Calculating VaR
A Review of Methods
CONTRIBUTOR
Don Brown
CoHead of Quantitative Analytics
TABLE OF CONTENTS
Have questions about calculating VaR?
Chapter 1
Introduction
Many firms now use ValueatRisk (“VaR”) for risk reporting. Banks need VaR to report regulatory capital usage under the Market Risk Rule, as outlined in the 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 “deltagamma” approach to value assets.
The VaR for a position or book of business can be defined as some threshold (in dollars) where the existing position, when faced with market conditions similar to some given historical period, will have P/L greater than with probability. Typically, is chosen to be or.
To compute this threshold , we need to
 Set a significance percentile , 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
After computing each position’s P/L, we sum the P/L for each scenario and then rank the scenarios’ P/L to find the k^{th} percentile (worst) loss.^{2} This loss defines our VaR T at the k^{th} percentile for observationperiod length n.
Determining what significance percentile k and observation length n to use is straightforward and is often dictated by regulatory rules, for example 99^{th} percentile 10day VaR is used for riskbased capital under the Market Risk Rule. Generating the scenarios and computing P/L under these scenarios is open to interpretation. We cover each of these in the next two sections, with their advantages and drawbacks.
Chapter 2
Generating Scenarios
To compute VaR, we first need to generate projective scenarios of market conditions. Broadly speaking, there are two ways to derive this set of scenarios^{3}
 Project future market conditions using a Monte Carlo simulation framework
 Project future market conditions using historical (actual) changes in market conditions
MONTE CARLO SIMULATION
Many 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 timeperiod (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 ratelevel 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 noncallable 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 SIMULATION
RiskSpan projects future market conditions by using actual (observed) day changes in market conditions over the lookback period. For example, if we are computing 10day VaR for regulatory capital usage under the Market Risk Rule, RiskSpan takes actual 10day 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 flighttoquality where risky assets tend to underperform riskfree 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
Chapter 3
Calculating Simulated P/L
With 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 “deltagamma”
 A full revaluation of each instrument using its marketaccepted technique for valuation
Market practitioners sometimes blend these two techniques, employing full revaluation where the valuation technique is simple (e.g. yield + spread) and using deltagamma where revaluation is more complicated (e.g. OAS simulation on MBS).
DELTAGAMMA P/L APPROXIMATION
Many market practitioners use a Taylor approximation or “deltagamma” 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 k^{th} scenario
Where ΔP is the simulated price change, Δx_{i} is the change in the i^{th} risk factor, and is the price delta with respect to the i^{th} 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 nonlinear, but not nonlinear 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 yieldtoprice, and these analytic solutions tend to be as fast as a firstorder Taylor approximation.
If the instrument is nonlinear, practitioners must use a higher order approximation which introduces secondorder partial derivatives. For an instrument with m riskfactors, we can approximate the price change in the k^{th} scenario by using the multivariate second order Taylor approximation
To simplify the application of the secondorder Taylor approximation, practitioners tend to ignore many of the crosspartial terms. For example, in valuing MBS under deltagamma, 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 shortcut raises a number of issues:
 It assumes that the crosspartials 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 secondorder mixed partials?^{8}
 For structured products, using a single, secondorder “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 secondorder approximation assumes that the second order derivatives are constant as rates change. For MBS, this is not true in general.
For example, in the graphs below we show a constantOAS price curve for TBA FNMA 30yr 3.5%, as well as a graph of its “DV01”, or first derivative with respect to rates. As you can see, the DV01 graph is nonlinear, implying that the convexity term (second derivative of the price function) is nonconstant, rendering a secondorder Taylor approximation a weak assumption. This is especially true for large moves in rate, the kind of moves that dominate the computation of the VaR.^{9}
In addition to the assumptions above, we occasionally observe that commercial VaR providers compute 1day VaR and, in the interest of computational savings, scale this 1day VaR by √10 to generate 10day VaR. This approximation only works if
 Changes in risk factors are all independently, identically distributed (no autocorrelation or heteroscedasticity)
 The asset price function is linear in all risk factors
In general, neither of these conditions hold and using a scaling factor of √10 will likely yield an incorrect value for 10day VaR.^{10}
RATIONALIZING WEAKNESS IN THE APPROXIMATION
With the weaknesses in the Taylor approximation cited above, why do some providers still use deltagamma VaR? Most practitioners will cite that the Taylor approximation is much faster than full revaluation for complex, nonlinear 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.
