# Simulation and evaluation of the distribution of interest rate risk

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## Abstract

We study methods to simulate term structures in order to measure interest rate risk more accurately. We use principal component analysis of term structure innovations to identify risk factors and we model their univariate distribution using GARCH-models with Student’s *t*-distributions in order to handle heteroscedasticity and fat tails. We find that the Student’s *t*-copula is most suitable to model co-dependence of these univariate risk factors. We aim to develop a model that provides low ex-ante risk measures, while having accurate representations of the ex-post realized risk. By utilizing a more accurate term structure estimation method, our proposed model is less sensitive to measurement noise compared to traditional models. We perform an out-of-sample test for the U.S. market between 2002 and 2017 by valuing a portfolio consisting of interest rate derivatives. We find that ex-ante Value at Risk measurements can be substantially reduced for all confidence levels above 95%, compared to the traditional models. We find that that the realized portfolio tail losses accurately conform to the ex-ante measurement for daily returns, while traditional methods overestimate, or in some cases even underestimate the risk ex-post. Due to noise inherent in the term structure measurements, we find that all models overestimate the risk for 10-day and quarterly returns, but that our proposed model provides the by far lowest Value at Risk measures.

## Keywords

Interest rate risk Principal component analysis Term structure Value at Risk## Mathematics Subject Classification

62H25 91G30## 1 Introduction

Interest rate risk is an important topic for both risk management and optimal portfolio management. The theory of measuring interest rate risk has evolved from dealing with sensitivity to risk factors, and immunization against risk, to a more modern view, where a better understanding of the actual risk factors is used in order to generate possible outcomes of risk factors. This paper takes the perspective of measuring interest rate risk by simulating future term structure scenarios, and ensuring that these properly describe the observed realizations. Interest rate risk has not always been properly connected to the term structure of interest rates. Bond duration, introduced by Macaulay (1938), was originally defined as bond sensitivity to discount factors. For the sake of simplicity, Macaulay (1938) chose to discount using yield to maturity, see Ingersoll et al. (1978) and Bierwag et al. (1983) for further details. A significant constraint when defining duration in terms of yield to maturity is that it cannot be used as a risk proxy unless the term structure is flat, as shown by Ingersoll et al. (1978). The interest rate risk was not properly linked to movements in the term structure until Fisher and Weil (1971) provided a proof of how to construct an immunized bond portfolio by choosing the duration of a portfolio equal to the investment horizon. This proof was carried out through a constant parallel shift of the term structure of forward rates. The view of interest rate risk as the sensitivity to certain perturbations in the term structure led to many suggestions of risk factors to be used in order to handle non-parallel risk, some of which we will describe in Sect. 4. However, modern risk concepts used in the Basel regulations are evolving towards measuring risk as outcomes of possible shocks to the risk factors. The way of measuring this risk is advancing, from Value at Risk (VaR), to the similar but coherent risk measure; expected shortfall (ES), see Chang et al. (2016) and the FRTB document from the Basel Committee on Banking Supervision (2014). The computation of ES usually requires simulation of potential outcomes of identified risk factors, although historical simulation might be an option. In this paper, we model risk through estimating the univariate distribution of risk factors and using copulas in order to model their co-dependence.

We choose to study the interest rate derivative market, instead of the bond market, where credit risk and liquidity cause additional problems for interest rate risk measurement. We use principal component analysis (PCA), which is a common, data driven, way of identifying risk factors in the interest rate market, see Litterman and Scheinkman (1991) and Topaloglou et al. (2002). We show that performing PCA on term structures estimated by bootstrapping and Cubic spline interpolation, results in problems of determining the systematic risk factors, which affects the results of the risk measurement. Instead, we wish to examine the possibility of improving risk measurement by utilizing an accurate method of measuring forward rate term structures, developed by Blomvall (2017). The well-established method proposed by Nelson and Siegel (1987) is also added for comparison. This method is commonly used in the bond market in order to mitigate noise problems, and dynamic extensions have been used in term structure forecasting by Diebold and Li (2006) and Chen and Li (2011). It has also been used for risk simulation, by Abdymomunov and Gerlach (2014) and Charpentier and Villa (2010).

