Calculating VaR: A Review of Methods
Fed and OCC regulations  and . Additionally, hedge funds now use VaR to report a unified risk measure across multiple asset classes. There are multiple approaches to VaR, so which method should we choose? In this brief paper, we outline a case for full revaluation VaR in contrast to a simulated VaR using a “delta-gamma” approach to value assets.Many firms now use Value-at-Risk (“VaR”) for risk reporting. Banks need VaR to report regulatory capital usage under the Market Risk Rule, as outlined in the
What is VaR?The VaR for a position or book of business can be defined as some threshold T (in dollars) where the existing position, when faced with market conditions similar to some given historical period, will have P/L greater than T with probability k. Typically, k is chosen to be 99% or 95%.
- Set a significance percentile k, a market observation period, and holding period n.1
- Generate a set of future market conditions (“scenarios”) from today to period n.
- Compute a P/L on the position for each scenario.
- Project future market conditions using a Monte Carlo simulation framework.
- Project future market conditions using historical (actual) changes in market conditions.
Monte Carlo SimulationMany commercial providers simulate future market conditions using Monte Carlo simulation. To do this, they must first estimate the distributions of risk factors, including correlations between risk factors. Using correlations that are derived from historical data makes the general assumption that correlations are constant within the period. As shown in the academic literature, correlations tend to change, especially in extreme market moves – exactly the kind of moves that tend to define the VaR threshold.4 By constraining correlations, VaR may be either overstated or understated depending on the structure of the position. To account for this, some providers allow users to “stress” correlations by increasing or decreasing them. Such a stress scenario is either arbitrary, or is informed by using correlations from yet another time-period (for example, using correlations from a time of market stress), mixing and matching market data in an ad hoc way. Further, many market risk factors are highly correlated, which is especially true on the interest rate curve. To account for this, some providers use a single factor for rate-level and then a second or third factor for slope and curvature of the curve. While this may be broadly representative, this approach may not capture subtle changes on other parts of the curve. This limited approach is acceptable for non-callable fixed income securities, but proves problematic when applying curve changes to complex securities such as MBS, where the security value is a function of forward mortgage rates, which in turn is a multivariate function of points on the curve and often implied volatility.
Historical SimulationRiskSpan projects future market conditions by using actual (observed) n-day changes in market conditions over the look-back period. For example, if we are computing 10-day VaR for regulatory capital usage under the Market Risk Rule, RiskSpan takes actual 10-day changes in market variables. This approach allows our VaR scenarios to account for natural changes in correlation under extreme market moves, such as occurs during a flight-to-quality where risky assets tend to underperform risk-free assets, and risky assets tend to move in a highly correlated manner. RiskSpan believes this is a more natural way to capture changing correlations, without the arbitrary overlay of how to change correlations in extreme market moves. This, in turn, will more correctly capture VaR.5
Calculating Simulated P/LWith the VaR scenarios defined, we move on to computing P/L under these scenarios. Generally, there are two methods employed
- A Taylor approximation of P/L for each instrument, sometimes called “delta-gamma.”
- A full revaluation of each instrument using its market-accepted technique for valuation.
Delta-Gamma P/L ApproximationMany market practitioners use a Taylor approximation or “delta-gamma” approach to valuing an instrument under each VaR scenario. For instruments whose price function is approximately linear across each of the m risk factors, users tend to use the first order Taylor approximation, where the instrument price under the kth VaR scenario is given by Making the price change in the kth scenario Where ΔP is the simulated price change, Δxi is the change in the ith risk factor, and is the price delta with respect to the ith risk factor evaluated at the base case. In many cases, these partial derivatives are approximated by bumping the risk factors up/down.6 If the instrument is slightly non-linear, but not non-linear enough to use a higher order approximation, then approximating a first derivative can be a source of error in generating simulated prices. For instruments that are approximately linear, using first order approximation is typically as good as full revaluation. From a computation standpoint, it is marginally faster but not significantly so. Instruments whose price function is approximately linear also tend to have analytic solutions to their initial price functions, for example yield-to-price, and these analytic solutions tend to be as fast as a first-order Taylor approximation. If the instrument is non-linear, practitioners must use a higher order approximation which introduces second-order partial derivatives. For an instrument with m risk-factors, we can approximate the price change in the kth scenario by using the multivariate second order Taylor approximation To simplify the application of the second-order Taylor approximation, practitioners tend to ignore many of the cross-partial terms. For example, in valuing MBS under delta-gamma, practitioners tend to simplify the approximation by using the first derivatives and a single “convexity” term, which is the second derivative of price with respect to overall rates. Using this short-cut raises a number of issues:
- It assumes that the cross-partials have little impact. For many structured products, this is not true.7
- Since structured products calculate deltas using finite shifts, how exactly does one calculate a second-order mixed partials?8
- For structured products, using a single, second-order “convexity” term assumes that the second order term with respect to rates is uniform across the curve and does not vary by where you are on the curve. For complex mortgage products such as mortgage servicing rights, IOs and Inverse IOs, convexity can vary greatly depending on where you look at the curve.
- Using a second-order approximation assumes that the second order derivatives are constant as rates change. For MBS, this is not true in general.
- Changes in risk factors are all independently, identically distributed (no autocorrelation or heteroscedasticity)
- The asset price function is linear in all risk factors
Rationalizing Weaknesses in the ApproximationWith the weaknesses in the Taylor approximation cited above, why do some providers still use delta-gamma VaR? Most practitioners will cite that the Taylor approximation is much faster than full revaluation for complex, non-linear instruments. While this seems true at first glance, you still need to:
- Compute or approximate all the first partial derivatives
- Compute or approximate some of the second partial derivatives and decide which are relevant or irrelevant. This choice may vary from security type to security type.