Application of Bayesian penalized spline regression for internal modeling in life insurance

Quang Dien Duong

doi:10.1007/s13385-019-00192-3

European Actuarial Journal

pp 1–41 | Cite as

Application of Bayesian penalized spline regression for internal modeling in life insurance

Authors
Authors and affiliations

Quang Dien Duong

Original Research Paper

First Online: 12 February 2019

Abstract

Solvency 2 requires insurance companies to compute a Best-Estimate of their Liabilities (BEL) as well as a Solvency Capital Requirement (SCR). Life insurance companies being in the business of selling participating contracts with financial guarantees have to rely on a Monte-Carlo approach to appropriately value their BEL, which is the source of a first Monte-Carlo statistical error. In addition, several insurance companies rely on a (partial) internal model to derive their SCR. In this context, insurance companies rely again on a Monte-Carlo approach to value their SCR, which is the source of a second Monte-Carlo statistical error. These later computations require evaluating the BEL several thousand times, which is not possible in practice, since one single Monte-Carlo BEL evaluation can be a computational burden. The BEL has therefore to be approximated by an analytic proxy function, which introduces an additional source of numerical approximation error. In this paper, we show how these three sources of error (statistical and numerical) are intrinsically related to one another. We show that to obtain the best possible SCR accuracy, the computing power invested in assessing the Monte-Carlo SCR should be directly related to that invested in computing the Monte-Carlo BEL. Interestingly, and to achieve these results, we introduce a novel proxy method, which is highly practical, modular, smooth and naturally relates the approximation errors to the Monte-Carlo statistical errors. Furthermore, our approach allows insurance companies to naturally and transparently start reporting confidence levels on their prudential reporting, which is not disclosed so far by insurance companies and would be a relevant information within solvency disclosures for the industry.

Keywords

Life Insurance Mathematics Bayesian penalized spline regression Solvency Capital Requirement

This is a preview of subscription content, to check access.

Notes

Acknowledgements

This work has been supported by the French National Agency for Research and Technology (ANRT) CIFRE 614/2015 and by PwC’s Risk and Value Measurement Services. The author gratefully acknowledges Prof. Agathe Guilloux, Prof. Olivier Lopez, in particular Dr. Jean Baptiste Monnier and Didier Riche for their time and discussions who helped develop this methodology. These collaborations led to a successful implementation of the ALM simulator. Special thanks also go to two anonymous referees for their insightful comments and valuable suggestions that significantly improve the paper.

Appendix A: Bayesian P-spline regression and Bayesian asymptotic confidence interval

In this section, a brief description of smoothing and penalized splines (or P-splines) will be presented. Some interesting additional information about smoothing splines and P-splines can be found for example in the work of Reinsch [34], Duchon [8], Green and Silverman [13], Hastie and Tibshirani [15], Eubank [10], Eilers and Marx [9] and Ruppert and Carroll [35].

Consider the regression model $Y_i=m(X_i)+\epsilon _i, i=1, \ldots , n$ where $\epsilon _i$ are the independent random variables with mean 0 and variance $\sigma ^2$. We assume that the design points $X_i \in [a,b]$ with $a,b < \infty$. As pointed out in Appendix D, for any regular functions m(x), we can always find a best spline approximation ${\tilde{m}}(x)$ of m(x) to minimize $\left\| m - {\tilde{m}}\right\| _{\infty }$. The error in approximating m(x) by ${\tilde{m}}(x)$ is usually negligible compared to the estimator error, thus in practice we estimate ${\tilde{m}}(x)$ instead of m(x). In the following, we denote by $B(x)= \{B_1(x),\ldots ,B_N (x)\}^T \in {\mathbb {R}}^N, N \le n$ a spline basis.

Smoothing splines

The key idea of this regression method is to approximate the target function m(x) by a natural cubic spline ${\tilde{m}}(x)$ with knots at the distinct $X_i$ values. The basis functions for natural cubic splines are

$$\begin{aligned} \{1, x, d_1(x) - d_{n-1}(x), \ldots , d_{n-2}(x) - d_{n-1}(x)\}, \end{aligned}$$

where $d_k(x) = \frac{(x-X_k)^3_{+} - (x-X_n)^3_{+}}{X_n-X_k}$ for $k=1,\ldots , n-1$.

A smoothing spline estimator arises as the minimizer of the penalized sum of squares,

$$\begin{aligned} \sum _{i=1}^n \left( Y_i - {\tilde{m}}(X_i)\right) ^2 + \lambda \int _a^b \left( {\tilde{m}}^{''}(x) \right) ^2 dx, \end{aligned}$$

(18)

for $\lambda > 0$.

The integral term in the previous expression is a roughness penalty with the smoothing parameter $\lambda$. We write the vector ${\tilde{m}} = ({\tilde{m}}(X_1 ),\ldots ,{\tilde{m}}(X_n))^T \in {\mathbb {R}}^n$. The theorem 2.1 in Green and Silverman [13] shows that there exists a $n \times n$ dimensional matrix ${\mathbf {K}}$ of rank $n-2$ such that the penalty term $\int _a^b ({\tilde{m}}^{''}(x))^2 dx$ can be written as ${\tilde{m}}^T {\mathbf {K}} {\tilde{m}}$. The smoothing spline estimate at the design points ${\hat{m}} = ({\hat{m}}(X_1),\ldots ,{\hat{m}}(X_n))^T \in {\mathbb {R}}^n$ is thus explicitly given by ${\hat{m}}= ({\mathbf {I}}+\lambda {\mathbf {K}} )^{-1} Y$ where $Y = (Y_1,\ldots ,Y_n)^T \in {\mathbb {R}}^n$. This regression method is however less practical when the number of design points n becomes large since it uses n knots.

