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MINIMIZING THE PROBABILITY OF LIFETIME RUIN: TWO RISKLESS ASSETS WITH TRANSACTION COSTS

Part of: Hamilton-Jacobi theories, including dynamic programming Optimality conditions

Published online by Cambridge University Press:  01 July 2019

Xiaoqing Liang  and
Virginia R. Young
Xiaoqing Liang
Affiliation:
Department of Statistics, School of Sciences Hebei University of TechnologyTianjin 300401, People’s Republic of China E-Mail: [email protected]
Virginia R. Young*
Affiliation:
Department of Mathematics University of Michigan Ann Arbor, MI, 48109, USA
*
E-Mail: [email protected]
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Abstract

We compute the optimal investment strategy for an individual who wishes to minimize her probability of lifetime ruin. The financial market in which she invests consists of two riskless assets. One riskless asset is a money market, and she consumes from that account. The other riskless asset is a bond that earns a higher interest rate than the money market, but buying and selling bonds are subject to proportional transaction costs. We consider the following three cases. (1) The individual is allowed to borrow from both riskless assets; ruin occurs if total imputed wealth reaches zero. Under the optimal strategy, the individual does not sell short the bond. However, she might wish to borrow from the money market to fund her consumption. Thus, in the next two cases, we seek to limit borrowing from that account. (2) We assume that the individual pays a higher rate to borrow than she earns on the money market. (3) The individual is not allowed to borrow from either asset; ruin occurs if both the money market and bond accounts reach zero wealth. We prove that the borrowing rate in case (2) acts as a parameter connecting the two seemingly unrelated cases (1) and (3).

Keywords

Probability of lifetime ruintransaction costsdeterministic controlriskless assetsC58C61G22G11

MSC classification

Primary: 49K10: Free problems in two or more independent variables
Secondary: 49K20: Problems involving partial differential equations 49L20: Dynamic programming method
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 847 - 883
Copyright
© Astin Bulletin 2019 

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References

Azcue, P. and Muler, N (2013) Minimizing the ruin probability allowing investments in two assets: A two-dimensional problem. Mathematical Methods of Operations Research, 77(2), 177–206.CrossRefGoogle Scholar
Azcue, P and Muler, N (2014) Stochastic Optimization in Insurance: A Dynamic Programming Approach. New York: Springer.CrossRefGoogle Scholar
Bayraktar, E., David Promislow, S. and Young, V.R. (2016) Purchasing life insurance to reach a bequest goal while consuming. SIAM Journal on Financial Mathematics, 7(1), 183–214.CrossRefGoogle Scholar
Bayraktar, E. and Young, V.R. (2007) Minimizing the probability of lifetime ruin under borrowing constraints. Insurance: Mathematics and Economics, 41(1), 196–221.Google Scholar
Bayraktar, E. and Young, V.R. (2016) Optimally investing to reach a bequest goal. Insurance: Mathematics and Economics, 70, 1–10.Google Scholar
Bayraktar, E. and Zhang, Y (2015a) Minimizing the probability of lifetime ruin under ambiguity aversion. SIAM Journal on Control and Optimization, 53(1), 58–90.CrossRefGoogle Scholar
Bayraktar, E. and Zhang, Y (2015b) Stochastic Perron’s method for the probability of lifetime ruin problem under transaction costs. SIAM Journal on Control and Optimization, 53(1), 91–113.CrossRefGoogle Scholar
Browne, S. (1997) Survival and growth with a liability: Optimal portfolio strategies in continuous time. Mathematics of Operations Research, 22(2), 468–493.CrossRefGoogle Scholar
Constantinides, G. (1986). Capital market equilibrium with transaction costs. Journal of Political Economy, 94(4), 842–862.CrossRefGoogle Scholar
Davis, M.H.A. and Norman, A.R. (1990). Portfolio selection with transaction costs. Mathematics of Operations Research, 15(4), 676–713.CrossRefGoogle Scholar
Delgado, F., Dumas, B. and Puopolo, GW. (2015) Hysteresis bands on returns, holding period and transaction costs. Journal of Banking and Finance, 57, 86–100.CrossRefGoogle Scholar
Hipp, C. and Plum, M. (2000) Optimal investment for insurers. Insurance: Mathematics and Economics, 27(2), 215–228.Google Scholar
Kallsen, J. and Muhle-Karbe, J. (2017) The general structure of optimal investment and consumption with small transaction costs. Mathematical Finance, 27(3), 659–703.CrossRefGoogle Scholar
Liang, X. and Young, V.R. (2018) Minimizing the probability of ruin: Two riskless assets with transaction costs and proportional reinsurance. Statistics and Probability Letters, 140, 167–175.CrossRefGoogle Scholar
Magill, M.J.P. and Constantinides, G. (1976) Portfolio selection with transaction costs. Journal of Economic Theory, 13(2), 245–263.CrossRefGoogle Scholar
Milevsky, M.A., Moore, K.S. and Young, V.R. (2006) Asset allocation and annuity-purchase strategies to minimize the probability of financial ruin. Mathematical Finance, 16(4), 647–671.CrossRefGoogle Scholar
Pestien, V.C. and Sudderth, W.D. (1985) Continuous-time red and black: How to control a diffusion to a goal. Mathematics of Operations Research, 10(4), 599–611.CrossRefGoogle Scholar
Shreve, S.E., Mete Soner, H. and Xu, G.-L. (1991). Optimal investment and consumption with two bonds and transaction costs. Mathematical Finance, 1(3), 53–84.CrossRefGoogle Scholar
Thonhauser, S. (2013) Optimal investment under transaction costs for an insurer. European Actuarial Journal, 3(2), 359–383.CrossRefGoogle Scholar
Vayanos, D. and Vila, J.-L. (1999) Equilibrium interest rate and liquidity premium with transaction costs. Economic Theory, 13(3), 509–539.CrossRefGoogle Scholar
Wang, T. and Young, V.R. (2012) Optimal commutable annuities to minimize the probability of lifetime ruin. Insurance: Mathematics and Economics, 50(1), 200–216.Google Scholar
Yang, Z. and Huang, L. (2004) Optimal portfolio strategies with a liability and random risk: the case of different lending and borrowing rates. Journal of Applied Mathematics and Computing, 15(1–2), 109–126.CrossRefGoogle Scholar
Young, V.R. (2004) Optimal investment strategy to minimize the probability of lifetime ruin. North American Actuarial Journal, 8(4), 105–126.CrossRefGoogle Scholar
Zariphopoulou, T. (1991) An optimal investment/consumption model with borrowing. Mathematics of Operations Research, 16(4), 802–822.Google Scholar
Zariphopoulou, T. (1992) Investment-consumption models with transaction fees and Markovchain parameters. SIAM Journal on Control and Optimization, 30(3), 613–636.CrossRefGoogle Scholar