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Explain how Monte Carlo simulation is used to compute VaR and ES, and pinpoint where it is strong and where it is weak.

FRM® Part I · Valuation and Risk Models · Calculating and Applying VaR · FRM1.VR.2.f

The idea

Monte Carlo simulation specifies a stochastic process for each risk factor, for example geometric Brownian motion with a covariance matrix. It then draws many random sets of factor values, revalues the whole portfolio in each trial, and ranks the resulting profits and losses. VaR is the percentile of the simulated distribution and ES the average of the tail trials. To get correlated draws, combine independent standard normals through a Cholesky factorisation. Its strengths are that it handles non-linear positions with full revaluation, and allows non-normal distributions, path-dependence and many risk factors. Its weaknesses are heavy computation, sampling error that falls only with the square root of the number of trials, and model risk, since the output is only as good as the assumed process and its parameters. The usual exam point is that simulation does not remove the need for assumptions.

Formulas

Correlated normal draw for two risk factors

Z₂ = ρ × e₁ + √(1 − ρ²) × e₂

ρ
the target correlation between the two factors
e₁, e₂
independent standard normal draws, with Z₁ = e₁

The two-factor case of a Cholesky factorisation. Using 1 − ρ² without the square root, or the raw e₂, breaks the correlation.

A practice question

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

Which sequence of steps best describes a Monte Carlo VaR calculation?

  1. Collect the actual changes in risk factors on each of the last N days and apply them to today's portfolio.Why not: This is historical simulation, which uses observed rather than simulated changes.
  2. Specify processes and correlations for the risk factors, simulate many sets of values, revalue in each, and take a P&L percentile. - correct
  3. Replace each option by its delta equivalent and compute the VaR analytically from a covariance matrix.Why not: This is the delta-normal approach, with no simulation.
  4. Draw random changes to the position sizes and record the resulting variation in portfolio value.Why not: Randomises holdings instead of risk factors; Monte Carlo VaR is about market uncertainty on the current portfolio.

Monte Carlo VaR needs a model for the risk factors. Each trial revalues the portfolio, and VaR and ES are read from the distribution of the simulated P&L.

A harder one

A Monte Carlo VaR using 10,000 trials has a standard error that the risk manager considers too large. Assuming the standard error is proportional to 1 / sqrt(N), the number of trials needed to halve it is closest to:

  1. 20,000Why not: Doubles the trials, but halving the error requires four times as many because error falls with the square root of N.
  2. 14,142Why not: Multiplies by the square root of 2, which would cut the error by only about 16 percent.
  3. 10,000Why not: Assumes the error of a quantile does not depend on the number of trials.
  4. 40,000 - correct

Error is proportional to 1/sqrt(N). Halving it needs sqrt(N) to double, so N must quadruple: 4 x 10,000 = 40,000. Sampling error falls slowly, which is a main drawback of simulation.

The hardest tier

A stock is modelled as geometric Brownian motion with drift 8% and volatility 30% a year. The price is 80 today. A simulation trial for a 10-day horizon (10/252 of a year) draws a standard normal shock of -2.0. The portfolio is long 50,000 shares. Which loss is correct? Use S(T) = S(0) x exp[(mu - sigma²/2) x dt + sigma x sqrt(dt) x shock].

  1. About USD 465,400Why not: Treats the move as arithmetic, S x (mu x dt + sigma x sqrt(dt) x shock), rather than lognormal.
  2. About USD 445,700 - correct
  3. About USD 433,000Why not: Adds sigma²/2 to the drift instead of subtracting it.
  4. About USD 147,800Why not: Uses a one-day horizon rather than 10 days.

Exponent = (0.08 - 0.045) x 0.03968 + 0.30 x 0.19920 x (-2.0) = 0.001389 - 0.11952 = -0.11813. S(T) = 80 x 0.8886 = 71.09. Loss per share 8.91, times 50,000 = USD 445,700.

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