In the light of the new, and the upcoming Basel regulations, we conclude that measuring interest rate risk through risk factor simulation is an important topic where literature is not very extensive. We also note that most papers evaluate different risk models using tail risk by studying the number of breaches in a VaR or ES setting, see e.g. Rebonato et al. (2005). While this is relevant for compliance of the regulations, it does not reflect the entire distribution, which is of importance for other applications, such as hedging or portfolio management. This paper thus focuses on the method of simulating term structure scenarios in order to accurately measure interest rate risk throughout the entire value distribution. We use VaR as a special case in order to illustrate differences in risk levels between the models. We find that three properties are of importance when measuring interest rate risk. First, there should be no systematic price errors caused by the term structure estimation method. Second, the distribution of the risk factors must be consistent with their realizations. Third, lower risk is preferable. This paper aims to develop models that satisfy all three properties, which proves to be a difficult task, especially for long horizons. This was also observed by Rebonato et al. (2005), where they had to rely on heuristic methods. In contrast, our aim is to use the extracted risk factors for risk measurement over different horizons. However, this paper does not focus on long term scenario generation over years or decades.

The paper is disposed as follows. In Sect. 2 we discuss how to estimate the term structures, using the data described in Sect. 3. We use PCA to extract systematic risk factors, and their properties are studied in Sect. 4. We use GARCH models together with copulas to model the evolution of these risk factors, which we describe in Sects. 5 and 6. The model evaluation is carried out by simulating term structure scenarios from different models, and using these scenarios to value portfolios consisting of interest rate derivatives. The values of the generated scenarios for each day are compared to portfolio values of the realized historical term structures. We describe this procedure together with the statistical tests used for comparison in Sect. 7. The results are presented in Sect. 8, followed by conclusions in Sect. 9.

## 2 Term structure estimation

Interest rate risk originates from movements in the term structure of interest rates. The term structure of interest rates is not directly observable, but has to be measured by solving an inverse problem based on observed market prices. However, these prices contain measurement noise, arising from market microstructure effects, as described by Laurini and Ohashi (2015). An important aspect of a well-posed inverse problem is that small changes in input data should result in small changes in output data. Hence, a term structure measurement method should not be sensitive to measurement noise in observed market prices.

Term structure measurement methods can be divided into exact methods and inexact methods. Exact methods will reprice the underlying financial instruments used to construct the curve exactly, and use some interpolation method to interpolate interest rates between the cash flow points belonging to those instruments. The fact that exact methods reprice all in-sample instruments exactly, makes them very sensitive to measurement noise in the market prices, as pointed out by Blomvall (2017). Inexact methods often assume that the term structure can be parametrized by some parametric function, and use the least square method to find these parameters. Inexact methods are less sensitive to measurement noise, but the limitations on the shape enforced by the parametric functions often lead to systematic price errors. Parametric methods include the parsimonious model by Nelson and Siegel (1987), and the extension by Svensson (1994). A way of extending interpolation methods to deal with noise is the recent application of kringing models by Cousin et al. (2016). This method can be adapted both for exact interpolation, or best fitting of noisy observations.

Blomvall (2017) presents a framework for measuring term structures though an optimization model that can be specialized to a convex formulation of the inverse problem. The framework is a generalization of previously described methods from literature, and is based on a trade-off between smoothness and price errors, which is accomplished by discretization and regularizations of the optimization problem. Blomvall and Ndengo (2013) compare this method to traditional methods, and show that it dominates all the compared traditional methods with respect to out-of-sample repricing errors. They also demonstrate a connection between in-sample price errors and interest rate risk, measured by the sum of variance in the curve. From this study, it is evident that low risk (variance) cannot be achieved in combination with low in-sample pricing errors. Each trading desk thus faces a decision of what level of risk, and what level of price errors should be acceptable according to laws and regulations.

The methodology in this paper will primarily be based around the generalized framework by Blomvall (2017), which we will use to measure daily discretized forward rate term structures directly. Two other term structure measurement methods will be used as reference methods in the comparison. The first one is the dynamic extension of Nelson and Siegel (1987), which was introduced by Diebold and Li (2006), and has been popular in literature of term structure forecasting. The other method is bootstrapping interest rates and interpolating spot rates using natural Cubic splines, which according to Hagan and West (2006) is commonly used by practitioners.

