# Disentangling Trend Risk and Basis Risk with Functional Time Series

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## Abstract

**:**

## 1. Introduction

## 2. Mortality Models for Multiple Populations

#### 2.1. Augmented Common Factor Model

- ACF0 Model:In the ACF0 model, we assume the three mortality indices ${K}_{t}$, ${k}_{t}^{\left(1\right)}$, and ${k}_{t}^{\left(2\right)}$ are independent. Under this assumption, ${K}_{t}$, ${k}_{t}^{\left(1\right)}$, and ${k}_{t}^{\left(2\right)}$ can be modeled by the following expression:$$\left\{\begin{array}{ccc}\hfill \Delta {K}_{t}& =& {\mu}^{c}+{\eta}_{t}^{c}\hfill \\ \hfill {k}_{t}^{\left(1\right)}& =& {\mu}^{\left(1\right)}+{\varphi}^{\left(1\right)}{k}_{t-1}^{\left(1\right)}+{\eta}_{t}^{\left(1\right)}\hfill \\ \hfill {k}_{t}^{\left(2\right)}& =& {\mu}^{\left(2\right)}+{\varphi}^{\left(2\right)}{k}_{t-1}^{\left(2\right)}+{\eta}_{t}^{\left(2\right)}\hfill \end{array}\right.,$$$$\left(\begin{array}{c}{\eta}_{t}^{c}\\ {\eta}_{t}^{\left(1\right)}\\ {\eta}_{t}^{\left(2\right)}\end{array}\right)\sim MVN(\overrightarrow{0},Q),\phantom{\rule{1.em}{0ex}}Q=\left(\begin{array}{ccc}{Q}_{11}& 0& 0\\ 0& {Q}_{22}& 0\\ 0& 0& {Q}_{33}\end{array}\right).$$There are eight parameters in this model, which include ${\mu}^{c},{\mu}^{\left(1\right)},{\mu}^{\left(2\right)},{\varphi}^{\left(1\right)},{\varphi}^{\left(2\right)},{Q}_{11},{Q}_{22}$, and ${Q}_{33}$.
- ACF1 Model:The ACF1 model extends the ACF0 model to incorporate the correlations between different sequences. As a result, the Q matrix is no longer diagonal and we have$$\left(\begin{array}{c}{\eta}_{t}^{c}\\ {\eta}_{t}^{\left(1\right)}\\ {\eta}_{t}^{\left(2\right)}\end{array}\right)\sim MVN(\overrightarrow{0},Q),\phantom{\rule{1.em}{0ex}}Q=\left(\begin{array}{ccc}{Q}_{11}& {Q}_{12}& {Q}_{13}\\ {Q}_{12}& {Q}_{22}& {Q}_{23}\\ {Q}_{13}& {Q}_{23}& {Q}_{33}\end{array}\right).$$Comparing to the ACF0 model, the number of unknown parameters increase from eight to eleven due to the addition of off-diagonal covariance parameters ${Q}_{12}$, ${Q}_{13}$, and ${Q}_{23}$.
- ACF2 Model:In the ACF2 model, we consider a more complicated autoregressive structure, the vector autoregressive (VAR) structure, for ${K}_{t}$, ${k}_{t}^{\left(1\right)}$, and ${k}_{t}^{\left(2\right)}$ sequences. We consider the simplest lag-1 vector autoregression model, the VAR(1) model, which can be expressed as follow:$$\left\{\begin{array}{ccc}\hfill \Delta {K}_{t}& =& {\mu}^{c}+{\varphi}_{11}\Delta {K}_{t-1}+{\varphi}_{12}{k}_{t-1}^{\left(1\right)}+{\varphi}_{13}{k}_{t-1}^{\left(2\right)}+{\eta}_{t}^{c}\hfill \\ \hfill {k}_{t}^{\left(1\right)}& =& {\mu}^{\left(1\right)}+{\varphi}_{21}\Delta {K}_{t-1}+{\varphi}_{22}{k}_{t-1}^{\left(1\right)}+{\varphi}_{23}{k}_{t-1}^{\left(2\right)}+{\eta}_{t}^{\left(1\right)}\hfill \\ \hfill {k}_{t}^{\left(2\right)}& =& {\mu}^{\left(2\right)}+{\varphi}_{31}\Delta {K}_{t-1}+{\varphi}_{32}{k}_{t-1}^{\left(1\right)}+{\varphi}_{33}{k}_{t-1}^{\left(2\right)}+{\eta}_{t}^{\left(2\right)}\hfill \end{array}\right.,$$$$\left(\begin{array}{c}{\eta}_{t}^{c}\\ {\eta}_{t}^{\left(1\right)}\\ {\eta}_{t}^{\left(2\right)}\end{array}\right)\sim MVN(\overrightarrow{0},Q),\phantom{\rule{1.em}{0ex}}Q=\left(\begin{array}{ccc}{Q}_{11}& {Q}_{12}& {Q}_{13}\\ {Q}_{12}& {Q}_{22}& {Q}_{23}\\ {Q}_{13}& {Q}_{23}& {Q}_{33}\end{array}\right).$$In this model, a fully vectorized autoregressive structure is used. Even in the simplest case, the VAR(1) structure, the number of unknown parameters increase to eighteen. For a more complicated autoregressive VAR(p) ($p>1$) structure, the number of unknown parameters would be $(18+9\times (p-1\left)\right)$.

