Author’s name: Meltem Firaz Student number: 0195421
Name of the instructor/supervisor: Joost Driessen
Date of submission: March 18, 2007
Department: International Financial Economics
Table of Contents
1 Introduction ………. ……… 3
2 Literature Review ……… 5
2.1 Risk adjusted profitability ………. 5
2.2 Measuring and calculating market risk ……….. 7
2.3 Value-at-Risk ……….. 8
2.4 Non-parametrical models ………. 9
2.4.1 Historical simulation ………. 9
2.4.2 Monte Carlo simulation ……….. 10
2.5 Parametrical unconditional models ………. 10
2.5.1 The variance-covariance approach ……… 10
2.5.2 Extreme value theory ……….. 11
2.6 Some shortcomings of Value-at-Risk ………… .……… 12
2.6.1 Expected shortfall ………. 12
2.6.2 VaR estimation over a longer horizon ………. 13
2.7 Stress testing ………. 13
2.7.1 Developing stress test scenarios ………. 14
2.7.2 Implementing stress testing ……… 15
2.7.3 Disadvantages of stress testing………. 15
2.8 Market risk capital calculations ……….. 15
3 Survey on economic capital ……… 18
3.1 Publications on economic capital ……… 18
3.2 A survey of economic capital practises in Dutch organizations ……… 21
4 Market risk capital calculations at ING BANK ………. 23
4.1 ING Bank’s model to calculate economic capital for market risk ………… ……….. 23
4.2 Market Risk Capital Attribution Model ………. 25
5 Assessment of the Market Risk Capital Model ……….. 32
6 Conclusion………. ……… 44
7 Appendix I ……….. 46
8 Appendix II ………. 62
1. Introduction
In recent years, financial markets have become more volatile, competition has increased and financial institutions have to take risks in order to earn returns. In today’s competitive financial markets, the importance of Shareholders’ Value maximization has strongly increased. In order to measure the amount of risk taken and the rate of return required to buffer against the risk, new measurement techniques have been developed during recent years. A method to achieve this is the so-called RAROC (risk adjusted return on capital). RAROC is a management framework for measuring risk-adjusted financial profitability of a bank and for providing a consistent view of profitability across its business units. RAROC is calculated as the ratio of risk adjusted return to economic capital.
Economic capital (EC) is a new concept and can be defined as the amount of capital that has to be withheld to safeguard against potential losses due to all possible risk, i.e. credit risk, market risk, operational risk, business risk, etc. EC was introduced in 1990s in sophisticated financial institutions and has only turn out to be considerably important in the last decade. The need for a concept such as EC has arisen more from practical than theoretical concerns, as financial institutions have strived to address business unit performance, price products with sufficient compensation for all risk, bring business unit managers’ objectives in line with shareholder-value maximisation and play a part in active credit portfolio management. Within many financial institutions, bonuses and promotions are influenced to a great extent by the RAROC hurdle rate attained. The continuance of projects or the choice between investments and divestments of business units may depend on whether they achieved the required RAROC hurdle rate. This gives the business units the incentive to minimize their risks while maximizing their profits.
The internal assessment of the capital that is required to safeguard against potential losses, is derived from the internal risk measurement models, by the use of statistical models, the difference between high quantile (loss threshold) of the P&L distribution and the expected value of the P&L, which is called unexpected loss. More specifically, EC is measured based on a pre-specified confidence interval founded on the solvency standard and debt rating over a given time period. Banks usually choose to calculate their EC over a one year time period. EC stands for a piece of key information that the board members and senior management use to manage the bank and determine their key strategies.
It is generally expected that all related risks faced by a banking group should be incorporated in the bank’s EC calculations. Ideally, EC presents an aggregated analysis of the bank’s risk position from individual business lines up to the consolidated group level. One should note that, EC estimates of different institutions cannot be fully compared since each institution has its own methodology to calculate the EC. The Basel committee gives the following definition for EC: Economic capital is the capital that a bank holds and allocates internally as a result of its own assessment of risk. It differs from regulatory capital, which is determined by supervisors on the basis of the Capital Accord (Basel Committee on Banking Supervision (2001e). There are substantial risks that are not incorporated by the regulatory framework; such as, interest rate risk in the banking book, business risk, in addition to external factors like business cycle effects. Therefore, the banks need to hold extra capital to support their risks beyond the ones stated by the regulator. EC is a more sophisticated measure of the amount of capital that is required to buffer against losses. Hence, the difference between these two figures can represent how misaligned the regulatory capital is.
The aim of this study is to research different methodologies to determine economic capital for market risk and assess the model used at ING Bank. On the basis of this research, ING might make enhancements to improve its current model. In my research, I will answer the following question: Is the current model to compute economic capital for market risk effective and reliable? What could be adjusted in order to enhance the effectiveness?
Since Value-at-Risk and stress testing are the only means by which economic capital for market risk is determined, in chapter 2, the concept of Value-at-Risk and stress testing will be explained. An overview of different
methodologies to calculate them together with their shortcomings will be presented. In addition, the most common methods used in the market risk capital calculations will be studied. Chapter 3 gives the results of the surveys on economic capital which are conducted by consultancy firms and an overview of economic capital implementation in a selected number of internationally active Dutch organizations. In Chapter 4, ING´s methodology to determine economic capital for market risk is thoroughly explained. Chapter 5, presents an assessment of the model used by ING and gives recommendations for improvement. Finally, Chapter 6 presents the conclusions drawn from this study.
2. Literature review
In this chapter, the concept of Value-at-Risk and stress testing will be explained; an overview of different methodologies to calculate them together with their shortcomings will be presented. In addition, the most common methods used in the market risk capital calculations will be studied. The subjects covered in this literature survey are based on information from standard textbooks such as Jorion (2000), van Lelyveld (2005), Dev (2004) and Crouch, Galai and Mark (2001).
