# 1.5 Risk Limits

In a context where risk taking is authorized, **risk limits** are bounds placed on that risk taking.

Suppose a pension fund hires an outside investment manager to invest some of its assets in intermediate corporate bonds. The fund wants the manager to take risk on its behalf, but it has a specific form of risk in mind. It doesn’t want the manager investing in equities, precious metals, or cocoa futures. It communicates its intentions with contractually binding investment guidelines. These specify acceptable investments. They also place bounds on risk taking, such as requirements that:

- the portfolio’s duration always be less than 7 years;
- all bonds have a credit rating of BBB or better.

The first is an example of a **market risk limit**; the second of a **credit risk limit**.

A risk limit has three components:

- a risk metric,
- a risk measure that supports the risk metric, and
- a bound—a value for the risk metric that is not to be breached.

At any point in time, a limit’s **utilization** is the actual amount of risk being taken, as quantified by the risk measure. Any instance where utilization breaches the risk limit is called a **limit violation**.

A bank’s corporate lending department is authorized to lend to a specific counterparty subject to a credit exposure limit of GBP 10MM. For this purpose, the bank measures credit exposure as the sum amount of outstanding loans and loan commitments to the counterparty. The lending department lends the counterparty GBP 8MM, causing its utilization of the limit to be GBP 8MM. Since the limit is GBP 10MM, the lending department has remaining authority to lend up to GBP 2MM to the counterparty.

A metals trading firm authorizes a trader to take gold price risk subject to a 2000 troy ounce delta limit. Using a specified measure of delta, his portfolio’s delta is calculated at 4:30 PM each trading day. Utilization is calculated as the absolute value of the portfolio’s delta.

###### 1.5.1 Market Risk Limits

For monitoring market risk, many organizations segment portfolios in some manner. They may do so by trader and trading desk. Commodities trading firms may do so by delivery point and geographic region. A hierarchy of market risk limits is typically specified to parallel such segmentation, with each segment of the portfolio having its own limits. Limits generally increase in size as you move up the hierarchy—from traders to desks to the overall portfolio, or from individual delivery points to geographic regions to the overall portfolio.

Exhibit 1.1 illustrates how a hierarchy of market risk limits might be implemented for a trading unit. A risk metric is selected, and risk limits are specified based upon this. Each limit is depicted with a cylinder. The height of the cylinder corresponds to the size of the limit. The trading unit has three trading desks, each with its own limit. There are also limits for individual traders, but only those for trading desk A are shown. The extent to which each cylinder is shaded green corresponds to the utilization of that limit. Trader A3 is utilizing almost all his limit. Trader A4 is utilizing little of hers.

Exhibit 1.1: A hierarchy of market risk limits is illustrated for a hypothetical trading unit. A risk metric—value-at-risk, delta, etc.—is chosen. Risk limits are specified for the portfolio and sub-portfolios based upon this. The limits are depicted with cylinders. The height of each cylinder corresponds to the size of the limit. The degree to which it is shaded black indicates current utilization of the limit. Fractions next to each cylinder indicate utilization and limit size. Units are not indicated here, as these will depend upon the particular risk metric used. Individual traders have limits, but only those for traders on desk A are indicated in the exhibit.

For such a hierarchy of risk limits to work, an organization must have a suitable risk measure to calculate utilization of each risk limit on an ongoing basis. Below, we describe three types of market risk limits, culminating withvalue-at-risk limits.

###### 1.5.2 Stop-Loss Limits

A **stop-loss limit** indicates an amount of money that a portfolio’s single-period market loss should not exceed. Various periods may be used, and sometimes multiple stop-loss limits are specified for different periods. A trader might be given the following stop-loss limits:

- 1-day EUR 0.5MM;
- 1-week EUR 1.0MM;
- 1-month EUR 3.0MM:

A limit violation occurs whenever a portfolio’s single-period market loss exceeds a stop-loss limit. In such an event, a trader is usually required to hedge material exposures—hence the name stop-loss limit. Stop-loss limits have shortcomings:

- Single-period market loss is a retrospective measure of risk. It only indicates risk after financial consequences of that risk have been realized.
- Single-period loss provides an inconsistent indication of risk. If a portfolio suffers a large loss over a given period, this is a clear indication of risk. If the portfolio does not suffer a large loss, this does not indicate an absence of risk!
- Traders cannot control the specific losses they incur, so it is difficult to hold them accountable for isolated stop-loss limit violations.

Despite their shortcomings, stop-loss limits are simple and convenient to use. Non-specialists easily understand stop-loss limits. A single proxy for risk—experienced loss—can be applied consistently across different types of exposures and with different trading strategies. Calculating utilization is as simple (or difficult, in some cases) as marking a portfolio to market. For these reasons, stop-loss limits are widely implemented by trading organizations.

