Insurance and Capital Markets

The Possibilities and Limits of Mathematical Risk Management

Authors: Rüdiger Frey und Uwe Schmock

In recent decades modern mathematical methods have entered the world of banking and insurance. In these areas – formerly the domain of lawyers and economists – mathematicians, physicists and other specialists with training in quantitative and mathematical methods have entered the fray. The main task for these quantitative analysts is the gauging and control of financial risk with which banks and other finance institutions see themselves confronted.

In contrast to banking, insurance companies have a long tradition in the assessment of pure insurance risk – traditionally the indisputed sphere of actuaries (insurance mathematicians). Nowadays, with the coming together of the areas of banking and insurance, as happens when financial institutions merge, capital and insurance risk must be looked at together. At a product level, for example, this trend has led to the advent of equity-tied life insurance and credit insurance, as well as high yield bonds to cover damages caused by earthquakes or other catastrophes (securitisation).

To accomplish their task, risk control managers use methods which have been developed in the areas of computer science and mathematics. Apart from modern stochastic techniques (probability theory and statistics) and simulation, mathematical tools exist to analyze financial data using numerical techniques from the theory of partial differential equations, and even methods arising from pure mathematics. In addition, due to the computerisation of the world of finance, some aspects of computer science also play a great role.

Heightened risk

The drive towards quantitative risk management has been triggered by a number of factors which have heightened the risk of greater losses for banks and other actors in capital markets. The dissolution of the Bretton Woods agreement in the 1970s went hand in hand with a strong increase in exchange rate fluctuations; stockbrokers talk of heightened volatility in this context. At the same time, due to the lowering of market barriers and technological development, the turnover of the capital markets has grown immensely. The daily turnover of shares on the New York stock exchange rose from 3.5 million in 1970 to 40 million in 1990, and in other markets the increase in business has been even stronger. At the same time, a new class of financial instruments, so-called derivatives, has prospered. Roughly speaking, derivatives are a way of betting on the future prices of basic values, such as bonds, shares or commodities. The most important derivatives are forwards, futures, swaps and options. Derivatives provide the actors in the capital markets with a handy tool to insure themselves against fluctuations in the price of basic securities (risk transfer) while lending themselves at the same time to speculation. The valuation of financial instruments is based on complex mathematical theory, for which the founders, Robert C. Merton and Myron S. Scholes received the Nobel Prize in Economics in 1997 (Fisher Black, who had worked with them had died). Derivatives are an excellent instrument for spreading risk and have become an irreplaceable weapon in the arsenal of modern capital markets. But, as a spectacular case in the early 1990s proved, risk transfer with derivatives is in itself not without risk.

Modern Risk Management Systems

In order to counteract the heightened risk potential in the capital markets – and under pressure from shareholders, the public, politicians and regulatory bodies – banks and insurers have begun in earnest to develop and implement modern quantitative risk management systems. The heart of such a system is the ascertaining of the profit and loss distribution. The developers try to pinpoint the probability of the appearance of certain profits and losses over a given period of time (a day, a month or a year, depending on intended application). To do this, the distribution of changes in the value of some core risk factors (such as share price indices and interest rate curves) have to be estimated and, subsequently, the value changes of an investment portfolio derived from the changed risk factors. In a second phase, risk measures can be calculated from the profit and loss distribution. The most well known of these measures, although not without its dissidents amongst researchers in Zurich, is the so-called Value-at-Risk (VaR). For the main part, such risk measures are used to judge the risk of a given investment portfolio and, subsequently, to calibrate the amount of risk capital financal institutions must set aside as security against loss. Setting a fixed ratio of equity aside is a way to ensure the stability of the entire finance system. The most important sort of risk which risk management systems must come to grips with is market risk (losses due to price changes of securities in the portfolio) and credit risk (losses caused by defaults of counterparties).

The development and implementation of quantitative risk management systems is not only a great scientific achievement, but also an organisational one; it is the most important step on the way to more stability in capital markets. However, modelling and quantifying financial risks is an extremely complex task, and systems that are already employed can be bettered by intensive collaboration between banks, regulators and researchers.

Changes of Risk Factors and Market Liquidity

Estimating changes in the weighting of risk factors still poses many challenging questions as far as mathematics and statistics are concerned and researchers at the ETH and the University of Zurich, especially those working at RiskLab, are on the cutting edge of research directed to answering these questions. For example, the problem of estimating the probability of the occurrence of huge losses. One knows from experience that such losses occur more often than predicted by the standard models, which are based on the normal distribution. In order to determine the probability of such events, RiskLab researchers apply extreme value theory, which has been used successfully to estimate the high water levels for dam building. A further core area of the Zurich risk management research is the modelling and estimation of dependencies between the various risk factors involved. This causes headaches amongst the practitioners of risk management because the investment portfolio of a bank or an insurance company is simultaneously beset by many different risk factors. One example of such a dependency is the relationship between market and credit risk. During periods of recession the number of bankruptcies rises and therefore heightens the credit risk of banks; at the same time, economic tendencies influence the value of a bank's stock portfolio.