Regression penalized splines or P-splines

The idea of penalized spline smoothing with basic functions can track back to O’Sullivan [31], see also Eilers and Marx [9] and Ruppert and Carroll [35] for additional information. Two common basis used in practical, up to our knowledges, are the B-spline basis (Eilers and Marx [9], De Boor [6]) and the truncated power basis (Ruppert and Carroll [35]). The B-spline basis is preferable for computation because of its better numerical properties. For formulation and theoretical study the truncated power basis is preponderant because of its simplicity.

The penalized spline model specifies that the estimate ${\tilde{m}}(x)$ is expressed as $B(x)^T \beta$ for some N-dimensional space. Hence, the estimation of ${\tilde{m}}(x)$ is equivalent to that of $\beta$. Let ${\mathbf {D}}$ be a fixed, symmetric, positive semidefinite $N \times N$ matrix, which is equivalent to the matrix ${\mathbf {K}}$ in the smoothing spline model. The penalized spline estimator ${\hat{\beta }}$ arises as the minimizer of

$$\begin{aligned} \sum _{i=1}^n \left( Y_i - B(X_i)\beta \right) ^2 + \lambda \beta ^T {\mathbf {D}} \beta . \end{aligned}$$

(19)

Define ${\mathbf {B}}$ the $n \times N$ matrix whose i-th row equals $B(X_i)^T$. The penalized spline estimator is thus written as ${\hat{\beta }}= ({\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}})^{-1} {\mathbf {B}}^T Y$. From this, we obtain:

$$\begin{aligned} E\left( {\hat{\beta }}\right) = \left( {\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}} \right) ^{-1} {\mathbf {B}}^T {\mathbf {B}} \beta , \end{aligned}$$

(20)

and the covariance of the estimate

$$\begin{aligned} {\mathbf {V}}_{{\hat{\beta }}} = \sigma ^2 \left( {\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}}\right) ^{-1} {\mathbf {B}}^T {\mathbf {B}} \left( {\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}}\right) ^{-1}. \end{aligned}$$

(21)

It follows that ${\hat{\beta }} \sim {\mathcal {N}} (E({\hat{\beta }}), {\mathbf {V}}_{{\hat{\beta }}})$. It is clear that $E ({\hat{\beta }}) \ne \beta$ except for $\beta = 0$ which leads to the difficulty in using this result for calculating confidence intervals.

In this paper, we choose the truncated p-polynomial basis as mentioned previously, i.e. $B(x)= (1,x,x^2,\ldots ,x^p,(x-\kappa _1)_{+}^p, \ldots ,(x-\kappa _K)_{+}^p )^T$ for $p \ge 1$ and ${\mathbf {D}}$ is the diagonal matrix $diag({\mathbf {0}}_{p+1},{\mathbf {1}}_K)$, indicating that only the spline coefficients are penalized. One can easily verify that $\int _a^b [{\tilde{m}}^{(p+1)}(x)]^2 dx =(p!)^2 \sum _{j=1}^K \beta _{j+p}^2$ by using the fact that the derivative of an indicator function is a Dirac delta function. Therefore, the Eq. (19) can be rewritten as

$$\begin{aligned} \sum _{i=1}^n \left( Y_i - {\tilde{m}}(x)\right) ^2 + \lambda ' \int _a^b \left[ {\tilde{m}}^{(p+1)}(x)\right] ^2 dx, \end{aligned}$$

(22)

which is a generalization of Eq. (18).

Bayesian analysis for penalized splines regression

The Bayesian analysis for penalized spline regression is an alternative approach to calculate the confidence intervals. The initial works for this approach is mainly due to Wahba [42] and Silverman [36]. The main idea behind this method is to partition the coefficient vector $\beta$ into the coefficients of the monomial basis functions of the truncated power functions by letting $\beta = (\beta ^T_1, \beta ^2_2)^T$ where $\beta _1 \in {\mathbb {R}}^{1+p}$ has an improper uniform prior density and $\beta _2 \in {\mathbb {R}}^{K}$ has a proper prior equal to $(\lambda /\sigma ^2)^{K/2}\exp {(-(\lambda /2\sigma ^2)\beta ^T_2 \beta _2 )}$.

Following the idea of Wood (section 4.8.1 in [44]), we obtain the Bayesian posterior covariance matrix for the parameter $\beta$:

$$\begin{aligned} {\mathbf {V}}_{\beta } = \left( {\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}}\right) ^{-1}\sigma ^2, \end{aligned}$$

(23)

and its corresponding posterior distribution:

$$\begin{aligned} \beta _{\mid Y} \sim {\mathcal {N}}\left( {\hat{\beta }}, {\mathbf {V}}_{\beta } \right) , \end{aligned}$$

(24)

where ${\hat{\beta }}= ({\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}})^{-1} {\mathbf {B}}^T Y$. The penalized least squares estimator is the mean of the posterior distribution of $\beta$. This posterior on $\beta$ induces a posterior on ${\tilde{m}}(\cdot )$ and then the posterior distribution of ${\tilde{m}} = ({\tilde{m}}(X_1), \ldots , {\tilde{m}}(X_n))^T$, i.e. ${\tilde{m}}_{\mid Y} \sim {\mathcal {N}}({\mathbf {A}}Y, \sigma ^2 {\mathbf {A}})$ with ${\mathbf {A}} = {\mathbf {B}}({\mathbf {B}}^T {\mathbf {B}} + \lambda {\mathbf {D}})^{-1} {\mathbf {B}}^T$.

