure usually slopes and √h becomes a first-order approximation - overstating risk when current volatility is below its long-run level and understating it when above.

GARCH(1,1) long-run variance

For σ_t² = ω + α(R(t-1)-μ)² + βσ(t-1)² with α+β < 1, the process is stationary and reverts to the long-run (unconditional) variance σ-bar² = ω/(1-α-β). The quantity α+β is the total persistence; the closer it is to 1, the slower the reversion. EWMA is the special case α+β = 1, which has no finite long-run variance.

GARCH multi-step variance forecast

The h-step-ahead conditional variance is Var_t(R(t+h)) = (1 - (α+β)^(h-1)) σ-bar² + (α+β)^(h-1) σ²(t+1|t), a weighted average of the long-run variance and the known next-period variance. At h = 1 it equals σ²(t+1|t); as h grows it reverts to σ-bar² at rate α+β. This is distinct from the h-period total variance, which is the sum of the per-step forecasts.

Value-at-Risk (VaR) at a horizon

VaR at coverage p is the negated left-tail quantile of the return distribution, reported as a positive loss: VaR(p) = -(μ + σ z(p)) for a location-scale model, with z(0.05) = -1.645 and z(0.01) = -2.326 for the normal. To move it to a horizon h you scale μ by h and σ by √h (flat volatility), then recompute; the 1% VaR always exceeds the 5% VaR.

Multi-Period Risk and the Square-Root-of-Time Rule FAQ

Exam move

Treat this chapter as two clean scaling laws plus one exception, because it almost always appears as the middle step of a larger VaR, ES or volatility question. First, be able to write the aggregation identity R(t+h,h) = Σ R(t+j) and derive from it that the mean scales with h and the variance is the sum of the one-step variances. Drill the square-root-of-time move until it is automatic: variance scales with h, volatility with √h, and to scale a VaR you scale μ by h and σ by √h then recompute - never scale volatility or VaR linearly in h. Practise the direction and sign checks out loud: VaR is a positive loss, the 1% VaR is larger than the 5% VaR, and a GARCH forecast that starts above its long-run level must decrease toward it. Then rehearse the exception: the √h rule needs flat volatility, so under GARCH use the multi-step forecast Var_t(R(t+h)) that mean-reverts to σ-bar² = ω/(1-α-β), and remember EWMA is the α+β = 1 special case with no reversion. Because Assignment 2 is written in the style of the final exam, its problems and the workshop exercises are the best rehearsal. The final exam is closed book and worth 40% of the unit; the exam duration is not published in the unit materials, so plan your time in proportion to the marks on each question and confirm the exact date and length on Moodle or the Monash exam timetable.

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