The Choice of Interest Rate Models and Its Effect on Bank Capital Requirements Regulation and Financial Stability

According to the Basel regulation banks may use internal risk models to measure interest rate risk and calculate regulatory capital requirements. Under its pillar II the Basel framework grants leeway to banks in their choice of these models. We therefore focus on how well interest rate models describe real interest rate movements empirically and which impact the model choice has on the economic value of bank equity during the financial crisis. Furthermore, we address the question how different choices of interest rate models affect the overall financial stability. To this end we estimate eight different interest rate models for three different currencies (USD, EUR, CHF) using the Generalized Method of Moments (GMM). Then we approximate the balance sheet of a typical Swiss bank during the financial crisis and run Monte Carlo simulations of the balance sheet using the estimated interest rate models. Our results show that the required economic value of equity for a bank varies considerably with the different choices of interest rate models. However, the interest rate models which are empirically best fitting do not imply aggregate financial stability. Thus, banks‟ choices of interest rate models to calculate regulatory capital requirements may have a crucial impact on overall financial stability.


Introduction
Recently, new standards on minimum capital requirements for market risk were established within the regulatory framework of Basel III by the Basel Committee on Banking Supervision (BCBS, 2016a).Furthermore, the Basel Committee puts special focus on the management of interest rate risk in the banking book, as it considers interest rate risk to be a material source of risk especially during times when interest rates worldwide normalize from historically low levels ten years after the financial crisis.When interest rates rise again from these low levels, even small relative changes in rates may have high absolute outcomes in terms of the economic value of bank equity.This is why inter alia the U.S. Federal Deposit Insurance Corporation (FDIC) highlights the importance of interest rate risk management (FDIC, 2009).Meanwhile also on an international scale greater guidance is given by the Basel Committee on banks" interest rate risk management.In particular, the Basel Committee now focuses on how banks develop interest rate shock and stress scenarios.But even in its revised principles on interest rate risk management, the Basel Committee still leaves leeway to banks regarding the use of internal models to calculate interest rate risk (BCBS, 2016b) (Note 1).Thus the question arises, which results the different choices of internally developed interest rate models may bear on regulatory capital and overall financial stability.To answer this question we consider a typical Swiss bank and its leeway of choice to report interest rate risk on the basis of internally developed models during the financial crisis.To be consistent, we use the then prevailing rules of Basel II.Hereafter, banks had to report to supervisors results of internal interest rate risk models expressed in terms of the economic value of equity relative to regulatory capital.To facilitate the monitoring of interest rate risk across the financial system, banks had to calculate a standardized interest rate shock.Supervisors gave particular attention to the sufficiency of capital of "outlier banks", whose sum of Tier 1 and Tier 2 capital declined by more than 20% as a result of a hypothetical 200 basis points interest rate shock (BCBS, 2007).
(USD, CHF, EUR).Following a simulation methodology as suggested in Estrella (2004) and Koopmann, Lucas, and Klaassen (2005), we run Monte Carlo simulations of a typical Swiss bank"s balance sheet to analyze bank-specific and systemic risks associated with the choice of different interest rate models.

