Interest-Rate Models for Cost-at-Risk Analysis

In the management of interest-rate risk on the central-government debt, the development in future interest costs is analysed under different assumptions regarding the level of interest rates. The analyses are performed using the Cost-at-Risk (CaR) model on the basis of simulations of the interest-rate development in an interest-rate model.

So far, the one-factor Cox-Ingersoll-Ross (CIR) model has been used to generate the interest-rate scenarios in the Cost-at-Risk model. On the basis of a principal component analysis of the development in the Danish yield curve, the interest-rate model has been extended to include two explanatory factors.

The extension improves the explanatory power of the model and allows for the decoupling of short-term and long-term interest rates that are observed historically for several periods. Simulation of the interest-rate development in the two-factor model indicates that the results are relatively robust to the choice of estimation period on an analysis horizon of 10 years.

10.1 RISK MANAGEMENT AND INTEREST-RATE MODELS

In the management of interest-rate risk on the government debt, the development in future interest costs is analysed under different assumptions regarding the level of interest rates. The analyses are performed using the Cost-at-Risk (CaR) model, which simulates the interest costs on central-government debt on the basis of the current debt portfolio, technical budgetary projections from the Danish Ministry of Finance, a strategy for future borrowing and an estimated development in interest rates.

In practice, interest-rate models are used to generate the interest-rate input to CaR. An interest-rate model is a mathematical formulation of the development in the yield curve. The latter expresses the relationship between term to maturity and the level of the interest rate on interest-bearing assets, typically government bonds.

Using an interest-rate model ensures that a large number of different interest-rate scenarios can be simulated and assigned a probability. In each scenario, interest costs for the debt portfolio can be calculated to give probability distributions around future interest costs. The risk measure absolute CaR is defined by the 95th percentile in the interest cost distribution, while the mean value is an expression of the expected interest costs. The risk measure relative CaR is defined as absolute CaR less the mean value and is thus an expression of the maximum increase in costs in relation to the mean value, with a probability of 95 per cent. On the basis of the central government's risk tolerance, the risk measures may be used to support the choice of duration target, cf. Chapter 8.

10.2 CRITERIA FOR CHOICE OF INTEREST-RATE MODELS

The probability distribution around the future interest costs is determined by the interest-rate model selected, and the assessment of trade-off between costs and risk is therefore closely linked to the model's properties.

The development in the interest costs for the government debt is simulated over a 10-year horizon, and generally government bonds are issued with several different maturities. These two circumstances are important in relation to the interest-rate models used, since it is necessary to model both the long-term dynamics in the general level of interest rates and the covariation between the individual interest rates:

The yield curve must fluctuate around a mean level (mean reversion) to prevent interest rates from rising or falling explosively over time.
The model should exclude negative interest rates.
The yield curve must exclude arbitrage opportunities so that the investor or borrower cannot achieve a risk-free gain.
Uncertainty, i.e. volatility, in the individual maturity segments must reflect the historically observed uncertainty.
The covariation between interest rates for different maturities must reflect the historically observed pattern.
The model must be well-known, documented and easy to communicate.
The model should be manageable. For instance, it must be possible to estimate the model's parameters, and to simulate yield curves within a reasonable time.

Macroeconomic and financial literature both include numerous interest-rate models that all meet several of the above requirements. It is not possible to identify a single, superior model, and any choice of interest-rate model will reflect an attempt to strike a balance between various considerations. Some models have attractive theoretical characteristics, but cannot be used in practice. However, the simplest models often give an unrealistic picture of the yield curve dynamics.

For previous analyses of different interest-rate models for CaR simulations, see Danish Government Borrowing and Debt 2001, Chapter 9.

Interest-rate modelling in the CaR model
So far, the one-factor Cox-Ingersoll-Ross (CIR) model has been used to generate the interest-rate scenarios in the CaR model. The one-factor CIR model is an affine term structure model. In such models, zero-coupon rates for all maturities are affine functions^[1] of a number of descriptive factors, cf. Box 10.1 .

COX-INGERSOLL-ROSS-MODELS

Box 10.1

In the one-factor CIR¹ model, the development in the short interest rate, r, is described by the stochastic differential equation

The equation shows that the change in the short interest rate over a short horizon depends on the sum of two movements.