Bayesian asymptotic confidence interval

Nychka [28] showed that if m(x) is estimated using a cubic smoothing spline for which the smoothing parameter is sufficiently reliably estimated that the bias in the estimates is a modest fraction of the mean squared error for m(x), then the average coverage probability (ACP)

$$\begin{aligned} \text {ACP} = \frac{1}{n} \sum _{i=1}^n {\mathbb {P}}\left( m(X_i) \in BI_{\alpha }(X_i) \right) , \end{aligned}$$

is very close to the nominal level $1-\alpha$, where $BI_{\alpha }(x)$ indicates the $(1-\alpha )100\%$ Bayesian interval for m(x) and $\alpha$ the significance level. This is due to the fact that the average posterior variance for the spline is similar to a consistent estimate of the average squared error and that the average squared bias is relatively small with respect to the total average squared error.

Marra and Wood [26] modified Nychka’s [28] approach to obtain the confidence interval for Generalized Additive Model (GAM), which is also applicable in the Bayesian penalized spline models. Here we will sketch the main steps to obtain confidence interval of variable width.

Given some constants $C_i$, which will be defined later, the primary purpose is to find a constant A, such that

$$\begin{aligned} \text {ACP} = \frac{1}{n}\sum _{i=1}^n {\mathbb {P}}\left( \left| {\hat{m}}(X_i) - {\tilde{m}}(X_i) \right| \le z_{\alpha /2} A/ \sqrt{C_i} \right) = 1-\alpha , \end{aligned}$$

(25)

where $z_{\alpha /2}$ is the $\alpha /2$ critical point from a standard normal distribution.

Letting ${\hat{m}} = ({\hat{m}}(X_1 ),\ldots ,{\hat{m}}(X_n ))^T= {\mathbf {B}} {\hat{\beta }}$ and ${\tilde{m}} = ({\tilde{m}}(X_1),\ldots ,{\tilde{m}}(X_n))^T= {\mathbf {B}} \beta$. We write the covariance of ${\hat{m}}$ as ${\mathbf {V}}_{{\hat{m}}} = {\mathbf {B}} {\mathbf {V}}_{{\hat{\beta }}} {\mathbf {B}}^T$ and the same as that of ${\tilde{m}}$, i.e. ${\mathbf {V}}_{{\tilde{m}}} = {\mathbf {B}}{\mathbf {V}}_{\beta }{\mathbf {B}}^T$. Define $b \equiv (b(X_1 ),\ldots ,b(X_n))^T=E({\hat{m}})- {\tilde{m}}$ and $v \equiv (v(X_1),\ldots ,v(X_n))^T= {\hat{m}}-E({\hat{m}})$. We have $v \sim {\mathcal {N}}(0,{\mathbf {V}}_{{\hat{m}}})$ following from multivariate normality of ${\hat{m}}$. Equivalently, the equation (25) can be written as:

$$\begin{aligned} \text {ACP}&= {\mathbb {P}} \left( \left| b(X_I) + v(X_I) \right| \le z_{\alpha /2} A/\sqrt{C_I} \right) \nonumber \\&= {\mathbb {P}} \left( \left| {\mathfrak {B}} + {\mathfrak {V}} \right| \le z_{\alpha /2} A \right) = 1-\alpha , \end{aligned}$$

(26)

where I is a random variable uniformly distributed on $\{1,2,\ldots ,n\}$, ${\mathfrak {B}}$ and ${\mathfrak {V}}$ are respectively the random scaled bias and random scaled variance defined as follows:

$$\begin{aligned} {\mathfrak {B}} = \sqrt{C_I}b(X_I) \quad \text {and} \quad {\mathfrak {V}} = \sqrt{C_I}v(X_I). \end{aligned}$$

This means that we need to know the distribution of ${\mathfrak {B}} + {\mathfrak {V}}$ in order to find the constant A.

Clearly, by definition, we have ${\mathbb {E}}({\mathfrak {B}}) = c^T{\mathbf {B}} ({\mathbf {F}}\beta - \beta ), {\mathbb {E}}({\mathfrak {V}}) = 0$ and $\text {var}({\mathfrak {V}}) = \text {tr}({\mathbf {C}}{\mathbf {V}}_{{\hat{m}}})/n$ where $c= (\sqrt{C_1},\ldots ,\sqrt{C_n} )^T, {\mathbf {F}}= ({\mathbf {B}}^T {\mathbf {B}}+\lambda {\mathbf {B}})^{-1} {\mathbf {B}}^T {\mathbf {B}}$ and ${\mathbf {C}}$ is the diagonal matrix $\text {diag} (C_1, \ldots , C_n)$. Since $v \sim {\mathcal {N}}(0, {\mathbf {V}}_{{\hat{m}}}), {\mathfrak {V}}$ is a mixture of normals. However, if we choose $C_i^{-1}= [{\mathbf {V}}_{{\hat{m}}}]_{ii}$, the random scaled variance ${\mathfrak {V}}$ then has a normal distribution. Since its distribution no longer depends on i, it implies the independence of ${\mathfrak {B}}$ and ${\mathfrak {V}}$.

