Optimal behavior strategy in the GMIB product

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Guaranteed Minimum Income benefit are variable annuities contract, which offer the policyholder the possibility to con- vert the guarantee level into an annuities income for life. This paper focuses on the optimal customer behavior assuming the maximization of the discounted expected future cash flows over the full life of the contract duration. Using convenient scaling properties of the contract value enables to reduce the complexity (dimension) of the problem and to characterize the policyholder’s decision as a function of the contract moneyness across four main choices: zero withdrawals, guaranteed withdrawals, lapse and the income period election. Sensitivities to key drivers such as the market volatility, the interest rate and the roll-up rate illustrate how crucial are not only the environment, but also the product design features, in order to ensure a fair and robust pricing for both customer and life insurer. In particular, the authors find that most empirical contracts are usually underpriced compared to mean optimal behavior pricing, which empirically translated into multiple updates of behavior assumptions and re-reserving by life insurers in the recent years.

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    • Fig. 1. Behavior of the account value and the different options for the GMIB benefit base when no withdrawals have been performed
    • Fig. 2. The withdrawals strategy γ ̃=γ/G as a function of time tand moneyness A/G
    • Fig. 3. The withdrawals strategy as a function of time t and moneyness A/G
    • Fig. A1. Policyholder optimal withdrawals amount as a function of the account value A and benefit base G for Product A and A-DB
    • Fig. A2. The withdrawals strategy as a function of time t and moneyness A/G
    • Fig. A3. The withdrawals strategy as a function of time t and moneyness A/G
    • Fig. A4. The withdrawals strategy as a function of time tand moneyness A/G for σ = 10%
    • Fig. A5. The withdrawals strategy as a function of time tand moneyness A/G for σ = 30%
    • Fig. A6. The contract value at inception as a function of the total fees α┴- for Product A with and without DB for r = 2%, 4%
    • Fig. A7. The contract value at inception as a function of the roll-up rate η for Product A and B with and without DB for r= 2%
    • Table 1. GMIB benefit base evolution for an allocation of $100,000 with partial withdrawals that does not affect the value of the guaranteed account
    • Table 2. Protected account value and GMIB behavior given a static withdrawals strategy and lifetime payments
    • Table 3. GMIB and GMDB behavior given policyholder’s withdrawals when the account value is less than the GMIB benefit base at the time of the first withdrawal
    • Table 4. GMIB and GMDB behavior given policy holder’s withdrawals when the account value is greater than the GMIB benefit base at the time of the first withdrawal
    • Table 5. Model parameters