Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Finance and Stochastics
Published in: Finance and Stochastics 1/2018

13-11-2017

Replicating portfolio approach to capital calculation

Authors: Mathieu Cambou, Damir Filipović

Published in: Finance and Stochastics | Issue 1/2018

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.
previous article Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs
next article An enlargement of filtration formula with applications to multiple non-ordered default times

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Appendix
Available only for authorised users
Footnotes
1
There is no such relation between \(C_{1}^{{\mathbb {P}}}\) and \(C_{2}^{{\mathbb {P}}}\). For example, let \(D_{T}/D_{1}\) be such that we have \({\mathbb {E}}^{\mathbb {P}} [D_{T}^{2}/D_{1}^{2}\mid{\mathcal {F}}_{1} ]=1/D_{1}^{2}\) and assume that \({\mathbb {E}}^{\mathbb {P}}[1/D_{1}^{2} ]=\infty\). Then \(C_{1}^{{\mathbb {P}}}=1\), but \(C_{2}^{{\mathbb {P}}}=\infty\). Conversely, assume that \(D_{T}=D_{1}\) and \({\mathbb {E}}^{\mathbb {P}}[D_{1}^{2} ]=\infty\). Then \(C_{2}^{{\mathbb {P}}}=1\), but \(C_{1}^{{\mathbb {P}}}=\infty\).
 
2
The computation of the initial asset–liability value is not the subject of this paper. It could be estimated by the same methods. The value of insurance liabilities in practice includes a risk margin that is determined as cost of future solvency capital for the asset–liability portfolio. For more details, we refer to Filipović [12, Sect. 1].
 
3
Formally, we assume that primary and sample random variables \(A(\omega)= A(\omega_{1})\) and \(A^{(j)}(\omega)= A^{(j)}(\omega_{2})\), etc., with \(\omega=(\omega_{1},\omega_{2})\), are modeled on a product space \(\Omega=\Omega'\times\Omega'\), \({\mathcal {F}}= {\mathcal {F}}'\otimes{\mathcal {F}}'\) equipped with product probability measures \({\mathbb {M}}= {\mathbb {M}}'\otimes{\mathbb {M}}'\).
 
4
While the rebalancing frequency of a real insurance asset–liability portfolio is adaptive, quarterly rebalancing for the long-term projection is a reasonable assumption.
 
Literature
1.
Acerbi, C., Tasche, D.: On the coherence of expected shortfall. J. Bank. Finance 26, 1487–1503 (2002) CrossRef
2.
Bauer, D., Bergmann, D., Kiesel, R.: On the risk-neutral valuation of life insurance contracts with numerical methods in view. ASTIN Bull. 40, 65–95 (2010) CrossRefMATHMathSciNet
3.
4.
5.
6.
7.
8.
Cont, R., Deguest, R., Scandolo, G.: Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10, 593–606 (2010) CrossRefMATHMathSciNet
9.
10.
Émery, M.: A discrete approach to the chaotic representation property. In: Azéma, J., et al. (eds.) Séminaire de Probabilités, XXXV. Lecture Notes in Math., vol. 1755, pp. 123–138. Springer, Berlin (2001) CrossRef
11.
Émery, M.: Chaotic representation property of certain Azéma martingales. Ill. J. Math. 50(1–4), 395–411 (2006) MATH
12.
13.
14.
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, Berlin (2011) CrossRefMATH
15.
16.
Glasserman, P.: Monte Carlo Methods in Financial Engineering. Applications of Mathematics (New York), vol. 53. Springer, New York (2004) MATH
17.
Gordy, M.B., Juneja, S.: Nested simulation in portfolio risk measurement. Manag. Sci. 56, 1833–1848 (2010) CrossRefMATH
18.
19.
Krätschmer, V., Schied, A., Zähle, H.: Comparative and qualitative robustness for law-invariant risk measures. Finance Stoch. 18, 271–295 (2014) CrossRefMATHMathSciNet
20.
21.
Natolski, J., Werner, R.: Mathematical analysis of different approaches for replicating portfolios. Eur. Actuar. J. 4, 411–435 (2014) CrossRefMATHMathSciNet
22.
Natolski, J., Werner, R.: Improving optimal terminal value replicating portfolios. In: Glau, K., et al. (eds.) Innovations in Quantitative Risk Management: TU München, September 2013, pp. 289–301. Springer, Cham (2015)
23.
24.
Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications (New York). Springer, New York (1995) CrossRefMATH
25.
Oechslin, J., Aubry, O., Aellig, M., Käppeli, A., Brönnimann, D., Tandonnet, A., Valois, G.: Replicating embedded options. Life Pensions Risk, February, 47–52 (2007)
26.
Pelsser, A., Schweizer, J.: The difference between LSMC and replicating portfolio in insurance liability modeling. Eur. Actuar. J. 6, 441–494 (2016) CrossRefMATHMathSciNet
27.
Metadata
Title
Replicating portfolio approach to capital calculation
Authors
Mathieu Cambou
Damir Filipović
Publication date
13-11-2017
Publisher
Springer Berlin Heidelberg
Published in
Finance and Stochastics / Issue 1/2018
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-017-0347-1

Other articles of this Issue 1/2018

Finance and Stochastics 1/2018 Go to the issue

OriginalPaper

Optimal liquidation under stochastic liquidity

OriginalPaper

An enlargement of filtration formula with applications to multiple non-ordered default times

OriginalPaper

Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

OriginalPaper

Financial equilibrium with asymmetric information and random horizon

OriginalPaper

Time-consistent stopping under decreasing impatience

OriginalPaper

Dynamic programming approach to principal–agent problems