2.1 Risk adjusted profitability
The measurement and management of risk is a difficult task which local regulators and financial institutions are faced with. Financial risk quantification and analysis have developed significantly over the last decades. When implementing a risk based profitability framework in a bank one deals with the following questions:
What are the main types of risk the bank is facing?
How much amount of capital is required to cover those risks?
How can the capital be translated into one profitability measure for the bank as a whole?
Many banks nowadays use economic profitability (EP) or risk adjusted return on capital (RAROC) in order to assess their risk adjusted profitability instead of relying on concepts such as return on equity (ROE) or return on capital (ROC).
Return on equity (ROE) is the ratio of earnings to own funds as they appear on the balance sheet. Although ROE can provide a general overall performance of an institution, it is insufficient in assessing an institution’s risk adjusted performance. Additionally, ROE is estimated at the level of the institution as a whole since accounting own funds are not allocated to business units within the organization. As a result, ROE is not an adequate measure to manage risks across business units within an institution.
To improve risk management, the concept of return on capital has been developed. Return on capital (ROC) is the ratio of earnings to capital charge. In comparison to ROE, return on capital is a better measure of profitability but still does not take into account the variability or the risk of earnings. Given the disadvantages of both of the concepts (ROE and ROC) a new methodology called RAROC is developed in the 1970s. The idea is to measure risk adjusted performance of an institution and to provide a consistent view of profitability across different business lines. RAROC is generally formulated in the following way:
(Revenues – expenses - expected losses + return on economic capital
+/- transfer values / prices) RAROC = Profit / Economic capital =
Economic capital
Risk is a concept that has several descriptions. A generally used definition within the framework of capital management is unexpected loss. Economic capital can then be described as a buffer against all possible unexpected losses. The quantitative measurement determines the potential maximum loss in value of assets and other exposures (or in another way the increase in value of liabilities) in a given time period at a pre-given confidence interval. The notion of unexpected loss is related to the decrease in market value not in accounting earnings.
RAROC ranks the business units by a performance rate on allocated economic capital rather than their performance in financial terms. The profitability of a business unit can be measured with respect to a minimum level of RAROC required by the bank, the so-called hurdle rate. The economic profit (EP) is defined by:
EP = Risk adjusted return – (h * Economic Capital)
The multiplication of the hurdle rate with the economic capital is generally referred to as the cost of capital or in other words the yearly interest one would pay for borrowing the amount of economic capital at the specified hurdle rate. The hurdle rate can be considered as the price of risk in the capital market from the shareholder perspective. It can also be derived from the following formulation:
h = ROE target * (Equity capital / EC)
Two different business units may have equal RAROC rates although the riskier unit generates higher income or loss, given by their individual economic profits. Therefore, while evaluating the performance of business units, economic profit is a better measure than RAROC as it counts true profits and losses.
Although RAROC and EP are better measures than simple financial ratios such as return on equity (ROE) and return on capital (ROC), they have their disadvantage as well. RAROC and EP are linear combinations of the risk adjusted return and the allocated economic capital. However, they cannot fully capture the volatility of the returns. This can mainly be attributed to the fact that the main aim of economic capital is to measure extreme loss events. Economic capital is as anticipated concentrated on the tail loss and cannot sufficiently reveal the volatility returns across all scenarios (See section 2.3 for more discussion on this topic).
Capital asset pricing model (CAPM) is the most extensively used approach which models the dependency between the market return and its uncertainty. Here, the uncertainty is considered as the standard deviation of the returns. More explicitly, the CAPM states that the required rate of return of a portfolio can be measured with reference to the past correlation between the portfolio and the market.
E(r i) = r f + [β i (E (r m) –r f)] (1) r f = risk free rate
r m = return on market
r m – r f = excess return of market over the risk free rate Î risk free rate
β i = beta of portfolio; a measure which shows how closely the stock is following the market, defined as the covariance of r i and r m divided by the variance of r m.
Assume that R is the return of portfolio P which consists of sub-portfolios P1, P2,…,Pn with returns as R1, R2,…,Rn. We also presume that economic capital is calculated and allocated to each portfolio. Economic capital of portfolio P is indicated as EC (P) and capital allocated to P1 is denoted by EC (Pi, P). Next, each sub-portfolio is scaled by the factor EC (P) / EC (Pi, P) such that same amount of capital is allocated to each sub-portfolio. The return of the sub-portfolio i is denoted by the following equation:
R’ i = [EC (P) / EC (Pi, P)] * R i (2) R = [EC (P1, P) / EC (P)] * R 1 + [EC (P2, P) / EC (P)] * R 2 + …+ [EC (Pn, P) / EC (P)] * R n (3) Following the CAPM, the expected return of the sub-portfolio P’i is defined as
ER = normalised mean of the portfolio return R The expected return of the un-scaled portfolio P i is:
[r f + (ER – r f) β’ i] EC (P) * [EC (Pi, P) / EC (P)] (5) [r f * EC (Pi, P)] + [(ER – r f) * EC (P) *β i] (6) The expected portfolio return ER can be replaced by a target return hurdle rate, h determined by the bank. Subtracting (6) from the E (R i) gives the profitability measure.
