Replicating portfolio approach to capital calculation

Abstract

The replicating portfolio (RP) approach to the calculation of capital for life insurance portfolios is an industry standard. The RP is obtained from projecting the terminal loss of discounted asset–liability cash flows on a set of factors generated by a family of financial instruments that can be efficiently simulated. We provide the mathematical foundations and a novel dynamic and path-dependent RP approach for real-world and risk-neutral sampling. We show that our RP approach yields asymptotically consistent capital estimators if the chaotic representation property holds. We illustrate the tractability of the RP approach by three numerical examples.

Appendix A: Proofs

Proof of Lemma 2.1

The first statement follows from the Cauchy–Schwarz inequality. As for the second, we note that

$$ \| {\mathbb {E}}^{\mathbb {Q}}[X \, | \, {\mathcal {F}}_{1} ] \|_{L^{1}({\mathbb {P}})} \le\| X D_{T}/D_{1} \| _{L^{1}({\mathbb {P}})} = \| \sqrt{D_{T}} X \sqrt{D_{T}}/D_{1} \|_{L^{1}({\mathbb {P}})} . $$

Applying the Cauchy–Schwarz inequality to the second and third expressions yields the claim for ${\mathbb {M}}={\mathbb {P}}$ and ${\mathbb {M}}={\mathbb {Q}}$, respectively. □

Proof of Theorem 3.7

The bound (3.6) follows from Lemma 2.1. The bound (3.7) follows similarly, where we write

$$ \| {\mathbb {E}}^{\mathbb {Q}}[{\epsilon^{\mathbb {M}}}\, | \,{\mathcal {F}}_{1}]-{\epsilon^{\mathbb {M}}} \|_{L^{1}({\mathbb {P}} )}\le \| {\epsilon^{\mathbb {M}}} \|_{L^{1}({\mathbb {P}})}+ \| {\mathbb {E}}^{\mathbb {Q}}[{\epsilon^{\mathbb {M}}}\, | \, {\mathcal {F}}_{1}] \| _{L^{1}({\mathbb {P}})} $$

for the ℙ-bound. □

Proof of Lemma 3.8

We have $K= {\mathrm{ES}}_{\alpha} [ {\mathbb {E}}^{\mathbb {P}}[ Z D_{T}/D_{1}\, | \,{\mathcal {F}}_{1}] ]$. Monotonicity with respect to stochastic order of expected shortfall yields (3.8); see Föllmer and Schied [14, Corollary 4.59]. Using subadditivity of expected shortfall, the other inequalities follow similarly. □

Proof of Theorem 4.2

The theorem follows from combining (2.1) and (2.2) with the LLN (4.2) and (4.4). □

Proof of Theorem 4.3

Assume $i=1$. Using the Lipschitz property (2.2) of expected shortfall, we derive

$$\begin{aligned} & \| K^{\mathbb {M}}_{1} - \widehat{K^{\mathbb {M}}_{1}} \|_{L^{2}({\mathbb {M}})} \\ &= \| v^{\mathbb {M}}-\widehat{v^{\mathbb {M}}} + {\mathrm{ES}}_{\alpha}[{\phi^{\mathbb {M}}_{A}}{}^{\top}A] - {\mathrm{ES}}_{\alpha} [ \widehat{\phi^{\mathbb {M}}_{A}}{}^{\top}A \, | \, {\mathcal {G}}] \| _{L^{2}({\mathbb {M}})}\\ & \le\|v^{\mathbb {M}}-\widehat{v^{\mathbb {M}}} \|_{L^{2}({\mathbb {M}})}+\frac{1}{1-\alpha} \sqrt {{\mathbb {E}} ^{\mathbb {M}}\Big[ {\mathbb {E}}^{\mathbb {P}}\big[ \big((\phi^{\mathbb {M}}_{A}-\widehat{\phi^{\mathbb {M}} _{A}})^{\top}A \big)^{2}\, \big| \, {\mathcal {G}}\big] \Big] }\\ &= \| v^{\mathbb {M}}-\widehat{v^{\mathbb {M}}} \|_{L^{2}({\mathbb {M}})}+\frac{1}{1-\alpha}\sqrt{ {\mathbb {E}}^{\mathbb {M}} \big[ (\phi^{\mathbb {M}}_{A}-\widehat{\phi^{\mathbb {M}}_{A}})^{\top}{\mathbb {E}}^{\mathbb {P}}[ A A^{\top}]\, (\phi^{\mathbb {M}} _{A}-\widehat{\phi^{\mathbb {M}}_{A}}) \big] }. \end{aligned}$$

The CLT (4.3) and (4.5) now yield the claim. The case $i=2$ follows similarly. □

Proof of Theorem 4.4

Denote by $\mu$ the ℙ-distribution of $X$. We claim that

$$\begin{gathered} \text{$\widehat{\mu}_{n}\longrightarrow\mu$\qquad weakly a.s.} \end{gathered}$$

(A.1)

Let $f(x)$ be a continuous function with compact support. Write

$$ \int f(x)\,d\widehat{\mu}_{n} = \frac{\frac{1}{n} \sum_{j=1}^{n} f(X^{(j)}) d{\mathbb {P}}/d{\mathbb {M}}^{(j)}}{\frac{1}{n}\sum_{k=1}^{n} d{\mathbb {P}}/d{\mathbb {M}}^{(k)}} . $$

By the law of large numbers, the numerator and denominator converge, with

$$ \frac{1}{n} \sum_{j=1}^{n} f(X^{(j)}) d{\mathbb {P}}/d{\mathbb {M}}^{(j)} \longrightarrow{\mathbb {E}}_{\mathbb {M}}[ f(X) d{\mathbb {P}}/d{\mathbb {M}}]={\mathbb {E}}_{\mathbb {P}}[f(X) ]\qquad\text{a.s.} $$

and

$$ \frac{1}{n}\sum_{k=1}^{n} d{\mathbb {P}}/d{\mathbb {M}}^{(k)} \longrightarrow{\mathbb {E}}_{\mathbb {M}}[d{\mathbb {P}}/d{\mathbb {M}} ]=1\qquad \text{a.s.} $$

Hence

$$ \int f(x)\,d\widehat{\mu}_{n} \longrightarrow\int f(x)\,d\mu\qquad\text{a.s.} $$

As we have $Xd{\mathbb {P}}/d{\mathbb {M}}\in L^{1}({\mathbb {M}})$ by assumption for expected shortfall $\rho={\mathrm{ES}}_{\alpha}$, the same convergence holds for $f(x)=|x|$ so that

$$\begin{gathered} \text{$\int|x|\,d\widehat{\mu}_{n}\longrightarrow\int|x|\,d\mu$\qquad a.s.} \end{gathered}$$

(A.2)

Because the space of continuous functions with compact support is separable, there exists a measurable $\Omega_{0}\in{\mathcal {F}}$ such that ${\mathbb {M}}[\Omega _{0}]=1$ and such that for each $\omega\in\Omega_{0}$, the desired properties (A.1) and (A.2), respectively, hold. The theorem now follows directly from (2.1) for $\rho={\mathrm{VaR}}_{\alpha}$, and by using arguments as in the proof of Krätschmer et al. [19, Theorem 2.6] for $\rho={\mathrm{ES}}_{\alpha}$. □