Portfolio volatility | \(\sigma \) (median deviations) | #Days as median | |
---|---|---|---|

Blomvall | 24.53 | 5.431 | 1955 |

Cubic spline | 29.09 | 12.94 | 1103 |

Nelson–Siegel | 24.73 | 13.82 | 730 |

The portfolio values can be found in the left panel of Fig. 1, and we note that all term structure estimation methods produce values that are seemingly consistent with each other most of the time. To further study the differences in portfolio values among these methods, without making any presumptions, we extract the median portfolio value out of the three methods for each day. One of the portfolios will thus equal the median value, and we assume that this value is the “correct” portfolio value. We then compare the value of each portfolio to this median to identify systematic price deviations, seen in the right panel of Fig. 1. By calculating the standard deviation of these series, we get a numeric estimate of the magnitude that each method deviates from the median value over the entire period. These values are presented in Table 1, together with portfolio volatilities and the number of days each method has attained the median value. We see that the Blomvall (2017) method has the lowest portfolio volatility and by far the lowest median deviation volatility, while it equals the median valuation most often. We interpret this as a sign of high consistency with market prices, which is one of our three desired properties when measuring risk. The Nelson and Siegel (1987) has low portfolio volatility but a high median deviation. This is due to its value drift away from the median value during long periods at a time, which we interpret this as a sign of systematic price errors due to the inflexible parametric nature of the model. In theory, the volatility of the portfolios can be divided into two unobservable components. The first, arising from movement in the systematic risk factors, and the second, arising from noise. The presence of both high portfolio values and high deviations for the Cubic spline method indicates that the additional portfolio volatility is caused by noise that is amplified by the interpolation method. These findings are also in line with the results by Blomvall and Ndengo (2013). We will later see how this noise affects the shape of the systematic risk factors.

## 3 Data

Interest rate swaps included in term structure measurement

Maturity (years) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Blomvall | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

Nelson–Siegel | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

Cubic spline | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

Forward rate agreements included in term structure measurement

FRA [M \(\times \) M] | 1 \(\times \) 4 | 2 \(\times \) 5 | 3 \(\times \) 6 | 4 \(\times \) 7 | 5 \(\times \) 8 | 6 \(\times \) 9 | 7 \(\times \) 10 | 8 \(\times \) 11 | 9 \(\times \) 12 | 12 \(\times \) 15 | 15 \(\times \) 18 | 18 \(\times \) 21 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Blomvall | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

Nelson–Siegel | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

Cubic spline | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

## 4 Systematic risk factors

*n*assets, where the price of an asset \(P_i({\varDelta }\xi )\) depends on the state of

*m*random risk factors \({\varDelta }\xi \in {\mathbb {R}} ^{m \times 1}\). A second order Taylor approximation of the asset value around the current risk levels gives us

A different, data-driven approach for identifying systematic risk factors is principal component analysis (PCA). This method can be used to approximate the covariance matrix of a dataset by finding a set of orthogonal components that maximize the variance. PCA was first applied to term structures of interest rates by Litterman and Scheinkman (1991). This study was later followed by Knez et al. (1994) and Bliss (1997), among others. An alternative approach for feature extraction is independent component analysis (ICA). While PCA aims to explain the second order information, i.e. the variance, ICA tries to maximize independence in higher orders as well. As a result, the factors obtained from ICA will be independent; while factors obtained from PCA only will be uncorrelated, unless the data follows a Gaussian distribution. In contrast, ICA assumes a non-normal distribution in order to be able to separate the independent sources since it maximizes non-Gaussianity of the components (Chen et al. 2007). A weakness of ICA is that the factors cannot be ordered in terms of importance, while PCA factors are ordered in terms of explained variance. In fact, the variance of the ICA factors cannot be determined in a unique way (Moraux and Villa 2003). Dealing with risk is directly related to dealing with variance, and since PCA factors are orthogonal, they also provide a natural interpretation (Molgedey and Galic 2001). While the less restrictive assumptions of ICA are appealing for non-Gaussian financial data, more recent applications of ICA to interest rates, especially of higher dimensionality, are missing in literature. Hence, PCA will be used to extract risk factors in this paper.

