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Erschienen in: European Actuarial Journal 2/2018

11.06.2018 | Original Research Paper

Optimal management of immunized portfolios

verfasst von: Riccardo Cesari, Vieri Mosco

Erschienen in: European Actuarial Journal | Ausgabe 2/2018

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Abstract

We generalize the contribution of Fong and Vasicek (Financ Anal J 39:73–78, 1983a; Innov Bond Portf Manag Durat Analy Immun 1983:227–238, 1983b; J Financ 39:1541–1546, 1984) by developing a risk-return optimization problem for immunized life insurance portfolios. The M2 measure of risk for arbitrary changes of the term structure of interest rates is used for a bond portfolio with duration matched to a given liability horizon. The immunization case becomes a “passive” strategy among an entire menu of active management decisions in which a partial risk minimization is exchanged for more return potential. As in the classical Markowitz (Portfolio Selection, New York, Wiley, 1959) approach, an efficient frontier at the given horizon provides the optimal tradeoff between risk and return. An empirical application to insurance companies shows that such a perspective may be proved useful to highlight which segregated funds can be re-positioned over the efficient frontier, at the chosen level of the firm’s risk appetite.

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Fußnoten
1
See Messmore [20] who shows that if E > 0 (the present value of assets greater than the present value of liabilities) the immunization of \(E\), i.e. \(D_{E} = 0\), implies \(D_{A} < D_{L}\). In the following we assume a fixed income portfolio with \(A = L\).
 
2
See Appendix 1 for a proof of the theorems. Without loss of generality we assume continuous compounding.
 
3
This multiple asset—single liability case is equivalent to the case of an asset only bond portfolio with target horizon H and target value \(\bar{L} = A(0)\exp \left( {\int \limits_{0}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right)\).
 
4
The duration measure was independently obtained by Hicks [13] with the name of “average period”. Earlier developments could be seen in Lidstone [16].
 
5
Clearly, interest rate dynamics implying constant shifts are not arbitrage-free. See Boyle [3].
 
6
As duration is linked to the first derivative of the price with respect to the interest rate, convexity represents the second derivative. It is easy also to show that \(M^{2} = - \frac{\partial D}{\partial r}\).
 
7
This multiple liability immunization has been generalized by Shiu [28].
 
8
The EIOPA calculation of the liability side in Fig. 1 does not take into account all the contract optionality available in different countries so that the country relative positions could be altered.
 
9
For instance, see the EIOPA official presentation of the 2014 Stress Test in EIOPA [6].
 
10
See EIOPA [7], section C, “Low Yield Module Description and Results”, paragraph 2, sub-paragraph 56.
 
11
See Martellini, Priaulet and Priaulet [19], ch. 6 and the references therein.
 
12
Apart from EIOPA Stress Tests, convexity effects are still ignored in many empirical applications as well as popular textbooks in banking and finance (e.g. Mishkin and Apostolos [21]). Moreover, under the Basel III Capital Requirement Regulation (CRR, in force since 1 January 2014), the Standardised Approach includes a calculation for the own funds requirement for the general risk on debt instruments only based on the duration. See European Banking Authority (EBA), Interactive Single Rulebook, article 340 in https://​www.​eba.​europa.​eu/​regulation-and-policy/​single-rulebook. A partial justification can be found in the relevance of the first order effect (duration) as documented in Schaefer [26] and in the empirical literature on Principal Components (Martellini, Priaulet and Priaulet [19], ch. 3) where the first factor (the parallel shift of the term structure) accounts for 50% to 90% of total variability of interest rate changes, especially in the long side of the yield curve.
 
13
Note that using (realistically) bonds instead of cash flows has no effect on the calculation of duration but it affects the calculation of risk. In fact, the duration of a portfolio of bonds is simply the “portfolio” (average) of bond durations; however, the M2 of a portfolio of bonds is the “portfolio” of M2 (within variance) plus the variance of durations (between variance). See the Appendix 4.
 
14
Note that Fong and Vasicek [10] assume, in the variance calculation, the asymptotic approximation: \(\mathop \sum \nolimits_{t = 0}^{H} M_{t}^{4} = \frac{{M_{0}^{4} }}{{H^{6} }}\mathop \sum \nolimits_{t = 0}^{H} (H - t)^{6} = \frac{{M_{0}^{4} }}{{H^{6} }}\frac{{H(H + 1)(2H + 1)(3H^{4} + 6H^{3} - 3H + 1)}}{42}\mathop \to \limits_{{for H\mathop \to \nolimits_{ } \infty }} \frac{H}{7}M_{0}^{4}\)
so that the variance is proportional to \(\frac{{M_{0}^{4} }}{7H}\). We do not use this asymptotic approximation.
 
