Monte Carlo VaR: Tail Percentiles, Horizons, and Path Simulation

Summary

The document outlines a Monte Carlo procedure for estimating portfolio Value at Risk, using correlated stock returns generated from a historical covariance matrix and a Cholesky factorization. Simulated terminal prices are used to revalue the portfolio; the resulting losses are sorted, and VaR is read from the chosen left-tail percentile. The response clarifies that VaR is a loss relative to the initial portfolio value at that percentile, and discusses selecting a horizon suited to the risk question. It cites common one-day and ten-day banking horizons, backtesting against historical exceedances, and longer horizons for some credit-risk applications.

For geometric Brownian motion, the terminal-price formula gives the horizon distribution directly, so intermediate path steps are unnecessary. Time stepping may be needed when the risk model lacks a tractable horizon solution or the portfolio includes path-dependent derivatives. The method depends on its distributional and covariance assumptions; the document does not develop calibration choices, non-normal shocks, or derivative valuation in detail.

Key ideas

• Monte Carlo VaR estimates portfolio values by simulating correlated asset returns and revaluing the holdings.
• Sort simulated outcomes and measure VaR as the loss at the selected lower-tail percentile.
• Choose the horizon to match the risk or regulatory question, then assess forecasts through exceedance backtests.
• A terminal-price simulation suffices when the process has a tractable horizon distribution.
• Intermediate time steps can matter for path-dependent instruments or processes without a direct horizon solution.

Tags

# callable bonds with FDM


# callable bonds with FDM












I am thinking of implementing a model to price callable bonds on a finite difference grid. I wonder how the Price to worst yield model will relate to it in terms of risks(or should do). What I expect is that with a simple yield to workout date there is only one single "best" call time which I can find by computing the min yield(often it is the first call date). And my model will yield a distribution of call dates but the mean of that should coincide with the workout date and as a result all risks are about the same?

How does it work when the best workout date is the first call date though? My model's distribution can have its mode on the first call date but not the mean and as a result don't I have larger risks with a stochastic model as duration would be larger in the model case?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.