*T*-forward measure, implying that changes in forward rates should be linked to the flow of information. Additionally, the risk factors of the central HJM model (Heath et al. 1992), are derived in terms of forward rates and forward rate volatility, which is why it would be ideal to use forward rates. However, performing a PCA on forward rate innovations requires forward rate data of high quality, in order to be able to extract realistic risk factors. Laurini and Ohashi (2015) provide an excellent review of the topic, and highlight the problem with measurement errors in forward rates due to market microstructure effects. They present a proposition that the effective number of risk factors should be the same, regardless of the choice of underlying interest type. The fact that this cannot be observed in literature provides strong evidence for the prevalence of measurement noise in the data. Instead of directly addressing this problem of term structure measurement, Laurini and Ohashi (2015) address the problem by introducing a technique for estimating the covariance matrix in the presence of measurement noise, called the long-run covariance matrix.

*C*denotes the \(n \times n\) covariance matrix of forward rates, we compute the vector of term structure volatilities for each discrete maturity point as the element-wise square root of the diagonal elements of

*C*

*n*eigenvectors, forming the matrix \(E_{f_n}\), and a vector containing their corresponding eigenvalues, \(\lambda _n\), we can compute the forward rate volatility explained by this approximation as

*A*,

*B*, which is the inverse of

*A*, and transforms spot rates to forward rates. Using

*B*, we can compute the forward rate volatility implied by the spot rate risk factors as

The result of these computations is illustrated in Fig. 4, and here it is evident that the term structures estimated from the Cubic spline method contain a lot of noise. The risk factors will, to a high degree, model this noise instead of the systematic risk in the market. This can be seen by comparing its forward rate volatility, \(\sigma _f\), to the volatility of the Blomvall (2017) model. Studying the spot rate volatilities, in the left panel of Fig. 4, one gets the impression that the amount of volatility for the longer maturities is the same for both models. Performing the analysis on spot rates does not get rid of the noise, but rather hides it, as seen by the implied forward rate volatility \(\sigma _{s_a}\). The volatility in forward rates is almost the same as when PCA is performed on the forward rate covariance matrix. This is due to the integrating property when converting forward rates to spot rates. For this reason, we study forward rate agreements rather than interest rate swaps, because of their price sensitivity to forward rates.

## 5 Volatility and scenario modeling

Volatility modeling of the equity market returns has been studied thoroughly, where several empirical properties have been documented, see Gavrishchaka and Banerjee (2006) and references therein. Equity returns have a very short-range autocorrelation, within time spans as small as a few minutes. Equity return volatility on the other hand, is found to be clustered with a long-range memory up to several months, while being mean-reverting. The distribution of returns is fat-tailed and leptokurtic at time frames up to a few days. Volatilities are also found to be negatively correlated with returns, called the leverage effect.

Tests for stylized facts from the equity market applied to the six most significant principal component of Blomvall (2017) data

Normality | Skewness | Heteroscedasticity | ||||
---|---|---|---|---|---|---|

Statistic |
| Statistic |
| Statistic |
| |

Shift | 0.978 | 6.83e\(-\)29 | 4.80 | 1.57e\(-\)06 | 76.5 | 1.73e\(-\)33 |

Twist | 0.887 | 0 | \(-\) 6.44 | 1.20e\(-\)10 | 518 | 4.54e\(-\)207 |

Butterfly | 0.847 | 0 | \(-\) 5.22 | 1.83e\(-\)07 | 216 | 5.80e\(-\)91 |

4th | 0.698 | 0 | 10.6 | 1.89e\(-\)26 | 322 | 4.22e\(-\)133 |

5th | 0.799 | 0 | \(-\) 2.18 | 0.0289 | 586 | 8.23e\(-\)232 |

6th | 0.836 | 0 | \(-\) 5.74 | 9.59e\(-\)09 | 366 | 2.80e\(-\)150 |

Deguillaume et al. (2013) empirically show that the interest rate volatility is dependent on the interest rate levels in certain regimes. For low interest rates, below 2%, and for high interest rates, above 6%, interest rate volatility increases proportionally to the interest rate level. For the regime of 2–6% however, the interest rate volatility does not seem to depend on the interest rate level. Although these are interesting findings, we choose to model interest rate volatility separated from interest rate levels. If the interest rate volatility is dependent on the current regime, a volatility model based on actual principal component changes will capture this behavior for these different regimes indirectly. This approach works regardless of the regime, even under negative interest rates. To model the observed stylized facts of the principal component series, we have implemented several models of the GARCH class, primarily based on the arch package for Python written by Sheppard (2017).

