Market Risk Measurement - VaR and Beyond › The VaR Framework

Before diving into calculation methods, we need to establish the formal definition of VaR and understand what it does and does not measure. At its heart, VaR addresses a fundamental question in risk management: given the inherent uncertainty in financial markets, what is the worst loss we should reasonably expect under normal conditions? The answer requires us to think probabilistically about potential outcomes rather than seeking a single deterministic prediction.

We cannot know exactly what loss will occur tomorrow. Instead, we can characterize the full distribution of possible losses and then identify a threshold that separates "normal" adverse outcomes from truly extreme events. This threshold becomes our VaR estimate. By specifying a confidence level, we acknowledge that losses beyond this threshold will sometimes occur, but we want such exceedances to be rare enough that we consider them exceptional rather than routine.

This formal definition uses the infimum (greatest lower bound) to handle distributions that may have discontinuities or discrete probability masses. For most practical purposes with continuous return distributions, you can think of VaR simply as the loss value that separates the worst $(1-\alpha)$ fraction of outcomes from the better $\alpha$ fraction. The definition ensures that we identify the precise point on the loss distribution where cumulative probability reaches our target confidence level.

VaR depends on three key parameters:

• Confidence level ($\alpha$): Typically 95% or 99%. Higher confidence levels produce larger VaR estimates but are harder to backtest due to fewer tail observations.
• Time horizon ($h$): Usually 1 day for trading desks or 10 days for regulatory capital calculations. Longer horizons capture more risk but assume portfolio composition remains static.
• Portfolio value: VaR is typically expressed in absolute currency terms, though it can also be expressed as a percentage of portfolio value.

The sign convention varies in practice. Some practitioners define VaR as a positive number representing potential loss, while others express it as a negative return. We follow the convention where VaR is positive, so a VaR of $10 million means the portfolio could lose up to $10 million.

Understanding how VaR connects to the underlying return distribution is needed for all calculation methods. This relationship links the abstract definition of VaR and the practical task of computing it from return data or models.

For a portfolio with current value $V$ and return $R$ over the horizon, the loss is $L = -V \cdot R$ (since negative returns create positive losses). This sign convention turns a return of -3% into a positive loss equal to 3% of the portfolio value. By expressing everything in terms of losses, we ensure that VaR comes out as a positive number that can be directly interpreted as potential monetary damage.

If returns follow a continuous distribution with cumulative distribution function $F_R$, then VaR at confidence level $\alpha$ corresponds to the $(1-\alpha)$ quantile of the return distribution:

$$\text{VaR}_{\alpha} = -V \cdot F_R^{-1}(1-\alpha)$$

where:

• $\text{VaR}_{\alpha}$: Value-at-Risk at confidence level $\alpha$
• $V$: current portfolio value
• $F_R^{-1}$: inverse cumulative distribution function of returns
• $1-\alpha$: probability level for the tail

To understand why VaR uses the $(1-\alpha)$ quantile rather than the $\alpha$ quantile, consider what we are seeking. For 99% VaR, we want the loss that is exceeded only 1% of the time. This corresponds to returns that fall below the 1st percentile of the return distribution. The inverse CDF $F_R^{-1}(0.01)$ gives us exactly this value: the return below which only 1% of outcomes fall. Multiplying by $-V$ converts this worst-case return into the corresponding dollar loss.

This relationship is the foundation for all VaR calculation methods. The methods differ in how they estimate the return distribution and its quantiles. Parametric methods assume a specific distributional form and estimate its parameters. Historical simulation uses the empirical distribution directly. Monte Carlo simulation generates a synthetic distribution from a specified stochastic model. Each approach represents a different answer to the question: "How do we characterize the distribution of portfolio returns?"