P (P i, P) = E (R i) – [r f * EC (Pi, P) + (h’ - r f)* EC (P) *β i] (7) Reminding one more time, the economic profit was defined by:
EP = Risk adjusted return – (h * EC) (8)
Risk adjusted return consists of factors like gross revenues, administration costs and expected loss. Presuming that these components are covered by the expected return of the portfolio, the economic profit can be formulated in a different way:
EP (P i, P) = E(R i) – h EC (P i, P) (9) The two scenarios below demonstrate the difference between the economic profit and implied CAPM profitability measures:
1. break even: h = r f Î EP = P 2. h ≥ r f : EP = E (R) – [h * EC (P) ]
P = E (R) – [h * EC (P) * β i]
The first scenario which sets the hurdle rate equal to risk free rate is likely to hold for banks with a low target rate such as state-owned banks, or banks with focus on retail. With this set-up, economic profit and CAPM implied profitability are equal to each other. The second scenario which has higher expectations arises only if either the risk free rates are too low or the hurdle rate defined by the bank is too high. This is likely to hold for the banks with strong M&A and trading divisions. With this set-up, the difference between the CAPM implied profitability and economic profit boils down to the allocation of the economic capital. A high hurdle rate means that the amount of economic capital is less important for profitability but more prominence is put on the overall earnings of the division.
This analysis shows that CAPM implied profitability measure is consistent with the economic profit measure if the bank’s required rate of return on capital is identical with risk free rate. As the bank sets a higher hurdle rate, the alternative profitability measure puts more emphasis on the earning’s volatility, a behaviour which is aligned with shareholder’s point of view.
2.2 Measuring and calculating market risk
Market risk is the risk of loss due to unfavourable market movements. Such losses take place when an unfavourable price movement causes the marked-to-market value of a position to decrease. Market risk can arise due to a large number of risk factors: fluctuations in interest rates, foreign exchange rates, equity prices, commodity prices. In addition, changes in volatility of these rates and prices that influence the values of options or other derivatives,
changes in correlations between these risk factors are also important market risk factors. This definition mostly applies to liquid active trading books.
To my knowledge, there is no article that has been published which describes different methodologies to calculate economic capital for market risk. This contrasts with the enormous amount of publications on the calculation of regulatory capital for market risk. This can probably be attributed to the fact that economic capital methodologies are developed either by banks themselves or by consultancy firms and this kind of information is not disclosed due to its confidentiality. However, we know from the market surveys (presented in chapter 3) that the most frequently used methodologies for allocating economic capital for market risk are Value-at-Risk and stress testing.
A number of methodologies can be applied to measure market risk. The most frequently used methodologies are Value-at-Risk (VaR) and stress testing. VaR is a statistical measure of the potential loss that could happen due to adverse market movements during a certain period (T, the time horizon) at a given probability level. Stress testing is another commonly used approach to assess market risk. It is a simple form of scenario analysis. Dissimilar to VaR, which is an objective evaluation based on the statistical examination of the past, stress testing is a subjective approach to measure market risk which depends on personal judgment. Next, the different approaches to calculate VaR and stress testing will be discussed in detail.
2.3 Value-at-Risk
There are diverse factors which makes the risk measurement of a portfolio complicated. First of all, the intricacy of the portfolio itself whose value may be dependent on numerous risk factors, the correlation between different instruments that make up the portfolio and at last the synthesis of all these risk factors into one single risk figure such as VaR or expected shortfall, which can detain all the necessary information about the risk profile of the portfolio.
The most common method to assess the risk of a portfolio is Value-at-Risk (Jorion, 2000). The purpose of VaR is to approximate the worst trading losses of a fixed portfolio during a specific time period for a given probability level P. Presuming that one day is chosen as a specific time period, the histogram of daily returns (relative changes in the value of the portfolio from day t to day t+1) is drawn from historic data. This distribution is then fitted to a parametric model from which a percentile for a given probability level P is attained. The standard assumption of VaR is that daily relative returns are random independent normally distributed. This assumption gives the VaR the advantage of fast implementation; however, it received lots of criticism in the recent years (see for example Artzner et al, 1997 & Embrechts et al, 1998).
In point of fact, there seems to be an agreement in the financial society that VaR cannot capture some of the crucial aspects of actual market risks. It has been empirically observed that the tails of the portfolio returns have normally more weight than what is forecasted by a perfect fit to a normal distribution (Duffie & Pan, 1997). Especially, the assumption of normality is identified to fail for large variations in the portfolio returns (Hull & White, 1998). These large fluctuations in the value of the portfolio are in fact the ones that VaR is trying to capture. Because VaR fails to accurately characterize the behaviour of the portfolio in worst case circumstances, alternative models to VaR have been developed (i.e., Eberlein & Keller, 1995). These models are more sophisticated models in a sense that they are able to accurately replicate the probability of extreme events. One of the features of these new models is to drop the assumption of normal distribution and to hypothesize a different distribution with a positive kurtosis and probably skewness which can characterize the real behaviour of the financial markets. The parameters of these non-normal distributions are found out by maximum probability estimation. In order to determine the parameters of the non-normal distribution, the analysis focuses on the distribution of the extreme events.
In the subsequent sections the most familiar models to characterize return distributions will be discussed. The first separation can me made between models assuming parametrical and non-parametrical distributions. Parametrical models are derived from predetermined analytical distribution function of which the parameters are estimated by means of empirical data. Non-parametrical models do not presume any analytical distribution and just use the empirical data. They can be non-normal and may not be independent over the time period.
Parametrical models can be split into conditional and unconditional models. Parametrical conditional models presume that the stochastic process that creates the time series data on returns is independent and identically distributed in time. Data are assumed to be randomly drawn from a single distribution. This is called homoscedasticity. The parametrical unconditional models presume that data are created by a stochastic method with time-varying volatility. The conditional distribution varies at each point in time and the volatility process is stochastic which is called heteroscedasticity. Next, non-parametrical and parametrical unconditional models will be studied in detail. Parametrical conditional models are out of the scope of this study.