The first movement is deterministic. If the interest rate is higher than its mean level, q , a falling interest rate is expected since k is positive. k indicates the speed at which the interest rate approaches its mean level. The first term thus ensures that the interest rate has a "mean reversion".

The second movement is stochastic, since W is a "Wiener process". Wiener processes are characterised by the change dW being normally distributed with the mean value 0 and the variance dt. The value s is called the volatility and scales the uncertainty with the level of the short interest rate. The higher the volatility or the level of the short interest rate, the greater the uncertainty concerning the change in interest rates in the following period.

To calculate zero-coupon rates on the basis of the process for the short interest rate, it is necessary to assume that the market is arbitrage free. In an arbitrage-free market it is not possible to achieve risk-free gains by purchasing and selling different bonds. Any expected return in excess of the short interest rate is therefore an indication that the investor is assuming an increased risk.

Assuming that there is no arbitrage², it can be demonstrated that the yield to maturity on a zero-coupon bond, y, with the maturity t in the one-factor CIR model is given by

A(t) and B(t) are positive functions of k, q and s , as well as an extra parameter, l , which determines the expected additional return on long-term relative to short-term zero-coupon bonds. The CIR model is an affine interest-rate model since the zero-coupon rate is an affine function – i.e. a linear term plus a constant – of the short interest rate.

The CIR model can be expanded to include several explanatory factors by letting the short interest rate be given as

If it is assumed that the two explanatory factors, x₁ and x₂, are independent, it can be demonstrated that the yield to maturity for a zero-coupon bond with the maturity t in the two-factor CIR model is given by

where A ₁(t) (A ₂(t)) and B ₁(t) (B ₂(t)) are as in the one-factor model, but calculated on the basis of the parameters from the two factor processes. In the two-factor model, the parameter space has been expanded to include two extra parameters, l₁ and l₂, which determine the expected additional return on long-term relative to short-term zero-coupon bonds. As in the one-factor model, it is seen that the zero-coupon rate is an affine function of the two explanatory variables. ³

The parameters l, l₁ and l₂ are referred to as risk premiums. In the one-factor model it can be shown that the expected additional return on a zero-coupon bond relative to the short interest rate is given by the bond's duration multiplied by its risk premium. A corresponding interpretation is not directly possible in the two-factor model, but here it applies that the risk premiums determine each factor's contribution to the expected additional return on a zero-coupon bond above the short rate of interest.

Parameter estimates for the two models are shown in the Table.

PARAMETER ESTIMATES, MONTHLY DATA FROM 1987
Parameter	k₁	k₂	q₁	q₂	s₁	s₂	l₁	l₂
One factor	0.24	-	0.047	-	0.11	-	-0.14	-
Two factor	0.024	0.59	0.032	0.029	0.048	0.091	-0.069	-0.036
Note:Parameter estimates based on methods described in Box 10.3 .

¹Cox, J. C., Ingersoll, J. E. and Ross, S.a., 1985, A Theory of the Term Structure of Interest Rates, Econometrica, vol. 53, no. 2, pp. 385-407.
²For an introduction to arbitrage theory and pricing, see Musiela, M. and Rutkowski, M., 1998, Martingale Methods in Financial Modelling, Springer Verlag .
³ Multiple-factor affine term structure models are treated in full generality in Duffie, D. and Kan R., 1996, A Yield-Factor Model of Interest Rates. Mathematical Finance vol. 6, no. 4, pp. 379-406.

In the one-factor model, the explanatory variable is the very short-term interest rate. This assumption implies that the entire yield curve is determined by the short interest rate. For instance, the 10-year yield is always at a certain level if the short interest rate is held unchanged in the model. In addition, interest rates with different maturities are perfectly correlated.

In practice, many different types of yield curves are observed at the same level of the short interest rate, and moreover interest rates with different maturities are not perfectly correlated. These circumstances indicate that it may be necessary to expand the model in order to facilitate the recreation of empirical characteristics.

On the basis of a principal component analysis of the development in the Danish yield curve, it is found that the variation in the level of interest rates can be explained satisfactorily by two independent factors, viz. level and slope, cf. Box 10.2 . Consequently, the one-factor CIR model has been expanded into a model with two explanatory factors.