#### 2.2. Product–Ratio Model

- The product model is defined by$$log\left({g}_{x,t}\right)={\mu}_{x}^{p}+\sum _{i=1}^{m}{B}_{i,x}{K}_{i,t}+{e}_{x,t},$$In the product model, the quantity log(${g}_{x,t}$) captures the average of (log-) mortality rates under different populations. The product of ${B}_{i,x}$ and ${K}_{i,t}$ is then used to reflect the evolution of (average) mortality at age x.
- The ratio model is defined by$$log\left({r}_{x,t}\right)={\mu}_{x}^{r}+\sum _{i=1}^{n}{b}_{i,x}{k}_{i,t}+{w}_{x,t}$$In the ratio model, the quantity log(${r}_{x,t}$) captures the deviation between the two populations. The product of ${b}_{i,x}$ and ${k}_{i,t}$ is then used to reflect the evolution of such deviation at age x.

## 3. Numerical Analysis

#### 3.1. Data

#### 3.2. Parameter Estimation

- The ACF ModelThe ACF model uses the common factor to capture the general mortality trend shared by all populations, and the population-specific factor to capture any deviation from the common trend. The estimated values of ${K}_{t}$ are therefore representing the co-movement of the mortality dynamic shared by the two countries, and ${B}_{x}$ representing the common age sensitivity to such co-movement.For the U.S. and Canada, although they are not identical, these two countries do share a great amount of similarities in economy, culture, living style, and healthcare system. As a result, the estimated levels ${a}_{x}^{\left(i\right)}$ as well as the population-specific factors ${b}_{x}^{\left(i\right)}$ and ${k}_{t}^{\left(i\right)}$ from the two countries are quite similar to each other, causing a vague distinction between trend risk and basis risk.
- The PR ModelThe PR model, on the other hand, decomposes trend risk and population basis risk by construction, meaning that these two risks are separated in the first place when the product model and ratio model are defined. As a result, the product model would be solely focusing on trend risk while the ratio model is solely focusing on basis risk, respectively, leading to a clear distinction between the two.
- –
- In the product model, the estimated ${\mu}_{p}$ represents the mean level of the general trend. It carries very similar shape to both of ${a}_{x}^{\left(1\right)}$ and ${a}_{x}^{\left(2\right)}$ in the ACF model. The estimated ${K}_{t}$ carries similar downward pattern as that in the ACF model, representing general mortality improvement overtime. For the estimated ${B}_{x}$, it presents similar peak (around age 70) and trough (around age 30) to those shown in the ACF model, with different magnitudes.
- –
- In the ratio model, the estimated ${\mu}_{r}$ describes the difference in the mean level between the two populations. It can be easily observed in the pattern of ${\mu}_{r}$ that the biggest difference between the two populations lies in the working age males (from 25 to 65). The gap closes out quickly toward older ages.Besides the difference in the mean level, the ratio model further decomposes the difference between the U.S. and Canada into two components, ${b}_{x}^{\left(1\right)}{k}_{t}^{\left(1\right)}$ and ${b}_{x}^{\left(2\right)}{k}_{t}^{\left(2\right)}$. The upward pattern of ${k}_{t}^{\left(1\right)}$ suggests that the differences between the two countries are getting bigger over time, and the pattern of ${b}_{x}^{\left(1\right)}$ shows such a gap is more significant in young adults around age 20. The remaining residuals not explained through the first component are then captured by the second component ${b}_{x}^{\left(2\right)}{k}_{t}^{\left(2\right)}$.

