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It is assumed that the financial market contains a risk-free asset and a risky asset whose price process satisfies the Stein\u2013Stein stochastic volatility model. To comply with regulatory standards and offer a risk management tool, we integrate the dynamic versions of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and worst-case CVaR (wcCVaR) constraints into the DC pension fund management model. The salary is assumed to be stochastic and characterized by geometric Brownian motion. In the dynamic setting, a CVaR\/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. By using the Lagrange multiplier method and the dynamic programming method to maximize the constant absolute risk aversion (CARA) utility of terminal wealth, we obtain closed-form expressions of optimal investment strategies with and without a VaR constraint. Several numerical examples are provided to illustrate the impact of a dynamic VaR\/CVaR\/wcCVaR constraint and other parameters on the optimal strategy.<\/jats:p>","DOI":"10.3390\/axioms13080543","type":"journal-article","created":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T11:23:46Z","timestamp":1723461826000},"page":"543","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimal Investment Strategy for DC Pension Plan with Stochastic Salary and Value at Risk Constraint in Stochastic Volatility Model"],"prefix":"10.3390","volume":"13","author":[{"given":"Zilan","family":"Liu","sequence":"first","affiliation":[{"name":"Faculty of Economics and Management, Hengyang Normal University, Hengyang 421002, China"},{"name":"School of Business, Hunan Normal University, Changsha 410081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huanying","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yijun","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Finance, Henan University of Economics and Law, Zhengzhou 450016, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ya","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Business, Hunan Normal University, Changsha 410081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0167-6687(00)00073-1","article-title":"Optimal management under stochastic interest rates: The case of a protected defined contribution pension fund","volume":"28","author":"Boulier","year":"2001","journal-title":"Insur. Math. Econ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0167-6687(02)00128-2","article-title":"Optimal investment strategies and risk measures in defined contribution pension schemes","volume":"31","author":"Haberman","year":"2002","journal-title":"Insur. Math. Econ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1016\/j.jedc.2005.03.009","article-title":"Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans","volume":"30","author":"Cairns","year":"2006","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.insmatheco.2014.12.006","article-title":"Mean-variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns","volume":"61","author":"Guan","year":"2015","journal-title":"Insur. Math. Econ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.insmatheco.2015.07.007","article-title":"Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk","volume":"64","author":"Wu","year":"2015","journal-title":"Insur. Math. Econ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4106","DOI":"10.1080\/03610926.2017.1367815","article-title":"Robust portfolio choice for a defined contribution pension plan with stochastic income and interest rate","volume":"47","author":"Sun","year":"2018","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.insmatheco.2018.05.005","article-title":"Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause","volume":"81","author":"Bian","year":"2018","journal-title":"Insur. Math. Econ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.insmatheco.2021.07.001","article-title":"Equilibrium investment strategy for a DC pension plan with learning about stock return predictability","volume":"100","author":"Wang","year":"2021","journal-title":"Insur. Math. Econ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2951","DOI":"10.1080\/02331934.2022.2081083","article-title":"Optimal DC pension investment with square-root factor processes under stochastic income and inflation risks","volume":"72","author":"Zhang","year":"2023","journal-title":"Optimization"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.insmatheco.2023.12.001","article-title":"Optimal investment in defined contribution pension schemes with forward utility preferences","volume":"114","author":"Ng","year":"2024","journal-title":"Insur. Math. Econ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.insmatheco.2016.05.014","article-title":"Optimal management of DC pension plan under loss aversion and Value-at-Risk constraints","volume":"69","author":"Guan","year":"2016","journal-title":"Insur. Math. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7980","DOI":"10.1080\/03610926.2021.1887236","article-title":"Robust optimal investment strategy of DC pension plans with stochastic salary and a return of premiums clause","volume":"51","author":"Yan","year":"2021","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.3934\/jimo.2021025","article-title":"Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria","volume":"18","author":"Chang","year":"2022","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.insmatheco.2009.01.005","article-title":"Optimal portfolios for DC pension plans under a CEV model","volume":"44","author":"Gao","year":"2009","journal-title":"Insur. Math. Econ."