## Tesi etd-12012008-153559 |

Thesis type

Tesi di dottorato di ricerca

Author

SILLA, SEBASTIANO

URN

etd-12012008-153559

Title

Valuing guaranteed annuity options using the principle of equivalent utility

Settore scientifico disciplinare

SECS-S/06

Corso di studi

MATEMATICA PER LE DECISIONI ECONOMICHE

Supervisors

**Relatore**Dott. Grasselli, Matheus R.

Parole chiave

- Longevity Risk
- Life Annuity
- Insurance
- Indifference Valuation
- Incomplete Markets
- Hamilton-Jacobi-Bellman equations.
- Guaranteed Annuity Option (g.a.o.)
- Expected Utility
- Annuitization
- Optimal Asset Allocation
- Optimal Consumption/Investment
- Retirement
- Stochastic Control

Data inizio appello

15/12/2008;

Consultabilità

Completa

Riassunto analitico

This thesis considers the valuation of guaranteed annuity options using an equivalent utility principle from the point of view of the policyholder. In this model I give an explicit form to the value functions involved in the indifference valuation. Also I offer a numerical implementation. For instance, in a setting where interest rates are constant, I find an explicit solution for the indifference problem, where the individual is described by a power (instantaneous) utility function. In this setting, I compare two strategies at the time of conversion, and two strategies at the moment when the policy is purchased. In the former, I assume that if the annuitant does not exercise the option, first she withdraws her policy's accumulated funds, and then seeks to solve a standard Merton's problem, under an infinite time horizon setting. In the latter strategy, I compare the agent's expected utility associated to a policy that embeds a guaranteed annuity option, and a policy that does not embed such an option. In order to accumulate the retirement funds, I assume in both cases a pure premium paid at a constant continuous stream. Regarding the optimal strategy, I am able to derive explicit solutions for a class of problems where finite horizon, bequest motive and power consumption utility are jointly considered.

The present research has as a primary objective to elaborate an utility indifference valuation model for guaranteed annuity options. The literature available until now considers both financial and actuarial approaches that have been used to evaluate and describe the nature of such options. On the contrary, the approach I present is able to embed the theory of the optimal asset allocation toward the end of the life cycle in the valuation of guaranteed annuity options. To my knowledge, the indifference approach I propose, is new and never developed before.

The main results show that the option's indifference value, both at the time when the policy is purchased and at the conversion time, depends on the difference between the guaranteed conversion rate $h$ and the market interest rate $r$. In line with the literature, at the time of conversion the agent will in general find advantageous to exercise the option. The dependency on $h$ and $r$ of the equivalent valuation also reveals that in periods characterized by high market interest rates, the value of the g.a.o. turns out to be very small. This model remains coherent if we compare the policyholder's point of view and the insurer's point of view, under an economic setting characterized by high interest rates.

The present model can be extended in order to consider a richer setting, concerning both the accumulation and the decumulation period. These ideas are suggested and described in the course of chapters that follow.

J.E.L. classification. D91; G11; J26.

The present research has as a primary objective to elaborate an utility indifference valuation model for guaranteed annuity options. The literature available until now considers both financial and actuarial approaches that have been used to evaluate and describe the nature of such options. On the contrary, the approach I present is able to embed the theory of the optimal asset allocation toward the end of the life cycle in the valuation of guaranteed annuity options. To my knowledge, the indifference approach I propose, is new and never developed before.

The main results show that the option's indifference value, both at the time when the policy is purchased and at the conversion time, depends on the difference between the guaranteed conversion rate $h$ and the market interest rate $r$. In line with the literature, at the time of conversion the agent will in general find advantageous to exercise the option. The dependency on $h$ and $r$ of the equivalent valuation also reveals that in periods characterized by high market interest rates, the value of the g.a.o. turns out to be very small. This model remains coherent if we compare the policyholder's point of view and the insurer's point of view, under an economic setting characterized by high interest rates.

The present model can be extended in order to consider a richer setting, concerning both the accumulation and the decumulation period. These ideas are suggested and described in the course of chapters that follow.

J.E.L. classification. D91; G11; J26.

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