15
Because of the profit participation mechanism, the liability cash-flows partly depend on the asset cash-flows, through the returns realized by the assets in the segregated funds. Usually, this effect is negligible.
 
16
Note that the second order (convexity) condition is always satisfied in the single liability case.
 
17
See for example, Smith [30], p. 1670, formula (9) or de La Grandville [5], p. 163, formula (19). With continuous compounding, the formula simplifies into M2 = C – D.
 
18
Bloomberg data include in the price P accrued interest (full price = clean price + accrued interest) and assume a variation of ± 1 basis point w.r.t. the yield implicit in the price P.
 
Literatur
1.
Zurück zum Zitat Abramowitz M, Stegun IA (eds) (1964) Handbook of mathematical functions, with formulas, graphs, and mathematical tables, 10th edn. National Bureau of Standards, USA, p 1972MATH Abramowitz M, Stegun IA (eds) (1964) Handbook of mathematical functions, with formulas, graphs, and mathematical tables, 10th edn. National Bureau of Standards, USA, p 1972MATH
2.
Zurück zum Zitat Albrecht P (1985) A note on immunization under a general stochastic equilibrium model of the term structure. Insur Math Econ 4:239–244MathSciNetCrossRef Albrecht P (1985) A note on immunization under a general stochastic equilibrium model of the term structure. Insur Math Econ 4:239–244MathSciNetCrossRef
3.
Zurück zum Zitat Boyle PP (1978) Immunization under stochastic models of the term structure. J Inst Actuaries 105:177–187CrossRef Boyle PP (1978) Immunization under stochastic models of the term structure. J Inst Actuaries 105:177–187CrossRef
4.
Zurück zum Zitat Cox JC, Ingersoll JE Jr, Ross SA (1979) Duration and the measurement of basis risk. J Bus 52(1):51–61CrossRef Cox JC, Ingersoll JE Jr, Ross SA (1979) Duration and the measurement of basis risk. J Bus 52(1):51–61CrossRef
5.
Zurück zum Zitat de La Grandville O (2001) Bond pricing and portfolio analysis. MIT Press, Cambridge de La Grandville O (2001) Bond pricing and portfolio analysis. MIT Press, Cambridge
7.
Zurück zum Zitat EIOPA (2014 b), Document EIOPA-BOS-14-203, 28 November 2014 EIOPA (2014 b), Document EIOPA-BOS-14-203, 28 November 2014
8.
Zurück zum Zitat Fisher L, Weil RL (1971) Coping with the risk of interest-rate fluctuations: returns to bondholders from naïve and optimal strategies. J Bus 44:408–431CrossRef Fisher L, Weil RL (1971) Coping with the risk of interest-rate fluctuations: returns to bondholders from naïve and optimal strategies. J Bus 44:408–431CrossRef
9.
Zurück zum Zitat Fong HG, Vasicek OA (1983a) The tradeoff between return and risk in immunized portfolios. Financ Anal J 39:73–78CrossRef Fong HG, Vasicek OA (1983a) The tradeoff between return and risk in immunized portfolios. Financ Anal J 39:73–78CrossRef
10.
Zurück zum Zitat Fong HG, Vasicek OA (1983b) Return maximization for immunized portfolios. In: Kaufman GG, Bierwag GO, Toevs A (eds) Innovations in bond portfolio management: duration analysis and immunization. JAI Press, London Fong HG, Vasicek OA (1983b) Return maximization for immunized portfolios. In: Kaufman GG, Bierwag GO, Toevs A (eds) Innovations in bond portfolio management: duration analysis and immunization. JAI Press, London
11.
12.
Zurück zum Zitat Grondin TM (1998) Portfolio Yield? Sure, but …, The society of actuaries, risk and rewards Letters, p 16 Grondin TM (1998) Portfolio Yield? Sure, but …, The society of actuaries, risk and rewards Letters, p 16
13.
Zurück zum Zitat Hicks JR (1939) Value and capital, 2nd edn. 1946, OUP, Oxford Hicks JR (1939) Value and capital, 2nd edn. 1946, OUP, Oxford
14.
Zurück zum Zitat Hull JC, White A (1990) Pricing interest-rate derivative securities. Rev Financ Stud 3:573–592CrossRef Hull JC, White A (1990) Pricing interest-rate derivative securities. Rev Financ Stud 3:573–592CrossRef
15.
