Filtered Historical Simulation (EWMA)

Daily equity returns are not identically distributed over time. Their volatility moves, and it moves persistently: turbulent days cluster together, and a market that was volatile last week is likely to be volatile tomorrow. An unconditional bootstrap ignores this. It draws from a pool that mixes 2008 and 2017 in fixed proportion regardless of what state the market is in when you press run.

Filtered historical simulation, introduced by Barone-Adesi, Giannopoulos and Vosper (1999), separates the two components. Write each return as a conditional volatility times a standardized shock. If the volatility model is doing its job, those shocks are close to IID even though the returns are not, which means they are the part of history it is legitimate to resample freely. Volatility is then reintroduced by a forward recursion that starts from today's level.

The practical consequence: run the same portfolio in a calm market and in a turbulent one, and FHS gives you materially different near-term distributions. The bootstrap gives you the same answer both times.

1. 1

   Move to log space

   Daily simple returns become log returns via log(1 + r). Everything downstream happens in log space, which is what guarantees positive gross returns at the end.

2. 2

   Remove the per-asset sample log mean

   Estimation-side demeaning uses the historical log mean, deliberately not the user's forward drift target: the residuals must not depend on a forward assumption. The difference is a per-asset constant, negligible for variance estimation, and disclosed in the artifact.

3. 3

   Estimate conditional volatility

   An EWMA recursion produces a conditional variance series for each asset, where each day's variance conditions only on strictly earlier observations.

4. 4

   Standardize

   Divide each demeaned log return by its conditional volatility to get the standardized residual matrix, centered with unit variance per asset.

5. 5

   Resample complete rows

   Draw whole cross-asset residual rows IID. Cross-sectional dependence is inherited exactly: a day when every holding was shocked together stays that way.

6. 6

   Rebuild volatility forward

   Run the same variance recursion forward from today's one-step-ahead variance, so simulated volatility evolves rather than staying frozen.

7. 7

   Return to simple returns

   Apply the drift normalization, then exp minus one. Gross returns are positive by construction and no clipping correction is needed.

For a drift-removed log-return series , the conditional variance follows

with the decay parameter defaulting to . Note the indexing: conditions on observations strictly before , so no day is standardized using its own realization. The recursion initializes at the full-sample variance, a declared and disclosed choice whose influence decays geometrically.

EWMA is the integrated special case of the more general GJR-GARCH form, with , , and . The engine stores both models in that single form so the forward simulation recursion vectorizes identically whichever one an asset ended up with.

The 0.94 decay is the RiskMetrics benchmark value. It is a reasonable starting point, not a validated production constant for Indian equities, and it is documented as such rather than presented as tuned.

Simulating in log space creates a subtlety that a naive implementation gets wrong. Setting the log mean equal to the desired arithmetic mean does not produce that arithmetic mean, because of Jensen's inequality: the expectation of an exponential is not the exponential of the expectation.

The engine therefore normalizes so the conditional expected gross return matches the declared daily arithmetic target exactly. The per-asset log mean is set to

where the expectation is taken over that asset's empirical residual pool, precomputed on a volatility grid. The correction is not the textbook , because the residuals are not Gaussian. Using the empirical cumulant generating function keeps the declared drift semantics exact even under fat-tailed, skewed residuals.

This is also why the engine must not add its ordinary additive simple-return drift shift on top of an FHS run. Drift and the per-path HAC mean perturbation are already normalized in, with the perturbation folded through the same transformation so positive support survives it. See Drift and estimation risk.

These are operability and confidence tiers. They are not proof of long-horizon calibration, and the labels are deliberately worded so they cannot be read that way.

During simulation the conditional volatility is clamped into around the long-run level. This is a declared guard against a runaway recursion, not a modeling feature, and like every other threshold on this page it is a versioned engineering constant that changes only on validation evidence.