We call ${\mathfrak {M}}$ the scaled average mean squared error which is given by:

$$\begin{aligned} {\mathfrak {M}} = \frac{1}{n}\sum _{i=1}^n C_i \left( {\hat{m}}(X_i) - {\tilde{m}}(X_i)\right) ^2. \end{aligned}$$

(27)

The mean squared error $E({\mathfrak {M}})$ can be then determined as follows:

$$\begin{aligned} E({\mathfrak {M}}) = \frac{1}{n}\text {Tr}\left( {\mathbf {B}}^T {\mathbf {C}}^2 {\mathbf {B}} {\mathbf {V}}_{{\hat{\beta }}}\right) + \frac{1}{n} \left\| {\mathbf {C}}{\mathbf {B}}\left( {\mathbf {F}} - {\mathbf {I}}\right) \beta \right\| ^2. \end{aligned}$$

(28)

By construction, we have ${\mathbb {E}}({\mathfrak {B}}+{\mathfrak {V}}) = {\mathbb {E}}({\mathfrak {B}})$ and $\text {var}({\mathfrak {B}}+{\mathfrak {V}}) = {\mathbb {E}}({\mathfrak {M}}) - {\mathbb {E}}({\mathfrak {B}})^2$. Nychka’s [28] simulation results showed that ${\mathfrak {B}}+{\mathfrak {V}}$ will be approximately normally distributed, provided that ${\mathfrak {B}}$ is small relative to ${\mathfrak {V}}$, i.e.

$$\begin{aligned} {\mathfrak {B}} + {\mathfrak {V}} \sim {\mathcal {N}}\left( {\mathbb {E}}({\mathfrak {B}}), {\mathbb {E}}({\mathfrak {M}}) - {\mathbb {E}}({\mathfrak {B}})^2 \right) . \end{aligned}$$

The expectations ${\mathbb {E}}({\mathfrak {B}})$ and $E({\mathfrak {M}})$ can be estimated by substituting $\beta$ by ${\hat{\beta }}$ which yields

$$\begin{aligned} \widehat{E({\mathfrak {B}})}=c^T {\mathbf {B}}({\mathbf {F}} {\hat{\beta }}- {\hat{\beta }})/n, \end{aligned}$$

and

$$\begin{aligned} \widehat{E\left( {\mathfrak {M}}\right) }&= \frac{1}{n}\text {Tr}\left( {\mathbf {B}}^T {\mathbf {C}}^2 {\mathbf {B}} {\mathbf {V}}_{{\hat{\beta }}}\right) + \frac{1}{n} \left\| {\mathbf {C}}{\mathbf {B}}\left( {\mathbf {F}} - {\mathbf {I}}\right) {\hat{\beta }}\right\| ^2 \nonumber \\&= 1 + \frac{1}{n} \left\| {\mathbf {C}}{\mathbf {B}}\left( {\mathbf {F}} - {\mathbf {I}}\right) {\hat{\beta }}\right\| ^2. \end{aligned}$$

(29)

Therefore, we have the approximate result

$$\begin{aligned} {\mathfrak {B}} + {\mathfrak {V}} \sim {\mathcal {N}} \left( \widehat{E({\mathfrak {B}})}, \widehat{E({\mathfrak {M}})} - \widehat{E({\mathfrak {B}})}^2 \right) , \end{aligned}$$

under the assumption that ${\mathfrak {B}}$ is small relative to ${\mathfrak {V}}$, i.e. $\overline{b^2} \ll \overline{v^2}$.

As the definition (26), it follows that $A = \sqrt{\widehat{E({\mathfrak {M}})} - \widehat{E({\mathfrak {B}})}^2}$ and then we obtain

$$\begin{aligned} {\hat{m}}(X_i )-\widehat{E({\mathfrak {B}})} \sqrt{\left[ {\mathbf {V}}_{{\hat{m}}}\right] _{ii}} \pm z_{\alpha /2} \sqrt{\left( \widehat{E({\mathfrak {M}})} - \widehat{E({\mathfrak {B}})}^2 \right) [{\mathbf {V}}_{{\hat{m}}}]_{ii}}, \end{aligned}$$

(30)

as the definition of $1-\alpha$ Bayesian asymptotic confidence intervals at the point $X_i$.

Finally, we use the estimator ${\hat{\sigma }}^2 = \frac{\left\| Y - {\mathbf {B}}{\hat{\beta }}\right\| ^2}{n-\text {Tr}({\mathbf {F}})}$ to estimate $\sigma ^2$.

Appendix B: Additive model and asymptotic confidence interval for each functional components

In this section, the additive model will be briefly presented. In this model the relation between the response $Y_i, i \in \{1,\ldots ,n \}$ and the d-explanatory variables $X_{1i},X_{2i},\ldots ,X_{di}$ is expressed through arbitrary univariate functions $f_j$ as follows:

$$\begin{aligned} Y_i = \beta _0 + \sum _{j=1}^d f_j\left( X_{ji}\right) + \epsilon _i, \end{aligned}$$

(31)

where the errors $\epsilon _i$ are independent and identically distributed with mean zero and the variance $\sigma ^2$. Several estimation strategies have been developed to fit such a model, e.g. backfitting algorithm (Hastie and Tibshirani [15]) as well as its asymptotic statistical properties (Opsomer and Ruppert [30], Opsomer [29] and Wand [43]), and a marginal integration approach (Linton and Nielsen [24]). Due to the computational expediency, the penalized splines regression to ordinary additive models (Hastie and Tibshirani [15]) have been widely used in practice. As consequent, we will apply this approach to estimate the excess loss function.

