Valuing State-Contingent Disability Cash Flows

Introduction

A disability contract can have several cash flows at different dates: a premium may be due only while the insured is active, while a benefit may be paid only if the insured is disabled. Valuation therefore requires more than discounting a list of amounts. For each amount, the actuary must identify the payment date, the relevant state at that date, and the probability of being in that state.

This tutorial values payments made at specified times and conditional on the state then occupied. It uses discrete payment dates and transition probabilities, which may be provided directly or obtained from a multiple-state model. It does not assume that an endpoint state reveals whether a transition occurred between payment dates.

Definition: Let $J_k$ denote the insured’s state at time $k$, where, for example, $A$ is active, $D$ is disabled, and $X$ is dead. If a payment of amount $c_k(s)$ is made at time $k$ when the state is $s$, its expected present value at time 0 is $$ v^k \sum_s P(J_k=s)c_k(s), $$ where $v=1/(1+i)$ for an effective annual interest rate $i$, and the sum includes all possible states. A payment that is not made in a state has amount zero in that state.

From State Probabilities to Expected Present Value

A discrete-time state model specifies the probabilities of moving from each current state to each possible state at the next payment date. For a homogeneous model, the one-period transition matrix $P$ has entry $p_{rs}=P(J_{k+1}=s\mid J_k=r)$ in row $r$, column $s$. Each row sums to 1. If $\boldsymbol{\alpha}_0$ is the row vector of initial state probabilities, then the state distribution at time $k$ is

\boldsymbol{\alpha}_k=\boldsymbol{\alpha}_0P^k.

For a model whose transition probabilities vary by period, multiply by the appropriate period-specific matrix at each step instead. Once the state probabilities are known, valuation follows by pairing each payment-date distribution with the cash-flow amounts conditional on state.

For example, if a premium $C_k$ is collected only in the active state at time $k$, its expected amount at that date is $C_kP(J_k=A)$. If a benefit $B_k$ is paid only in the disabled state, the expected benefit is $B_kP(J_k=D)$. These are expected amounts before discounting; each must be discounted from its own payment date.

Model conditions: The state probabilities must describe the insured’s state at the actual payment time. The cash flow must be fully determined by the state (or states) specified in the payment rule. If payment depends on an event during an interval, such as whether disability began at any time in that interval, endpoint state probabilities alone may not determine the payment probability. That rule requires the appropriate event or transition probabilities.

Why the Expected-Present-Value Formula Holds

Let $C_k$ be the amount paid at time $k$, determined by the state then occupied. Write $I_{k,s}=1$ if $J_k=s$, and $I_{k,s}=0$ otherwise. Then the payment can be written as $C_k=\sum_s c_k(s)I_{k,s}$. Its present value is $Z=\sum_k v^kC_k$. By linearity of expectation,

E[Z] =\sum_k v^kE[C_k] =\sum_k v^k\sum_s c_k(s)E[I_{k,s}] =\sum_k v^k\sum_s c_k(s)P(J_k=s).

The indicator $I_{k,s}$ has expectation $P(J_k=s)$, which gives the final equality. This argument does not require payments at different dates to be independent. It does require the probabilities and payment conditions to be matched correctly at each date.

Worked Example: Active-State Premiums and Disabled-State Benefits

A hypothetical annual disability contract begins with an active insured at time 0. The states are active $A$, disabled $D$, and dead $X$. The annual transition probabilities are given by the following matrix, with rows and columns ordered $A,D,X$:

P= \begin{pmatrix} 0.80&0.15&0.05\\ 0.20&0.70&0.10\\ 0&0&1 \end{pmatrix}.

Death is absorbing. The insured pays a premium of $1,000 at times 0, 1, and 2 if active at the payment time. A benefit of $10,000 is paid at times 1 and 2 if disabled at the payment time. Use an effective annual interest rate of $i=5\%$. Find the expected present values of premiums and benefits, and the expected net benefit outgo (benefits less premiums).

Method. Find the state probabilities at each payment date, then multiply each conditional amount by its state probability and discount it. The model assumes the stated transition probabilities apply in each year, conditional on the current state; in particular, the dead state remains dead.

At time 0, the insured is certainly active, so $P(J_0=A)=1$. After one year, the probability of being active is 0.80 and the probability of being disabled is 0.15.

At time 2, the probability of being active is the sum of the probabilities of the two paths that end active:

$$ P(J_2=A)=(0.80)(0.80)+(0.15)(0.20)=0.64+0.03=0.67. $$

Similarly, the probability of being disabled at time 2 is

$$ P(J_2=D)=(0.80)(0.15)+(0.15)(0.70)=0.12+0.105=0.225. $$

The remaining probability is death: $P(J_2=X)=1-0.67-0.225=0.105$. The time-2 probabilities sum to 1, as required.

The premium present value has three terms. The time-0 premium is certain; the later premiums are payable only in state $A$:

\begin{aligned} EPV(\text{premiums}) &=1{,}000+\frac{1{,}000(0.80)}{1.05} +\frac{1{,}000(0.67)}{(1.05)^2}\\ &=1{,}000+761.9047619+607.7097506\\ &=2{,}369.6145125. \end{aligned}

The benefit is payable only in state $D$ at times 1 and 2:

\begin{aligned} EPV(\text{benefits}) &=\frac{10{,}000(0.15)}{1.05} +\frac{10{,}000(0.225)}{(1.05)^2}\\ &=1{,}428.5714286+2{,}040.8163265\\ &=3{,}469.3877551. \end{aligned}

Thus the expected net benefit outgo is $3{,}469.3877551-2{,}369.6145125=1{,}099.7732426$. Rounded to the nearest cent, the premium present value is $2,369.61, the benefit present value is $3,469.39, and the net benefit outgo is $1,099.77. Subtracting the rounded displayed values gives the same net amount.