PRINCIPAL COMPONENT ANALYSIS OF THE DANISH YIELD CURVE	Box 10.2
Principal component analysis is a statistical method that can be used to explain the covariance in a multi-dimensional system on the basis of independent factors.¹ In connection with interest-rate modelling, the multi-dimensional system comprises zero-coupon yields for different maturities, i.e. a cross-section of the yield curve. If the individual variables in the system are closely correlated, a large part of the variance can be explained on the basis of relatively few independent factors. This is precisely the case for yields with different maturities. A principal component analysis of the development in the Danish yield curve from 1987 to 2005 shows that around 66 per cent of the variance can be explained on the basis of one factor, while two independent factors explain around 88 per cent of the variance. The impact of the factors on yields with different maturities can be deduced as "factor loadings", cf. the Chart.
FACTOR LOADINGS FOR PRINCIPAL COMPONENT ANALYSIS OF DANISH YIELD CURVE, 1987-2005

Note: The analysis was performed on the basis of changes in the level of interest rates. The impact on a shock to the factors indicates the influence from the factor on the month-on-month change in the level of interest rates.
Broadly speaking, the first principal component affects interest rates with different maturities in the same way and can therefore be interpreted as a parallel shock to the yield curve. This indicates that the component determines the level of the yield curve. The second principal component determines the slope of the yield curve, since long-term and short-term interest rates are not affected in the same way.
¹ See e.g. Golub, B.W. and Tilman. L.M., 2000, Risk Management Approaches for Fixed Income Markets. John Wiley & Sons. Inc.

Outside the class of affine term structure models, forward-rate models^[2] and market models^[3] are widely used. In the forward-rate models, the development in the entire forward curve, rather than the short interest rate alone, is specified. Market models are closely related to forward-rate models, but are based on observable, discretely compounded market interest rates rather than continuously compounded interest rates. Both types of model allow more flexible specification of the volatility structure, as well as exact recreation of the current yield curve in the model. These two properties are especially important when pricing interest-rate derivatives, and particularly the market models have become prevalent in the financial sector. For long-term simulation and risk management, however, forward-rate and market models are less appropriate as they are relatively less suitable for estimation based on historical data and difficult to implement.

10.3 HISTORICAL INTEREST-RATE DYNAMICS IN THE MODELS

Initially, the ability of the one-factor and two-factor models to describe the historical development in interest rates in Denmark over a period of 18 years is investigated. In both models, the parameters are estimated on the basis of monthly data for the period 1987-2005, cf. Box 10.3, and the development in the model-based interest rates is then compared with the actual interest rates in the period.

ESTIMATION OF CIR MODELS

Box 10.3

So far, the one-factor model has been estimated on the basis of a procedure developed by Overbeck and Rydén (1997)¹. The method results in closed-form estimators on the basis of a time series for the 3-month interest rate, which ensures simple implementation. The risk premium cannot be estimated solely on the basis of a time series for the short interest rate, but is subsequently found on the basis of the yield curve's average slope over the estimation period.

In the general two-factor model, it is not possible to identify the underlying factors with observable interest rates, which impedes the estimation. The model can e.g. be estimated using a Kalman filter² where the unobservable factors are derived simultaneously with the estimation of the parameters.

The Kalman filter is a general method for estimation of unobservable variables on the basis of observable data, where the relationship between the two data sets is known. The development in the observable data is summarised in a "measurement equation", while the development in the unobservable variables is summarised in a transition equation. The measurement equation is given by the affine relationship between factors and yields, which can be vectorised as follows:

The vector y contains a cross-section of the yield curve as zero-coupon rates with different maturities (3 months, 2 years, 5 years, 10 years and 15 years), the matrix A ( B ) contains the A functions ( B functions) described in Box 10.2, and the vector x contains the unobservable factors. The last term is a measurement error indicating that the model-based interest rates always deviate from the actual interest rates when the number of factors is lower than the number of zero-coupon rates. The transition equation is given by the conditional mean value and variance of the underlying factors. ³ Closed-form expressions of the conditional mean and variance are available since the transitional distribution in a CIR process is a "noncentral Chi ² distribution". The transition equation can be vectorised as follows:

The vector x contains the factors, while the matrixes C and D are selected so that the conditional mean value and variance of x are in accordance with the distribution of the underlying factors. The mean value and the variance in the transition equation are therefore consistent with the underlying interest-rate model.