Like the univariate case, each of the functional components $f_j$ is modeled as ${\mathbf {B}}^T\beta _j$ a degree $p_j$ penalized spline estimator with smoothing parameters $\lambda _j$ and ${\mathbf {B}}_j$ a $n \times (p_j+K_j)$ matrix whose i-th row is $(X_{ji},X_{ji}^2,\ldots ,X_{ji}^{p_j},(X_{ji}-\kappa _{j1})^{p_j}_{+},\ldots ,(X_{ji}-\kappa _{jK_j})^{p_j}_{+})$. However, for the identifiability reasons, we replace $f_j(\cdot )$ by $f_j(\cdot )-1/n \sum _{i=1}^n f_j (X_{ji})$ which leads to the condition on $f_j(\cdot )$ of the form $\sum _{i=1}^n f_j(X_{ji}) = 0$. Therefore, the estimate of $\beta _0$ is ${\hat{\beta }}_0= {\bar{Y}} \equiv \frac{1}{n} \sum _{i=1}^{n} Y_i$ which is independent of $X_{ji}$’s and the matrix ${\mathbf {B}}_j$ is adjusted to ${\mathbf {B}}_j^{*}= \frac{1}{n}({\mathbf {I}}-{\mathfrak {1}}.{\mathfrak {1}}^T ) {\mathbf {B}}_j$ where ${\mathbf {I}}$ is the identity matrix and ${\mathbf {1}}$ is a $n \times 1$ column of ones. This adjustment is called the “centering effect”. Letting $Y^{*}= (Y_1-{\hat{\beta }}_0,\ldots ,Y_n-{\hat{\beta }}_0)^T, \ddot{{\mathbf {B}}}= [{\mathbf {B}}_1^{*},\ldots ,{\mathbf {B}}_d^{*}]$ and ${\mathbf {D}}_{\lambda }=\text {blockdiag}_{1 \le j \le d} (\lambda _j {\mathbf {D}}_j )$ with $\lambda _j > 0$. We then have the estimate of $\beta ^T= (\beta _1^T,\ldots ,\beta _d^T )$ is given by

$$\begin{aligned} {\hat{\beta }} = \left( \ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}} + {\mathbf {D}}_{\lambda }\right) ^{-1} \ddot{{\mathbf {B}}}Y^{*} \in {\mathbb {R}}^{\sum _{j=1}^d (p_j+K_j+1)}. \end{aligned}$$

(32)

From this we derive the estimate of $\beta _j, {\hat{\beta }}_j={\mathbf {P}}_j {\hat{\beta }}$ where ${\mathbf {P}}_j= [0,\ldots ,{\mathbf {1}}_{(p_j+K_j+1)},\ldots ,0] \in {\mathbb {R}}^{(p_j+K_j+1) \times \sum _k (p_k+K_k+1)}$.

Regarding the Bayesian asymptotic confidence interval for each of the functional components, the derivation is analogue as for the univariate case. Namely, assume that $\beta _0$ has an improper uniform prior distribution and the prior for $\beta _j$ is given by $(\lambda _j/\sigma ^2)^{K_j/2} \exp {(-(\lambda _j/2\sigma ^2)\beta ^T_j \beta _j )}$ for $j = 1, \ldots , d$. By using Bayes rule, it has been shown [44] that

$$\begin{aligned} \beta _{\mid Y} \sim {\mathcal {N}} \left( {\hat{\beta }}, {\mathbf {V}}_{\beta }\right) , \end{aligned}$$

where ${\mathbf {V}}_{\beta } = (\ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}}+{\mathbf {D}}_{\lambda })^{-1} \sigma ^2$. Thanks to the Eq. (32), it is then routine to show that ${\mathbf {V}}_{{\hat{\beta }}} = (\ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}} + {\mathbf {D}}_{\lambda })^{-1} \ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}}(\ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}}+ {\mathbf {D}}_{\lambda })^{-1} \sigma ^2$. It follows immediately that the variance of ${\hat{f}}_j$ is ${\mathbf {V}}_{{\hat{f}}_j}=({\mathbf {B}}^{*}_j {\mathbf {P}}_j ) {\mathbf {V}}_{{\hat{\beta }}} ({\mathbf {B}}^{*}_j {\mathbf {P}}_j )^T$. The $1-\alpha$ Bayesian asymptotic confidence interval computation for ${\hat{f}}_j(X_{ji})$ is similar to that can be found in Appendix A.4. Routine manipulation then results in

$$\begin{aligned} {\hat{f}}_j \left( X_{ji}\right) - \widehat{E({\mathfrak {B}}_j)}\sqrt{\left[ {\mathbf {V}}_{{\hat{f}}_j} \right] _{ii}} \pm z_{\alpha /2} \sqrt{\left( \widehat{E({\mathfrak {M}}_j)} - \widehat{E({\mathfrak {B}}_j)}^2 \right) \left[ {\mathbf {V}}_{{\hat{f}}_j} \right] _{ii}}, \end{aligned}$$

(33)

as the definition of $1-\alpha$ Bayesian asymptotic confidence interval for ${\hat{f}}_j(X_{ji})$. Here, we denoted $\widehat{E({\mathfrak {B}}_j)} = \frac{1}{n}c^T {\mathbf {B}}^{*}_j {\mathbf {P}}_j ( \ddot{{\mathbf {F}}} {\hat{\beta }} - {\hat{\beta }} )$ with $c = ([{\mathbf {V}}_{{\hat{f}}_j}]^{-1/2}_{11},\ldots ,[{\mathbf {V}}_{{\hat{f}}_j}]^{-1/2}_{nn} )$ and $\ddot{{\mathbf {F}}} = (\ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}} + {\mathbf {D}}_{\lambda })^{-1}\ddot{{\mathbf {B}}}^T \ddot{{\mathbf {B}}}, \widehat{E({\mathfrak {M}}_j)} = 1 + \frac{1}{n}\left\| {\mathbf {C}}_j {\mathbf {B}}^{*}_j {\mathbf {P}}_j (\ddot{{\mathbf {F}}} {\hat{\beta }} - {\hat{\beta }})\right\|$ with ${\mathbf {C}}_j = \text {diag}([{\mathbf {V}}_{{\hat{f}}_j}]^{-1}_{11},\ldots ,[{\mathbf {V}}_{{\hat{f}}_j}]^{-1}_{nn})$. Readers can refer to [26] for more details.