*p*,

*o*,

*q*) volatility process is given by

*t*-distribution, which is scaled by the volatility from (Eq. 13). Our findings suggest that the drift term is of very limited importance compared to the volatility, and that the constants become insignificant, as we will see in Table 6 later. Autoregressive processes are commonly used for the evolution of risk factors in the term structure forecasting literature. Define the AR(

*p*)-model as

*i*. Autoregressive models of order \(p=1\) are popular, since an autocorrelation is usually observed at lag one for the different principal components. However, there is no inherent mechanic in the fixed income market to justify this behavior. This is rather an effect of term structures adapting to measurement noise and returning back to the previous shape the following day. As a result of this, the use of autoregressive models to model general movements in principal components can be questioned. We believe that this autocorrelation effect should be prevented by the term structure measurement method directly, by utilizing information from adjacent days. This is, however, not yet integrated in the Blomvall (2017) framework, nor in any other term structure estimation method. We find that negative autocorrelations are present in the principal components, shown in Table 5. Even though this indicates that an AR(1) would improve our results, we have not been able to show any improvements by including autoregressive processes, which is why we do not present any AR results. We note that principal components based on Blomvall (2017) term structures have less autocorrelation than Cubic splines, especially in the most significant component, which is another indication that Cubic splines contain more noise.

Autocorrelation for the ten most significant principal components for different methods, together with the proportion of the total variance explained by each eigenvector

Shift | Twist | PC3 | PC4 | PC5 | PC6 | PC7 | PC8 | PC9 | PC10 | |
---|---|---|---|---|---|---|---|---|---|---|

Blomvall, | 0.02 | \(-\) 0.32 | \(-\) 0.22 | \(-\) 0.29 | \(-\) 0.34 | \(-\) 0.40 | \(-\) 0.37 | \(-\) 0.46 | \(-\) 0.40 | \(-\) 0.45 |

PC, forward | 64.7% | 17.8% | 10.2% | 3.3% | 2.6% | 1.1% | 0.2% | 0.0% | 0.0% | 0.0% |

Cubic spline, | \(-\) 0.40 | \(-\) 0.12 | \(-\) 0.25 | \(-\) 0.42 | \(-\) 0.41 | \(-\) 0.48 | \(-\) 0.33 | \(-\) 0.37 | \(-\) 0.40 | \(-\) 0.26 |

PC forward | 28.0% | 19.0% | 14.8% | 7.9% | 7.5% | 6.6% | 5.6% | 3.3% | 2.7% | 1.5% |

Cubic spline, | 0.07 | \(-\) 0.45 | \(-\) 0.06 | \(-\) 0.30 | \(-\) 0.36 | \(-\) 0.43 | \(-\) 0.48 | \(-\) 0.36 | \(-\) 0.37 | \(-\) 0.42 |

PC spot | 70.4% | 19.6% | 5.1% | 1.4% | 0.8% | 0.6% | 0.5% | 0.4% | 0.3% | 0.3% |

### 5.1 Simulation of term structures

*t*, and let \(E_n\) denote the matrix of the

*n*most significant eigenvectors from the PCA. If \(y_t\) represents the simulated shocks for these

*n*risk factors between the time point

*t*and \(t+1\), the forward rate term structure dynamics are given by

### 5.2 Model fitting and calibration

*p*,

*o*,

*q*), the constant and the order of lag of the autoregressive model, as well as the choice of residual distribution among normal and Student’s

*t*. Different risk factors may have different model orders and use different distributions. The parameters attained for in-sample dataset can be found in Table 6, and we can observe that the same GARCH(1, 1)-model using a Student’s

*t*distribution has been selected across all models. We also note that the drift term \(\mu \) is insignificant for all the displayed parameters. The risk factors are estimated once, using 6 years of in-sample data between 1996 and 2001. The model parameters and copulas are recalibrated every 2 years, using new data but keeping the data window of 6 years fixed.