2.4 Non-parametrical models
2.4.1 Historical simulation
Historical simulation is doubtless the simplest non-parametrical model (Crouch et al, 2001). There is no assumption made about the complex organization of the markets. As an alternative, the historical behaviour of the portfolio is observed during the last few years. The daily percentage changes in the market parameters are computed and next these changes are applied to the current portfolio to estimate the profit and losses.
The risk of a portfolio can be assessed by using the unconditional probability distribution of historical returns. It is presumed that we have a restructured series of the values of the negative of the log returns.
r t = - log [S t+1 / S t ] ≈ - [(S t +1 –S t ) / S t ]
where S t are the values of the theoretical portfolio which has the same constant composition as the actual portfolio. The log returns can be approximated to relative returns if only the time horizon is short. The approximation can only be applied to daily returns. The negative sign is applied in order that losses emerge on the right hand side of the probability distribution.
The advantage of this approach is its simplicity since it does not require simulations or development of an analytical model. In addition, it can easily incorporate non-linear instruments; for instance, options. The major disadvantage of this approach is the availability of data. For some of the instruments, enough historical data may not be available. Another issue is that the further we go with the historical data to the past, the less relevant the data becomes in comparison to today’s market. On the one hand, we would like to add as much as possible data to be able to count for the rare events, i.e., heavy tails. In contrast, it is not actually relevant to build current risk estimates on very old data. Assume that, the current practice is to use five years historical data to estimate risk of our portfolio. If there was a very rare event that occurred on a particular day which resulted in large losses, exactly after five years later this big market jump will not be included in our dataset. This will also result in a very big shoot in our VaR estimate from one day to the next day. Therefore, risk estimation with historical simulation approach may result in instable results.
2.4.2 Monte Carlo simulation
Another well known non-parametric approach to estimate VaR is Monte Carlo simulation which is a more sophisticated methodology compared with historical simulation (Crouch et al, 2001). This approach starts with identifying the most important market factors. Secondly, a joint distribution of these market factors is built based on historical data or data based on economic scenarios. Next, the simulation is performed mostly with a large number of scenarios and profits & losses are estimated for each scenario at the end of each period. Finally, the profit & loss figures are ordered and the 5% quantile (or 1%, depending on the confidence interval chosen) of the worst results is the VaR estimate.
Monte Carlo simulation has several advantages of its own. The most important pro of this model is that the results of the simulation can be enhanced to a large extent by applying a large number of simulated scenarios. Portfolios which include non-linear instruments, such as options, can be estimated with this method. Besides, it is possible to track path-dependence since the entire market process is simulated instead of the final result only (Crouch et al, 2001).
One major drawback of this method is that the simulation converges to the true value as 1/ √N. Here, N is the total number of simulated paths. As a result, in order to increase the accuracy factor by 10, 100 more simulations must be performed. When Monte Carlo simulations are performed, it is advised to create trajectories for each type of market parameter and next estimate every particular instrument along these trajectories. One major problem here is the pricing of each instrument because this is very time consuming. However, this problem can be alleviated to a large extent when all the instruments are split into several categories. This is called variance reduction and is founded on the grounds that the instruments that we want to categorize have similar risk features. An additional problem with Monte Carlo approach is that the joint distribution of many market parameters must be estimated. This is especially important when the market parameters are extremely correlated (Crouch et al, 2001).
2.5 Parametrical unconditional models
2.5.1 The variance - covariance approach
Variance - covariance approach is the most commonly used parametrical model based on the assumption that returns are normally distributed.
It is generally assumed that the portfolio returns behave according to the Black-Scholes model and that daily returns are normally distributed. The probability density function of the normal distribution is given by:
where σ is defined as the standard deviation and μ as the expected value (mean).
With the variance covariance approach, historical data is used to measure the major parameters such as means, standard deviations, correlations, etc. The overall distribution of these market parameters is determined by using the historical data. When the market value of the portfolio depends linearly on its underlying parameters, the distribution of the parameters is normal as well. Hence, the 5% quantile proportionate to VaR can be measured at 1.65 times the standard deviation which is below the mean (for the 95% confidence interval, 2.33 times the σ).
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There are several advantages of the variance covariance approach. The first one is that for many market parameters all of the relevant data is well known. Secondly, the methodology is simple, flexible and widely used. In addition, it also facilitates the addition of particular scenarios and also allows the examination of the sensitivity of the results regarding the parameters (Carol & Leigh, 1997).
The major drawback of this approach is the assumption of normal distribution of the market parameters. Therefore, when a substantial portion of the portfolio is non-linear (i.e., options), it is not possible to use this approach anymore. Another drawback relates to the foreign exchange. When a target zone is used, this approach can only be used if the exchange rate is in the middle of the band. When the exchange rate is close to the margin, it cannot be relied on variance covariance approach any further; since the distribution of changes would be far from normal in this situation (Carol & Leigh, 1997).
As the returns of the risk factors are influenced by approximately infinite factors, it is assumed that they can be depicted by normal distribution. However, assessment with empirical data showed that this is not always true. Extreme events appear more to happen than predicted by normal distribution. This is the so called fat-tails effect (Huisman, et al, 1998). This problem can be alleviated to a certain degree by using the Student-t distribution.
The student t distribution depends on a single parameter v, the degrees of freedom. As the degrees of freedom goes to infinity, the student t distribution converges to normal distribution.
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The main market information used in the variance covariance model is included in the variance covariance matrix. All of these parameters are usually generated from historical data. Here, the question of how long time interval to use for historical correlations comes up. Basle approach requires using at least a year of historical data for the calculation of correlations.