Literature Review
Research suggests that there is no generally accepted model to determine interest rate risk (Kuritzkers & Schuermann, 2007).Consequently, the Basel Committee allows to measure interest rate risk with different approaches (BCBS, 2016b).One approach is to understand interest rate risk as earnings at risk.Another approach is to measure interest rate risk by changes in economic value of bank equity (Note 2).In general the Basel Committee defines interest rate risk in the banking book as current or prospective risk to the bank"s capital and earnings arising from adverse movements in interest rates that affect the bank"s banking book positions.The source of interest rate risk is hence a change in the present value and timing of future cash flows and therefore a change in the value of a bank"s underlying assets and liabilities.Moreover, changes in interest rates affect a bank"s earnings, i.e. its net interest income.
Due to the variety of interest rate risk measurement approaches different results exist on the question, whether banks reserve the appropriate amount to cover these risks.For example Esposito, Nobili, and Ropele (2015) find that Italian banks" interest rate risk exposure was noncritical over the period 2008 to 2012 by using a simplified measurement methodology for interest rate risk introduced by the Bank of Italy (2006).By contrast, Memmel, Seymen, and Teichert (2017) find for a set of German banks during the period 2005 to 2014, that bank managers tend to increase interest rate risk when operating income falls below a certain threshold.The authors thereby underscore that a prolonged low interest rate environment is counterproductive from a financial stability perspective.In this context Chaudron (2016) finds that Dutch banks, who received government help during the financial crisis, took on greater interest rate risk.Posner (2014) argues that bank regulators failed to give banks an adequate incentive to increase capital during the financial crisis.He explains that not using cost-benefit analysis to determine capital requirements may have contributed to the financial crisis.
Therefore, Alessandri and Drehmann (2010) as well as Kretzschmar, McNeil, and Kirchner (2010) suggest integrated models to calculate capital adequacy for banks.The authors argue that correlations between the different risk types, such as credit risk and interest rate risk, play a crucial role in calculating appropriate capital reserves.Cerrone, Cocozza, Curcio, and Gianfrancesco (2017) develop an internal measurement system for interest rate risk.On the basis of a sample of Italian banks the authors show that the current Basel regulation on interest rate risk needs to be improved from a financial stability perspective.Miller, Olson, and Yeager (2015) develop a forecasting model for bank failures and find out that Tier 1 leverage ratio is the most accurate distress signal for large bank holding companies.
Our contribution to the literature is twofold.We first highlight problems from a bank management perspective in choosing the right interest rate model.Then we discuss from a financial stability perspective, that the freedom of choice between different interest rate models to calculate regulatory bank capital requirements may negatively impact the overall financial system stability.

Methodology
Our research methodology comprises three steps.In the first step we set up a balance sheet of a typical Swiss bank with actual economic and currency exposures during the financial crisis.In the second step we calibrate stochastic processes for interest rates and stock returns.The interest rate models are estimated and benchmarked according to Chan, Karolyi, Longstaff, and Sanders (1992) for Swiss Francs (CHF), U.S. Dollars (USD) and Euros (EUR).To calibrate the models we use the Generalized Method of Moments (GMM).Stock returns follow a standard Geometric Brownian Motion, for which the parameters will be estimated.The third and final step of our methodology consists of a Monte Carlo simulation of assets and liabilities of the bank.The simulations highlight the consequences of the choice of different interest rate models on the economic value of equity capital of the bank.The simulations are run for three interest rate term structures, namely a normal, a flat and an inverse term structure.

Structure of the Bank Balance Sheet
In order to define a representative balance sheet of a Swiss bank, we use data from the central bank, the Swiss National Bank.Assets: 70% are loans, approximated as bonds with a constant duration of 2 years and 30% are stocks (SNB, 2007b).Liabilities: Between 12% and 15% are equity and between 88% and 85% are short-term liabilities with a duration of 1 year (SNB, 2007a(SNB, , 2007c) (Note 3).
Credit risk, market and operational risks are incorporated by calculating average exposures to these risk categories of representative public and private sector banks in Switzerland with regard to their balance sheet totals (SNB, 2007a) (Note 4).In addition, as the annual report of Credit Suisse (CS, 2008) indicates, large Swiss banks had the following currency exposures during the financial crisis: 41% in Swiss Francs, 34% in US Dollars, 25% in Euros.To find out how different magnitudes of equity impact financial stability, we perform two simulation runs with an initial equity of 12% and an initial equity of 15%.This leads to the following representative balance sheets (Note 5): Figure 1.Bank balance sheets with 12% equity and 15% equity

