﻿ Projections and profitability of unit-linked products

# Projections and profitability of unit-linked products

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Speaker(s): Kristian Juul Schomacker (Edlund A/S)

Calculation of cash flows for unit-linked products is a common actuarial task. We propose a multistate model to treat simultaneously biometric risk and policyholder options. Within the model, we get insight into profitability for the insurance company and understand the effect on profitability of management actions using input from stochastic scenarios.
Projections and cash flow calculations for unit-linked products with lifetime uncertainty is treated in Jensen & Schomacker (2015). Here we extend the model to include surrender and disability. More specifically the model includes the states surrender, active, dead, and disabled allowing for reactivation (transitioning from disabled to active).
A crucial ingredient for cash flows in unit-linked products is the insured's deposit. Rolling the deposit forward is formulated using Thiele differential equations. For a classical guaranteed product, this is a basic exercise in ordinary differential equations with given (guaranteed) terminal conditions and an equivalence initial condition for the state of initiation. For a unit-linked product, it is much more involved. Here the terminal condition is replaced by a link between the uncertain terminal values for the different states, since e.g. the terminal value of the contract in the disability state is given as a function of the terminal value of the contract in the active state, which again is a function of the financial scenario. Thus, a system of equation with side conditions at different sides occur, a so-called forward-backward differential equation. This is interesting from a mathematical point of view as well as a numerical point of view.
We present the general problem of the forward-backward system, discuss the various assumptions it takes to disentangle this forward-backward problem, and present relevant numerical illustrations. The economic scenarios inputted directly affect the size of the deposit in all states of the contract. This is reflected in the numerical illustrations.