Abstract
In this paper, we pursue the optimal reinsurance-investment strategy of an insurer who can invest in both domestic and foreign markets. We assume that both the domestic and the foreign nominal interest rates are described by extended Cox-Ingersoll-Ross (CIR) models. In order to hedge the risk associated to investments, rolling bonds, treasury inflation protected securities and futures are purchased by the insurer. We use the dynamic programming principles to explicitly derive both the value function and the optimal reinsurance-investment strategy. As a conclusion, we analyze the impact of the model parameters on both the optimal strategy and the optimal utility.
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Allayannis G, Ofek E (2001) Exchange rate exposure, hedging, and the use of foreign currency derivatives. J Int Money Financ 20(2):273–296. https://doi.org/10.1016/S0261-5606(00)00050-4
Amin KI, Jarrow RA (1991) Pricing foreign currency options under stochastic interest rates. J Int Money Financ 10(3):310–329
Bai L, Guo J (2008) Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance: Mathematics and Economics 42(3):968–975
Brennan MJ, Cao HH (1997) International portfolio investment flows. J Financ 52(5):1851–1880 https://doi.org/10.1111/j.1540-6261.1997.tb02744.x
Brennan MJ, Xia Y (2002) Dynamic asset allocation under inflation. J Financ 57(3):1201–1238. https://doi.org/10.1111/1540-6261.00459
Browne S (1995) Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of Ruin. https://doi.org/10.1287/moor.20.4.937
Cox JC, Ingersoll JrJE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica: Journal of the Econometric Society 53(2):385–407. http://www.jstor.org/stable/1911242
Duffie D, Kan R (1996) A yield-factor model of interest rates. Math Financ 6(4):379–406
Gu A, Guo X, Li Z, Zeng Y (2012) Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insurance: Mathematics and Economics 51(3):674–684. https://doi.org/10.1016/j.insmatheco.2012.09.003
Guan G, Liang Z (2014a) Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework. Insurance: Mathematics and Economics 57(1):58–66. https://doi.org/10.1016/j.insmatheco.2014.05.004
Guan G, Liang Z (2014b) Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks. Insurance: Mathematics and Economics 55(1):105–115. https://doi.org/10.1016/j.insmatheco.2014.01.007
Jarrow R, Yildirim Y (2003) Pricing treasury inflation protected securities and related derivatives using an HJM model. J Financ Quant Anal 38(2):337–358. https://doi.org/10.2307/4126754
Jorion P (1989) Asset allocation with hedged and unhedged foreign stocks and bonds. J Portf Manag 15(4):49–54. https://doi.org/10.3905/jpm.1989.409221
Levy H, Sarnat M (1970) Internatioal diversification of investment portfolios. Am Econ Assoc 60(4):668–675. http://www.jstor.org/stable/1818410
Li Z, Zeng Y, Lai Y (2012) Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model. Insurance: Mathematics and Economics 51(1):191–203. https://doi.org/10.1016/j.insmatheco.2011.09.002
Liang Z, Yuen KC, Guo J (2011) Optimal proportional reinsurance and investment in a stock market with Ornstein–Uhlenbeck process. Insurance: Mathematics and Economics 49(2):207–215. https://doi.org/10.1016/j.insmatheco.2011.04.005
Lioui A, Poncet P (2002) Optimal currency risk hedging. J Int Money Financ 21(2):241–264
Nawalkha SK, Beliaeva NA, Soto GM (2007) Dynamic term structure modeling: the fixed income valuation course. Wiley, Hoboken
Promislow SD, Young VR (2005) Minimizing the probability of ruin when claims follow brownian motion with drift. North American Actuarial Journal 9(3):110–128. https://doi.org/10.1080/10920277.2005.10596214
Schmidli H (2001) Optimal proportional reinsurance policies in a dynamic setting. Scand Actuar J 2001(1):55–68. https://doi.org/10.1080/034612301750077338
Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5(2):177–188. https://doi.org/10.1016/0304-405X(77)90016-2
Yi B, Viens F, Li Z, Zeng Y (2015) Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria. Scand Actuar J 2015(8):725–751. https://doi.org/10.1080/03461238.2014.883085
Acknowledgements
This research has been carried out with funding provided by the Alexander von Humboldt Foundation, under the programme financed by the German Federal Ministry of Education and Research entitled German Research Chair No 01DG15010 and by the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement No. 318984-RARE. The second author gratefully acknowledge the support of Natural Science Foundation of China under grant agreement No. 71532001.