Another exciting group of problems arise from market liquidity. Current risk management systems usually assume perfect market liquidity, which means that it is implicitly taken for granted that banks can sell quite large positions on the market within a very short period of time without influencing the market price. When it comes down to it, this is often not true, especially in times of market turbulence. Naturally, banks are aware of this and have, over time, developed practical procedures to protect themselves from losses arising from market illiquidity. Nevertheless, a systematic analysis of causes and effects of market illiquidity is of the utmost importance, especially for the valuation of options. Precise mathematical models are difficult to develop in this setting because both economical and psychological aspects play a great role in the liquidity of important markets. Nevertheless, RiskLab tries, within the bounds of mathematical accuracy, to determine the effects of lack of market liquidity on risk management systems.

RiskLab-Project: Modelling of Long-term Financial Risk

Funds managed by institutional investors in Switzerland has long passed the billion-franc-limit. In 1999 the banks alone were managing funds of 3.4 billion Swiss francs (about 2.1 billion US$). Reliable forecasts are essential for fund managers and their clients and these have to be weighted with the probabilities of a (possibly negative) development of their funds or portfolios. These funds can be diversified internationally and include thousands of shares, bonds, derivatives and other finance instruments. The development, therefore, of practical, mathematically consistent methods for forecasting the long-term development and assessing the inherent risk of such a fund is an important job. Existing models, such as «RiskMetrics», lead to fairly accurate forecasts for short-term market risk, say for the coming two or three weeks – this is, after all, what they were developed for. However, these models reveal severe weaknesses when they are applied to longer periods of time, such as the typical annual time span, necessary for the strategic investments of institutional investors. While models for short periods of time can presume that the composition of a portfolio will remain unchanged, such a presumption is not realistic for longer time spans; loans are repaid, options mature and portfolios are rebalanced to take market developments into account while conforming to investment directives.

The aim of the RiskLab project «Modelling of Long-term Financial Risk» is to develop a theoretically well understood and empirically founded concept with which to estimate the long-term financial risk of a strategically oriented portfolio.

As no single method in the pertinent literature appears to be better than any other, it is necessary to examine suitable proposals for models for various areas of use. In order to determine how well which model forecasts long-term value trends, RiskLab researchers proceed as follows. First of all they concentrate on each individual structural component of an investment fund, such as share prices or exchange rates, and, taking historical data into account, they determine the best model and its best calibration. Then the entire fund is modelled by pulling together all the single components, taking possible dependency structures into account. At this point they are able to create synergy by drawing on another area of RiskLab research where the modelling of dependencies in a general frame is being examined more closely. A major part of the project is to more accurately forecast the influence of investment strategy on the composition of an investment portfolio and, therefore, also its influence on the development of value and financial risk.

Economical Aspects

Above all, RiskLab works on the mathematical aspects of integrated financial risk management. The improvement in risk management systems naturally throws up important economic questions. Some examples of this are the efficient allocation of capital, performance measurement and bonus systems. It is a well-known fact that wrongly-tailoured bonus systems might tempt traders, insurance agents and fund managers to accept greater risk than their employers would wish. In addition, when examining the cause and effect of a lack of liquidity, or when deciding how finance and insurance markets should be organised, economical aspects must be put firmly at the centre. In order to combine all the relevant aspects, researchers at RiskLab, from the finance industries, the mathematics department of the ETH, the economics faculty of the University of Zurich and the Swiss Institute for Banks and Finances at the University of St. Gall are working ever closer together.

RiskLab: Finance Competence Centre at the ETH Zurich

From its beginnings, RiskLab was conceived as an inter-university research institute open to collaboration with other universities and research centres, with participants concentrating on strictly "pre-competitional" applied research in the area of integrated risk management for the finance and insurance. The institute, founded in 1994 is currently financed by the ETH Zurich, the two biggest Swiss banks (the Credit Suisse Group and the UBS AG) and by SwissRe. A team of international, first-class young researchers – some of them postgraduate – work in RiskLab. Applied scientific research in the area of finance and insurance mathematics, and above all, in risk management, is mostly carried out in the format of projects in close collaboration with its partners from the financial sector and further researchers from a number of universities and scientific centres. Currently, these include the Universities of Zurich, Lausanne, St. Gall, and the INRA-Institute in Sophia-Antipolis, France. Academic methods are applied to questions which arise in the real world and emerge from the close co-operation between the researchers and their partners in the finance industry.

Relative Daily Changes in the DAX and VAR Forecasts


The graph shows the relative daily changes in the value of the German stockmarket index (DAX), losses have a positive sign. The bottom line shows the value at risk for the next day, which losses should not exceed with a probability of 95%; the top line is the corresponding 99% threshold. These thresholds were calculated using a method developed at the ETH Zurich, by drawing heavily on time series models and extreme value theory. Where the limits were exceeded (as should happen on 5% or 1% of trading days, respectively) this is marked with a circle and a triangle, respectively; in this way we are able to backtest the model. Current prognosis for these limits using the «Riskometer» for the DAX, the Dow Jones, the S&P index, as well as historical volatility data and statistical analyses are available online under

Authors' affiliation:

Dr. Rüdiger Frey was assistant professor at the Swiss Banking Institute, University of Zurich from March 1999 until February 2002.
Dr. Uwe Schmock was research director of RiskLab, Department of Mathematics, at the ETH Zurich from October 1999 until November 2001.

Further information:

Finance and Insurance Mathematics at the ETH Zurich:
Swiss Banking Institute:

Bulletin, Magazin der Eidgenössischen Technischen Hochschule Zürich, No. 279, November 2000,
and Unimagazin, Die Zeitschrift der Universität Zürich, No. 3, October 2000, pages 56–59.

(Translation commissioned by RiskLab, links set by RiskLab)
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