The variance covariance matrix has its own drawbacks as well. First of all, because of very high degree of correlation, even a small flaw in the data can easily cause the correlations to deteriorate. The second drawback relates to the fixing of the time window which is also mentioned in historical simulation approach. Any big market jump generates an unexpected change in all key parameters when the window moves. This problem can be alleviated with an appropriate weighting scheme or the so-called exponentially weighted covariance matrix. This approach is founded on the fact that volatilities and correlations change over the time and hence the most recent data gives the most appropriate information about the future returns. Therefore, more weight is given to the recent data whereas the older data receives less weight (Carol & Leigh, 1997).
2.5.2 Extreme value theory
The VaR obtained by the above described three approaches (Historical simulation, Monte Carlo Simulation, Variance-Covariance) is a benchmark about the risk measure and therefore can provide no insight into the risk that might occur when the tails of the distribution change. The tails are the parts of the distribution which capture the most exceptional and catastrophic events, so are the most interesting areas to risk managers and regulators. Practically, risk managers are more concerned about the largest potential loss in an unanticipated disaster. Regulators are also
concerned about the market conditions during a financial catastrophe because the capital put aside by banks must be enough to cover the largest losses such that banks can stay in business even after a great financial crisis. As a result, how to deal with extreme events is very important in risk management.
In financial markets, extreme price movements occur during stock market crashes, bond market failures, foreign exchange crisis, all of which corresponds to extraordinary periods. In addition, extreme price movements could also occur during ordinary periods when markets correct themselves. Extreme value approach covers market conditions varying from usual market environments which are considered by the usual VaR approaches to the financial disasters which are the focal point of stress testing. To apply the extreme value approach in practice, a parametric model derived from “extreme value theory (EVT)” is built up to calculate the VaR of the portfolio (i.e. see Longin, 1996).
The estimation of VaR using extreme value approach entails a number of steps (Longin, 2000). The first step involves the selection of the return frequency which is usually influenced by liquidity and portfolio risk. The second step requires the extent of the chosen period or block length which is the number of days from which the smallest return is picked. In order to do so, the entire period is split into non-overlapping sub-periods each including n observations. Christoffersen et al. (1998) put forward that 10 to 15 trading days are essential for independent and identically distributed observations. On the other hand, Longin’s (2000) study of S&P 500 Index returns recommends that 21 trading days (period of a month) is proper.
The third step involves selecting the minimum returns from the non-overlapping sub-periods. From the first n observations of returns R1, R2,…,Rn, the smallest observation indicated by Rn,1 is picked. From the next n observations of returns, again the second smallest observation Rn,2 is chosen. From T return observations (T=n*N) a time series Rn.1, Rn,2, …,Rn,N consisting of N observations of smallest returns is obtained. The fourth step is to estimate the parameters of the limiting distribution of the smallest returns. The parameters can be estimated by the maximum likelihood method. The largest returns (maxima series) are constructed with the same method.
2.6 Some shortcomings of Value-at-Risk
2.6.1 Expected shortfall
It is a well-known fact that Value-at-Risk methods might be inadequate to measure portfolio’s risk under market stress. VaR models are mostly built on the assumption of normal asset returns and do not work under extreme price fluctuations. VaR models ignore the fat tailed properties of actual returns and undervalue the probability of severe price movements. By definition, VaR measures only percentiles of profit-loss distributions and as a result disregards any loss beyond the tail level (Krause, 2003).
Far-out-of-the money option is one of the examples of assets for which tail risk can be serious since these types of assets may show infrequent but very large losses. Investors can influence the return distribution of these types of assets which would cause the tail risk to become fat and sides thin. This would result in a lower Value-at-Risk while the level of underlying risk is not reduced. This problem can also not be alleviated by increasing the confidence level while measuring VaR since investors might construct positions that would result in extreme losses beyond the new confidence level.
Artzner et al (1997) proposed the “expected shortfall” concept to lessen the inherent problems of VaR. In literature, expected shortfall is defined as conditional expectation of loss provided that the loss is beyond the VaR level. The expected shortfall at the 100 (1-α) % is defined by the following:
ES α (Z) = E [ Z | Z ≥ VaR α (Z) ]
Expected shortfall methodology has several advantages when compared with VaR. Expected shortfall can capture nonlinearities, i.e., options and non normality in asset returns. Expected shortfall is a more conservative methodology than VaR and gives higher results. In addition, VaR is not sub-additive1 while expected shortfall well. Expected shortfall is without difficulty optimized by means of the linear programming approach, whereas VaR cannot be.
2.6.2 VaR estimation over a longer horizon
Most of the literature on Value-at-Risk deals with the estimation of risk over a daily horizon. However, there may be many situations in which we need to estimate value-at-risk over a longer horizon, for instance, a week or a month. The square-root t rule is the most common industrial practice for estimating the VaR over a horizon period of t days. The t days VaR of a position is computed as follows according to the square-root t approach.
VaR (t) = VaR (1 day) * √ t
Such time scaling is widely used across the industry and recommended by the Basel Committee on Banking Supervision for horizons up to 10 days. Unfortunately, this approach is extremely unreliable and often results in overestimations of VaR. The reason is as follows (Christoffersen et al, 1998). In reality, market returns are not identically and independently distributed; but show a strong effect of volatility clustering. When applied to the quantiles, it is a well known fact that time scaling (square-root t approach) requires returns not only be identically and independently distributed; but also normal. On the other hand, it is also well known that returns exhibit excess kurtosis, i.e., they are fat tailed.