Design of the Monte Carlo Simulation Model
The economic value of equity of a bank is the result of interest rate movements.Future interest rate movements are described in different stochastic models.We analyze consequences of the choice of the stochastic model on the resulting economic value of equity capital of the bank.The methodology we use is a simulation of the assets and liabilities which are interest rate sensitive (Figure 1).The simulations are run for periods of 120 months.For these ten years, the economic values of equity are compared to the equity requirement of the Basel regulation.We set the balance sheet total equal to 100 monetary units and thus assume that bank management will not change the structure of the balance sheet during the simulation horizon.The balance sheet positions are determined at the beginning of the simulation in 0 t  and change only on the basis of the stochastic processes involved in the simulation.The market value of equity at any given point in time t (120 months) is given as the difference between assets and liabilities: The Basel regulation requires determining the minimum equity capital on the basis of two provisions: a) Pillar 1: Credit, market and operational risks must be covered by the sum of risk-weighted assets for credit risk, as well as the exposures on market and operational risks: b) Pillar 2: Interest rate risks in the banking book are controlled by a provision which stipulates that equity capital must not be reduced by more than 20 percent after a standardized interest rate shock.A standardized interest rate shock is defined for G10 currencies by a 200 basis points parallel up-or downward shift of the yield curve or a change of the 1 st or the 99 th percentile of interest rate changes in the past five years on a one-year holding period (BCBS, 2007) .
Based on these two provisions we define the minimum equity capital requirement for interest rate risk in the banking book as a function of the respective market values (MV):

MV stocks MV bonds MV bonds Equity
MV stocks MV bonds MV short term liabilities During the simulation runs, the minimum capital requirement for keeping the bank solvent is defined as the maximum value between the two capital requirement provisions, i.e. max (Equity min1 ; Equity min2 ).If the market value of equity MV(equity t ) drops below the higher value of Equity min1 or Equity min2 , the bank is considered to be insolvent, i.e. defaulted.The simulations are based on the calibrations of stochastic interest rate processes.The calibrations are conducted for all three currency exposures (Figure 1).Each of the currency positions are included by 10,000 simulation runs for 120 months.The simulated balance sheet positions are connected by means of the Cholesky transformation.That is, the random number generation process accounts for the correlations between the bond, stock and short-term liability positions in the respective currency.Compounding effects are offset among the balance sheet positions.The market values of equity are calculated as residual factors according to (1).Then the resulting magnitudes of equity are compared to the Basel capital requirements as defined in (2) and (3).This again, is done in each of the 10,000 simulation runs.This enables us to analyze the equity demanded by the regulator for each scenario, as well as the actual equity situation of the bank under the 10,000 simulated scenarios.

Estimating Stochastic Interest Rate Models
Chan et al. (1992) estimate and compare eight continuous-time models of the short-term riskless rate using the Generalized Method of Moments (GMM).The authors find that the best models in describing the dynamics of (the short-term) interest rate are the ones, which reflect in their formulae that the true volatility of interest rates depends on the level of the riskless rate.We re-estimate and rank these eight models according to their GMM-based empirical performance by using similar data up to the mid of the financial crisis.But we extend the aforementioned analysis by two aspects: First, we use longer time periods.Second, we use three different currencies.Chan et al. (1992) point out that many single and multifactor term structure models imply a dynamic behavior of the short-term riskless rate r which follows this stochastic differential equation: Table 1 shows the term structure models used in the literature and their relationship to the basic stochastic differential equation (4).
Table 1.Interest rate model definitions Brennan and Schwartz (1982), Dietrich- Campbell and Schwartz (1986), Sanders and Unal (1988), estimate the parameters of the continuous-time model using this discrete-time econometric specification: Following Hansen (1982), Chan et al. (1992) and others, we also use the Generalized Method of Moments (GMM) to test (5) as a set of over-identifying restrictions on a system of moment equations.The advantages of this approach are that the GMM-method does not require a normal distribution of interest rate changes.Only stationarity and ergodicity for the distribution of interest rate changes are required.Another advantage is that the GMM estimators and their standard errors are consistent even in cases where the disturbances are conditionally heteroskedastic.This facilitates the temporal aggregation problem which arises from the estimation of a continuous-time process with discrete-time data.
Let  be the parameter vector with elements  Cox, 1975) 0 Under the null hypothesis that the restrictions implied in (5) are true, the expectation of vector   . Using a time series of length T observations, the GMM procedure replaces . In the next step the parameter estimates are chosen, which minimize the quadratic form is a positive-definite weighting matrix (Note 8).For the nested interest rate models detailed in Table 1, the GMM estimates of the over-identified parameter subvector of  depend on the choice of T W . Hansen (1982) shows that choosing results in the GMM estimator for  with the smallest asymptotic covariance matrix.Defining an estimator of this covariance matrix as   0 S  , the asymptotic , where   0 D  is the Jacobian evaluated at the estimated parameters.This covariance matrix is used to test the significance of the individual parameters.The minimized value of the quadratic form  under the null hypothesis that the model is true with degrees of freedom equal to the number of orthogonality conditions net of the number of parameters to be estimated.The 2  measure provides a goodness-of-fit test for the model, which indicates misspecification when the value of the statistic is high.