Mean value and model parameters for the GARCH-process including degrees of freedom for the Student’s *t*-distribution together with their corresponding *p* values for the in-sample period

\(\mu \) | \(\omega \) | \(\alpha \) | \(\beta \) | \(\nu \) | ||
---|---|---|---|---|---|---|

Blomvall, PC forward | Shift | \(-\) 0.00094 (0.27) | 2.6e−05 (0.069) | 0.041 (0.00048) | 0.94 (0) | 6.7 (2.1e−10) |

Twist | \(-\) 4.7e−05 | 4e−05 | 0.24 | 0.66 | 5.1 | |

(0.89) | (0.031) | (0.0019) | (7.6e−09) | (7e−11) | ||

Butterfly | 0.00018 | 2.5e−06 | 0.069 | 0.93 | 2.9 | |

(0.36) | (0.18) | (0.10) | (1e−88) | (3.7e−31) | ||

Cubic spline, PC forward | Shift | \(-\) 0.00046 (0.31) | 0.00032 (3.8e−05) | 0.73 (2e−06) | 0.22 (0.0095) | 3.0 (3.6e−24) |

Twist | \(-\) 0.001 | 6.9e−05 | 0.055 | 0.9 | 5.5 | |

(0.23) | (0.064) | (0.012) | (1.7e−95) | (6.3e−12) | ||

Butterfly | 0.00027 | 0.00015 | 0.31 | 0.51 | 4.6 | |

(0.58) | (0.35) | (0.18) | (0.22) | (2.4e−10) | ||

Cubic spline, PC spot | Shift | \(-\) 0.00067 (0.36) | 1.8e−05 (0.15) | 0.045 (0.0038) | 0.94 (0) | 5.7 (1.7e−13) |

Twist | \(-\) 7.9e−05 | 5.3e−05 | 0.57 | 0.077 | 2.5 | |

(0.53) | (0.0096) | (0.0086) | (0.12) | (3.1e−22) | ||

Butterfly | 0.00012 | 3.4e−07 | 0.069 | 0.93 | 5.8 | |

(0.35) | (0.087) | (0.00019) | (0) | (1.8e−11) | ||

Nelson–Siegel | \(\beta _0\) | \(-\) 1.9e−05 | 3.5e−07 | 0.54 | 0.46 | 3.1 |

(0.36) | (0.0017) | (6.3e−05) | (4e−05) | (5.9e−24) | ||

\(\beta _1\) | 1.7e−05 | 1.8e−08 | 0.076 | 0.92 | 4.0 | |

(0.42) | (0.19) | (0.061) | (2.7e−102) | (5.1e−22) | ||

\(\beta _2\) | 1.8e−05 | 6.2e−08 | 0.050 | 0.95 | 3.0 | |

(0.71) | (0.42) | (0.027) | (2e−218) | (2.6e−44) |

*t*-distributed residuals have been selected over normally distributed ones, which also holds for less significant risk factors. These findings also confirm the stylized facts presented earlier.

## 6 Modeling co-dependence via copulas

We have thus far described univariate models for the principal component time series. By construction, these series are uncorrelated for the in-sample data. If the input data were also normally distributed, they would be independent. As argued previously, the principal component series are not normally distributed, and hence not independent. In addition, the out-of-sample data used in the test will still be correlated to some degree. According to Kaut and Wallace (2011), copulas are able to capture such dependencies in non-normally distributed principal components, and we therefore apply copulas to deal with these dependencies.

*t*copulas are not suitable for modeling this complex dependence structure. Instead, the class of Archimedean couplas, including Frank and Gumbal copulas, could capture this structure. Modeling two distinct interest rate levels differs from modeling the movement of several risk factors. We thus had very limited success using Archimedean copulas, where they worked at all for this high dimensional data, using the R-package by Hofert et al. (2017). The copulas are calibrated using principal component changes normalized by the volatility from our volatility models, since they are to be used to generate dependent random numbers used in the volatility process. The inverse cumulative distribution function for the chosen distribution of each risk factor is then used together with the copula, in order to generate dependent scenarios. An in-sample estimation of different copulas, using the GARCH models described in Sect. 5, is performed in order to find the best copula for each dataset. The likelihood value of each copula is compared within each dataset, using the Bayesian Information Criterion (BIC) to account for the number of parameters for each copula. The results are displayed in Table 7, and since the Student’s

*t*-copula provides the best information for all datasets, we choose it to model the co-dependence across all models.