2.7 Stress-testing
In recent years, stress testing became a key tool for the large financial institutions for measuring the sensitivity of their businesses to extreme changes in market parameters. Similar to VaR methodology, stress tests assess how the value of the portfolio of an institution will be affected by sharp movements in, say, stock prices, interest rates, exchange rates or commodity prices. Dissimilar to VaR, which is an objective evaluation based on the statistical examination of the past, stress testing is a subjective approach to measure market risk which depends on personal judgment. Stress tests are intended to answer such type of questions:
If the stock market falls by 25 percent, by how much would the value of my portfolio change?
If the euro appreciates against US dollar by 30%, how much would the value of my portfolio rise or fall? In conducting stress tests, banks assess the impact of such sharp changes in markets across all their business lines and trading portfolios. In the literature, two most common types of stress tests can be found: stress test scenarios and sensitivity stress tests. Stress test scenarios assess the impact of synchronized severe movements in several asset prices on a firm’s portfolio, for instance simultaneous extreme movements in equity prices, interest rates, foreign exchange rates and interest rate spreads. The scenarios can be based on extreme events that occurred in the past (historical scenarios) or on probable market events that may happen in the future (hypothetical scenarios). On the other hand, a sensitivity stress test gauges the impact of a sharp price or yield changes on a
1 A risk measure is said to be sub-additive when the risk of the total position is less than or equal to the sum of the risk of individual portfolios.
portfolio’s value. One good example to sensitivity stress tests which is commonly practised by risk managers in the industry is the impact of parallel shift in the yield curve.
Stress tests can be seen as complements to the firm’s other risk management tools, mainly Value-at-Risk models. VaR is a statistical measure of the potential loss that could happen due to adverse market movements during a certain period (T, the time horizon) at a given probability level. Because VaR models use historical market data (volatilities and correlations), they have inadequate capacity to capture the risks of extreme market events, especially type of market events in which asset prices move dissimilar to historical norms. Because of this incapability of VaR, Basel Committee requires banks to perform regular stress tests as one of the seven requirements to use internal models.
2.7.1 Developing stress test scenarios
The development of proper stress test scenarios can be considered as both art and science. Large banks mostly follow the scenarios specified by supervisors. The Basle Committee on Banking Supervision refers to several likely scenarios such as the equity market crash of 1987, the bond market crisis of 1994, Asian market crisis in 1997, etc.
Berkowitz (1999) explains in his paper how the stress test scenarios should be identified. He splits the scenarios into the following categories:
1. Simulating shocks which are most probable to happen than historical observation suggests. 2. Simulating shocks that have never happened.
3. Simulating shocks that reflect the likelihood that statistical patterns could fail in some circumstances. 4. Simulating shocks that reflect some kind of structural break that could take place in the future.
The second category could be thought as a special case, although extreme, of the first grouping. The third group illustrates structural breaks across the world, for instance asset correlations increasing during a crisis. The fourth grouping depicts the initiation of a systemic structural break over time, such as change from fixed exchange rates regime to floating exchange rate regime.
In an historical scenario, the actual historical prices and rates which are observed during the crisis time are applied to the current value of the portfolio. There can be no rejection on the likelihood of historical scenarios since they occurred in real life. Unfortunately, there are not many historical scenarios that can be used in stress testing, as they happen very rarely by definition and it may be difficult to attain consistent data for many new instruments which were not introduced yet then in the markets. In addition, it cannot be assured that the historical events that can be used for stress testing are applicable for the instruments in a firm’s portfolio. Once it is decided to include a historical event in stress testing, it is required to specify the start and end dates of the scenario. The vagueness arises at this point since not all of the effected markets show the spikes at the same days. The chosen dates may have remarkable impacts on the stress test results.
On the other hand, hypothetical scenarios correspond to a different type of stress scenario. A hypothetical scenario involves identifying particular changes in all relevant prices and rates to be used in the scenario by the designer of the scenario. In theory, an infinite number of hypothetical stress test scenarios can be produced however; the task of identifying a fully articulated and sensible scenario is very complicated. The first step in developing hypothetical stress scenarios is to specify plausible risky future market developments. Next, it should be specified how the chosen scenario will manifest itself in key market prices and rates. After that, it should be identified how these changes will spread across other markets, whether usual arbitrage relationships will be valid then or will be abused because of the illiquidity of the markets.
2.7.2 Implementing stress testing
Once the stress scenarios are chosen, the next question arises regarding the best approach to implement the scenarios. One such issue is the method in which stress price shocks should be identified, i.e., absolute or percentage shocks. While this may seem as an unimportant point, the impact on stress test results can be significant. A second issue which may arise is the selection of proxies for the shocks to rates which have no available market data during the certain stress test period. If the stress tests are to be approaching, then they must incorporate price shocks even for those instruments which were not available during the time of the actual event. This situation is most likely for emerging market instruments, however can even be relevant for developed market instruments where traded instruments, say, interest rate swaps or foreign exchange future tenors have extended over the years. The most common approach under these conditions is to link the no-existing market rate of the instrument into another related instrument which is known to behave similar. A less popular, but occasionally the necessary solution is to specify hypothetical shocks.
2.7.3 Disadvantages of stress testing
Stress tests have several problems as well. Perhaps, the most important one is that they are subjective since they are based on personal judgement. Therefore, the results of the stress testing depends significantly on the choice of the scenarios and thus on the skills of the designer. This subjectivity also makes it complicated about how to evaluate the results objectively and causes apparent problems for the management, supervisory regulators and other parties who are interested in understanding the stress testing procedure of a firm.
Another problem is related with the interpretation of the stress test results since they give us no idea about the plausibility of the underlying scenarios. As Berkowitz (1999) states it in his study “We have some loss numbers but who is to say whether we should be concerned about them?”