Data Description
For the calibration of the interest rate models, the short-term riskless rate of interest is proxied by a one-month interest rate.Ball and Torous (1999), Duffee (1996) and others argue that the idiosyncratic variation of U.S. Treasury yields increased since the early 1980ies.Therefore one should not rely exclusively on Treasury yields in calibrating models of short-term interest rate dynamics for the U.S. market.Following Ball and Torous (1999) and to ensure international comparability of the short-term interest rate dynamics, we use one-month rates drawn from the London Euro-currency market.Each of these rates is a London interbank rate denominated in a given currency.In particular, we use monthly observations of the one-month Euro-Dollar, Euro-Mark/Euro and Euro-Swiss Franc middle rates taken from Datastream over the sample period February 1981 to November 2009, giving 346 observations for each series (Note 9).
The equity markets are proxied with the "FTSE All-World Index Series" (in prices), which is the successor index series of the "FT Actuaries / Goldman Sachs International Indexes" used in Roll (1992) (Note 10).The sample is also taken from Datastream and coversas in the case of the interest rate datathe time period from February 1981 to November 2009, giving the 346 mentioned observations.We base our analysis on this time span, because it comprises several historical all time highs and lows in interest rates and excludes the abnormal low interest rate period after the financial crisis.Figure 2 provides a graphical overview and descriptive statistics on the data sample.
This sample is used for two purposes.First, the interest rate data is used to estimate and benchmark different interest rate models.Second, the data on both capital market segments (interest rates and stock market returns) are used to calibrate the stochastic processes.Then the Monte Carlo simulation can be started.The unconditional average level of the one-month interest rates for the Swiss Franc is 3.46% with a standard deviation of 2.60%.For the US-Dollar the values are 5.93% with a standard deviation of 3.54%.The Euro (German Mark) shows a mean interest rate of 4.93% with a standard deviation of 2.51%.Table 2 shows the autocorrelations in the interest rate levels, which decay slowly.With regard to the month-to-month changes of the respective time series, the autocorrelations are small.They are neither consistently positive nor negative (except for the German interest rate).This gives evidence that short-term interest rates are stationary.The balance sheet exposures in Figure 1 give an impression of the overall business risk of the bank, if the correlations between the various risk factors are considered.Table 3 shows that significant risk reduction effects in terms of negative correlations are given between Swiss interest rates and the national and international stock market exposures.However, no risk reduction effect can be obtained in the international term-transformation business of the bank, as interest rates are highly correlated between the three markets.