The general principle behind the estimation is that the measurement equation can be used to calculate a theoretical yield curve based on an estimate of the unobservable factors and parameter values. If the theoretical yield curve deviates from the actual yield curve, the reason must be that either the parameter values or the factors deviate from the true values. The Kalman filter ensures optimum derivation of the estimates for the underlying factors within the class of linear estimators.

By minimising the difference between the actual and calculated yield curves over the estimation period, it is possible to derive both the parameter values and a time series for the factors. In practice, this difference is minimised using an approximative maximum likelihood procedure.

¹ Overbeck, L. and Rydén, T., 1997, Estimation in the Cox-Ingersoll-Ross Model, Econometric Theory, vol. 13, pp. 430-461.
² See e.g. Harvey , A.C., 1990, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press.
³ See e.g. Duan, J. C. and Simonato, J. G., 1995, Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter, Centre Interuniversitaire de Recherche en Analyse des Organisations.

Both models can explain the development at the short end of the yield curve, here represented by the 6-month interest rate, cf. Chart 10.3.1. Since the explanatory factor in the one-factor model is the 3-month interest rate, it is not surprising that the model-based 6-month interest rate matches the actual level of interest rates. The difference between the 3- and 6-month interest rates is generally very limited.

6-MONTH YIELD, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.3.1

Note: Continuously compounded zero-coupon yields. Source: Danmarks Nationalbank and own calculations.

As stated, the two-factor model is estimated on the basis of a cross-section of the yield curve, and not just a single interest rate. In the modelling of the historical development in the 6-month interest rate, this does not result in major differences relative to the one-factor model.

Considering interest rates with longer maturities, it becomes clear that the one-factor model overestimates or underestimates the level of interest rates over longer periods of time, cf. Chart 10.3.2. The two-factor model, on the other hand, can model the level and dynamics in both short-term and long-term interest rates.

10-YEAR YIELD, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.3.2

Note:Continuously compounded zero-coupon yields. Source:Danmarks Nationalbank and own calculations.

Identification of the descriptive factors
The primary reason for the one-factor model’s limited explanatory power at the long end of the yield curve is that the only explanatory variable in the model has been defined as the short interest rate. If the yield curve for a given level of the short interest rate deviates from the "normal scenario", it will be very difficult for the model to capture this.

The improved fit provided by the two-factor model can initially be ascribed to the increased number of parameters in the model. Further insight into the dynamics in the models is achieved by analysing the influence of the factors on the formation of interest rates. In the models, the impact of the factors on the individual interest rates is determined on the basis of factor loadings. Mathematically, factor loadings are calculated as

which corresponds to the change in the zero-coupon rate with term to maturity t for a small change in the ith factor, x_i. Unlike factor loadings from the principal component analysis presented above, which are solely based on the covariance of the data set, the factor loadings defined here depend on the interest-rate model selected.

Depicting factor loadings as a function of the maturity gives insight into the formation of interest rates in the models, cf. Chart 10.3.3.

FACTOR LOADINGS, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.3.3

In the one-factor model, the short interest rates increase one-to-one with positive shocks to the factor, and the effect diminishes with maturity. A positive shock thus always increases the general level of interest rates, with a diminishing effect over maturities. If the yield curve is normal, i.e. increasing in the remaining term to maturity, a flatter yield curve in the one-factor model will always coincide with an increase in interest rates, whereas a steeper yield curve will always coincide with falling interest rates. As previously stated, the long-term interest rates will thus not change in the one-factor model, unless the short interest rate also changes.

In the two-factor model, a shock to one of the factors affects all interest rates, with a marginally increasing impact over maturities. The second factor, however, diminishes substantially over maturities and thus primarily influences interest rates with shorter maturities. More specifically, factor 1 can be identified with level and factor 2 with slope, which was precisely the case in the model-independent principal component analysis. The fact that level and slope can be managed independently is the primary reason for the increased flexibility and higher explanatory power in the two-factor model.

