VAR veterans Benedict Roth and Alan Layng describe the rapid implementation of a risk model for emerging market equity trading

What kind of risk management tools are needed by a new business trading equities in the global emerging markets? This question faced us in 1996, when we were working in just such an environment. In response, we developed a new value-at-risk model, aimed specifically at measuring the risks of trading emerging market equities.

The model, based on historic simulation, was different both in design and approach from the off-the-shelf packages then available. And because of the time and resource constraints inevitable in a new firm, implementation was carried out on a minimal budget, over a period of only three months. But the new design proved to be sound and the model was later extended from equities to convertible bonds. In fact, the VAR approach described below is not only a reliable one, but an easy one to generalise from vanilla to exotic products. It could be significantly more straightforward to implement -- either in small or large trading institutions -- than traditional models such as RiskMetrics.

Why run a VAR model?

The main reason we chose to implement a VAR-based risk measure was the company's focus -- not unreasonably, in a new operation -- on daily trading profit and loss. VAR models attempt to forecast the size of potential trading profits and losses and to estimate -- at some fixed level of confidence -- how bad those losses could be. In this sense, VAR is more universal than other risk measurement methods: it is widely applicable across different trading activities and easily understood throughout a company's management structure. Furthermore, in an environment where risk capacity is limited, VAR encourages traders and management to adopt efficient risk strategies.

Applicability across trading strategies and traded markets. A VAR risk measure expressed in daily profit and loss terms -- where VAR of $10 million indicates potential day-on-day losses of $10 million -- can serve as a common denominator between different traded markets and trading activities or strategies. Without a common risk measure, different traders would run their books under different and incompatible sets of position and concentration limits specific to their businesses. Meaningful comparison between businesses would be difficult when dealing with simple cash equities and nearly impossible if option products with exposure to volatility skew and term structure were introduced. For example, figure 1 illustrates volatilities in different equity markets in the second half of 1997 and shows that average daily price changes in the most volatile ones were three times as large as in the quieter ones.

A Turkish equity trading book -- all other things being equal -- might have generated position-taking gains three times as large as those generated by an Indian trading book, and would have run three times the risk of position-taking losses. Furthermore, because the Indian global depository receipt (GDR) market can be shorted, whereas short-selling in Turkey is difficult, the Indian book-runner could have been aggressively positioned in single stocks yet broadly hedged against general market fluctuations, while the Turkish one would always have been long.

Within a given risk profile and revenue budget, the Indian book could therefore have been much larger. In a framework of risk limits based on position sizes, the two books -- given similar revenue budgets -- would have run under very different structures. But under a VAR measure, the two books could run under similar risk limits, and at the end of the year their profitability could be compared objectively on a risk/return basis in spite of the difference in their trading style and prevailing market conditions.

Of course, we never stopped reporting and controlling position sizes in addition to VAR. Position reports contain valuable data and, in times of crisis, when market movements may spike beyond the levels predicted by VAR models or many markets may crash simultaneously, simple measures such as global net long positions can be more useful than calculated VAR. In other words, we always used a mix of different tools -- including stress tests of disaster scenarios -- to report and control risk. But the VAR measure was the only one of these that was ubiquitous across the books and at every level of the company.

Risk-efficient trading strategies. The existence of tight VAR limits, applied widely, and looser position-size limits motivates the search for strategies that make efficient use of risk: diversification, hedging (where appropriate) and arbitrage. This search is motivated not only at the trading book level but also at the management level because, just as traders can increase their return on risk by intelligent hedging or diversification of their books, managers can do the same by diversifying their exposure to individual markets, trading teams and strategies.

So, as managers plan their business and financial strategy, use of a VAR-based risk measure encourages healthy diversification and leads to the enhancement of return on risk over the company as a whole. As regulatory capital begins to be linked directly to VAR, this motivation for efficient risk strategies will go all the way up to the boardroom.

Stock prices or equity indexes?

Given the company's need for a VAR model that accurately assessed the risk taken on arbitrage and on hedged positions, one of the earliest decisions we made was to work as far as possible with the price history of the instruments actually held on the trading book, and not with equity indexes. This was a major departure from many models then available off-the-shelf, which expressed equity positions in terms of underlying indexes only.

For example, a $10 million holding in the Brazil Fund, a New York-listed country fund with a beta of 70% with respect to the local stock index would be represented in a traditional VAR model by a $7 million holding in the index. If that long position were hedged with a short position in the index or in a beta-equivalent holding of another stock, most off-the-shelf models would consider the position to be flat and calculate a risk figure equal to zero. This would be wrong.