Results of the GMM-Estimations of Interest Rate Models
Table 4 reports the parameter estimates, asymptotic t-statistics, and GMM minimized criterion χ 2 -values for the unrestricted model and for each of the eight nested models.Since the unrestricted model represents an exactly identified system, the minimized GMM criterion value is exactly equal to zero.The unrestricted model for the Euro-Swiss Franc middle rate differs from the originally estimated model in Chan et al. (1992) insofar, as the parameter γ is significantly below 1, since here γ = 0.3766.Hence, we cannot confirm that the conditional volatility of the real interest rate movements would be highly sensitive to the level of the short-term rate.As Chan et al. (1992), we find only weak and insignificant evidence of mean reversion as measured by the parameter β in the unrestricted version of the model.In the case of the Euro-Swiss Franc middle rate from the London Euro-currency market, the χ 2 -tests for goodness-of-fit suggest that six models are misspecified.These models are: Merton (1973), Vasicek (1977), Dothan (1978), GBM of Black and Scholes (1973), Brennan-Schwartz (1979) andCIR VR (1980).All these six models have χ 2 -values in excess of 6 and can thus be rejected at the 95% confidence level.Only the CEV model by Cox (1975) and the CIR SR (1985) model fit the actual interest rate movements (during Feb 81 and Nov 09) well.The χ 2 -values and p-values are low and indicate that these two models cannot be rejected even on the 90% confidence level.In terms of the significance of the interest rate variance, we obtain similar results as in Chan et al. (1992), but we cannot confirm that the ranking may be done according to the γ-values.
Turning to the estimation results for the Euro-Dollar middle rates from the London Euro-currency market in Table 5, we can see better conformity with the results of Chan et al. (1992).

Table 5. GMM-estimation of the US EURO-$ 1 MTH -MIDDLE RATE
The estimation horizon for t r , the annualized monthly Euro-Dollar middle rates from the London Euro-currency market, is from February 1981 to November 2009 (346 observations).The parameters are estimated by the Generalized Method of Moments with t-statistics in parentheses.Tests evaluate overidentified restrictions imposed by alternative models on the unrestricted model.The χ 2 test statistics are reported with p-values in parentheses and associated degrees of freedom (d.f.).The existence of a unit root is rejected according to the Augmented Dickey-Fuller test with intercept and drift (lag-length 4, Akaike criterion).The parameters are estimated from the following discrete-time system of equations: For the case of the unrestricted model we find a γ-value of 1.1654 (Chan et al. (1992) find 1.4999).This confirms that the conditional volatility of the interest rate process is sensitive to the level of the short-term rate in the U.S.-market.We also find comparable (although insignificant) t-statistics for interest rate volatility, σ 2 , and our estimates for β confirm mean reversion in the unrestricted model.In contrast to the findings of Chan et al. (1992), our values for the intercept, mean reversion and volatility are significantly smaller, though.This finding may be explained by the generally lower and falling interest rate level during the time period of our study.As Chan et al. (1992) report for the U.S. market, we find that the Merton (1973), Vasicek (1977), and CIR SR (1985) models are misspecified and can be rejected on the 90% confidence level.Furthermore, the best-performing model for our sample is clearlyas in the Chan et al. (1992) casethe Brennan-Schwartz (1979) model which has the lowest χ 2 -value.A well-known fact is that the sensitivity of GMM-estimations to alternative specifications of the length of time series is large.This leaves room for the question, which time horizon will be adequate.If the regulated banks choose different time frames (which results in alternative specifications of the interest rate models), the stability of the financial system could either be enhanced by naive diversification or endangered by (unintended) system-wide risk clustering.