BIC values for in-sample fitting of different copulas to different datasets

Blomvall, PC forward | Cubic spline, PC forward | Cubic spline, PC spot | Nelson–Siegel | |
---|---|---|---|---|

Normal Copula | 456.1 | 566.3 | 768.4 | 679.6 |

t Copula | 1075 | 1675 | 1638 | 708.2 |

## 7 Back-testing and evaluation procedure

We now wish to evaluate which method is most consistent with the realizations that can be observed in the market during our out-of-sample period. Since we cannot directly observe term structures in the market, we compare it to the ones estimated using the different methods. The drawback of this approach is that it will make the realization, and thus this evaluation procedure, dependent on the term structure method used. Furthermore, it is problematic to evaluate time series of distributions of high dimensional term structures, against high dimensional realizations of term structures. Since term structures are primarily used to price interest rate instruments, we use portfolio values in order to reduce the dimensionality of the evaluation problem. By pricing a randomized portfolio using our generated term structure scenarios and comparing them to the historical realizations, we reduce the problem to evaluating distributions of portfolio values, compared to realized values. Besides evaluating the fit of the distribution, we will also study portfolio volatilities and VaR to get a grasp of the cost of using the different methods for measuring interest rate risk.

We use forward rate agreements (FRA) from the U.S. market to make up the portfolio as described earlier. Our portfolio tries to mimic an unhedged book by containing a randomized, normally distributed cash flow for each day. To be able to study interest rate risk over the entire maturity span up to 10 years, we use the market quoted FRA contracts up to 21 \(\times \) 24 months, and in addition, we synthesize contracts for longer maturities spanning up to 117 \(\times \) 120 months spaced with 3 months in between. We do this in order to capture the risk in the entire maturity span studied in Figs. 2, 3, 4, 5, 6, and 8. These figures highlight the problems to correctly estimate forward rate risk in Cubic spline term structures. We have set up trades to the quoted market rates when available, or the Blomvall (2017) forward rate, making the initial value of the synthetic contract close to zero. We keep only one single cashflow per day in order to minimize the amount of cashflows to handle when simulating over longer time periods, as described below. This is achieved by trading a normally distributed random amount in all maturities for all trading days during 3 months prior to the out-of-sample starting date. During the back-test, the longest contract is traded in order to maintain this structure. This results in a portfolio containing approximately 2500 instruments, which are constantly replaced as they roll out of maturity.

The back-testing system is built in C++, using a custom built portfolio management system for QuantLib (Ametrano and Ballabio 2000), which is scripted in Python. Since the portfolio for a given day is constant across all scenarios, the portfolio valuation procedure can be simplified. By gathering all cash flows from the instruments making up the portfolio into one composite instrument, and pricing this single instrument across all scenarios, a dramatic speed up is achieved. This makes it possible to simulate 2000 scenarios within approximately 2.5 s per model, using a standard desktop computer.

*t*is the time between \(t_1\) and \(t_2\), measured as a year fraction by the day-counting convention used in the market. When fixing rates differs among scenarios, portfolio composition will not only differ between days, but also between scenarios. This makes the valuation procedure substantially more computationally heavy.

### 7.1 Statistical test

*t*. Our goal should be to maximize the log-likelihood function of all

*n*independent observations \(x_t, t=1,\ldots ,n\),

*s*simulated portfolio scenario values of time

*t*, follow the target distribution

*h*, in each \(x_t^\theta \). The kernel density estimator \({\hat{f}}(x)\) is given by the normed sum of these bells, or more formally

*x*(Silverman 1986).

## 8 Simulation results

*t*-copula.

### 8.1 Back-test using a 1-day forecasting horizon

Results for the 1-day-horizon, where \({\bar{\delta }}\) is the difference in likelihood value between the Blomvall (2017) model and the compared model

Compared Model | T | \({\bar{\delta }}\) | |
---|---|---|---|

Blomvall, PC spot | 0.463 | \(-\) 0.00939 | 0.496 |

Cubic spline, PC forward | nan | \(-\)inf | 0 |

Cubic spline, PC spot | 68.3 | \(-\)1.15 | 1.42e−16 |

Const. Cubic spline, PC spot | nan | \(-\)inf | 0 |

Nelson–Siegel | 0.799 | 0.0160 | 0.371 |

### 8.2 Back-test using a 10-day forecasting horizon

Results for the 10-day simulation horizon, where \({\bar{\delta }}\) is the difference in likelihood value between the reference model and the Blomvall (2017) model

Compared model | T | \({\bar{\delta }}\) | |
---|---|---|---|

Cubic spline, PC spot | 25.3 | \(-\) 0.766 | 4.95e−07 |

Nelson–Siegel | 14.4 | \(-\) 0.0907 | 0.000148 |

Studying the fit of our models in Fig. 14 we start to notice a slight overestimation of the risk in the Blomvall (2017) model and the Nelson and Siegel (1987) model, especially for positive returns. The Cubic spline model displays a more severe overestimation of the risk. Turning once again to the statistical test, we ran a separate test where simulations were spaced equal to the simulation horizon of 10 days to ensure that there is no dependence between observations, as required by the test. The results can be seen in Table 9, where we find that the model by Blomvall (2017) produces significantly higher likelihood values than both the Cubic spline model and the dynamic Nelson and Siegel (1987) model.