Zurück zum Zitat Ingersoll JE Jr, Skelton J, Weil RL (1978) Duration forty years later. J Financ Quant Anal 13:627–650CrossRef Ingersoll JE Jr, Skelton J, Weil RL (1978) Duration forty years later. J Financ Quant Anal 13:627–650CrossRef
16.
Zurück zum Zitat Lidstone GJ (1893) On the approximate calculation of the values of increasing annuities and assurances. J Inst Actuaries 31:68–72MathSciNet Lidstone GJ (1893) On the approximate calculation of the values of increasing annuities and assurances. J Inst Actuaries 31:68–72MathSciNet
17.
Zurück zum Zitat Macaulay FR (1938) Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the United States since 1856. NBER, New York Macaulay FR (1938) Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the United States since 1856. NBER, New York
18.
Zurück zum Zitat Markowitz, H. (1959), Portfolio selection: efficient diversification of investments, Cowles Foundation Monograph n. 16, New York, Wiley Markowitz, H. (1959), Portfolio selection: efficient diversification of investments, Cowles Foundation Monograph n. 16, New York, Wiley
19.
Zurück zum Zitat Martellini L, Priaulet P, Priaulet S (2003) Fixed-income securities. Valuation, risk management and portfolio strategies. Wiley, Chichester (UK) Martellini L, Priaulet P, Priaulet S (2003) Fixed-income securities. Valuation, risk management and portfolio strategies. Wiley, Chichester (UK)
20.
Zurück zum Zitat Messmore TE (1990) The duration of surplus. J Portf Manag 16(2):19–22CrossRef Messmore TE (1990) The duration of surplus. J Portf Manag 16(2):19–22CrossRef
21.
Zurück zum Zitat Mishkin FS, Apostolos S (2011) The economics of money, banking and financial markets, 4th edn. Pearson, Toronto Mishkin FS, Apostolos S (2011) The economics of money, banking and financial markets, 4th edn. Pearson, Toronto
22.
Zurück zum Zitat Montrucchio L, Peccati L (1991) A note on Shiu–Fisher–Weil immunization theorem. Insur Math Econ 10:125–131MathSciNetCrossRef Montrucchio L, Peccati L (1991) A note on Shiu–Fisher–Weil immunization theorem. Insur Math Econ 10:125–131MathSciNetCrossRef
24.
Zurück zum Zitat Redington FM (1952) Review of the principles of life-office valuations. J Inst Actuaries 78(350):286–340 Redington FM (1952) Review of the principles of life-office valuations. J Inst Actuaries 78(350):286–340
25.
Zurück zum Zitat Samuelson PA (1945) The effect of interest rate increases on the banking system. Am Econ Rev 35(1):16–27MathSciNet Samuelson PA (1945) The effect of interest rate increases on the banking system. Am Econ Rev 35(1):16–27MathSciNet
26.
Zurück zum Zitat Schaefer SM (1984) Immunization and duration: a review of theory, performance and applications. Midl Corpor Financ J Fall 2(2):41–59 Schaefer SM (1984) Immunization and duration: a review of theory, performance and applications. Midl Corpor Financ J Fall 2(2):41–59
29.
Zurück zum Zitat Shiu ESW (1990) On Redington’s theory of immunization. Insur Math Econ 9:171–175CrossRef Shiu ESW (1990) On Redington’s theory of immunization. Insur Math Econ 9:171–175CrossRef
30.
Zurück zum Zitat Smith DJ (2010) Bond portfolio duration, cash flow dispersion and convexity. Appl Econ Lett 17(2010):1669–1672CrossRef Smith DJ (2010) Bond portfolio duration, cash flow dispersion and convexity. Appl Econ Lett 17(2010):1669–1672CrossRef
31.
Zurück zum Zitat Vasicek OA (1977) An equilibrium characterization of the term structure. J Financ Econ 5:177–188CrossRef Vasicek OA (1977) An equilibrium characterization of the term structure. J Financ Econ 5:177–188CrossRef
Metadaten
Titel
Optimal management of immunized portfolios
verfasst von
Riccardo Cesari
Vieri Mosco
Publikationsdatum
11.06.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
European Actuarial Journal / Ausgabe 2/2018
Print ISSN: 2190-9733
Elektronische ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-018-0174-6

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