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Appendix
Appendix
1.1 A.1 The proof of Theorem 2
Proof
Suppose \(V(t,{r_{n}^{d}},{r_{n}^{f}},i^{d},y)\) hasthe form of (27). Substituting (27) into (26) yields the following differential equation for \(h(t,{r_{n}^{d}},{r_{n}^{f}})\):
From (37), we may assume that \(h(t,{r_{n}^{d}},{r_{n}^{f}})\) canbe separated by independent variables and write\(h(t,{r_{n}^{d}},{r_{n}^{f}})=\exp \{{r_{n}^{d}} q_{1}(t)+{r_{n}^{f}} q_{2}(t)+q_{3}(t)\}\) with boundary conditions q1(T) = q2(T) = q3(T) = 0. Substituting \(h(t,{r_{n}^{d}},{r_{n}^{f}})\), Λt, σd, σf, and σI into (37), we obtain
where
Since (38) is true for all \({r_{n}^{d}}\) and\({r_{n}^{f}}\), it follows one must have
For j = 1, 2,qj(t) is a Riccati equation with constant coefficients. Then using Nawalkha et al. (2007), qj(t) is explicitly given by
where \(\alpha _{j}=\frac {M_{j}+\sqrt {{M_{j}^{2}}-4N_{j} R_{j}}}{2}\),\(\beta _{j}=\frac {M_{j}-\sqrt {{M_{j}^{2}}-4N_{j} R_{j}}}{2}\). Since
Let us observe that (41) makes sense if γ > 1, so that we have \({M_{j}^{2}}-4N_{j} R_{j}>0(j = 1,2)\). Integrating both sides of (40) from t to T and using the fact that q3(T) = 0, we can obtain (30). □
1.2 A.2 The Proof of Theorem 3
Proof
Since supremum (26) is attained at u∗(t), applying the first order maximum condition, we have
Then
where\(\boldsymbol {{\Sigma }}=\boldsymbol {\sigma }_{t} \boldsymbol {\sigma }_{t}^{\prime }\). Substitute (27) and (28) into the last equation to obtain
Equation (31) follows by substituting σt, Λt, σd, σf and σI into (44).
In order to obtain the optimal strategy u∗(t), we need to find the SDE satisfied by Y∗(t) corresponding to u∗(t). Now, substituting (44) into (24), we have
where
\(\boldsymbol {D}_{1}=\frac {1}{\gamma }\!\big (\!\frac {\lambda \mu _{1} \theta }{\sqrt {\lambda \mu _{2}}},\sqrt {{k_{1}^{d}} {r_{n}^{d}}(t)\,+\,{k_{2}^{d}}}(\!{\lambda _{r}^{d}}\,+\,(\!\gamma \,-\,1\!)\sigma _{I_{1}}^{d}\,-\,q_{1}\!(t)\!),\sqrt {{k_{1}^{f}} {r_{n}^{f}}(t)\,+\,{k_{2}^{f}}}(\!{\lambda _{r}^{f}}\,-\,q_{2}\!(t)\!),\lambda _{Q},{\lambda _{S}^{d}},{\lambda _{S}^{f}},{\lambda _{I}^{d}}\,+\,(\!\gamma \,-\,1\!)\sigma _{I_{2}}^{d},{\lambda _{I}^{f}}\!\big )'\). Y∗(t) can also be obtained by using Y∗(t) = X∗(t) + G(t). In order toget Y∗(t), let us introduce an auxiliary process {Z(t)}t ∈ [0, T] representing the insurance risk and satisfying:
Observing (17), let us assume that Y∗(t) is of the following form:
Using Itô’s formula, we get
where
By the uniqueness of the solution of SDE, we can obtain (33)–(35).□
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Guo, C., Zhuo, X., Constantinescu, C. et al. Optimal Reinsurance-Investment Strategy Under Risks of Interest Rate, Exchange Rate and Inflation. Methodol Comput Appl Probab 20, 1477–1502 (2018). https://doi.org/10.1007/s11009-018-9630-7
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DOI: https://doi.org/10.1007/s11009-018-9630-7
Keywords
- Optimal reinsurance-investment strategy
- Foreign exchange market
- Extended CIR
- Stochastic inflation
- Dynamic programming principle