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wang, Y., Xu, X., and Zhang, J. (2022). Optimal investment strategy for DC pension plan with deposit loan spread under the CEV model. Axioms, 11.","DOI":"10.3390\/axioms11080382"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.insmatheco.2011.10.013","article-title":"Portfolio selection problem with multiple risky assets under the constant elasticity of variance model","volume":"50","author":"Zhao","year":"2012","journal-title":"Insur. Math. Econ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.insmatheco.2014.05.004","article-title":"Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework","volume":"57","author":"Guan","year":"2014","journal-title":"Insur. Math. Econ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3168","DOI":"10.1080\/03610926.2019.1586938","article-title":"Optimal investment strategy for a DC pension plan with mispricing under the Heston model","volume":"49","author":"Ma","year":"2020","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/s40314-019-0832-6","article-title":"An alternative form to calibrate the correlated Stein-Stein option pricing model","volume":"38","author":"He","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"115662","DOI":"10.1016\/j.cam.2023.115662","article-title":"Analytically pricing European options with a two-factor Stein-Stein model","volume":"440","author":"Lin","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent measures of risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1093\/rfs\/14.2.371","article-title":"Value-at-risk-based risk management: Optimal policies and asset prices","volume":"14","author":"Basak","year":"2001","journal-title":"Rev. Financ. Stud."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"\u010cerb\u00e1kov\u00e1, J. (2006). Worst-case VaR and CVaR. Operations Research Proceedings, Springer.","DOI":"10.1007\/3-540-32539-5_128"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1287\/opre.1080.0683","article-title":"Constructing risk measures from uncertainty sets","volume":"57","author":"Natarajan","year":"2009","journal-title":"Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1317","DOI":"10.1016\/S0165-1889(03)00116-7","article-title":"Optimal portfolios under a value-at-risk constraint","volume":"28","author":"Yiu","year":"2004","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1287\/opre.1070.0433","article-title":"Optimal dynamic trading strategies with risk limits","volume":"56","author":"Cuoco","year":"2008","journal-title":"Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.insmatheco.2016.09.011","article-title":"Optimal reinsurance under dynamic VaR constraint","volume":"71","author":"Zhang","year":"2016","journal-title":"Insur. Math. Econ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1016\/j.cam.2016.10.001","article-title":"Portfolio selection based on a benchmark process with dynamic value-at-risk constraints","volume":"313","author":"Zhang","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2084","DOI":"10.3934\/math.2024104","article-title":"Optimal investment of DC pension plan under a joint VaR-ES constraint","volume":"9","author":"Dong","year":"2024","journal-title":"AIMS Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0167-6687(03)00153-7","article-title":"Optimal investment strategies in the presence of a minimum guarantee","volume":"33","author":"Deelstra","year":"2003","journal-title":"Insur. Math. Econ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s11766-013-3087-9","article-title":"Optimal investment for the defined-contribution pension with stochastic salary under a CEV model","volume":"28","author":"Zhang","year":"2013","journal-title":"Appl. Math.-A J. Chin. Univ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1007\/s11424-016-3171-3","article-title":"Time-consistent investment strategy for DC pension plan with stochastic salary under CEV model","volume":"29","author":"Li","year":"2016","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.jedc.2018.01.023","article-title":"Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility","volume":"88","author":"Zeng","year":"2018","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1262","DOI":"10.3934\/jimo.2021228","article-title":"Optimal portfolios for the DC pension fund with mispricing under the HARA utility framework","volume":"19","author":"Liu","year":"2023","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1080\/14697680701213868","article-title":"Portfolio optimization under the value-at-risk constraint","volume":"7","author":"Pirvu","year":"2007","journal-title":"Quant. Financ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1287\/opre.1080.0684","article-title":"Worst-case conditional value-at-risk with application to robust portfolio management","volume":"57","author":"Zhu","year":"2009","journal-title":"Oper. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/j.insmatheco.2010.06.002","article-title":"Optimal investment\u2013reinsurance policy for an insurance company with VaR constraint","volume":"47","author":"Chen","year":"2010","journal-title":"Insur. Math. Econ."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/543\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:34:36Z","timestamp":1760110476000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/543"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,10]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080543"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080543","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,10]]}}}