Simulation Results
As we have seen, the explanatory power of the models depends on the currency.It therefore makes sense not to choose one model and use it for all three currencies.Appropriate models should better be chosen for each currency and then combined to a set.In our Monte Carlo simulations of the assets and liabilities of the bank (Figure 1) we combine the eight interest models to Interest Model Sets (IMS) according to their rank as shown in Table 7.The risks associated with selecting models of different explanatory power can then better and more accurately be assessed.In total, we consider four Interest Model Sets, denoted by IMS 1, IMS 2, IMS 3, and IMS 8.These sets are defined by forming the four best combinations of models for each of the three currencies and they have the highest explanatory power: The tables in the appendix contain the descriptive statistics on the performance of the different Interest Model Sets (IMS) after 60 simulation periods and 120 simulation periods, respectively.The empirically best performing model set is IMS 1.In spite of the high empirical power (which is certainly positive), the set however shows the largest number of defaults after 60 and 120 simulation periods (which is negative).IMS 1 is closely followed by the empirically worst performing model set IMS 8.The second best performing model set IMS 2 displays the smallest number of defaults after 60 and 120 simulation periods (positive for financial stability).It is closely followed by the third-best performing model set IMS 3 (Note 11).Therefore, from a financial stability perspective, IMS 2 would be strongly favorable.
IMS 1 and IMS 8 as well as IMS 2 and IMS 3 are similar regarding the number of defaults during all the simulation runs, whereas the former sets IMS 1 and IMS 8 always generate more defaults than the latter.It becomes evident that the worst performing sets (IMS 1 and IMS 8) contain the Brennan-Schwartz (1979) and the Vasicek (1977) models, which take mean-reversion and an intercept into account.The Vasicek (1977) model does not reflect conditional volatility scaling, and the Brennan-Schwartz (1979) model reflects volatility scaling only to a very small extent.
IMS 2 and IMS 3 abstain from mean-reversion and intercept ( 0, 0   ) and they scale the conditional volatility with a higher interest rate level (see Dothan (1978) and GBM (1973)).We can conclude: the lack of mean reversion and intercept combined with a high conditional volatility scaling are the winning attributes of models and sets of models.
The importance to select an optimal IMS is underlined in Table 9 by the low p-values of the Kruskal-Wallis test.
Since the p-values are very close to zero, we conclude, that at least one distribution of the actual equity distributions is significantly different from the others.This points out the importance of the choice of Interest Model Sets (IMS), as financial system stability strongly depends on the equity distributions generated by the sets of models.Another result is the robustness of our simulations in terms of the interest rate term structure.Different term structures have only minor effects on the number of defaults.The maximum difference in the number of defaults after 120 periods is less than 0.3% between the three term structure types "normal", "flat" and "inverse".A normal structure generates the lowest and a flat term structure the highest amount of defaults.From a regulator"s point of view the most important outcome is consequently this fact: An increase of a bank"s equity from 12% to 15% cuts the default rate in half for any given IMS.Although an increase of the equity requirement by some 3% appears to be a drastic demand, it would contribute strongly to a more stable financial system.
Our results disclose the complexity of the decisions, which must be made by the management of a bank.Following the shareholder value approach one would suggest to choose either IMS 3 in the short run or IMS 8 in the long run, as these two models give the highest expected economic value of equity.But a viable consideration for the management may also be to minimize the regulatory capital requirements in order to enhance the profitability of the bank.According to this argument, managers may prefer IMS 2 in the short run and in the long run, as this model set produces a minimum target equity capital requirement.Unfortunately, this interest rate model produces also the second highest number of defaults in the financial system during the 10,000 simulation runs.
We also report that the distributions of market values of equity significantly change during the course of the simulations.This observation is made for all interest rate model sets.In the short-run of 5 years the distribution of the market values of equity is tight, while it becomes increasingly right-skewed and wider in the long run after 10 years.Hence, the probability that outlier banks occur clearly increases over time.However, for 12% (15%) equity at the starting point, overall default rates are below 16% (6%) for the five-year horizon and they reach 20% (10%) for the worst IMS in the ten-year horizon.
Summing up, all stochastic models of interest rates and sets of such models show that even a little bit more equity strongly enhances the solvency of a bank and increases financial system stability.But there is no relation between empirical performance of an interest rate model or sets of these models and the resulting financial system stability.Some models or sets of models are performing well regarding the empirical description of interest rate movements, but they show many defaults in the simulation runs.Other well-accepted models make the bank look safer, but they are poorer in there empirical power to explain real movements of interest rates.The freedom to choose the interest model in the Basel regulation consequently appears to be questionable.This is the case, because the differences among the models regarding empirical power and the demand of equity are large and there is no model, which would be superior in all categories.

