Tontines with mixed cohorts

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.