Disability Income Reserves With Recovery

Why Disability Income Reserves Need More Than One State

A disability income contract has different future cash flows depending on whether the insured is currently active or disabled. An active insured may enter disability, while a disabled insured may recover. Mortality can end coverage from either state. As a result, one reserve value is not enough: the reserve must be specified for each living state, and transitions between those states affect how it changes over time.

This tutorial uses a hypothetical finite-term contract with continuous premiums and disability income. All forces are constant within an example, and all monetary flows are measured per year. The model is a continuous-time Markov model: conditional on the current state, transition times have exponential waiting-time distributions with the stated constant intensities.

Write \(p_A\) and \(p_D\) for premium income rates in the active and disabled states, respectively, and \(b\) for the continuous disability income rate while disabled. There is no continuous benefit while active, no lump-sum payment on incidence or recovery, and no benefit on death. Thus the reserve in the dead state is zero. If the contract ends at time \(T\) with no maturity payment, the terminal conditions are \(V_A(T)=V_D(T)=0\).

These assumptions matter. A death benefit would change the mortality transition term, and a lump-sum payment on recovery would change the recovery term. Neither is included here.

Thiele Equations for the Disability Model

For a living state, the reserve changes with interest, the continuous net cash flow, and the possible movement to another state. With premiums treated as income and benefits as outgo, the equations for this model are:

Equivalently, after collecting terms,

The incidence term in the active equation depends on the difference between the disabled-state and active-state reserves. Incidence moves the policy from \(A\) to \(D\), so the change in liability at that transition is \(V_D-V_A\). Recovery similarly uses \(V_A-V_D\). Mortality leads to a state with zero reserve under the stated assumptions.

Why the Transition Terms Have This Form

A short-interval argument explains the equation and its signs. Suppose the policy is in state \(i\) at time \(t\). Let \(g_i\) be its net outgo rate during the interval, equal to benefit outgo minus premium income. Over a short interval of length \(h\), a transition from \(i\) to \(j\) occurs with probability \(\lambda_{ij}h+o(h)\), while remaining in \(i\) has probability \(1-\sum_j\lambda_{ij}h+o(h)\). Discounting one interval at force \(\delta\) gives, to first order,

Use \(V_i(t+h)=V_i(t)+V_i'(t)h+o(h)\), subtract \(V_i(t)\), divide by \(h\), and let \(h\) decrease to zero. The result is

Since \(g_i\) is benefits minus premiums, \(-g_i\) is premium income minus benefits. This gives the displayed disability equations. The argument assumes finite transition intensities and no lump-sum transition payments; a lump-sum payment at a transition would need to be added to the transition cash flow.

A Useful Closed Form When the Forces Are Symmetric

For transparent hand calculations, suppose incidence and recovery have the same intensity \(k\), mortality has the same intensity \(\mu\) in both living states, and the forces remain constant until the end of a term of length \(T\). Define \(c_A=p_A\) and \(c_D=p_D-b\), the premium income less continuous benefit outgo in each state. The equations become

The sum of the two reserve components evolves with coefficient \(q-k=\delta+\mu\), and their difference evolves with coefficient \(q+k=\delta+\mu+2k\). Define

The finite-term reserves at time zero are therefore

To justify the finite-term expression, write \(M\) for the displayed coefficient matrix and \(c=(c_A,c_D)^\mathsf{T}\). The solution of \(V'=MV+c\) with \(V(T)=0\) gives \(V(0)=-\int_0^T e^{-Ms}c\,ds\). The sum and difference directions are eigenvectors of \(M\), with eigenvalues \(\delta+\mu\) and \(\delta+\mu+2k\). Integrating in those two directions gives the formulas above. They apply only under the stated symmetric, constant-force assumptions; unequal incidence, recovery, or mortality generally requires solving the full coupled system.

Common Sign and Boundary Checks

A useful check is to inspect each transition as a change in conditional liability. In the active equation, incidence contributes \(-\alpha(V_D-V_A)=\alpha(V_A-V_D)\). In the disabled equation, recovery contributes \(-\rho(V_A-V_D)=\rho(V_D-V_A)\). For example, writing the recovery contribution as \(+\rho(V_A-V_D)\) is the wrong sign: it reverses the effect of moving from disabled to active.

Also check that the reserve being calculated is conditional on the current state. The active reserve does not include a disability benefit immediately just because the contract covers disability; it includes the expected future cost of disability after allowing for the incidence intensity. The disabled reserve includes the current continuous income outgo, as well as the possibility of recovery and mortality.

• Use the remaining term in the terminal condition, and set both living-state reserves to zero at expiry only when there is no maturity payment.
• Use annual intensities and annual cash-flow rates consistently with time measured in years.
• Include a premium only in the state where it is payable; a waiver during disability means \(p_D=0\).
• Check that mortality leads to zero reserve only under the no-death-benefit assumption.

Disability income reserves are a direct application of multi-state reserve principles: the reserve depends both on cash flows in the current state and on the reserve consequences of entering another state. This setup is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.