Neither of these tasks are computationally simple for complex, pathdependent securities which are found in many portfolios. Further, the choice of which secondorder terms to ignore has to be supported by documentation to satisfy regulators under the Market Risk Rule.
Even after approximating partials and making multiple, qualitative assessments of which secondorder terms to include/exclude, we are still left with error from the Taylor approximation. This error grows with the size of the market move, which also tends to be the scenarios that dominate the VaR calculation. With today’s flexible cloud computation and ultrafast, cheap processing, the Taylor approximation and its computation of partials ends up being only marginally faster than a full revaluation for complex instruments.^{11}
With the weaknesses in Taylor approximation, especially with nonlinear instruments, and the speed and cheapness of full revaluation, we believe that fully revaluing each instrument in each scenario is both more accurate and more straightforward than having to defend a raft of assumptions around the Taylor approximation.
Chapter 4
Conclusion
With these points in mind, what is the best method for computing VaR? Considering the complexity of many instruments, and considering the comparatively cheap and fast computation available through today’s cloud computing, we believe that calculating VaR using a historicalscenario, full revaluation approach provides the most accurate and robust VaR framework.
From a scenario generation standpoint, using historical scenarios allows risk factors to evolve in a natural way. This in turn captures actual changes in risk factor correlations, changes which can be especially acute in large market moves. In contrast, a Monte Carlo simulation of scenarios typically allows users to “stress” correlations, but these stresses scenarios are arbitrary which may ultimately lead to misstated risk.
From a valuation framework, we feel that full revaluation of assets provides the most accurate representation of risk, especially for complex instruments such as complex ABS and MBS securities. The assumptions and errors introduced in the Taylor approximation may overwhelm any minor savings in runtime, given today’s powerful and cheap cloud analytics. Further, the Taylor approximation forces users to make and defend qualitative judgements of which partial derivatives to include and which to ignore. This greatly increasing the management burden around VaR as well as regulatory scrutiny around justifying these assumptions.
In short, we believe that a historical scenario, fullrevaluation VaR provides the most accurate representation of VaR, and that today’s cheap and powerful computing make this approach feasible for most books and trading positions. For VaR, it’s no longer necessary to settle for secondbest.
References

Federal Register, “Code of Federal Regulations, Title 12, Vol.1, Part 3, Subpart F,” 2014.

F. Diebold, A. Hickman, A. Inoue and T. Schuermann, “Scale Models,” Risk, pp. 104107, 1998.
ENDNOTES
^{5} For example, a bank may have positions in two FX pairs that are poorly correlated in times normal times and highly negatively correlated in times of stress. If a 99%ile worstmove coincides with a stress period, then the aggregate P/L from the two positions may offset each other. If we used the overall correlation to drive a Monte Carlo simulated VaR, the calculated VaR could be much higher.
^{6} This is especially common in MBS, where the first and second derivatives are computed using a secantline approximation after shifting risk factors, such as shifting rates ± 25bp
^{7} For example, as rates fall and a mortgage becomes more refinancible, the mortgage’s exposure to implied volatility also increases, implying that the crosspartial for price with respect to rates and vol is nonzero.
^{8} Further, since we are using finite shifts, the typical assumption that ƒ_{xy} = ƒ_{yx} which is based on the smoothness of ƒ(x,y) does not necessarily hold. Therefore, we need to compute two sets of cross partials, further increasing the initial setup time.
^{9} Why is the second derivative nonconstant? As rates move significantly, prepayments stop rising or falling. At these “endpoints,” cash flows on the mortgage change little, making the instrument positively convex like a fixedamortization schedule bond. In between, changes in prepayments case the mortgage to extend or shorten as rates rise or fall, respectively, which in turn make the MBS negatively convex.
^{10} Much has been written on the weakness of this scaling, see for example [7]
^{11} For example, using a flexible computation grid RiskSpan can perform a full OAS revaluation on 20,000 MBS passthroughs using a 250day lookback period in under one hour. Latticesolved options are an order of magnitude faster, and analytic instruments such as forwards, European options, futures and FX are even faster.