# The Impact of Sovereign Yield Curve Differentials on Value-at-Risk Forecasts for Foreign Exchange Rates

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Theory and Methods

#### 3.1. Functional Principal Components

#### 3.2. Econometric Model

**Definition 1**(ARMA(1,1)FunX-logGARCH(1,1)FunX process).

- ${r}_{t}$ is the FX rate EURUSD
- ${z}_{t}^{\mathrm{US}}(\xb7)$ is the sovereign rate curve for the US
- ${z}_{t}^{\mathrm{EUR}}(\xb7)$ is the sovereign rate curve for EUR.
- ${x}_{t}(\xb7)={z}_{t}^{\mathrm{EUR}}(\xb7)-{z}_{t}^{\mathrm{US}}(\xb7)$

**Remark**

**1.**

- Estimation of the curved valued process ${x}_{t}$ via an orthonormal FPC expansion:$${\widehat{x}}_{t}=\widehat{\mu}+\sum _{k=1}^{K}{\widehat{\xi}}_{k,t}{\widehat{\gamma}}_{k},$$
- Estimation of the ARMA-FunX parameters using the scores ${\widehat{\xi}}_{k,t}$ for $k=1,\cdots ,K$ and $t=1,\cdots ,\left|\mathcal{T}\right|$ from Step 1 and the return data by Gaussian QML.
- Gaussian QML estimation of the GARCH-FunX parameters using the scores ${\widehat{\xi}}_{l,t}$ for $l=1,\cdots ,L$, $t=1,\cdots ,\left|\mathcal{T}\right|$ from Step 1 and the estimated errors from Step 2.

**Remark**

**2.**

- $|\alpha +\beta |<1$
- $|\gamma +\delta |<1$

- We force past volatility to influence present volatility positively, so we choose $\gamma >0.$ (see Francq et al. (2013).)
- Past errors should positively influence present volatility, leading to the choice $\delta >0.$

**armaxfilter**from the MFE toolbox by Kevin Sheppard. The conditional working distribution of the logarithmic GARCH(1,1)-FunX is given by:

## 4. Results

#### 4.1. Model Fit

#### 4.2. VaR Backtesting

- Firstly, we have the unconditional coverage (uc) test, which assumes the independence of the violations and tests the hypothesis that the empirical percentage of violations is equal to the expected p.
- The independence test (ind) checks for the independence of violations or detects clustering, respectively.
- Finally, there is the conditional coverage (cc) test that compares the empirical percentage of violations and the expected percentage as the unconditional coverage test does, but considers a possible dependence structure of the violations. We may treat it as a combination of the former two tests.
- The statistics ${\mathrm{LR}}_{\mathrm{uc}}$ and ${\mathrm{LR}}_{\mathrm{ind}}$ for the uc test and the ind test are ${\chi}^{2}$-distributed with one degree of freedom, whereas the ${\mathrm{LR}}_{\mathrm{cc}}$, the one for the cc test, is ${\chi}^{2}$-distributed with two degrees of freedom.

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | “Observed” yield curves are actually estimates obtained from observed bond prices. In the present paper, as in almost all of the literature (see for example Diebold and Li (2006)), we treat the yield curve data as if they had been observed directly. |

2 | To simplify notation, we write ${x}_{t}$ instead of ${x}_{t-1}$. |

3 | Traces of this assumption are scattered all over the Internet, but we restrain from quoting web pages. |

4 | Note that we will work with the full models in the following, as explained in Section 4.2. |

5 | The peculiarity of having a higher logL for the nested model in comparison to the full model of Table 3 arises due to using a two-step procedure instead of estimating jointly. |

6 | Although, since then, various alternative backtests have been established as, e.g., in Ziggel et al. (2014) or Wied et al. (2016). However, such new approaches would deviate too much from the core idea of the present paper, which is why we stick to the classical procedure of Christoffersen (1998). |

7 | We also take daily differences to ensure the stationarity of our yield curve process. |

**Figure 3.**Yield curves for EUR (

**left**) and the US (

**right**). Parallels to the y-axis mark the maturities from the dataset.

**Table 1.**Parameter estimates, bootstrapped means and bootstrapped 95%-confidence intervals using 10,000 simulated paths of length 2903.