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The frequent use of simple equity indexes -- to represent positions in equities that might be complex -- may reflect the fixed-income focus of many of the original VAR model developers. Or it may reflect the focus of some trading firms on naked long or short positions in their markets. In our case, where arbitrage and relative value trading in equities were seen to be important parts of the business mix from the beginning, it was clearly inappropriate. There could be no substitute for genuine equity price data, and that meant the construction of an equity price data bank.

The price history data bank was to cover equities, equity indexes, convertible bonds, closed-end funds, interest rates and currencies -- in fact, any market data important to the trading book. It was not created specifically for the risk model but, rather, as a more general analytical resource for traders, sales people and analysts.

At the beginning, the price data bank was fed with two years' historic data from Reuters and this initial data upload allowed users to obtain value from the system immediately after it went live. As time went on, most data items were maintained automatically by carrying out a daily scan of the live feeds but some data -- for example, closing statistics for instruments traded off-exchange -- were always maintained by hand or updated via specialised interfaces from pricing tools used by traders. The existence of these off-exchange price histories, which in some cases were not even available on Bloomberg, made the data bank unique.

Technical architecture

The data bank was implemented in Microsoft's SQL Server database running on Windows NT. This turned out to be a powerful, cost-effective and flexible architecture. Along with the database, a library of data access functions was created, which could be called from C or Visual Basic programs, or from Excel spreadsheets, and which imposed a modern object-oriented view on the time series so that they could be retrieved or manipulated as objects without SQL calls.

Any generic calculations that might have been required by the risk model but also by other users -- for example, generating a series of day-on-day price changes from an instrument's price history, or calculating historic volatilities -- were implemented in the function library. Calculations that were computationally intensive were also implemented in the library. The rest of the model-specific logic remained relatively simple and the first version of the risk model was in fact implemented in only a couple of days of Visual Basic programming, using Excel for data input and output. A later version of the model read the trading positions directly from the back-office records via a universally available Microsoft protocol, called open database connectivity, and used Excel only for static data input and logs of model results. Eventually these, too, could have been located on the data server, leaving a robust, stand-alone application in Visual Basic interfacing to position data, static data and model results all held on secure central databases.

The simplicity of the technical architecture -- model-specific logic in Visual Basic combined with spreadsheets for data input/output, a database for time series and a generic function library for technical or performance-critical calculations -- facilitated speedy model implementation. Such an implementation could be carried out even more rapidly today using commercial products such as Xenomorph, which reproduce in off-the-shelf packages much of the technical architecture and many of the data feed interfaces described above.

Historic simulation

Another factor that enabled us to get a risk model running quickly was our decision to use historic simulation of previous day-on-day price changes to estimate VAR. Historic simulation had three key benefits:

• given the vast number of instruments that might potentially be traded, and the existence of the price data bank, it was far easier to implement than a traditional model based on variances and cross-correlations;
• given the unusual nature of many 'real-world' distributions of returns, the direct use of the distributions observed historically gave more useful results, and was less susceptible to data errors, than a variance/cross-correlation estimate based on an assumption of normality; and
• as the trading book grew, the calculation time needed for the historic simulations would grow only in proportion, with no undue expansion of size or calculation time if trading became more active.

Each of these benefits, and the contrast with other VAR approaches such as those originally used by RiskMetrics and Bloomberg's risk model, is discussed in more detail below.

Early VAR analysis. Early VAR models would have used the price database to generate variances and cross-correlations for day-on-day price changes of each of the instruments on the trading book. To the extent that foreign instruments held on the trading book were funded in dollars -- our core reporting and operating currency globally -- the implied foreign exchange exposures would have been included in a traditional variance/cross-correlation model as well. Furthermore, given the dynamic composition of the trading book, with individual equity positions coming in and out all the time, the variances and correlations would have had to be refreshed every day.

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It became clear at an early stage that this traditional approach would eventually produce enormous correlation matrices. If the traders were to hold 1,000 instruments or currencies on their books, the matrix would have more than 500,000 unique cells. While advanced numerical techniques exist for processing these matrices efficiently, we were not aware at that time of readily available software implementations of these techniques. And the sole purpose of the correlation matrices -- to model the distribution of profit and loss movements in the portfolio as a whole -- could be achieved more easily by working directly with the equity prices themselves, as we describe below.