#### 3.3. The Time-Series Process

#### 3.3.1. Sample Cross-Correlation Matrix

#### 3.3.2. AICs and Likelihood Ratio Test

#### 3.3.3. Term Structure of Correlation

- ACF0 vs. ACF1Let us first focus on ACF0 model and ACF1 model. For these models, the only difference is the specification of Q matrix. To be more specific, In ACF0, the off-diagonal elements of Q matrix are zeros, while in ACF1, those elements are all non-zeros. The term structures of correlation under ACF0 and ACF1 have similar shape. However, a 10% increase is observed when the model does incorporate the off-diagonal covariance elements.
- ACF2 vs. ACF1 or ACF0The mortality indices in the ACF2 model follows a VAR(1) model where additional parameters are used to capture the cross-correlation among different period effects. Comparing to ACF1 and ACF0, a dramatic change of shape (the black dotted line) is observed for the ACF2, reflecting those additional interactions.
- PR0 vs. PR1 vs. PR2Within the PR framework, the resulting term structures of correlation from the three models carry two features. The first feature is related to the magnitudes of the lines, with the values being greater for more complex models. The second feature is related to the shape of the lines, where all three lines yield very similar patterns. This finding is consistent with the observations we made in Section 3.3.1 and Section 3.3.2. It also justifies the argument made by Hyndman et al. (2013) that the latent factors are uncorrelated in the PR model.
- PR vs. ACFSince the two models have different model specification, the shape of the curve may not be the same. For the PR model, the three variants yield similar term structure of correlations, while for the ACF model, the ACF0 variant and ACF1 variant have omitted the cross-correlation between different latent states, and thereby not adequately fitting the data and leading to the worst term structure of correlations. When the full VAR structure is used in the ACF2 variant, its term structure of correlation would be similar to those in the PR framework.

## 4. Hedge Performance

#### 4.1. Basic Set Up

#### 4.2. Illustration of Hedge Performance

- The liability is a life annuity sold to individuals aged 70 (${x}_{0}=70$) 20 years from now (${T}_{d}=20$). The liability is linked to population 1, the U.S. males. The maximum age is set to be $w=100$.
- The q-forward has a reference age of 65 (${x}_{h}=70$) and will matures 20 years from now (${T}_{h}=20$). The q-forward is linked to population 2, the Canadian males.
- The interest rate is assumed to be 1% per annum.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Partial Derivatives of Liability and Hedging Instrument

