Set the age-weighted, volatility-weighted, correlation-weighted and filtered historical simulation methods side by side, noting how they differ and what they share.

FRM® Part II · Market Risk Measurement and Management · Non-Parametric Approaches · FRM2.MR.2.c

The idea

Basic historical simulation weights every observation equally and uses returns as they occurred. The four refinements make the sample more relevant to current conditions in different ways. Age-weighted simulation changes the probability weights. Observation 1 is the most recent and λ is a decay factor below 1, so observation i gets this weight, Weight = λ^(i−1) × (1 − λ) ÷ (1 − λ^n). VaR is then read from the cumulative weights of the ranked losses. Volatility-weighted simulation rescales each return by current volatility divided by the volatility at the time, keeping equal weights. Correlation-weighted simulation adjusts the returns so their covariance structure reflects current correlations. Filtered historical simulation fits a model such as GARCH, standardises returns into residuals, resamples those residuals and rescales them by forecast volatility, which also allows multi-day paths. What they share is that all use actual historical observations and make no parametric assumption about the shape of the return distribution.

Formulas

Age weight of an observation

w(i) = λ^(i − 1) × (1 − λ) ÷ (1 − λⁿ)

λ: the decay factor, below 1
i: the age of the observation, 1 for the most recent
n: the number of observations in the window

VaR is the loss at which cumulative weight, ranked from the largest loss, first reaches the tail probability. Omitting the scaling term makes the weights too large.

Volatility-weighted historical return

rescaled return = rₜ × σnow ÷ σₜ

rₜ: the historical return on day t
σnow: the current volatility forecast
σₜ: the volatility on day t

Equal probability weights are kept. A modest loss on a calm day can become the worst rescaled scenario, so re-rank after rescaling.

A practice question

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

Four refinements of basic historical simulation are in use. Which one leaves the historical returns themselves unchanged and gives each observation an unequal probability weight that declines with its age?

Age-weighted historical simulation - correct
Volatility-weighted historical simulationWhy not: Rescales returns by the ratio of current to then-prevailing volatility and keeps equal probability weights.
Correlation-weighted historical simulationWhy not: Adjusts the returns so their covariance structure matches current correlations; the weights stay equal.
Filtered historical simulationWhy not: Standardises returns with a GARCH-type model and rescales the residuals by forecast volatility, which changes the returns, not the weights.

Age-weighted simulation changes the weights (lambda^(i-1)(1 - lambda)/(1 - lambda^n)) and leaves the returns alone. The other three change the returns while keeping equal weights.

A harder one

An age-weighted historical simulation uses n = 100 days, decay factor lambda = 0.98, and weight w_i = lambda^(i-1) x (1 - lambda)/(1 - lambda^n), where i = 1 is the most recent day. The six largest losses (USD million) and their ages i are: 6.0 (i = 60), 5.2 (i = 3), 4.8 (i = 25), 4.5 (i = 8), 4.1 (i = 90), 3.8 (i = 40). Ranking losses from largest, the 95% VaR is the loss at which the cumulative weight first reaches or exceeds 5%. The 95% VaR is closest to:

USD 4.1 millionWhy not: Gives each observation a weight of 1%, which is the equally weighted answer (the fifth largest loss) and ignores the age weights.
USD 4.5 million - correct
USD 4.8 millionWhy not: Stops at the last loss before cumulative weight reaches 5% (4.3%) instead of the loss that takes it above 5%.
USD 6.0 millionWhy not: Omits the (1 - lambda)/(1 - lambda^n) scaling, so the first weight alone (0.98^59 = 0.30) exceeds 5%.

The scaling constant is 0.02/(1 - 0.98^100) = 0.0231. Weights from the largest loss: 0.70%, 2.21%, 1.42%, 2.00%. Cumulative: 0.70%, 2.91%, 4.33%, 6.34%, so the 5% mark is crossed at 4.5. Equal weights would give 4.1.

The hardest tier

A volatility-weighted historical simulation uses 20 daily observations and rescales each return by (current volatility)/(volatility on that day). Current daily volatility is 1.5%. The four worst raw returns and the volatility at the time are: -5.0% (volatility 2.5%), -4.0% (1.25%), -2.0% (0.5%) and -3.5% (2.0%). After rescaling, all other observations are losses of less than 2.5%. Taking the 95% VaR as the worst of the 20 rescaled outcomes, the 95% VaR is closest to:

5.0%Why not: Reads the worst raw return and ignores the rescaling, which changes the ranking.
6.0% - correct
4.8%Why not: Picks the rescaled loss of the second-worst raw return and misses that the calm-day loss rescales higher still.
3.0%Why not: Rescales only the worst raw return (-5.0% x 1.5/2.5) and does not check the other observations.

Rescaled returns: -5.0 x 1.5/2.5 = -3.0%; -4.0 x 1.5/1.25 = -4.8%; -2.0 x 1.5/0.5 = -6.0%; -3.5 x 1.5/2.0 = -2.625%. A modest loss on a very calm day becomes the worst scenario at today's volatility, so VaR is 6.0%.

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