Historic simulation. Rather than attempting variance/cross-correlation analysis, we employed simulation. We used the price database to simulate the historic profit and loss swings that would have been generated by the trading positions under examination. The total daily profit and loss swings -- illustrated in figure 2 for a sample Brazilian portfolio of equities and American Depository Receipts (ADRs) -- include all necessary correlation effects by virtue of being based on a set of correlated price histories, one for each instrument or currency.

It would then have been possible to calculate the standard deviation of the total profit and loss distribution and, assuming this distribution to be normal, to use the standard deviation to calculate confidence intervals. But, following our original simulation approach and our decision to work with real price histories as we found them, we decided to work with the distribution of profit and loss changes as found too, without any assumption of normality. Given simulation over 100 trading days, which generated a distribution of profits and losses, we estimated the fifth percentile of the underlying distribution simply as the fifth-worst loss, the tenth percentile as the tenth-worst loss and so on. If the underlying distribution of portfolio profit and loss swings were normal, the results given by the variance/cross-correlation approach and the two simulation-based approaches given here would converge. But, if the underlying distribution of portfolio profit and loss itself had fat or thin tails -- perhaps as the result of the presence of a few large positions that themselves exhibited unusual behaviour -- our pure simulation approach would be more reliable.

Robustness. A further attraction of the pure simulation approach was its lack of susceptibility to data errors. Many price data sets contain error points and each individual erroneous price creates two erroneous day-on-day price changes: one before the error point and one after it. Under the simulation approach, bad prices have no impact on the ninety-fifth percentile as long as there are no more than two bad ones per hundred. The anomalous price changes they produce simply fall outside the ninety-fifth percentile. In a classical variance/cross-correlation analysis, on the other hand, even a single bad data point can create significant problems because classical statistics are based on squared deviations from the mean and spurious extreme points tend to have an influence on the model results out of proportion to their number.

Implementation parameters

As the implementation progressed, we had to make a number of tactical decisions. Some of these were specific to the historic simulation approach and others were more general.

Data surrogates. The most common problem with historic simulation is a lack of historic data. Some instruments -- such as ADRs -- may be thinly traded, or traded off-exchange, and historic data may not be easily available. Others may be newly issued. A third important category -- particularly in a new operation or in one that is growing rapidly -- consists of instruments traded for the first time by the trading desk before their inclusion in the price database.

We operated in all these cases with substitution rules, coded into the model's static data and updated as and when required. For example, ADR and GDR price history was often modelled with:

• data from the underlying local stocks;
• data from a more widely traded class of stock issued by the same company, for less liquid equity issues; or, failing these approaches,
• equity indexes as in a traditional model.

Equity indexes were most often used on a temporary basis while the price database extended its scope to follow the activities of the trading desk.

Mirroring of upward and downward price changes. Another problem specific to users of historic simulation is the fact that the distribution of historic returns is not symmetric. In a rising market, the upward movements are larger and more frequent than the downward ones and the model, unless adjusted, attributes more risk to short positions than to long ones. The opposite happens in a falling market.

We configured the model to treat long and short exposures consistently: in other words, sharp price spikes either upwards or downwards increased the VAR attributed both to long and short positions. This was achieved by 'mirroring' the historic distribution: an equal and opposite price change was created for each historic price change and the simulation was carried out on the combined time series. Intuitively, this is similar to the use of an antithetic variable in a Monte Carlo simulation or to running through the time series twice: once forwards and once backwards. It is consistent with a theoretical model that assumes price changes to be symmetrically distributed around a risk-free rate.

Confidence interval. The confidence interval used to estimate possible trading losses was 95%, in common with many other VAR models. A 95% confidence interval indicates that trading profit and loss should exceed VAR approximately once in 20 trading days, or once a month. We found the slogan "Once a month on average" easy to communicate and to understand throughout the company. It was also easy to test after only a few months' experience of running the model. Given the need to monitor the performance of the model internally, this was a key factor.

Historic data horizon. Notwithstanding the Basle Committee on Banking Supervision's recommendations for the use of two years' history, and the existence of at least two years' history in the data bank, we held a strong view that our VAR model should be responsive to changing market conditions and should reflect rising market volatility quickly.

Trimming down a firm's risk profile after a major crash is easy -- albeit often ineffective -- but it is more difficult to develop a responsiveness to the rising volatility that often prefigures a crash. The presence of a volatility-sensitive model, updated daily and directly linked to trading limits, was seen as a powerful tool to this end. A three-month data horizon, the shortest one practical, was therefore used. This ensured that the VAR figure, on the ninety-fifth percentile, would begin to be affected after only three unusual trading days. Figure 3 overleaf shows how these figures would have emerged in practice during 1997 had the Brazilian equity/ADR portfolio described above been held on the trading book for that period. VAR rose very rapidly during autumn 1997 in line with market volatility.

