With the advancements of medical technology and the improvements in quality of life, the demand for innovative retirement products designed to address increasing longevity risks has been growing in recent decades. Tontines and tontine-like products, where the insurers and policyholder share longevity risks, are being explored as an alternative to annuities. As of now, homogeneous policyholders are often assumed in the design process of this mortality-pooling product type. Inspired by the work of Milevsky M. A. & Salisbury T. S. [(2016). Equitable retirement income tontines: mixing cohorts without discriminating. ASTIN Bulletin 46(3), 571–604] in which heterogeneous cohorts are considered, we also extend the tontine products to heterogeneous policyholders. Different from the method employed in Milevsky M. A. & Salisbury T. S. [(2016). Equitable retirement income tontines: mixing cohorts without discriminating. ASTIN Bulletin 46(3), 571–604], we establish the explicit expression of optimal withdrawal rates under the actuarial fairness budget constraint held for one single cohort. In addition, we propose a numerical procedure to achieve approximate fairness among the cohorts by choosing participation rates (or the share prices) properly.

随着医疗技术的进步和生活质量的提高，近几十年来，对旨在应对日益增加的长寿风险的创新退休产品的需求不断增长。保险公司和投保人分担长寿风险的联合养老保险和类似联合养老保险的产品正在被探索作为年金的替代品。截至目前，在这种死亡率汇集产品类型的设计过程中，经常假设保单持有人是同质的。受到 Milevsky MA 和 Salisbury TS [(2016) 工作的启发。公平的退休收入联合养老保险：混合队列而不歧视。ASTIN Bulletin 46(3), 571–604] 其中考虑了异质群组，我们还将联合保险产品扩展到异质保单持有人。与 Milevsky MA & Salisbury TS [(2016) 中采用的方法不同。公平的退休收入联合养老保险：混合队列而不歧视。ASTIN Bulletin 46(3), 571–604]，我们在为一个单一队列持有的精算公平预算约束下建立了最佳提款率的明确表达。此外，我们提出了一个数值程序，通过正确选择参与率（或股价）来实现群组之间的近似公平。