The increased flexibility of the two-factor model is clearly seen around the currency crisis in 1992, when short-term interest rates rose considerably, while long-term interest rates were virtually unaffected. In the two-factor model, the level of the short interest rates is raised through an increase in the slope factor, cf. Chart 10.3.4. Long-term interest rates are also affected, but to a far lesser extent, and the impact is neutralised via a small decrease in the level factor, cf. Chart 10.3.5.

HISTORICAL DEVELOPMENT IN FACTORS, TWO-FACTOR MODEL, 1987-2005	Chart 10.3.4

YIELD CURVE, APRIL-OCTOBER 1992	Chart 10.3.5

Source: Danmarks Nationalbank and own calculations.

In summary, it is seen that the two-factor model enables the decoupling of short-term and long-term interest rates that is observed historically in several periods. Since the shape of the yield curve is significant to the trade-off between long-term and short-term borrowing, it is appropriate that the interest-rate model applied does not from the outset specify a clear correlation between changes in short-term and long-term interest rates.

Modelling of average interest rates, percentiles and correlations
A further examination of the models' ability to explain the empirical characteristics of the yield curve can be performed by comparing descriptive statistical values for the model-based and empirical yield curves.

In the one-factor model, the average level of interest rates is underestimated by 70-120 basis points for the period 1987-2005, while the volatility of the level of interest rates is overestimated for short maturities and underestimated for long maturities, cf. Chart 10.3.6. The volatility of the interest rates is represented by the 5th percentiles for the yield curve, indicating the upper and lower limits of a confidence band within which the yield curve will stay with a probability of 90 per cent.^[4] All other things being equal, higher volatility entails greater fluctuations in interest rates and thus a wider confidence band.

AVERAGE YIELD CURVE AND PERCENTILES, 1987-2005, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.3.6

Note: Continuously compounded zero-coupon yields. Source: Danmarks Nationalbank and own calculations.

The extra factor in the two-factor model provides for significantly better concordance between the model-based and empirical percentiles, even though the model consistently overestimates the 95th percentile (upper limit of the band) by around 60 basis points. The mean-yield curve in the two-factor model is identical to the empirically observed average-yield curve.

In the same way, the correlation structure in the data can be analysed. The central variable is taken to be the 6-month interest rate, and the correlation between this interest rate and interest rates with longer maturities is calculated on the basis of data and the two models, cf. Chart 10.3.7.

CORRELATION STRUCTURE FOR THE DANISH ZERO-COUPON CURVE,1987-2005	Chart 10.3.7

Note: Correlation calculated on the basis of the level of interest rates. Source: Danmarks Nationalbank and own calculations.

In the one-factor model, the correlation across maturities equals one per definition. The two-factor model permits decoupling of the individual interest rates, but for longer maturities the correlation is overestimated by around 7 per cent in relation to the empirical correlation structure. The increased flexibility of the two-factor model thus allows substantially better modelling of the covariation between yields with different maturities than the one-factor model.

10.4 ESTIMATION PERIODS AND SIMULATIONS

One of the criteria for selecting interest-rate models is that the yield curve fluctuates around a constant level of interest rates. This entails that the long-term mean in the model by and large corresponds to the observed average level of interest rates over the estimation period, cf. above.

The above analyses were performed on the basis of interest-rate data for the period 1987-2005. Over this period, the 10-year yield fell steadily from around 12 per cent to around 3.5 per cent. A shorter estimation period would therefore result in a lower long-term mean yield level in the models, cf. Chart 10.4.1.

LONG-TERM MEAN-YIELD CURVES, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.4.1

Note: Long-term is defined on the basis of the mean-yield levels for the factors.

In the same way, the standard deviation is reduced around the long-term mean yield level when a shorter estimation period is applied, cf. Chart 10.4.2.

STANDARD DEVIATION AROUND LONG-TERM YIELD CURVE, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.4.2

Note: Long-term is defined on the basis of the mean-yield levels for the factors.

The choice of estimation period may therefore affect the trade-off between costs and risk in CaR. For instance, a model estimated on the basis of a shorter estimation period may result in a lower long-term mean yield level and a narrower fluctuation band around this level. This naturally raises the question of which estimation period should be used in connection with model simulations.

Simulation over a 10-year horizon
In the CaR model, the level of interest rates is simulated 10 years ahead, and the models' properties must therefore be compared over this hori zon. It cannot be determined beforehand whether a 10-year analysis horizon would give the same results as the long-term levels outlined above.