#### Appendix A.1. The ACF Model:

#### Appendix A.2. The PR Model:

## Appendix B. Simulation Procedure

- Simulate M sample paths of ${K}_{{t}_{0}+T},{k}_{1,{t}_{0}+T}$ and ${k}_{2,{t}_{0}+T}$. We denote these simulated paths as ${\omega}_{j}$, for $j=1,2,\dots ,M$. Accordingly, for each simulated path ${\omega}_{j}$, the values of the corresponding state variables are ${K}_{{t}_{0}+T}\left({\omega}_{j}\right),{k}_{1,{t}_{0}+T}\left({\omega}_{j}\right)$ and ${k}_{2,{t}_{0}+T}\left({\omega}_{j}\right)$.
- For each simulated path ${\omega}_{j}$, $j=1,2,\dots ,M$, we project the life table at time ${t}_{0}+T$, based on the simulated ${K}_{{t}_{0}+T}\left({\omega}_{j}\right),{k}_{1,{t}_{0}+T}\left({\omega}_{j}\right)$ and ${k}_{2,{t}_{0}+T}\left({\omega}_{j}\right)$. The methodology of how we project the life table is summarized as follows:
- 1.
- Based on ${K}_{{t}_{0}+T}\left({\omega}_{j}\right),{k}_{1,{t}_{0}+T}\left({\omega}_{j}\right)$ and ${k}_{2,{t}_{0}+T}\left({\omega}_{j}\right)$, we first compute the central death rate of an individual aged x, ${m}_{x,{t}_{0}+T}\left({\omega}_{j}\right)$:
- –
- ACF Model$$\begin{array}{c}\hfill log\left({m}_{x,{t}_{0}+T}^{\left(1\right)}\left({\omega}_{j}\right)\right))={a}_{x}^{\left(1\right)}+{B}_{x}{K}_{{t}_{0}+T}\left({\omega}_{j}\right))+{b}_{x}^{\left(1\right)}{k}_{{t}_{0}+T}^{\left(1\right)}\left({\omega}_{j}\right))\\ \hfill log\left({m}_{x,{t}_{0}+T}^{\left(2\right)}\left({\omega}_{j}\right)\right))={a}_{x}^{\left(2\right)}+{B}_{x}{K}_{{t}_{0}+T}\left({\omega}_{j}\right))+{b}_{x}^{\left(2\right)}{k}_{{t}_{0}+T}^{\left(2\right)}\left({\omega}_{j}\right))\end{array}$$
- –
- PR Model$$\begin{array}{ccc}\hfill log\left({m}_{x,{t}_{0}+T}^{\left(1\right)}\left({\omega}_{j}\right)\right)& =& ({\mu}_{x}^{p}+{\mu}_{x}^{r})+{B}_{x}{K}_{{t}_{0}+T}\left({\omega}_{j}\right)+{b}_{1,x}{k}_{1,{t}_{0}+T}\left({\omega}_{j}\right)+{b}_{2,x}{k}_{2,{t}_{0}+T}\left({\omega}_{j}\right),\hfill \\ \hfill log\left({m}_{x,{t}_{0}+T}^{\left(2\right)}\left({\omega}_{j}\right)\right)& =& ({\mu}_{x}^{p}-{\mu}_{x}^{r})+{B}_{x}{K}_{{t}_{0}+T}\left({\omega}_{j}\right)-{b}_{1,x}{k}_{1,{t}_{0}+T}\left({\omega}_{j}\right)-{b}_{2,x}{k}_{2,{t}_{0}+T}\left({\omega}_{j}\right).\hfill \end{array}$$

- 2.
- Then we compute the death probability ${q}_{x,{t}_{0}+T}\left({\omega}_{j}\right)$, based on the value of ${m}_{x,{t}_{0}+T}\left({\omega}_{j}\right)$. Throughout this study, we assume constant force of mortality. Therefore, ${q}_{x,{t}_{0}+T}\left({\omega}_{j}\right)$ can be computed through the following equation:$${q}_{x,{t}_{0}+T}\left({\omega}_{j}\right)=1-exp(-{m}_{x,{t}_{0}+T}\left({\omega}_{j}\right)),$$$${p}_{x,{t}_{0}+T}\left({\omega}_{j}\right)=1-{q}_{x,{t}_{0}+T}\left({\omega}_{j}\right).$$
- 3.
- We use ${}_{s}{\mathrm{P}}_{x,{t}_{0}+T}\left({\omega}_{j}\right)$ to denote the probability that an individual aged x at time ${t}_{0}+T$ survives to age $x+s$, for a particular path ${\omega}_{j}$. The survival probability is computed as$${}_{s}{\mathrm{P}}_{x,{t}_{0}+T}\left({\omega}_{j}\right)=\prod _{k=1}^{s}{p}_{x+k-1,{t}_{0}+T}\left({\omega}_{j}\right).$$