Conclusion
In this article we analyze interest rate models for banks regulated under the Basel framework.We calibrate eight representative interest rate models and find that the criterion of best empirical performance does neither induce financial system stability in the short nor in the long run.Models which make a bank look safer, however, are poor in their power to describe real movements of interest rates.
We consider a typical Swiss Bank and allow combinations of different interest rate models for the three currencies USD, CHF and EUR, which determine the assets and liabilities of the hypothetical bank.Interest rate models and sets of models with no mean-reversion parameters but with conditional volatility scaling factors would be preferable from a regulatory point of view, since they exhibit a minimum of defaults during the simulation runs.The empirically mediocre performing model sets IMS 2 and IMS 3 surprisingly deliver the best results regarding financial stability.This is true for horizons of both five and ten years.
Financial stability crucially depends on the objectives of the bank management, i.e., whether shareholder value will be maximized or profitability enhanced through low equity holdings.We observe that the objective may be viable from a business perspective, namely enhancing the profitability of a bank, but that this objective turns out to be contrary to the goal of financial stability.Thereby, even small increases in bank equity have strong positive effects on financial stability.Our simulations with 12% of equity and 15% of equity respectively confirm that the new capital requirements in Basel III with up to 13% equity must be considered as absolutely necessary.
Our results also show that management objectives can be linked directly to the performance of the aggregate financial system, if regulatory provisions allow for an internal capital adequacy process, such as the one stipulated in the Basel framework.The paradox is that the regulatory framework on interest rate risk legally outsources the decision of risk model choice to the individual agents of the economy (banks).Our work shows that this can lead to more profitable banks, but it may also have negative effects on the overall stability of the financial system.Hence, regulators have to be particularly careful in judging the internal risk models banks choose.Individual first-best solutions are not the first-best solutions for the financial system.
For the recent result of the negotiations in the Basel Committee on Banking Supervision on the use of internal risk models our study also indicates, that choosing a benchmark model and defining a variability range to set limits to the model results is a viable way to foster financial stability.Using output floors to set these limits is practical to control model results.Moreover, cutting back on the complexity of bank internal risk models and thereby enhancing transparency in setting bank capital requirements is necessary in the future.Further studies could therefore focus on the effect of different output floor designs on financial stability, as well as on the development of models to calculate bank capital requirements, which capture uncertainty and risk and are still tractable in practice.

Notes
Note 1. Banks are expected to implement the new standards on interest rate risk in the year 2018 based on information as of 31.December 2017.In comparison to the Basel II standard, the new Basel III standards on interest rate risk in the banking book request six prescribed interest rate shock scenarios, as well as frequent model tests and sensitivity analyses.The amount of total capital to be reserved remains at 8 percent.Hence, the economic outcomes under the old and the new standard are comparable.
Note 3. The values of 12% and 15% were chosen to represent the region of the Basel II/BIS-conform Tier 1 year-end capital ratios of Credit Suisse (16.3% in 2009 and13.3% in 2008) and UBS (15.4% in 2009 and11.0% in 2008) during the financial crisis (cp.UBS and Credit Suisse annual reports 2008 and 2009).
Note 4. Risk weights are based on the aggregated balance sheet total weights of UBS, Credit Suisse and Canton Bank of Zurich as of the respective annual reports 2008, (14% credit risk, 2% market risk, 2% operational risk).
Note 5.The Basel III capital requirements are accompanied by extensive reporting requirements.The Handbook on Securities Statistics, published by The Bank of International Settlement, demands a four-dimensional reporting on debt securities holdings (BIS 2010).Banks have to classify their debt security holdings with respect to residence (issuer), currency, maturity and interest rate (fixed/variable).
Note 7.For ease of exposition, longer maturities are extrapolated from simulated monthly rates in terms of a constant liquidity premium factorpositive in the normal term structure, negative for an inverse term structure and zero in the flat term structure.
Note 8. Minimizing   T J  with respect to  is equivalent to solving the homogeneous system of equations,  with respect to  .In the case of the unrestricted model, the parameters are identified.

 
T J  is zero for all choices of   T W  .Note 9. Datastream files ECUS$1M, ECWGM1M, ECSWF1M, respectively.Note 10.Datastream files WIUSAM$, WIWGRML, WISWITL, respectively.Note 11.The only small exceptions are the two 15%-equity simulations for inverse and flat interest rate curves for the period 60 to 120, where IMS 8 generates the largest number of defaults and IMS 1 performs second worst.