ARMA-GARCH | ARMAFunX-GARCHFunX | 2y-ARMAX-GARCHX | |||||||
---|---|---|---|---|---|---|---|---|---|

Parameters | Estimate | Mean | Confidence Interval | Estimate | Mean | Confidence Interval | Estimate | Mean | Confidence Interval |

${\alpha}_{0}$ | $1.12\times {10}^{-5}$ | $3.34\times {10}^{-5}$ | $[-5.12\times {10}^{-4},6.06\times {10}^{-4}]$ | $1.21\times {10}^{-5}$ | $1.59\times {10}^{-5}$ | $[-1.71\times {10}^{-4},2.22\times {10}^{-4}]$ | $2.02\times {10}^{-5}$ | $2.08\times {10}^{-5}$ | $[-3.24\times {10}^{-4},3.65\times {10}^{-4}]$ |

$\alpha $ | $6.09\times {10}^{-1}$ | $1.51\times {10}^{-3}$ | $[-9.37\times {10}^{-1},9.41\times {10}^{-1}]$ | $5.82\times {10}^{-1}$ | $5.10\times {10}^{-1}$ | $[-1.99\times {10}^{-1},8.34\times {10}^{-1}]$ | $4.05\times {10}^{-1}$ | $4.03\times {10}^{-1}$ | $[3.09\times {10}^{-1},4.91\times {10}^{-1}]$ |

$\beta $ | $-6.09\times {10}^{-1}$ | $-1.90\times {10}^{-3}$ | $[-9.43\times {10}^{-1},9.38\times {10}^{-1}]$ | $-6.11\times {10}^{-1}$ | $-5.41\times {10}^{-1}$ | $[-8.57\times {10}^{-1},1.68\times {10}^{-1}]$ | $-4.23\times {10}^{-1}$ | $1.62\times {10}^{-3}$ | $[-9.67\times {10}^{-2},1.02\times {10}^{-1}]$ |

${b}_{1}$ | − | − | − | $-1.48\times {10}^{-3}$ | $-1.47\times {10}^{-3}$ | $[-2.33\times {10}^{-3},-5.94\times {10}^{-4}]$ | $-1.28\times {10}^{-2}$ | $-1.48\times {10}^{-3}$ | $[-8.93\times {10}^{-3},6.21\times {10}^{-3}]$ |

${b}_{2}$ | − | − | − | $-1.46\times {10}^{-4}$ | $-1.80\times {10}^{-4}$ | $[-1.91\times {10}^{-3},1.48\times {10}^{-3}]$ | − | − | − |

${b}_{3}$ | − | − | − | $6.80\times {10}^{-4}$ | $6.77\times {10}^{-4}$ | $[-1.91\times {10}^{-3},3.27\times {10}^{-3}]$ | − | − | − |

$\omega $ | $-2.10\times {10}^{-2}$ | $-3.21\times {10}^{-2}$ | $[-7.81\times {10}^{-2},-6.78\times {10}^{-4}]$ | $-2.25\times {10}^{-2}$ | $-5.12\times {10}^{-2}$ | $[-8.26\times {10}^{-2},-1.40\times {10}^{-3}]$ | $-4.48\times {10}^{-2}$ | $-6.91\times {10}^{-2}$ | $[-1.32\times {10}^{-1},-1.62\times {10}^{-2}]$ |

$\gamma $ | $2.15\times {10}^{-2}$ | $2.11\times {10}^{-2}$ | $[1.54\times {10}^{-2},2.69\times {10}^{-2}]$ | $2.05\times {10}^{-2}$ | $1.86\times {10}^{-2}$ | $[1.27\times {10}^{-2},2.46\times {10}^{-2}]$ | $2.22\times {10}^{-2}$ | $1.97\times {10}^{-2}$ | $[1.38\times {10}^{-2},2.60\times {10}^{-2}]$ |

$\delta $ | $9.73\times {10}^{-1}$ | $9.72\times {10}^{-1}$ | $[9.64\times {10}^{-1},9.80\times {10}^{-1}]$ | $9.74\times {10}^{-1}$ | $9.73\times {10}^{-1}$ | $[9.66\times {10}^{-1},9.83\times {10}^{-1}]$ | $9.70\times {10}^{-1}$ | $9.70\times {10}^{-1}$ | $[9.60\times {10}^{-1},9.81\times {10}^{-1}]$ |

${c}_{1}$ | − | − | − | $6.50\times {10}^{-2}$ | $6.61\times {10}^{-2}$ | $[2.85\times {10}^{-2},1.06\times {10}^{-1}]$ | $-2.06\times {10}^{-1}$ | $-2.05\times {10}^{-1}$ | $[-5.48\times {10}^{-1},1.22\times {10}^{-1}]$ |