Appendix C: Upper bound of the probabilities of deviation

1. Let us denote $S_j = \{|{\hat{\phi }}_j (x^{(\nu )}_j ) - \phi _j (x^{(\nu )}_j ) |> {\varDelta }^{(\nu )}_{j,\alpha } \}$ and ${\tilde{S}}_j = \{|{\hat{h}}_J (x^{(\nu )}_J ) - h_J (x^{(\nu )}_J ) |> {\tilde{{\varDelta }}}^{(\nu )}_{J,\alpha } \}$. Since $\{S_j \}$ and $\{ {\tilde{S}}_J \}$ are mutually independent, it implies that (by De Morgan’s law)

$$\begin{aligned} {\mathbb {P}}\left( \left( \bigcup _j S_j \right) \bigcup \left( \bigcup _J {\tilde{S}}_J \right) \right)&= 1 - {\mathbb {P}}\left( \left( \bigcap _j S^{c}_j \right) \bigcap \left( \bigcap _J {\tilde{S}}^{c}_J \right) \right) \\&= 1 - \left( \prod _j {\mathbb {P}}(S^c_j)\right) \times \left( \prod _J {\mathbb {P}}({\tilde{S}}^c_j)\right) , \end{aligned}$$

which asymptotically tends to $1-(1-\alpha )^{d(d+3)/2}$ as ${\varGamma } \rightarrow \infty$. With this the equation (14) is then a straightforward consequence of the relation

$$\begin{aligned} {\mathbb {P}}\left( \left|{\widehat{\phi }}(x^{(\nu )}) - \phi (x^{(\nu )}) \right|> \sum _{j=1}^d {\varDelta }^{(\nu )}_{j,\alpha } + \sum _J {\tilde{{\varDelta }}}^{(\nu )}_{J,\alpha } \right) \le {\mathbb {P}}\left( \left( \bigcup _j S_j \right) \bigcup \left( \bigcup _J {\tilde{S}}_J \right) \right) , \end{aligned}$$

thanks to the Eq. (13).

2. For a given value of ${\varGamma }$, we call $r^{*} \equiv r({\varGamma })$ the positive constant mentioned in Assumption (3). Furthermore, we assume that $B(x^{*}, r^{*}) \subset {\varOmega }$ for large enough ${\varGamma }$.

For notional simplicity, let us denote $\delta \phi (x, y) = |{\hat{\phi }}(y) - \phi (x) |$. We have

$$\begin{aligned} {\mathbb {P}}\left( \delta \phi (x^{*}, x^{*}_{({\varGamma })})> {\varDelta }(\alpha , {\varGamma }) + Lr^{*} \right)&= {\mathbb {P}}\left( \delta \phi (x^{*}, x^{*}_{({\varGamma })})> {\varDelta }(\alpha , {\varGamma }) + Lr^{*} \mid x^{*}_{({\varGamma })} \in B(x^{*},r^{*})\right) {\mathbb {P}}\left( x^{*}_{({\varGamma })} \in B(x^{*},r^{*}) \right) \\&\quad +\,{\mathbb {P}}\left( \left|{\hat{\phi }}(x^{*}_{({\varGamma })}) - \phi (x^{*}) \right|> {\varDelta }(\alpha , {\varGamma }) + Lr^{*} \mid x^{*}_{({\varGamma })} \notin B(x^{*},r^{*})\right) {\mathbb {P}}\left( x^{*}_{({\varGamma })} \notin B(x^{*},r^{*}) \right) \\&\le {\mathbb {P}}\left( \delta \phi (x^{*}_{({\varGamma })}, x^{*}_{({\varGamma })}) > {\varDelta }(\alpha ,L) \mid x^{*}_{({\varGamma })} \in B(x^{*},r^{*}) \right) + \xi (r^{*},d){\varGamma }^{-\gamma (r^{*},d)} \end{aligned}$$

since $|{\hat{\phi }}(x^{*}_{({\varGamma })}) - \phi (x^{*}) |\le |{\hat{\phi }}(x^{*}_{({\varGamma })}) - \phi (x^{*}_{({\varGamma })}) |+ |\phi (x^{*}_{({\varGamma })}) - \phi (x^{*}) |$ and

$$\begin{aligned} {\mathbb {P}}\left( \left|\phi (x^{*}_{({\varGamma })}) - \phi (x^{*}) \right|> Lr^{*} \mid x^{*}_{({\varGamma })} \in B(x^{*},r^{*}) \right) = 0 \end{aligned}$$

thanks to Assumption (1).

Finally, by applying Assumption (3), we obtain

$$\begin{aligned} {\mathbb {P}}\left( \delta \phi (x^{*}, x^{*}_{({\varGamma })}) > {\varDelta }(\alpha , {\varGamma }) + Lr^{*} \right) \le \left[ 1 - (1-\alpha )^{d(d+3)/2} \right] + \xi (r^{*},d){\varGamma }^{-\gamma (r^{*},d)}. \end{aligned}$$

Appendix D: Best approximation by splines

Let us first introduce two functional spaces.