Convertible bonds

When the risk model was first released, it could not handle convertible bonds, so the relevant risk calculations were carried out in other ways. However, the convertible bond book carried significant equity price risk at times and the company's overall, consolidated VAR could be calculated only by a model that processed the convertible positions and the vanilla equity positions simultaneously. So extending the model to convertibles was a high priority.

To do this, we needed to solve two problems. The first was the translation of historic day-on-day stock price returns into prospective changes in the market value of traded positions. In the cash equity world, this translation is automatic and implicit. A stock price movement of 3% would produce a change in position value of 3%. The translation is so simple and universal that many model users would not even be aware that they were carrying it out. Indeed, it would be easy to confuse historic simulation of stock price movements -- which the VAR model would appear to be carrying out -- with estimation of potential future portfolio returns, which is the model's real task.

However, in the derivatives or convertible world, the translation between stock price changes and portfolio returns is more complicated. For example, the same historic price movement of 3%, which appeared in the time series perhaps one month before the date on which the VAR model was run, could produce a shift in the value of a convertible bond anywhere between 0% and 3% if it occurred again, depending on the actual stock price at run-time and the market value of the fixed-income element of the bond.

So the first step in extending the VAR model to convertibles was to make use of the bond valuation models to translate stock price returns into bond price returns. This was carried out dynamically whenever the model was run. In fact, on a portfolio basis, small stock price movements generated larger VAR results and large price movements generated smaller VAR results. This is because hedged convertible positions tend to be long in gamma and to benefit from price volatility.

The second step was more complicated. Convertible bonds -- particularly emerging market convertibles -- are notorious for not trading at 'fair value' as modelled, and changes in the spread over or under the fair value can be a large part of the trading profit and loss account. To recognise this source of profit and loss, the VAR model back-tested the theoretical values of each convertible -- with historic stock prices -- against the historic prices at which the bonds traded. Changes in position values resulting from day-on-day changes in the theoretical/historic spread were then added into the simulated profit and loss series and thus into the VAR figures.

Roll-out

When the risk model was first demonstrated to desk heads, they found the notion that position riskiness and limits could rise and fall in response to market conditions new and interesting. The model gained credibility from the speed of its roll-out: the head of global trading and sales -- who had joined the company from a gigantic global broking and trading house -- commented that his previous institution had VAR but that "they had about 10 people working on it". A comparison of recorded VAR against the following day's trading profit and loss, based on the first six months' live operation, found that the model had hit the ninety-second, rather than the ninety-fifth, percentile and, while this represented a slight under-reporting of real trading risk, the result was sufficiently close to be very encouraging.

We not only used the VAR model for trading risk management but also for balance-sheet and business planning and financial forecasting. Used in this context, the model was a critical link between trading revenue budgets and target position sizes and balance sheets. It injected clarity and reality into senior management discussions about trading revenue budgets and company risk tolerances.

Further applications

The simulation approach described in this paper has several advantages. It is easy to apply to an equity or to an equity option market maker where the number of underlyings is large. Since there is no need to build correlation matrices, calculation time is moderate and manageable even when the number of positions on the book explodes. Furthermore, it makes no assumptions about the distribution of the underlying returns and is relatively unsusceptible to data error.

The simulation approach also produces economies of implementation when consolidation takes place across trading desks or trading rooms. In an organisation with many separate trading systems, full consolidation of trade data -- in particular derivatives trade data -- may be difficult or impossible. Even if such consolidation can be achieved, the system carrying the consolidated trades may lack the associated valuation models and valuation parameters, and thus be forced to accept delta or delta/gamma approximations of possible profit and loss swings rather than the swings themselves.

Under a simulation approach, trade consolidation can become an optional extra. Owners of exotic trading systems with complex valuation models, difficult delta/gamma profiles such as barriers or digitals, or complex trade structures, can submit for consolidation a vector of simulated profit and loss swings as calculated on their own systems. The calculations concerned -- as described above -- are straightforward and much easier to implement in practice in local systems than carrying out a whole-sale consolidation, from vanilla to exotic, in one central system. Simulation can therefore achieve, with distributed trade data, something that may be impossible by other means.

We suspect -- in view of these advantages -- that historic simulation is currently under-utilised as a tool for calculation of VAR. Perhaps, with the advent of cheaper and more efficient mechanisms for storing and manipulating price time series, this will change. If so, the risk management community as a whole would gain.