Appendix
Table A1.Summary statistics of simulation results after 5 and 10 years (t=60 and 120 periods) 12% Equityinterest rate term structure: normal

Balance Sheet Assets Liabilities Balance Sheet Assets Liabilities standard
Geometric Brownian motion of the form dS t /S t = μd t + σdW t , where μ represents the drift, σ is the volatility of the stock, and dW t is the increment of a Wiener process.

Table 2 .
Autocorrelation coefficients Autocorrelation coefficients of the annualized monthly Euro-CHF/USD/EUR middle rates from the London Euro-currency market from Feb./1981 to Nov./2009 (N=346 observations).The variable rt denotes the yield maturing in one month and rt+1-rt is the associated monthly yield change, ρj denotes the autocorrelation coefficient of order j.

Table 3 .
Correlations of balance sheet risk factorsOne-month London Euro-currency market interest rates p.a.

Table 4 .
GMM-estimation of the SWITZERLAND EU-FRC-1M -MIDDLE RATEThe estimation horizon for rt, the annualized monthly Euro-Swiss Franc middle rates from the London Euro-currency market, isFebruary 1981 to November 2009 (346 observations).The parameters are estimated by the Generalized Method of Moments with t-statistics in parentheses.Tests evaluate overidentified restrictions imposed by alternative models on the unrestricted model.The χ 2 test statistics are reported with p-values in parentheses and associated degrees of freedom (d.f.).The existence of a unit root is rejected according to the Augmented Dickey-Fuller test with intercept and drift (lag-length 4, Akaike criterion).The parameters are estimated from the following discrete-time system of equations:

Table 6 .
GMM-estimation GERMANY EU-MARK 1M -MIDDLE RATE , the annualized monthly Euro-Mark/Euro middle rates from the London Euro-currency market, is from February 1981 to November 2009 (346 observations).The parameters are estimated by the Generalized Method of Moments with t-statistics in parentheses.Tests evaluate over-identified restrictions imposed by alternative models on the unrestricted model.The χ 2 test statistics are reported with p-values in parentheses and associated degrees of freedom (d.f.).The existence of a unit root is rejected according to the Augmented Dickey-Fuller test with intercept and drift (lag-length 4, Akaike-criterion).The parameters are estimated from the following discrete-time system of equations: t r

Table 6
Chan et al. (1992)for the German Euro-Mark/Euro middle rate.These are similar to the results of the Swiss case in the unrestricted version of the model.Again we cannot find a significant connection between the conditional volatility of the interest rate process and the level of the short-term yield.There is only weak and insignificant evidence of mean reversion (measured by β in the unrestricted version of the model).Thus we have to reject theMerton and CIR VR (1980)model on the 95% confidence level.Our finding is that the best performing models are the CEV model byCox (1975)followed by the CIRSR (1985)model.As in the case of Switzerland and in contrast toChan et al. (1992), we estimate a γ-value below 1 with γ = 0.7067.Summarizing, we rank the different results of the parameter estimation for stochastic interest rate processes in the three different currencies according to the χ 2 -test (Table7):

Table 7 .
Performance ranking of interest rate models in different currencies

Table 8 .
Interest rate model sets (IMS)

Table 9 .
Kruskal-Wallis test of interest model sets (IMS) with respect to the distribution of actual equity capital, MV t(Equity)

Table A2 .
Summary statistics of simulation results after 5 and 10 years (t=60 and 120 periods) 12% Equityinterest rate term structure: flat

Table A3 .
Summary statistics of simulation results after 5 and 10 years (t=60 and 120 periods) 12% Equityinterest rate term structure: inverse

Table A4 .
Summary statistics of simulation results after 5 and 10 years (t=60 and 120 periods) 15% Equityinterest rate term structure: normal

Table A5 .
Summary statistics of simulation results after 5 and 10 years (t=60 and 120 periods) 15% Equityinterest rate term structure: flat

Table A6 .
Summary statistics of simulation results after 5 and 10 years (t=60 and 120 periods) 15% Equityinterest rate term structure: inverseCopyrightsCopyright for this article is retained by the author(s), with first publication rights granted to the journal.This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).