${c}_{2}$ | − | − | − | $-8.93\times {10}^{-2}$ | $-9.12\times {10}^{-2}$ | $[-1.60\times {10}^{-1},-2.38\times {10}^{-2}]$ | − | − | − |

${c}_{3}$ | − | − | − | $1.07\times {10}^{-1}$ | $1.01\times {10}^{-1}$ | $[-5.41\times {10}^{-2},2.56\times {10}^{-1}]$ | − | − | − |

**Table 2.**Parameter estimates, bootstrapped means and bootstrapped 95%-confidence intervals using 10,000 simulated paths of length 2903 restricting the models from Table 1 to significant parameters only.

ARMA-GARCH | ARMAFunX-GARCHFunX | 2y-ARMAX-GARCHX | |||||||
---|---|---|---|---|---|---|---|---|---|

Parameters | Estimate | Mean | Confidence Interval | Estimate | Mean | Confidence Interval | Estimate | Mean | Confidence Interval |

${\alpha}_{0}$ | $3.18\times {10}^{-5}$ | $3.56\times {10}^{-5}$ | $[-4.44\times {10}^{-4},5.05\times {10}^{-4}]$ | $3.19\times {10}^{-5}$ | $3.27\times {10}^{-5}$ | $[-3.50\times {10}^{-4},4.06\times {10}^{-4}]$ | $3.11\times {10}^{-5}$ | $3.04\times {10}^{-5}$ | $[-3.97\times {10}^{-4},4.55\times {10}^{-4}]$ |

$\alpha $ | − | − | − | − | − | − | $5.98\times {10}^{-3}$ | $5.68\times {10}^{-3}$ | $[-3.51\times {10}^{-2},4.50\times {10}^{-2}]$ |

$\beta $ | − | − | − | − | − | − | − | − | − |

${b}_{1}$ | − | − | − | $-1.21\times {10}^{-3}$ | $-1.21\times {10}^{-3}$ | $[-2.06\times {10}^{-3},-3.63\times {10}^{-4}]$ | − | − | − |

${b}_{2}$ | − | − | − | − | − | − | − | − | − |

${b}_{3}$ | − | − | − | − | − | − | − | − | − |

$\omega $ | $-1.78\times {10}^{-2}$ | $-2.41\times {10}^{-2}$ | $[-7.00\times {10}^{-2},5.07\times {10}^{-3}]$ | $-2.21\times {10}^{-2}$ | $-3.04\times {10}^{-2}$ | $[-8.08\times {10}^{-2},1.33\times {10}^{-3}]$ | $-2.37\times {10}^{-2}$ | $-3.60\times {10}^{-2}$ | $[-8.24\times {10}^{-2},-2.31\times {10}^{-3}]$ |

$\gamma $ | $2.43\times {10}^{-2}$ | $2.57\times {10}^{-2}$ | $[1.93\times {10}^{-2},3.24\times {10}^{-2}]$ | $2.33\times {10}^{-2}$ | $2.37\times {10}^{-2}$ | $[1.73\times {10}^{-2},3.05\times {10}^{-2}]$ | $2.11\times {10}^{-2}$ | $2.06\times {10}^{-2}$ | $[1.51\times {10}^{-2},2.65\times {10}^{-2}]$ |

$\delta $ | $9.70\times {10}^{-1}$ | $9.68\times {10}^{-1}$ | $[9.58\times {10}^{-1},9.76\times {10}^{-1}]$ | $9.71\times {10}^{-1}$ | $9.70\times {10}^{-1}$ | $[9.60\times {10}^{-1},9.78\times {10}^{-1}]$ | $9.73\times {10}^{-1}$ | $9.72\times {10}^{-1}$ | $[9.64\times {10}^{-1},9.81\times {10}^{-1}]$ |

${c}_{1}$ | − | − | − | $5.47\times {10}^{-2}$ | $5.37\times {10}^{-2}$ | $[1.36\times {10}^{-2},9.31\times {10}^{-2}]$ | − | − | − |

${c}_{2}$ | − | − | − | $-1.16\times {10}^{-1}$ | $-1.17\times {10}^{-1}$ | $[-1.84\times {10}^{-1},-4.77\times {10}^{-2}]$ | − | − | − |