###### 1.5.3 Exposure Limits

**Exposure limits** are limits based upon an exposure risk metric. For limiting market risk, common metrics include: duration, convexity, delta, gamma, and vega. Crude exposure limits may also be based upon notional amounts. These are called **notional limits**. Many exposure metrics can take on positive or negative values, so utilization may be defined as the absolute value of exposure.

Exposure limits address many of the shortcomings of stop-loss limits. They are prospective, indicating risk before its financial consequences are realized. Also, exposure metrics provide a reasonably consistent indication of risk. For the most part, traders can be held accountable for exposure limit violations because they largely control their portfolio’s exposures. There are rare exceptions. A sudden market rise may cause a positive-gamma portfolio’s delta to increase, resulting in an unintended delta limit violation. For the most part, utilization of exposure limits is easy to calculate. There may be analytic formulas for certain exposure metrics. At worst, a portfolio must be valued under multiple market scenarios with some form of interpolation applied to assess exposure.

Exposure limits pose a number of problems:

- At higher levels of portfolio aggregation, exposure limits can multiply. While a trader who transacts in 30 stocks might require 30 delta limits, the entire trading floor he works on might transact in 3,000 stocks, requiring 3,000 delta limits.
- Different exposure limits may be required to address dissimilar exposures or different trading strategies. For example, delta might need to be supplemented with duration or rho to address yield curve risk. It might need to be supplemented with gamma or vega to address options risk.
- Custom exposure limits may be required to address specialized trading strategies such as cross-hedging, spread trading or pairs trading that reduce risk by taking offsetting positions in correlated assets. In such contexts, any delta limits must be large to accommodate each of the offsetting positions. Being so large, they cannot ensure reasonable hedging consistent with the intended trading strategy.
- With the exception of notional limits, non-specialists do not easily understand exposure limits. For example, it is difficult to know what might be a reasonable delta limit for an electricity trading desk if you don’t have both a technical understanding of what delta means and practical familiarity with the typical size of market fluctuations in the electricity market.

###### 1.5.4 Value-at-Risk Limits

Value-at-risk is used for a variety of tasks, but supporting risk limits is its quintessential purpose. When risk limits are measured in terms of value-at-risk, they are called **value-at-risk limits**. These combine many of the advantages of exposure limits and stop-loss limits.

Like exposure metrics, value-at-risk metrics are prospective. They indicate risk before its economic consequences are realized. Also like exposure metrics, value-at-risk metrics provide a reasonably consistent indication of risk. Finally, as long as utilization is calculated for traders in a timely and ongoing manner, it is reasonable to hold those traders accountable forvalue-at-risk limit violations. As with exposure limits, there are rare exceptions. Consider a trader with a negative gamma position. While she is responsible for hedging the position on an ongoing basis, it is possible that a sudden move in the underlier will cause an unanticipated spike in value-at-risk.

As with stop-loss limits, non-specialists may intuitively understandvalue-at-risk limits. If a portfolio has 1-day 90% USDvalue-at-risk of 7.5MM, a non-specialist can understand that such a portfolio will lose less than USD 7.5MM an average of 9 days out of 10. As with stop-loss limits, a single limit can suffice at each level of portfolio aggregation—at the position level, trader level, trading desk level, sub-portfolio level and portfolio level. Andvalue-at-risk limits are uniformly applicable to all sources of market risk and all trading strategies. Of course, for value-at-risk, such generality is theoretical. The ability of a particular value-at-risk measure to address the market risk associated with specific instruments or trading strategies depends on the generality and sophistication of that particular value-at-risk measure.

This brings us to the drawbacks ofvalue-at-risk limits:

- Depending on the level of generality and/or sophistication required of value-at-risk measures, they can be difficult to implement. This book is a testament to (and hopefully a palliative for) the potentially complicated analytics value-at-risk measures require.
- Utilization of somevalue-at-risk limits may be computationally expensive to calculate. While value-at-risk can be calculated in real time or near-real time for many portfolios, it may take minutes or hours to calculate for others.
- While most risk metrics entail some model risk or potential for manipulation, the complexity of value-at-risk measures make them particularly vulnerable.

The last point was illustrated in the aftermath of JPMorgan’s 2012 “London Whale” trading scandal. It came to light that bank employees had manipulated portfolio valuations, undermining stop-loss limits. They had also replaced a value-at-risk measure with a rudimentary spreadsheet, which further understated risk.

###### 1.5.5 Summary Comparison

Exhibit 1.2 summarizes the strengths and weakness of stop-loss, exposure, andvalue-at-risk limits.

Exhibit 1.2: Comparison of stop-loss, exposure, andvalue-at-risk limits.