Definition 1

(Polynomial Spline Space${\varPhi }^{a,b}_s$) Letting $\kappa _l$ for $l \in \{1,\ldots ,K\}$ be K-interior knots satisfying the condition $a= \kappa 0 \le \kappa _1 \le \ldots \le \kappa _K \le \kappa _{K+1}=b$. We define ${\varPhi }^{a,b}_s$ the space of functions whose element is a polynomial of at most degree p on each of the intervals $[\kappa _l,\kappa _{l+1} )$ for $l=0,1,\ldots ,K$ and is $p-1$ continuously differentiable on [a, b] if $p \ge 1$.

Definition 2

(Empirically Centered Polynomial Spline Space${\overline{{\varPhi }}}^{a,b}_s$) Given the design points $(x_1,\ldots ,x_n )\in [a,b]^n$, a polynomial spline space is centered if for every $g \in {\varPhi }^{a,b}_s$ the following identity holds:

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n g(x_i)=0. \end{aligned}$$

We denote by ${\overline{{\varPhi }}}^{a,b}_s$ the Empirically centered polynomial spline space.

According to de Boor (p. 149 in [6]), for every $\phi (x) \in C^{p+1}([a,b])$, there exists a constant $c>0$ and a spline function $\phi ^{*}(x) \in {\varPhi }^{a,b}_s$, such that $\left\| \phi -\phi ^{*}\right\| _{\infty } \le c \left\| \phi ^{(p+1)}\right\| _{\infty } \delta ^{p+1}$ with $\delta = \max _{1 \le l \le K}(\kappa _{l+1}-\kappa _l)$.

Given the design points $(x_1,\ldots ,x_n )\in [a,b ]^n$, we assume furthermore that

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n \phi (x_i) = 0. \end{aligned}$$

By defining $\phi ^{**}(x) = \phi ^{*}(x) - \frac{1}{n}\sum _{i=1}^n \phi ^{*}(x_i) \in {\overline{{\varPhi }}}^{a,b}_s$, it is straightforward to show that there exists a positive constant $c'$ such that $\left\| \phi -\phi ^{**}\right\| _{\infty } \le c' \left\| \phi ^{(p+1)}\right\| _{\infty } \delta ^{p+1}$.

Appendix E: Asymptotic distribution of empirical quantiles

Theorem 1

Let$X_1,\ldots ,X_n$ben-real valued observations with unknown distribution functionF, pbe a real number defined in the interval$[0,1], x_p=x_p(F)$be thepth-percentile ofFand$F_n(x)$be the empirical distribution function.

If we suppose thatFis continuous and differentiable at$x_p$of derivation$f(x_p)$, then

$$\begin{aligned} \sqrt{n} \left( x_p(n) - x_p\right) \rightarrow {\mathcal {N}}\left( 0, \frac{p(1-p)}{f^2(x_p)} \right) \end{aligned}$$

(34)

as$n \rightarrow \infty$.

This is a well-known result for the asymptotic convergence of empirical quantiles. We will thus omit the proof here. For any further and detailed information, the interested reader can refer to the section 3.9.21 in [40].