- For each projected life table, we calculate the value of the life annuity ${V}_{L}\left({\omega}_{j}\right)$, which is$${V}_{L}\left({\omega}_{j}\right)=\sum _{s=1}^{w-{x}_{0}}{e}^{-r\xb7s}{}_{s}{\mathrm{P}}_{{x}_{0},{t}_{0}+{T}_{d}}^{\left({P}_{L}\right)}\left({\omega}_{j}\right),$$

## Notes

1 | Data source: Human Mortality Database (HMD). |

2 | For an expanded data set, the same MLE method can be used. |

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**Figure 1.**Parameter estimates of the Augmented Common Factor model fitted to the U.S. and Canadian males with ages from 20 to 100 and years from 1950 to 2019.

**Figure 2.**Parameter estimates of the product–ratio model fitted to the U.S. and Canadian males with ages from 20 to 100 and years from 1950 to 2019.

**Figure 3.**The term structure of correlation between ${q}_{x,t}^{\left(1\right)}$ and ${q}_{x,t}^{\left(2\right)}$, $x=65$.

**Table 1.**Sample cross-correlation matrix of the estimated sequence of ($\Delta {K}_{t},{k}_{t}^{\left(1\right)},{k}_{t}^{\left(2\right)}$) under the ACF model and the PR model. The symbol “.” means insignificance (at $\alpha =1\%$ significance level), “+” means positive significance.

ACF Model | P/R Model | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\Delta {\mathit{K}}_{\mathit{t}}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{1}\right)}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{2}\right)}$ | $\Delta {\mathit{K}}_{\mathit{t}}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{1}\right)}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{2}\right)}$ | $\Delta {\mathit{K}}_{\mathit{t}}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{1}\right)}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{2}\right)}$ | $\Delta {\mathit{K}}_{\mathit{t}}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{1}\right)}$ | ${\mathit{k}}_{\mathit{t}}^{\left(\mathbf{2}\right)}$ |