This absence of probabilities leads to another problem. There are two sets of risk estimates however no way of combining them. The first one is based on statistical probabilistic estimates (VaR) and the second type provides loss numbers based on what if questions (stress tests). Therefore, we must work with these estimates independently.
The methodology of stress testing is also still in its immaturity and the different methodologies that are used are still open to question. For instance, a very common stress testing methodology is to stress several portfolios with some pre-defined parameters and derive the firm’s exposure consequently. This approach is very simple and also very easy to implement, however it does not take into account the correlations between the stressed prices and other prices.
Another disadvantage of stress testing is the fact that it is very difficult to backtest obtained results since neither the completeness nor the consistency of the results obtained by stress tests can be scientifically evaluated. 2.8 Market risk capital calculations
Market risk capital is the capital which is held to guarantee creditors, with some general accord level of probability that adequate reserves exist to meet an entity’s commitments to creditors. Market risk capital must be distinguished from regulatory capital for market risk. Regulatory capital is only held to assure the regulator’s requirements and cannot be completely associated with corporate objectives. Regulators are concerned about maintaining the stability of the financial system. Modified in 1996, Basel Banking Accord specifically addresses market risk and provides different methods for calculating the regulatory capital for market exposures. Banks are allowed to choose methods to determine the amount of regulatory capital that they should assign for market risk. They could choose a simplified
“bucket approach” like the one that had been applied for credit risk or provided that they meet some requirements or implement a more advanced internal approach.
The internal models approach is a very important step in the course of alignment of regulatory capital with economic capital calculations. Banks are allowed to implement internal models approach for determining their regulatory capital charge provided that the below listed conditions are met:
The bank’s risk management system is theoretically sound,
There are adequate number of personnel who are capable of using sophisticated models,
Backtesting of the internal model is done regularly and the results tend to confirm the model’s ex ante measurements,
The bank performs stress testing on a regular basis.
The internal models approach used for determining the bank’s regulatory capital charge is based on a Value-at-Risk calculation with 99% one-tailed confidence level by means of an instant price shock which is corresponding to a 10-day movement in prices. Banks are also allowed to benefit from correlation between different types of risk categories when determining VaR. Banks are also expected to use different risk factors in their computation of VaR. For instance, for interest rates, yield curves and spreads; for equities, index volatility and beta equivalent measures. In order to determine the regulatory capital charge of the bank, VaR is multiplied with a factor, not below 3, which is specified by the supervisor, based on the backtesting performance of the internal risk assessment model. Besides, a specific risk capital charge may be added on the scaled-up VaR figure.
On the other hand, market risk capital is the amount of risk contributed by specific business units of a larger entity. Given this definition, risk capital also provides an objective method for the entity to gauge the risk adjusted performance of different business units of the entity and hence take part in more informed portfolio optimisation decision making.
The Basel standard approach is the most straightforward and less advanced method to calculate regulatory capital; however, it gives some gradation and suggestion that more capital should be held for riskier assets. On the other hand, the internal models approach suggests that many banks have already created models that have even advanced gradations and more precisely reflect the capital that is necessary for the market risk. Most of the firms that use internal models for reporting their regulatory capital for market risk apply models that aim to progress further on the spectrum of modelling unlikely events in order to calculate their market risk capital.
In order to calculate the economic capital for market risk, banks can base their computations on Value-at-Risk, on stress testing or a combination of both. To originate the market risk capital from the Value-at-Risk number some modifications have to be made (van Lelyveld, 2005).
Confidence interval conversion: depending on the target rating of the bank, some modifications have to be made to the confidence interval as a result. For that reason, the hypothetical value distribution from which the VaR is computed has to be fitted to a well-known theoretical distribution (such as normal, lognormal, student-t, etc). Instead, scaling to extreme confidence intervals can be attained by applying Extreme Value Theory (EVT).
Fat tails correction: Conditional on the used distribution, an adjustment that takes into consideration the fat tails may be required.
Time-scaling: VaR is generally calculated at 1 day holding period. On the other hand, economic capital calculation is based on a one-year time period. Therefore, the holding period of 1 day used in the calculation of VaR must be scaled up to a one year risk estimate. This adjustment presumes that daily distributions are statistically independent. When this adjustment does not hold, inaccuracy in the computation of longer economic capital horizons can be quite significant.
Management intervention: Large losses can be realized in some form of management intervention to alleviate the losses. Monte Carlo simulations can be used to determine the risk mitigation provided by the management intervention.
3. Survey on economic capital
3.1 Publications on economic capital
Since economic capital concept has gained great significance during the last decade, a number of surveys have been undertaken recently to document current industry practices. Some surveys have been conducted by consultancy firms. One of the surveys conducted by a consultancy firm is the PricewaterhouseCoopers (PWC) 2005 survey. Further on, several other surveys have been undertaken by Chief Risk Officers forum, Capital Market Risk Advisors 2001, Economic Capital subgroup of the Societies of Actuaries Risk Management Task Force 2004, and by a UK group of actuaries in 2003.
Economic capital models can be used for a number of purposes. At present, economic capital is mainly assessed for internal use, in order to allocate capital across business units or to evaluate the performance of business units. Besides, economic capital models are sometimes used as input for (risk-adjusted) pricing or limit setting. Occasionally, economic capital models are used to improve credit rating or for diversification benefits from M&A decisions. An external use of EC models is to communicate with shareholders. A recent survey of PricewaterhouseCoopers / Economic Intelligence Unit, October 2005 reveals the business drivers to adopt economic capital modelling.
If your organization has adopted, or have plans to adopt economic capital, what are the
reasons?
