23 December 2015
Author: Brandon Davies
Arguably the most surprising lesson for risk managers from the financial crisis was that tail risk – the extremely unlikely event that just about everyone ignored – could actually happen. Suddenly it became clear that we no longer lived, if we ever had, in a well behaved, if volatile, world.
So regulators and regulated are focusing on the subject as they have never done before. But there are real dangers in defining, measuring and managing tail risk particularly if regulators jump to conclusions and then attempt to measure tail risk before fully understanding it.
At present the Basel Committee is focusing on implementing regulation based on the Fundamental Review of the Trading Book (see Box 1) which, amongst other changes, will replace the Stressed VaR based calculations of market risk in the so called Basel 2.5 based regulation with an Expected Shortfall (ES) based calculation.
Regulators are also focusing on the results of bank stress tests where the results for different banks have surprised regulators (see Box 2) leading to concerns that the risk weight calculations, in stress, cannot be trusted. And the recent problems of Citi Bank (and others) in getting approval for their stress test results from the Federal Reserve has also caused banks to try to understand the way regulators interpret these results (see Box 3)
The difficulties in measuring tail risk are nothing new. When a quarter of a century ago the decisions was taken to replace the then jumbled world of customer exposure limits, duration mismatch limits and outright (nominal) position limits, a measure of mean-variance (VaR) and a measure of tail dependency (ES) were both considered (see Box 4). VaR was seen at that time as a usable and reasonably reliable measure of risk in “normal” circumstances whilst ES was seen as very dependent on the choice of the distribution used to project future outcomes.
- The graph above contrasts the VaR at the 95% for a normal distribution with the VaR at the 99% for both a normal and Generalised Pareto Distribution (GDP) and an Expected Shortfall measure for the GDP. Note the GPD 99% is approx. 3x that of the normal distribution and the ES of the GDP is approx 7x that of the normal 99%.
The decision to choose VaR as a measure of risk implied a definition of risk that in many ways was quite unsatisfactory. VaR is a constrained measure and so looks at risk as variance measured at some percentile from the mean (average) outcome.
Given this, the constraint on the outcome made for a very much more simple measure of risk than we would need were we to look for the most extreme outcomes.
When anyone thinks of risk they usually focus on some absolutely bad outcome epitomized say by the risk of death, which is a pretty absolute measure of risk! ES is clearly a more appropriate way of measuring extreme outcomes and such outcomes are what most people would think of as risk.
The VaR was chosen because there was no reliable way of predicting the shape of the tail of risk distributions, something that is vital if the computation of ES is to have any practical use.
However, there would have been problems for the financial sector whichever system had been chosen. In some industries the likelihood of future events can be judged on a proper analysis of what has happened in the past. This probabilistic methodology is very powerful in natural sciences and has led to much better understanding of natural processes and to risk control through improvements in design and manufacturing processes.
In finance, future outcomes are much more difficult to forecast, as events in markets are not driven by the fundamental laws of nature but by individuals and corporate behaviors that are perceived to be dynamic and conditional – that is they change as circumstances change and may change in seemingly irrational (unpredictable) ways.
Arguably seemingly irrational behaviour can be rational. If one takes a simple example of momentum trading, which is a common strategy, it is evident that buying a financial asset such as an equity stock on the basis that everyone else appears to be buying is rational. So, if an investor follows the trend and buys, the price will rise and if it is then sold at the “right time” the investor will benefit. It may also be seen as irrational as unless the stock’s current price is a reflection of future discounted earnings the investor will over-pay for the stock and its price will fall and money will be lost.
Momentum investing and value investing are common, if different, investment strategies and result in different behaviors; but one investor may apply both strategies at different times and in different market circumstances.
The financial world, which is driven by changing behaviors of market participants, is very different from the natural worldI, it is far more dynamic and unpredictable.
This makes it particularly difficult to predict rare and extreme outcomes as such forecasts are likely to be based on very few observations in a financial world where it is known that unpredictable changes in behavior will create these outcomes. Moreover in financial markets there are good reasons to doubt that past behaviour is a good guide to future outcomes. This is because the undesirable effects of extreme outcomes will lead to changes in the physical operation of markets (the change from OTC to CCP clearing for derivatives trades) and the regulations governing them (Basel II to Basel III) which are designed to ensure “it never happens again”.
In these circumstances the choice of any parametric distribution to predict extreme outcomes in markets looks rather like a guessing game with the odds stacked against the player. The temptation is to give up and predict we are all doomed – but it would be the wrong conclusion for there is a way through this problem.
To start with, it is important to accept that that extreme outcomes cannot be predicted in the world of finance. Any attempts to torture data from past extreme events to predict the next extreme event will fail, because the data does not have the information that risk managers are trying to extract from it. In essence past performance is not a guide to future outcome.
So what in practice can be done? Above all it is necessary to reject the notion that it is impossible to manage what cannot be measured.
