State-Dependent Insurance

Study guide: Exam ALTAM

State-Dependent Insurance is a long-term insurance or annuity contract whose payments depend on the state a life occupies in a multiple state model: an annuity paid while the life is in a state, a lump sum paid on a move between states, and premiums paid only in some states. Disability income, long-term care, critical illness and CCRC contracts are all of this kind, and they are priced with the equivalence principle and reserved with a policy value for each state.

$\bar{a}^{ij}_x = \int_0^\infty e^{-\delta t}\;{}_tp^{ij}_x\,dt$

$\bar{A}^{ij}_x = \int_0^\infty e^{-\delta t} \sum_{k \neq j} {}_tp^{ik}_x\,\mu^{kj}_{x+t}\,dt$

${}_tV^{(i)} = \text{EPV}_t(\text{benefits}) - \text{EPV}_t(\text{premiums}) \text{ given } Y(x+t) = i$

Symbols: Following Dickson, Hardy & Waters (chapter 8), $\bar{a}^{ij}_x$ is the EPV of 1 a year paid continuously while the life is in state $j$ , for a life now aged $x$ in state $i$ , and $\bar{A}^{ij}_x$ is the EPV of 1 paid on every move into state $j$ . A benefit on one particular move $k \to j$ keeps only that term of the sum. A term version, such as $\bar{a}^{ij}_{x:\overline{n}|}$ , stops the integral at $n$ , and the annual versions replace the integral by sums such as $\ddot{a}^{ij}_x = \sum_{k \geq 0} v^k\,{}_kp^{ij}_x$ .
Present value random variables: The present value of a sickness benefit of $B$ a year for $n$ years is $B\int_0^n e^{-\delta t}\,\mathbb{1}\{Y(x+t) = 1\}\,dt$ , the discounted time spent sick. Its expected value is $B\,\bar{a}^{01}_{x:\overline{n}|}$ , because the expectation of the indicator at time $t$ is ${}_tp^{01}_x$ .
Premiums: Premiums are usually paid only while the life is healthy and are waived while it is sick, so the premium annuity is $\bar{a}^{00}_{x:\overline{n}|}$ , which counts healthy time after a recovery too. Monthly premiums and benefits come from the annual values by Woolhouse's formula.
Policy values: Each state has its own policy value, and a move between states changes the reserve the insurer needs. Thiele's equations connect the state policy values through the sums at risk ${}_tV^{(j)} - {}_tV^{(i)}$ .

Death Benefit Paid From Either State In a permanent disability model, a healthy life has constant intensities $\mu^{01} = 0.012$ of disablement and $\mu^{02} = 0.008$ of death, and a disabled life dies with intensity $\mu^{12} = 0.04$ . A whole life policy on a healthy life pays $100,000 at the moment of death, whether the life is healthy or disabled when it dies, and $\delta = 0.05$ . Find the EPV.
Answer Death can come from state 0 or from state 1, so the EPV is $100{,}000\left(\mu^{02}\,\bar{a}^{00}_x + \mu^{12}\,\bar{a}^{01}_x\right)$ . With constant intensities, $\bar{a}^{00}_x = 1/(0.05 + 0.02)$ , and $\bar{a}^{01}_x$ is the healthy annuity times $\mu^{01}$ times the disabled annuity $1/(0.05 + 0.04)$ .
 $\begin{align*} \bar{a}^{00}_x &= 14.285714 \\ \bar{a}^{01}_x &= \frac{0.012}{(0.07)(0.09)} \\ &= 1.904762 \\ \text{EPV} &= 100{,}000\left[0.008(14.285714) + 0.04(1.904762)\right] \\ &= 100{,}000(0.114286 + 0.076190) \\ &= 19{,}047.62 \end{align*}$ 
As a check, a benefit paid on death from any living state is worth $1 - \delta\left(\bar{a}^{00}_x + \bar{a}^{01}_x\right) = 1 - 0.05(16.190476) = 0.190476$ per unit. Deaths of disabled lives add two-thirds to the value of deaths from the healthy state, because disabled lives die five times as fast.

Disability Income Premium and Policy Values by State A disability income policy is issued to a healthy life aged 40 and runs to age 65. It pays a benefit of $30,000 a year continuously while the insured is sick, and level premiums are paid continuously while the insured is healthy. The model is healthy–sick–dead with recovery, and $\delta = 0.05$ . You are given $\bar{a}^{00}_{40:\overline{25}|} = 13.0116$ , $\bar{a}^{01}_{40:\overline{25}|} = 0.45542$ , and at age 55, $\bar{a}^{00}_{55:\overline{10}|} = 7.0488$ , $\bar{a}^{01}_{55:\overline{10}|} = 0.36866$ , $\bar{a}^{10}_{55:\overline{10}|} = 0.03687$ and $\bar{a}^{11}_{55:\overline{10}|} = 7.3806$ . Find the annual premium, then the policy values at time 15 for an insured who is healthy and for one who is sick.
Answer A sick insured who recovers starts paying premiums again, so the sick-state policy value has a premium term.
 $\begin{align*} P &= \frac{30{,}000(0.45542)}{13.0116} \\ &= 1{,}050.03 \\ {}_{15}V^{(0)} &= 30{,}000(0.36866) - 1{,}050.03(7.0488) \\ &= 11{,}059.80 - 7{,}401.45 \\ &= 3{,}658.35 \\ {}_{15}V^{(1)} &= 30{,}000(7.3806) - 1{,}050.03(0.03687) \\ &= 221{,}418.00 - 38.71 \\ &= 221{,}379.29 \end{align*}$ 
The insurer holds about $3,700 for a healthy policyholder and about $221,400 for a sick one. Falling sick at age 55 therefore costs the insurer about $217,700 in extra reserve, even though no lump sum is paid.

How it appears on the exam

ALTAM asks candidates to define the present value random variables of state-dependent insurance and annuity benefits and to calculate their EPVs, from given intensities, transition probabilities or annuity values.

Premiums and policy values: ALTAM then asks for the premium by the equivalence principle for state-dependent life insurance, long-term health insurance and CCRCs, and for the policy value in each state. Euler's method on Thiele's equations, or a profit test by state, can follow.

Payment frequency: A question can give annual values and ask for monthly cash flows, which is where the two-term and three-term Woolhouse approximations come in.

Exam traps

The premium annuity: Premiums waived during sickness are valued with $\bar{a}^{00}$ , not with the survival annuity $\bar{a}^{00} + \bar{a}^{01}$ . Using the survival annuity understates the premium.

Recovery and premiums: In a model with recovery, a sick life's policy value subtracts the EPV of the premiums it will pay after recovering, $P\,\bar{a}^{10}$ . Leaving it out overstates the sick-state policy value.

Benefit on a state versus on a move: A sickness annuity uses ${}_tp^{01}_x$ , while a lump sum on falling sick uses ${}_tp^{00}_x\,\mu^{01}_{x+t}$ . Mixing them values a different benefit from the one described.

The starting state: $\bar{a}^{01}_x$ and $\bar{a}^{11}_x$ are different annuities. A policy value for a sick life starts in state 1, so every EPV in it has 1 as its first index.