References

1.
Aerts M, Claeskens G, Wand MP (2002) Some theory for penalized spline generalized additive models. J Stat Plan Inference 103:455–470MathSciNetzbMATHGoogle Scholar
2.
Bauer D, Bergmann D, Reuss A (2009) Solvency II and nested simulations—a least square Monte Carlo approach. In: Proceedings of the 2010 ICA congressGoogle Scholar
3.
Bauer D, Reuss A, Singer D (2012) On the calculation of the solvency capital requirement based on nested simulations. ASTIN Bull 42:453–499MathSciNetzbMATHGoogle Scholar
4.
Bengio Y, Grandvalet Y (2004) No unbiased estimator of the variance of k-fold cross-validation. J Mach Learn Res 5:1089–1105MathSciNetzbMATHGoogle Scholar
5.
Beutner E, Pelsser A, Schweizer J (1993) Theory and validation of replicating portfolios in insurance risk management. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2557368. Accessed 20 June 2018
6.
de Boor C (2001) A practical guide to splines, revised edn. Springer, New YorkzbMATHGoogle Scholar
7.
Devineau L, Loisel S (2009) Construction of an acceleration algorithm of the nested simulations method for the calculation of the Solvency II economic capital. Bulletin Français d’ActuariatGoogle Scholar
8.
Duchon J (1977) Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Schempp W, Zeller K (eds) Constructive theory of functions of several variables. Lecture notes in mathematics, vol 571. Springer, Berlin, HeidelbergGoogle Scholar
9.
Eilers PHC, Marx BD (1996) Flexing smoothing with B-splines and Penalties. Stat Sci 11:89–102zbMATHGoogle Scholar
10.
Eubank RL (1999) Nonparametric regression and spline smoothing, 2nd edn. Marcel Dekker, New YorkzbMATHGoogle Scholar
11.
Filipovic D (2009) Term-structure models: a graduate course. Springer Finance Textbooks, New YorkzbMATHGoogle Scholar
12.
Giraud C (2014) Introduction to High-Dimensional Statistics. CRC Monographs on Statistics and Applied Probability. Chapman and Hall, Boca RatonGoogle Scholar
13.
Green PJ, Silverman BW (1994) Nonparametric regression and generalized linear models: a roughness penalty approach. Chapman and Hall, LondonzbMATHGoogle Scholar
14.
Harrison JM, Pliska SR (1981) Martingales and stochastic integrals in the theory of continuous trading. Stoch Process Appl 11:215–260MathSciNetzbMATHGoogle Scholar
15.
Hastie T, Tibshirani R (1990) Generalized additive models. Chapman and Hall, New YorkzbMATHGoogle Scholar
16.
Hong LJ, Juneja S, Liu G (2017) Kernel smoothing for nested estimation with application to portfolio risk measurement. Oper Res 65:657–673MathSciNetzbMATHGoogle Scholar
17.
Kemp M (1993) Market consistency. Wiley Finance, New YorkGoogle Scholar
18.
Koursaris A (2011) Improving capital approximation using the curve-fitting approach. Barrie and HibbertGoogle Scholar
19.
Koursaris A (2011) The advantages of least squares Monte Carlo. Barrie and Hibbert. http://www.barrhibb.com/documents/downloads/The_Advantages_of_Least_Squares_Monte_Carlo.pdf. Accessed 22 June 2018
20.
Koursaris A (2011) A least squares Monte Carlo approach to liability proxy modelling and capital calculation. Barrie and Hibbert. http://www.barrhibb.com/documents/downloads/Least_Square_Monte_Carlo_Approach_to_Liability_Proxy_Modelling_and_Capital_Calculation.pdf. Accessed 22 June 2018
21.
Koursaris A (2011) A primer in replicating portfolios. Barrie and Hibbert. http://www.barrhibb.com/documents/downloads/Primer_in_Replicating_Portfolios.pdf. Accessed 22 June 2018
22.
KPMG (2015) Technical Practices Survey 2015 Solvency II. https://assets.kpmg.com/content/dam/kpmg/pdf/2016/04/TPS_2015.pdf. Accessed 22 June 2018
23.
Lan H, Nelson BL, Staum J (2007) A confidence interval procedure for expected shortfall risk measurement via two-level simulation. Oper Res 58:1481–1490MathSciNetzbMATHGoogle Scholar
24.
Linton O, Nielsen JP (1995) A kernel method of estimating structured non-parametric regression based on marginal integration. Biometrika 82:93–100MathSciNetzbMATHGoogle Scholar
25.
Longstaff FA, Schwartz ES (2001) Valuing American options by simulations: a simple least-squares approach. Rev Financ Stud 14(1):113–147Google Scholar
26.
Marra G, Wood SN (2012) Coverage properties of confidence intervals for generalized additive model components. Scand Stat Theory Appl 39:53–74MathSciNetzbMATHGoogle Scholar
27.
Natolski J, Werner R (2014) Mathematical analysis of different approaches for replicating portfolios. Eur Actuar J 4(2):411–435MathSciNetzbMATHGoogle Scholar
28.
Nychka D (1988) Bayesian confidence intervals for smoothing splines. J Am Stat Assoc 83:1134–1143MathSciNetGoogle Scholar
29.
Opsomer JD (2000) Asymptotic properties of back-fitting estimators. J Multivar Anal 73:166–179MathSciNetzbMATHGoogle Scholar
30.
Opsomer JD, Ruppert D (1997) Fitting a bivariate additive model by local polynomial regression. Ann Stat 25:186–211MathSciNetzbMATHGoogle Scholar
31.
O’Sullivan F (1986) A statistical perspective on ill-posed inverse problems. Stat Sci 1:505–27MathSciNetzbMATHGoogle Scholar
32.
Pelsser A, Schweizer J (2016) The difference between LSMC and replicating portfolio in the insurance liability modeling. Eur Actuar J 6:441–494MathSciNetzbMATHGoogle Scholar
33.
Pliska SR (1997) Introduction to mathematical finance: discrete time models. Blackwell, MaldenGoogle Scholar
34.
Reinsch C (1971) Smoothing by spline functions II. Numer Math 16:451–454MathSciNetzbMATHGoogle Scholar
35.
Ruppert D, Carroll RJ (2000) Spatially adaptive penalties for spline fitting. Aust N Z J Stat 42:205–233Google Scholar
36.
Silverman BW (1985) Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J R Stat Soc Ser B Stat Methodol 47:1–52zbMATHGoogle Scholar
37.
Stentoft L (2004) Convergence of the least square Monte-Carlo approach to American option valuation. Manage Sci 50(9):1193–1203zbMATHGoogle Scholar
38.
Stone CJ (1985) Additive regression and other nonparametric models. Ann Stat 13(2):689–705MathSciNetzbMATHGoogle Scholar
39.
Teuquia ON, Ren J, Planchet F (2014) Internal model in life insurance: application of least squares Monte-Carlo in risk assessmentGoogle Scholar
40.
van der vaart AW, Wellner J (1996) Weak convergence and empirical process. Springer Series in Statistics. Springer, New YorkzbMATHGoogle Scholar
41.
Vidal EG, Daul S (2009) Replication of insurance liabilities. Riskmetrics J. 9:79–96Google Scholar
42.
Wahba G (1983) Bayesian confidence intervals for the cross validated smoothing spline. J R Stat Soc Ser B Stat Methodol 45:133–150MathSciNetzbMATHGoogle Scholar
43.
Wand MP (1999) Central limit theorem for local polynomial backfitting estimators. J Multivar Anal 70:57–65MathSciNetzbMATHGoogle Scholar
44.
Wood SN (2006) On confidence intervals for generalized additive models based on penalized regression splines. Aust N Z J Stat 48:445–464MathSciNetzbMATHGoogle Scholar

Copyright information

Authors and Affiliations

Quang Dien Duong
- 1
- 2
Email authorView author's OrcID profile

1.Laboratoire de Probabilités, Statistique et ModélisationPierre and Marie Curie UniversityParisFrance
2.Risk and Value Measurement ServicesPricewaterhouseCoopersNeuilly-sur-Seine cedexFrance

Application of Bayesian penalized spline regression for internal modeling in life insurance

Abstract

Keywords

Notes

Acknowledgements

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article