This is though not a reason to abandon attempts at measurement. But it is a reason not to put faith in any one measure; and, far from being a reason to throw up our hands and accept our fate, it is actually a reason to prize intellectual curiosity and the quest for insight into financial risk.
Tail risk modeling requires risk managers to decide how they are going to describe the tail of the distribution. There are currently three intellectually accepted ways to do this.
Firstly, the current focus of tail risk modeling is on scenario analysis, which has the advantage of being easily understood, especially by bank boards who are responsible for signing off on the “risk appetite” of their organizations and by regulators who in the US and Euro Area are placing great reliance on this methodology.
A scenario does not, however, have any statistical probability; it is instead a guess at what could happen and how this would affect the risk profile of the organization. In implementation at least three scenarios should be used, one based on an historic stress, one on a reverse stress scenario (thus triggering a recovery and resolution plan) and one based on a stress scenario devised by the board and senior management of the bank.
The second approach is that proposed by Rebonato and Denev (“Coherent Asset Allocation and Diversification in the Presence of Stress Events” April 2011) – they assign subjective probabilities using Bayesian net technology.
This has the advantage of polling experts for their judgment to specify the relationships between asset portfolios in exceptional circumstances. As a result a debate is created, though of course there is no guarantee of agreement as a result. Moreover there is a substantial problem in choosing the experts. If they are too alike, they may simply represent one view of the world; if they are too dissimilar, there may be no agreement.
The third approach is to use a copula based on a modified Student T distribution, as proposed by Boryana Racheva-lotova and Dr Stoyan Stoyanov of FinAnalytica. In this instance available statistical evidence is combined with limited judgment to implement a parametric approach to the problem under which a fat tailed distribution is specified that is assumed to represent the likely result from the dynamic conditional correlation process set up by the crisis. “Buy side” investors have used this methodology extensively but banks have used it much less.
The heavy reliance of regulators on scenario analysis has, understandably, caused banks to focus on this methodology. But the challenges in defining, managing and measuring tail risk make reliance on any one technique potentially dangerous and a “triangulation” of the risk by use of all three techniques would result in a much better approach.
Triangulation will not give us the truth about tail risk for there is none to be had, but it would provide a better basis for investigating uncertainty about the shape of the tail of risk distributions.
Indeed rather than picture risk outcomes as a parametric distribution with a long and fat tail it is better to think of events that trigger tail risk as starting a dynamic and unpredictable process. This may drive us to outcomes we could not imagine if we pictured tail risk outcomes as static results derived from any parametric distribution. We may rightly worry about where the process may take us, but we would be much less surprised by it.
“We are seeing things that were 25 standard deviation events, several days in a row” – David Viniar (Goldman’s CFO FT 15th Aug 07)
The Fundamental Review of the Trading Book – Box 1
The Basel Committee’s “The Fundamental Review of the Trading Book” is designed to result in changes to the emergency trading book rules known as Basel 2.5.
The key conclusions include:
- A revised boundary between the trading book and banking book, creating a less permeable and more objective boundary that remains aligned with banks’ risk management practices, and reduces the incentives for regulatory arbitrage.
- A revised risk measurement approach and calibration, shifting the measure of risk from value-at-risk to expected shortfall (ES) so as to better capture “tail risk”, and calibration based on a period of significant financial stress.
- Incorporating the risk of market illiquidity by introducing “liquidity horizons” in the market risk metric and an additional risk assessment tool for trading desks with exposure to illiquid, complex products.
- A revised standardised approach that is sufficiently risk-sensitive to act as a credible fallback to internal models, and is still appropriate for banks that do not require sophisticated measurement of market risk.
- A revised internal models-based approach, including a more rigorous model approval process and more consistent identification and capitalisation of material risk factors. Hedging and diversification recognition will also be based on empirical evidence that such practices are effective during periods of stress.
- A strengthened relationship between the standardised and the models based approaches by a closer calibration of the two approaches. This requires banks to calculate the standardised approach, and publicly disclose standardised capital charges, on a desk-by-desk basis.
- A closer alignment between the trading book and the banking book in the regulatory treatment of credit risk. This involves a differential approach to securitisation and non-securitisation exposures.
The Committee is also considering the merits of introducing the standardised approach as a floor or surcharge to the models-based approach. However, it will only make a final decision on this issue following a comprehensive Quantitative Impact Study, after assessing the impact and interactions of the revised standardised and models based approaches.
Can Risk Weighted Assets (RWAs) be Trusted – Box 2
Basel II enables advanced banks to use their own internal models to calculate the risk weights used to determine their banking book capital requirements. Concerns have been raised about the accuracy and variability of these risk weights and the models used to calculate them and the Basel Committee and the European Banking Authority are now investigating RWA variability.
The Basel study observed that, in general, banks are fairly consistent in the way they rank order exposures, but the following factors contributed to RWA differences.