${c}_{3}$ | − | − | − | − | − | − | − | − | − |

**Table 3.**logL, AIC and BIC, corresponding to the full models from Table 1.

Model | logL | AIC | BIC |
---|---|---|---|

ARMA-GARCH | 10,826 | −21,639 | −21,604 |

ARMAFunX-GARCHFunX | 10,868 | −21,711 | −21,640 |

2y-ARMAX-GARCHX | 10,841 | −21,666 | −21,619 |

**Table 4.**logL, AIC and BIC, corresponding to the restricted models from Table 2.

Model | logL | AIC | BIC |
---|---|---|---|

ARMA-GARCH | 10,8325 | −21,656 | −21,632 |

ARMAFunX-GARCHFunX | 10,863 | −21,712 | −21,670 |

2y-ARMAX-GARCHX | 10,829 | −21,649 | −21,619 |

**Table 5.**VaR prediction performance with a window size of 500 days for ARMA-GARCH, ARMAFunX-GARCHFunX and 2y-ARMAX-GARCHX for $1-p\in \{99\%,97.5\%,95\%\}$. Bold numbers denote significant values at the $5\%$-level.

Model | p | % Viol. | ${\mathbf{LR}}_{\mathbf{uc}}$ | ${\mathbf{LR}}_{\mathbf{ind}}$ | ${\mathbf{LR}}_{\mathbf{cc}}$ |
---|---|---|---|---|---|

ARMA-GARCH | $1\%$ | $1.46\times {10}^{-02}$ | $\mathbf{4.42}\times {\mathbf{10}}^{\mathbf{00}}$ | $1.03\times {10}^{00}$ | $5.49\times {10}^{00}$ |

$2.5\%$ | $2.79\times {10}^{-02}$ | $7.84\times {10}^{-01}$ | $6.19\times {10}^{-01}$ | $1.46\times {10}^{00}$ | |

$5\%$ | $5.20\times {10}^{-02}$ | $1.99\times {10}^{-01}$ | $4.15\times {10}^{-02}$ | $3.48\times {10}^{-01}$ | |

ARMAFunX- | $1\%$ | $1.50\times {10}^{-02}$ | $\mathbf{5.21}\times {\mathbf{10}}^{\mathbf{00}}$ | $1.10\times {10}^{00}$ | $\mathbf{6.34}\times {\mathbf{10}}^{\mathbf{00}}$ |

GARCHFunX | $2.5\%$ | $2.83\times {10}^{-02}$ | $1.02\times {10}^{00}$ | $\mathbf{3.96}\times {\mathbf{10}}^{\mathbf{00}}$ | $5.04\times {10}^{00}$ |

$5\%$ | $5.07\times {10}^{-02}$ | $2.82\times {10}^{-02}$ | $\mathbf{7.21}\times {\mathbf{10}}^{\mathbf{00}}$ | $\mathbf{7.34}\times {\mathbf{10}}^{\mathbf{00}}$ | |

2y-ARMAX-GARCHX | $1\%$ | $1.58\times {10}^{-02}$ | $\mathbf{6.96}\times {\mathbf{10}}^{\mathbf{00}}$ | $1.22\times {10}^{00}$ | $\mathbf{8.21}\times {\mathbf{10}}^{\mathbf{00}}$ |

$2.5\%$ | $2.83\times {10}^{-02}$ | $1.02\times {10}^{00}$ | $5.66\times {10}^{-01}$ | $1.65\times {10}^{00}$ | |

$5\%$ | $5.20\times {10}^{-02}$ | $1.99\times {10}^{-01}$ | $1.23\times {10}^{00}$ | $1.53\times {10}^{00}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fink, H.; Fuest, A.; Port, H.
The Impact of Sovereign Yield Curve Differentials on Value-at-Risk Forecasts for Foreign Exchange Rates. *Risks* **2018**, *6*, 84.
https://doi.org/10.3390/risks6030084

**AMA Style**

Fink H, Fuest A, Port H.
The Impact of Sovereign Yield Curve Differentials on Value-at-Risk Forecasts for Foreign Exchange Rates. *Risks*. 2018; 6(3):84.
https://doi.org/10.3390/risks6030084

**Chicago/Turabian Style**

Fink, Holger, Andreas Fuest, and Henry Port.
2018. "The Impact of Sovereign Yield Curve Differentials on Value-at-Risk Forecasts for Foreign Exchange Rates" *Risks* 6, no. 3: 84.
https://doi.org/10.3390/risks6030084