Lag 0 | |||||||||||

1.0000 | 0.1797 | 0.2752 | + | . | . | 1.0000 | –0.0405 | 0.3563 | + | . | . |

0.1797 | 1.0000 | 0.9426 | . | + | + | –0.0405 | 1.0000 | 0.0180 | . | + | . |

0.2752 | 0.9426 | 1.0000 | . | + | + | 0.3563 | 0.0180 | 1.0000 | . | . | + |

Lag 1 | |||||||||||

0.0668 | 0.2995 | 0.3395 | . | . | . | 0.2952 | –0.0118 | 0.2574 | . | . | . |

0.1238 | 0.9612 | 0.8934 | . | + | + | –0.0062 | 0.9484 | –0.0333 | . | + | . |

0.2453 | 0.9377 | 0.9688 | . | + | + | 0.3023 | 0.0684 | 0.8464 | . | . | + |

Lag 2 | |||||||||||

0.3110 | 0.3811 | 0.3538 | . | . | . | 0.2163 | –0.0193 | 0.3509 | . | . | . |

0.0895 | 0.8990 | 0.8371 | . | + | + | –0.0487 | 0.9092 | –0.0683 | . | + | . |

0.2414 | 0.9178 | 0.9348 | . | + | + | 0.2803 | 0.1131 | 0.8269 | . | . | + |

Lag 3 | |||||||||||

0.1718 | 0.3854 | 0.3461 | . | . | . | 0.0640 | –0.0482 | 0.3242 | . | . | . |

0.0817 | 0.8311 | 0.7815 | . | + | + | –0.0924 | 0.8635 | –0.1339 | . | + | . |

0.2139 | 0.8959 | 0.9005 | . | + | + | 0.2535 | 0.1753 | 0.7405 | . | . | + |

Lag 4 | |||||||||||

0.0070 | 0.4049 | 0.3551 | . | + | . | 0.1223 | –0.0878 | 0.3519 | . | . | . |

0.0786 | 0.7685 | 0.7310 | . | + | + | –0.1899 | 0.8075 | –0.1704 | . | + | . |

0.1759 | 0.8736 | 0.8643 | . | + | + | 0.1557 | 0.2055 | 0.6686 | . | . | + |

Lag 5 | |||||||||||

0.1093 | 0.3565 | 0.2957 | . | . | . | 0.0949 | –0.0788 | 0.3239 | . | . | . |

0.0840 | 0.7078 | 0.6851 | . | + | + | –0.2548 | 0.7553 | –0.2155 | . | + | . |

0.1583 | 0.8397 | 0.8243 | . | + | + | 0.1600 | 0.2280 | 0.6172 | . | . | + |

ACF0 | ACF1 | ACF2 | PR0 | PR1 | PR2 | |
---|---|---|---|---|---|---|

$\mathcal{N}$ | 8 | 11 | 18 | 8 | 11 | 18 |

$\mathcal{L}$ | –338.5748 | –297.5167 | –285.9478 | –144.2575 | –141.3853 | –135.7845 |

AIC | 693.1496 | 617.0334 | 607.8957 | 304.5149 | 304.7706 | 307.5689 |

${\mathit{H}}_{0}:\mathit{\theta}={\mathit{\theta}}_{0}$ | ${\mathit{H}}_{1}:\mathit{\theta}={\mathit{\theta}}_{1}$ | |
---|---|---|

Restricted Model | Unrestricted Model | p-Value |

ACF0 | ACF1 | 0.0000 |

ACF0 | ACF2 | 0.0000 |

ACF1 | ACF2 | 0.0016 |

PR0 | PR1 | 0.1247 |

PR0 | PR2 | 0.0756 |

PR1 | PR2 | 0.1301 |

**Table 4.**The value of hedging effectiveness (HE) in the longevity hedge under the ACF framework (left panel) and the PR framework (right panel), using different combinations of simulation model and hedge calibration model.

Simulation | Simulation | ||||||||
---|---|---|---|---|---|---|---|---|---|

Calibration | Model | ACF0 | ACF1 | ACF2 | Calibration | Model | PR0 | PR1 | PR2 |

ACF0 | 52.6756% | 61.7846% | 77.8786% | PR0 | 68.7645% | 68.2410% | 73.0943% | ||

ACF1 | 52.6756% | 61.7846% | 77.8786% | PR1 | 68.7645% | 68.2410% | 73.0943% | ||

ACF2 | 47.1669% | 53.5626% | 70.2547% | PR2 | 68.6478% | 68.1193% | 72.9632% | ||

Max diff. = 30.7117% | Max diff. = 4.9750% |

**Table 5.**The notional amounts of q-forward in the longevity hedge under the ACF framework (left panel) and the PR framework (right panel), using different combinations of simulation model and hedge calibration model.

Calibration | Simulation | Calibration | Simulation | ||||||

Model | ACF0 | ACF1 | ACF2 | Model | PR0 | PR1 | PR2 | ||

ACF0 | 182.9765 | 182.9765 | 182.9765 | PR0 | 146.9599 | 146.9599 | 146.9599 | ||

ACF1 | 182.9765 | 182.9765 | 182.9765 | PR1 | 146.9599 | 146.9599 | 146.9599 | ||

ACF2 | 141.4138 | 141.4138 | 141.4138 | PR2 | 146.5070 | 146.5070 | 146.5070 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Li, J.S.-H.
Disentangling Trend Risk and Basis Risk with Functional Time Series. *Risks* **2023**, *11*, 208.
https://doi.org/10.3390/risks11120208

**AMA Style**

Liu Y, Li JS-H.
Disentangling Trend Risk and Basis Risk with Functional Time Series. *Risks*. 2023; 11(12):208.
https://doi.org/10.3390/risks11120208

**Chicago/Turabian Style**

Liu, Yanxin, and Johnny Siu-Hang Li.
2023. "Disentangling Trend Risk and Basis Risk with Functional Time Series" *Risks* 11, no. 12: 208.
https://doi.org/10.3390/risks11120208