2 13 18 20 20 22 34 38 41 43 51 51 51 65 68 0 10 20 30 40 50 60 70 80 Other Improve M&A decisions Set risk-adjusted compensation Enhance external reporting Improve credit rating Improve investor relations Improve picing policies Meet regulatory capital requirements Assess risk-adjusted product profitability Assess risk-adjusted customer Improve capital adequacy Assess risk-adjusted business unit Set risk limits Define risk appetite Improve strategic planningChart 1: Source: PricewaterhouseCoopers 2005, Economic intelligence Unit survey
Please note that total do not add up to 100 because respondents could choose more than one answer.
Implementing an economic capital model in an organization is a lengthy and costly process. Once more, in a recent survey from PricewaterhouseCoopers, it was disclosed that organizations from Europe and Asia-Pacific use economic capital to a greater extent across the firm than the American organizations. In addition, 37% of American organizations indicated that they have no plans to implement economic capital in comparison to 27% of European and 19% of Asia Pacific organizations.
Does your organization currently use or have plans to adopt EC? 35 27 12 20 7 12 37 22 20 10 32 19 17 14 19 0 5 10 15 20 25 30 35 40 45 We already use EC across the enterprise
We have no plans to introduce EC
We plan to introduce EC w ithin the next
five years
We already use EC in certain units
We plan to introduce EC w ithin the next
year
Europe Americas Asia-Pacific
Chart 2: Source: PricewaterhouseCoopers 2005, Economic intelligence Unit survey
Please note that total do not add up to 100 because respondents could choose more than one answer.
As mentioned previously, an external use of economic capital modelling is to communicate with shareholders. The Economic Intelligence Unit survey of PricewaterhouseCoopers revealed that two thirds of the organizations do not report their economic capital results to outside audiences. And from those who do, only a few report the figures to analysts or agency presentations.
Does your organization report EC results externally (other than to regulators) and if so, how?
34 40 25 15 17 17 4 44 40 24 8 12 8 8 35 27 25 25 13 11 8 0 10 20 30 40 50 We do not report the figures externally and have no plans to We do not report the figures externally yet but plan to We include them in our annual report We include them in analysts presentations We include them in rating agency presentations We include them in our quarterly financial results Other Europe Americas Asia-Pacific
Chart 3 Source: PricewaterhouseCoopers 2005, Economic intelligence Unit survey. Please note that total do not add up to 100 because respondents could choose more than one answer.
In another survey conducted by Capital Market Risk Advisors (CMRA) amongst 55 financial institutions all over the world, respondents indicated that the economic capital is allocated as follows among different types of risk.
Overall response
62% 19% 12% 7% Credit Risk Market Risk Operational Risk OtherLarge global banks
49% 21%
30%
Credit Risk Market Risk Operational Risk & Other
Chart 4 & 5: Source: Capital Market Risk Advisors, Economic Capital Survey Overview, 2001
According to the investigation of CMRA, the most frequently used methodologies for allocating economic capital for market risk are as follows.
Methodologies to calculate EC for market risk
29% 28% 20% 16% 7% Value-at-Risk (VaR) Multiple of VaR Stress Testing Combinations of the methods above No response
3.2 A survey of economic capital practises in Dutch organizations
In order to see how the situation has evolved since the surveys mentioned in the previous section, an overview of economic capital implementation in a selected number of internationally active Dutch organizations is presented. In particular, the annual reports of ABN-AMRO, Fortis, ING Bank and Rabobank are studied.
The study of large Dutch financial conglomerates’ annual reports has revealed a wealth of information about the current practice of economic capital models and uncovered a number of commonalities. The findings are as follows:
All of the institutions are in possession of an economic capital model which produces an economic capital amount estimated to be adequate to cover unexpected losses at some pre-defined confidence level.
The institutions are using their economic capital model in order to measure performance or for capital allocation and external reporting. The significance of each of these purposes may vary by institution.
The institutions have conceptually consistent models in place for the calculation of economic capital for different type of risks. An example to this is the implementation of VaR models for the calculation of economic capital for market risk. The short-term VaR figures are mostly scaled up to the appropriate horizon in order to come to an appropriate estimate of the annual VaR.
All of the institutions apply a time horizon of one year. Confidence levels vary from 99.95% to 99.99% which reflects the target ratings of the institutions
Economic capital model implementation of the institutions is not in steady-state but going through a rapid change. One of the important external pressures in this process is the approaching implementation of Basel II.
Economic capital is first determined for each type of risk for the whole organization and after that the results per risk type are combined to get an overall economic capital figure. The figures below show the share of each type of risk in economic capital of the four Dutch international conglomerates.
EC ABN-AMRO per risk type
16% 3% 12% 69% Credit & Co untry Risk M arket Risk Operatio nal Risk B usiness Risk
EC Fortis per risk type 2,10% 4,70% 5,91% 38,14% 49,15% Credit Risk A LM Risk B usiness Risk Trading Risk Event Risk
Chart 8: Source: Annual report Fortis 2005
EC ING Bank per risk type
15% 12%
26% 47%
Credit Risk M arket Risk Operatio nal Risk B usiness Risk
Chart 9: Source: Annual report ING Group 2005. Please note that EC figures presented in the pie-chart above is only for the ING Bank. EC for insurance business of ING is not included in the figures presented above.
EC Rabobank per risk type
50% 4% 13% 10% 23% Credit Risk M arket Risk Operatio nal Risk B usiness Risk Insurance Risk
4 Market risk capital calculation at ING BANK
4.1 ING Bank’s model to calculate economic capital for market risk
4.2 Risk Capital Attribution Model
5 Assessment of the Market Risk Capital Model
6 Conclusions
7 Appendix I
8 Appendix II
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