In order to illustrate the significance of the chosen estimation period for CaR analyses performed in 2005, the yield curve is simulated 10 years ahead on the basis of the yield curve at end-2005, cf. Chart 10.4.3.

MEAN-YIELD CURVES ON A 10-YEAR HORIZON, ONE-FACTOR (LEFT) AND TWO-FACTOR (RIGHT)	Chart 10.4.3

Note: 90-per-cent confidence bands indicated with dashed lines.

In the one-factor model, there are substantial differences between the mean-yield curves and particularly the confidence bands based on the two estimation periods. In the two-factor model, the expected yield curves and the 90-per-cent confidence bands deviate by around 30-80 basis points, depending on the term to maturity.

I forbindelse med de senere års CaR-analyser er enfaktormodellen, ud over at være blevet estimeret på historisk data, blevet delvist kalibreret til Finansministeriets renteskøn. Ved eksplicit at lægge begrænsninger på det forventede fremtidige renteniveau i modellen reduceres afhængigheden af den valgte estimationsperiode.

In addition to estimating CaR on the basis of historical data, the one-factor model has in recent years been calibrated on the basis of the interest-rate forecasts from the Danish Ministry of Finance. By explicitly setting limits to the expected future level of interest rates in the model, the dependence on the selected estimation period is reduced.^[5]

The fairly small deviations in the two-factor model indicate that the results are relatively robust to the choice of estimation period if the analysis horizon is limited to 10 years. In view of the difference between the mean-yield curves and the standard deviations in the long term, this result may seem surprising. However, a further analysis of the parameter estimates reveals that the speed at which the level factor approaches its mean level, k 1 , is halved in the period 1987-2005 relative to 1997-2005. Since the level factor is very low in 2005, this means that the higher level of interest rates in the period 1987-2005 is only achieved for very long analysis horizons. Consequently, the choice of estimation period becomes less significant in the two-factor model.

Based on a principle of prudence, it has been decided to use the parameter set from the period 1987-2005 for analyses in the CaR model based on the two-factor model. Between the two estimation periods, this gives the highest mean yield level and the greatest volatility in interest rates.

10.5 CaR-ANALYSIS

In connection with the determination of the duration band for 2006, the Government Debt Management Office has, for the first time, applied the two-factor model to CaR simulations. In order to examine whether the risk profile for the selected strategy changes with the introduction of the new interest-rate model, the basic scenario with a duration of 3.0 years ± 0.5 year has been recalculated using the one-factor model, cf. Chart 10.5.1.

DISTRIBUTION OVER INTEREST COSTS, 2007 (LEFT) AND 2010 (RIGHT)	Chart 10.5.1

Note: The calculations are based on the future issuance strategy as described in Chapter 4. The year 2007 has been chosen since swaps concluded in 2006 are not fully reflected in the risk profile until 2007. Source: Own calculations in the CaR model.

The expected interest costs in 2007 are close to identical in the two models, while the cost distribution based on the two-factor model is wider than the distribution based on the one-factor model. This means that the probability of observing high interest costs is greater in the two-factor model. Consequently, the short-term risk, i.e. absolute CaR, is assessed to be higher in the two-factor model than in the one-factor model.

In the longer term, here represented by 2010, the distribution of costs based on the two-factor model is displaced leftwards in relation to the distribution based on the one-factor model. Both the expected interest costs and absolute CaR are reduced relative to simulations in the one-factor model.

[1] Linear plus a constant.
[2] See Heath, D., Jarrow, R. and Morton A., 1992, Bond Pricing and the Term Structure of Interest Rates: A New Methodology, Econometrica, vol. 60, no. 1, pp. 77-105.
[3] See Brace, A., Gatarek, D. and Musiela M., 1997, The Market Model of Interest Rate Dynamics, Mathematical Finance, vol. 7, no. 2, pp. 127-155.
[4] The confidence band is two-sided, which means that the highest and lowest 5 per cent of observations are excluded.
[5] This method is not directly comparable with the model implementation described here, which is solely based on historical data. The calibration procedure is described in Danish Government Borrowing and Debt 2003, Chapter 11.

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