Studying the VaR for this commonly used simulation horizon, seen in Fig. 15, we note that the tail risk is overestimated by all models at this simulation horizon. Due to some unrealistic scenario values of the Nelson and Siegel (1987), we have use median values instead of average values of VaR. These measures are substantially lower for the Blomvall (2017) model, especially for the lowest percentile. This is a result of the model generating excessive noise in addition to the actual risk, something that would unnecessarily increase capital requirements under Basel regulations.

### 8.3 Long-term back-testing

### 8.4 Towards accurate measurements of interest rate risk

To accurately model interest rate risk, it is necessary to ensure that the model does not capture noise. It is evident from the right panel of Fig. 4 that, measuring interest rate curves using cubic splines, introduces noise that may be hidden in the PCA of spot rates. Even though adding more realistic modeling of the principal components for the Cubic spline models significantly improves the results (Figs. 11, 12), it is obvious that improved modeling will not overcome the fact that noise is being modeled. As a direct consequence, the Cubic spline models manage to overestimate the inherent interest rate risk ex-ante, while it simultaneously underestimates the tail risk ex-post for the daily horizon (Fig. 12). The inherent property that Cubic spline models cannot deal with measurement noise that creates unrealistic interest rates for the 10-day and quarterly simulation horizons. The consequence of this is very high VaR measures (Figs. 15, 18).

The Nelson and Siegel (1987) model creates less accurate prices (Fig. 1), but simultaneously improve the risk measurement both ex-ante and ex-post compared to the Cubic spline models (Fig. 12). However, for longer horizons, also the Nelson and Siegel (1987) model tend to increase the risk significantly (Figs. 15, 18), indicating that this model also models noise to a large extent. Relying on a formulation of the inverse problem that is more resilient to measurement noise, as the one suggested by Blomvall (2017), allows for decreased VaR and expected shortfall measures. This results in lower capital requirements for the FRTB, while having control over the realized risk.

## 9 Conclusions

This paper concerns modeling of the entire distribution of interest rate risk by modeling its systematic risk factors. We find GARCH-models useful for modeling of the volatility of principal component series. We also find that the Student’s *t*-copula is the most suitable copula for modeling co-dependence between these principal component series. Based on these findings, we construct models that are independent of the underlying term structure measurement methods and work in all regimes, including negative interest rates. After establishing the interconnection between the measurement of interest rate risk, and the measurement of term structures, we identify three important properties. First, there should be no systematic price errors caused by the term structure measurement method. Second, the distribution of the risk factors must be consistent with their realizations. Third, lower risk is preferable. We investigate these three properties and seek a model which satisfies them all. Scenario volatility of different models are studied in order to compare their level of risk. The ability to generate scenarios that are consistent with realized outcomes, is studied by valuing a portfolio containing interest rate derivatives, and compare scenario values with realized outcomes. In order to reveal any weakness of the model, we choose forward rate agreements to make up the portfolio. These instruments are sensitive to changes to the forward rates, which is an inherent problem for traditional interest rate measurement methods. We conclude that our proposed model, based on Blomvall (2017) term structures, accurately reflects 1-day-ahead the risk in in the portfolio, while obtaining the lowest volatility and VaR levels. For the 10-day horizon, Cubic spline models and the Nelson and Siegel (1987) model exhibit further problems caused by the volatility arising from noise being scaled by the increased time horizon, leading to unnecessarily increases in capital requirements under Basel regulations. For long-term simulations over a quarter, the Blomvall (2017) produces the by far best results. We conclude that the term structures by the Blomvall (2017) model contain the least amount of noise, but that the method could be improved in order to further reduce the amount of noise. We believe that this would further reduce the measured risk, as well as improve the fit of the risk distribution for longer simulation horizons.

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