- PDs and LGDs are much less consistent than exposures across banks
- Differences in regulatory approach e.g. the portion of assets on Standardised
- The treatment of defaulted assets e.g. do provisions raise RWAs and/ or lower exposure?
- The portfolio mix strongly contributes to RWA variance in some IRB models.
- Supervisory practices also lead to risk weight variation; some supervisors top-up underestimation of RWAs in Pillar 1, others in Pillar 2.
- RWA variability is also reflected in management practices. Such practices include: the use of netting agreements, the pace of movement of assets onto IRB, and increasing the scope of models.
In response Stefan Ingves (Chairman of the Basel Committee on Banking Supervision) said ;
“At the heart of this problem is a question of whether, for regulatory purposes, banks have too much freedom in their modelling choices, so we are looking at whether, and how far, greater constraints on the modelling practices of banks are needed. To make a more direct impact, we are also examining the role of floors and benchmarks within the regulatory framework, and considering whether they should have a greater role to play. Finally, we now have the leverage ratio as a backstop to the risk-based regime. And the case for a leverage ratio will only grow stronger if risk-weight variability is not adequately dealt with.”
This statement clearly illustrates a problem with the move from reliance on a mean-variance measure of risk to a measure of tail risk. As tail risk is heavily dependent on changes to correlations both within and across banks asset portfolios accurate measures of tail risk are going to require banks to be able to measure these changing correlations. It is, however, highly unlikely that banks will hold identical assets within their asset portfolios and even less likely that they will hold identical portfolios. The asset and portfolio correlations will produce very different outcomes to their measures of tail risk.
It seems inevitable that the move of risk measures away from mean-variance to tail risk will weaken passed moves towards risk-based regulation.
For regulatory purposes we must either settle for:
- A simple “look up” table approach where assets are allocated a “weight” drawn from a table of weights as in Basel I, or
- A standard and relatively simple model (such as the Gordy model) as has been used extensively under Basel II, or
- A very detailed approach where each banks data gathering structure as well as its risk management model, including the specification of the dynamic processes that drive correlations within and across asset portfolios, are agreed with the bank regulators.
This latter approach is a herculean task where simply agreeing a modeling methodology will not be easy.
In practice it seems most likely that regulators will settle for capital calculations based on a simple standard model (such as the Gordy model), backed up for SIFI and GSIFI banks, at least, by an institution specific model approach that is reviewed by the regulators under Pillar 2 and where banks are required to make public disclosure of their data, models, methodologies and results under Pillar 3.
Citigroup Fails Stress Test – Box 3
The Federal Reserve Board announced in March that Citigroup (ranked third by assets in the U.S.) was the only one among the nation’s Big Six that did not pass the stress tests.
The Fed argued that, while on their own each deficiency was not critical enough to warrant an objection, taken together they raised sufficient concerns regarding the overall reliability of Citigroup’s capital planning process.
This conclusion shows the subjective nature of a stress test as opposed to a model-based approach to measuring risk. In its rebuff of Citigroup’s capital plan, the Fed singled out shortfalls in the bank’s financial projections in “material parts of the firm’s global operations.”
The debate between VaR and ES – Box 4
Value-at-Risk (VaR) is a measure of losses due to “normal” market movements which means it ignores extreme outcomes. By ignoring the tail risks, VaR creates an incentive to take excessive but remote risks. Consider an investment in a coin-flip. A $100 bet on tails at even money creates a VaR to a 99 per cent threshold of $100 as you will lose that amount 50 per cent of the time, which obviously is within the threshold. In this case the VaR will equal the maximum loss.
Compare that to a bet where you offer 127 to 1 odds on $100 that heads won’t come up seven times in a row. You will win more than 99.2 per cent of the time, which exceeds the 99 per cent threshold. As a result, your 99 per cent VaR is zero even though you are exposed to a possible $12,700 loss. In other words, an investment bank wouldn’t have to put up any capital to make this bet.
In contrast, Expected Shortfall is a measure that takes into account the fact that tail risks and extreme events happen frequently in the real world. It also accepts that extreme events initiated by exogenous shock feed upon themselves. And it takes into account the incentives to “game” the risk measurement system and that extreme contingent risk (e.g. out of the money options) may create reward and no VaR measured risk.
Regulators are being persuaded to move towards an ES-based system by what happens in markets, where prices are not normally distributed and empirical studies show that asset price distributions exhibit “fat tails” (leptokurtosis).
The mathematics of such distributions imply far higher capital requirements than normal distributions and that analysis of the tails is more important than that of the variance from the mean – this means an ES measure is more relevant than VaR.
ES also invariably produces a far higher capital requirement than VaR which suggests the latter provides a gross under estimate of risk and capital.
However ES also ignores the probability, that the rare tail risk data observations are far less stable observations than the common data observations around the mean average of a distribution. Projections of risk based on an assumed distribution will be stable but if the tail observations are the result of dynamic and conditional processes such projections may prove to be very accurate wrong numbers.
