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Show how a VaR model's sensitivity analysis is carried out and weigh its potential benefits against its difficulties.

FRM® Part II · Market Risk Measurement and Management · Validating Bank Holding Companies’ Value-at-Risk Models for Market Risk · FRM2.MR.6.b

The idea

Sensitivity analysis re-runs the VaR model while changing one input or assumption at a time. Examples are the confidence level, the holding period, the data window, the EWMA decay factor, the correlation matrix and the volatility estimate. The analyst records how far VaR moves. Its benefit is that it shows which assumptions drive the number, where model risk is concentrated, and how fragile the result is to choices that have no single correct value. Its difficulties are practical. A model has many parameters, plausible ranges must be judged, and there is no objective threshold for an acceptable change. Effects also interact, so moving inputs one at a time can miss combined shocks. VaR is not additive across inputs. Sensitivity analysis shows dependence, not correctness, and it does not replace backtesting.

Formulas

EWMA variance update

σnew² = λ × σold² + (1 − λ) × r²

λ
the decay factor
σold
the previous volatility estimate
r
the latest return

A higher λ weights the old variance more, so a shock moves VaR less. Swapping the weights is the usual error.

Parametric VaR with zero mean

VaR = z × σ × V

z
the standard normal quantile for the confidence level, 2.326 at 99%
σ
the standard deviation of returns over the horizon
V
the portfolio value

Normal returns with zero mean. VaR scales with the quantile z, not with the confidence level itself.

A practice question

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

Which is a main benefit of carrying out sensitivity analysis on a VaR model?

  1. It shows which assumptions and parameters the VaR depends on most, revealing where model risk is concentrated. - correct
  2. It produces a statistically exact confidence interval around the true VaR.Why not: Sensitivity analysis shows how VaR moves with inputs; it does not give a sampling confidence interval.
  3. It identifies which parameter value is the correct one to use.Why not: It reveals dependence, not correctness; it cannot say which choice is right.
  4. It removes the need for backtesting because it shows the model is accurate.Why not: Sensitivity analysis does not compare the model with realised outcomes, so it cannot replace backtesting.

The value of the exercise is diagnostic: large movements flag assumptions that need justification. It neither measures accuracy nor chooses parameter values.

A harder one

A bank's parametric VaR model assumes normal returns with zero mean. The portfolio is worth USD 250 million with daily volatility of 1.2%. As a sensitivity run, the validator changes the confidence level from 95% (z = 1.645) to 99% (z = 2.326) at the one-day horizon. The new VaR is closest to:

  1. USD 5.14 millionWhy not: Scales the 95% VaR (USD 4.93 million) by 99/95; VaR scales with the normal quantile, not with the confidence level itself.
  2. USD 6.98 million - correct
  3. USD 7.73 millionWhy not: Uses 2.576, the two-sided 99% critical value; a one-day VaR is a one-tail quantile (2.326).
  4. USD 22.07 millionWhy not: Also scales by the square root of 10, which changes the holding period; the run changed only the confidence level.

VaR = z x sigma x value = 2.326 x 1.2% x 250 = USD 6.98 million, up from 1.645 x 1.2% x 250 = USD 4.93 million. The sensitivity is the ratio 2.326/1.645 = 1.41.

The hardest tier

A bank uses EWMA to estimate daily volatility, updating variance = lambda x previous variance + (1 - lambda) x² return. Yesterday's volatility estimate was 1.0% and today's return is -3.0%. For a USD 200 million position, a validator runs the 99% one-day normal VaR (z = 2.326) with lambda = 0.94 and again with lambda = 0.97. Which result is correct?

  1. About USD 5.66 million at 0.94 and USD 5.18 million at 0.97: the higher decay factor gives the lower VaR. - correct
  2. About USD 5.18 million at 0.94 and USD 5.66 million at 0.97: the higher decay factor weights today's shock more heavily.Why not: Reverses the effect: a higher lambda gives more weight to the old variance, so the shock is absorbed more slowly.
  3. About USD 13.58 million at 0.94 and USD 13.77 million at 0.97.Why not: Swaps the weights, putting lambda on the new squared return instead of the previous variance.
  4. About USD 6.89 million at 0.94 and USD 5.77 million at 0.97.Why not: Forgets to take the square root of the variance, using 1.48% and 1.24% as if they were volatilities.

Lambda 0.94: variance = 0.94 x 1 + 0.06 x 9 = 1.48, sigma = 1.217%, VaR = 2.326 x 1.217% x 200 = 5.66 million. Lambda 0.97: variance = 0.97 + 0.27 = 1.24, sigma = 1.114%, VaR = 5.18 million. The decay factor choice moves VaR by about 9%.

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