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Models

Making VAR More Flexible

Mark Garman, president of Financial Engineering Associates, explains how a new methodology, delta value-at-risk, can help traders and risk managers use VAR more effectively for strategic decision-making.

In every risk manager's tool kit, there is available a value-at-risk (VAR) system that acts as the trash compactor for the entire company, compressing diffuse, firm-wide risk information into a single number. At cutting-edge institutions, however, risk managers want something more versatile than this one-dimensional number. A simple VAR number may indicate that you have a risk problem, but doesn't tell you how to solve it. Ideally VAR should provide actionable information and help companies answer the following questions: how do individual trades affect total firm-wide VAR? How much will a hypothetical hedge decrease total VAR? Which trade, out of several, will have the greatest effect on firm-wide VAR?

VAR, as it is currently practiced at most banks, cannot provide this information for a variety of reasons. First, VAR's single-number approach is designed to reduce the market's natural complexity; it is impossible to "reverse engineer" VAR to, for example, recover the cash flows that were fed into the original VAR number or reassemble the individual trades from the cash flows. Second, VAR even at its most reductive is still a time-consuming process for institutions with very large portfolios. This means that if you want to look at how a particular trade will affect your firm-wide VAR, you must run a baseline VAR on an existing portfolio, add the new trade to your portfolio, rerun VAR for the entire firm, and take the difference between the two.

How can you turn your plain-vanilla VAR report into a robust decision-support tool? Well, at Financial Engineering Associates (FEA) we have developed a new methodology to address these questions and more. Known generically as "DelVAR," this new approach focuses on simplifying the calculation of incremental changes of firm-wide VAR. This allows us to minimize the horsepower required to run VAR and related functions such as risk-based trading limits in real time.

Deep Background

Of course, to understand DelVAR, it is first necessary to consider how standard VAR-type analysis works. In order to ensure VAR calculations speed along at a good clip, it is necessary to break down all the trades in your portfolio into their component cash flows which, in turn, are netted according to their maturities. These cash flows are then run through a variance/covariance matrix. That matrix is generated according to a number of assumptions that the user makes at the outset. The basic assumptions include the time period over which P&L will vary, how different market factors are correlated, how volatility numbers are generated, and the confidence or probability that the resulting VAR figure will indeed accurately reflect the effect of actual market fluctuations on P&L.

Calculated in this fashion, standard VAR makes it quite difficult to determine the effect of any single "what-if" or actual trade on the total portfolio. The only way to do this under the old methodology is to split the new trade into its component cash flows, then fold everything into the existing portfolio and recalculate VAR for the total portfolio. The difference between the "before" VAR and the "after" VAR is what we call the incremental VAR of the single trade.

Enter DelVAR, Stage Left

DelVAR eliminates the problems described above by calculating not only total VAR, but also an additional quantity that we will call "gradient" VAR, the cash flow direction in which VAR is increasing the fastest. Gradient VAR can be determined by plotting a sort of VAR "terrain map" of cash flow combinations. The cash flow combinations with lowest VAR appear in the center of the map, with oval bands extending upwards towards their outer limits as VAR increases. The total VAR of the portfolio can be identified as the terrain "height" of a cash flow "point" on this map.

In the chart at right we have mapped two simple cash flows, one representing U.S. dollar flows with a certain maturity and the other representing Deutsche mark flows with some maturity. (In a real-life situation, of course, the terrain map would likely be N-dimensional and more complex.) From point "P," which represents the portfolio's cash flows, it is possible to extrapolate the gradient VAR, which is represented by the white arrow. This is the cash flow direction in which VAR increases most quickly.

In order to evaluate a new trade's effect on VAR, all you must do is compare the direction of the line between the point representing the trade's cash flows on the terrain map and point "P" to the direction of the gradient VAR line. If the direction of this new line is within 90 degrees of the gradient VAR line, this particular deal will increase the portfolio's total VAR. If the angle is greater than 90 degrees, the deal will decrease total VAR.

Putting DelVAR into practice

How can DelVAR provide actionable information while simultaneously reducing time and energy spent on portfolio-wide calculations? Consider the following suggested procedure: at the designated start of a day-long trading period, calculate both total VAR and DelVAR (i.e. the "gradient" VAR). As new trades are executed (or considered, as the case may be), plot their cash flows and compare these against the initial DelVAR. This process generates an approximate incremental VAR without the arduous recalculation of portfolio-wide VAR.

Drawbacks to Consider

The incremental VAR resulting from the calculations described above is only an approximation because the system does not recalculate total VAR every time a trade is registered. Generally speaking, however, the approximation will be quite close to the actual incremental VAR, except under some conditions. The DelVar method is vulnerable to two such conditions: extremely large trades that represent a considerable portion of the total portfolio, and cases in which the sum of the daily accumulation of trades is a large fraction of the whole portfolio. Both could throw off the DelVAR method because point "P"-or the total VAR-would have been calculated without these large, "influential" deals. For firms that occasionally have large trades or ultra-high-volume trading, it may make sense to recalculate total VAR and DelVAR after such periods.

Dare to Compare

One of the most important uses of DelVAR is to compare how a variety of different trades (or hedges, in the case of corporates) could potentially affect the total VAR of a portfolio. In this case, however, it is critical to remember that large trades will always affect VAR more than small trades no matter what they are. The trick here is to factor out the size effect by measuring the potential impact of different deals on total VAR by examining incremental VAR per unit of trade. To this end, you can choose to measure deal size in terms of any of the following: the cash flows it generates; its individual VAR; its potential return; its market price; the capital it requires; or its notional value. Whatever size measure you choose must be maintained throughout the comparisons.

Once you have determined how you are going to measure size (and have selected a standard trade "unit") you can divide each trade's incremental VAR by the number of units to arrive at incremental VAR per unit. (You can also simply calculate incremental VAR for a single unit.) Using this method you can compare the relative "riskiness" of various trades.

I believe DelVAR increases computational efficiency while giving users more "proactive" decision support information than standard VAR alone. In other words, it can help your trash compactor become a multi-function appliance that can also suggest new recipes for lowering risk.

Outlook, FEA's value-at-risk add-in software, contains VARDelta, FEA's version of DelVAR (patent pending).

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