Common mistake avoided. The time-2 premium term is not discounted once: it is paid at time 2, so its denominator is $(1.05)^2$. Also, the term is $1{,}000(0.67)/(1.05)^2=607.7097506$, or $607.71 to the nearest cent—not $607.7029. The unrounded terms should be retained until the final rounding.

Valuing Cash Flows from a Mixed Initial Population

The initial state need not be known with certainty. If a group contains insureds in more than one state at time 0, represent that composition by an initial probability vector. The same approach values expected cash flows per insured in the group. State-contingent amounts at each date are weighted by the group’s probability of occupying the corresponding state.

Worked Example: A Mixed Active-and-Disabled Cohort

A hypothetical cohort has initial state probabilities $\boldsymbol{\alpha}_0=(0.70,0.30,0)$ for states $A,D,X$. Its annual transition matrix is

P= \begin{pmatrix} 0.80&0.15&0.05\\ 0.25&0.65&0.10\\ 0&0&1 \end{pmatrix}.

A $300 premium is due at time 0 and again at time 1 only for insureds active at the respective payment time. Benefits of $4,000 at time 1 and $5,000 at time 2 are payable only to insureds disabled at those dates. Use $i=4\%$ effective annually. Calculate the expected present values per initial insured and the net benefit outgo.

At time 0, the active probability is 0.70. At time 1, the state probabilities are

\begin{aligned} P(J_1=A)&=(0.70)(0.80)+(0.30)(0.25)=0.635,\\ P(J_1=D)&=(0.70)(0.15)+(0.30)(0.65)=0.300,\\ P(J_1=X)&=(0.70)(0.05)+(0.30)(0.10)=0.065. \end{aligned}

At time 2, the disabled probability is

$$ P(J_2=D)=(0.635)(0.15)+(0.300)(0.65)=0.09525+0.195=0.29025. $$

For a check, the active probability is $(0.635)(0.80)+(0.300)(0.25)=0.583$, and the death probability is $1-0.583-0.29025=0.12675$. The three probabilities sum to 1.

The time-0 premium is expected to be $300(0.70)=210$. The time-1 premium is expected to be $300(0.635)=190.50$. Therefore,

EPV(\text{premiums})=210+\frac{190.50}{1.04} =210+183.1730769=393.1730769.

The time-1 expected benefit is $4{,}000(0.300)=1{,}200$, and the time-2 expected benefit is $5{,}000(0.29025)=1{,}451.25$. Hence,

EPV(\text{benefits}) =\frac{1{,}200}{1.04}+\frac{1{,}451.25}{(1.04)^2} =1{,}153.8461538+1{,}341.7622 =2{,}495.6084.

The net benefit outgo is $2{,}495.6084-393.1730769=2{,}102.4353$. To the nearest cent, the premium present value is $393.17, the benefit present value is $2,495.61, and the net benefit outgo is $2,102.44.

Method and trap. The initial 30% disabled group does not pay the time-0 premium, and it is not automatically still disabled at time 1. Applying the time-0 composition to later payments would ignore recovery, further disability transitions, and death. Propagate the distribution to each payment date before valuing that date’s cash flow.

Different Cash Flows Can Apply After Recovery

A state-contingent valuation can include both benefits and premiums in a contract where a disabled insured may recover. A premium payable after recovery is valued using the active-state probability at its payment date, not the probability of having been active initially. A benefit payable while disabled is valued using the corresponding disabled-state probability.

Worked Example: Premiums After Recovery and Benefits While Disabled

An insured is disabled at time 0. In each year, the transition probabilities from $A$ are 0.85 to $A$, 0.10 to $D$, and 0.05 to $X$; from $D$ they are 0.20 to $A$, 0.70 to $D$, and 0.10 to $X$. Death is absorbing. A $600 premium is paid at times 1 and 2 if active at the payment date. A benefit of $2,400 is paid at time 1 and $2,700 at time 2 if disabled at the payment date. Use an effective annual interest rate of 5%. Find the present values and net benefit outgo.

The initial state is $D$ with probability 1. At time 1, the distribution is $P(J_1=A)=0.20$, $P(J_1=D)=0.70$, and $P(J_1=X)=0.10$. At time 2,

\begin{aligned} P(J_2=A)&=(0.20)(0.85)+(0.70)(0.20)=0.17+0.14=0.31,\\ P(J_2=D)&=(0.20)(0.10)+(0.70)(0.70)=0.02+0.49=0.51,\\ P(J_2=X)&=(0.20)(0.05)+(0.70)(0.10)+0.10=0.18. \end{aligned}

These probabilities sum to 1. The premiums are valued using the active probabilities at times 1 and 2:

\begin{aligned} EPV(\text{premiums}) &=\frac{600(0.20)}{1.05}+\frac{600(0.31)}{(1.05)^2}\\ &=114.2857143+168.7074830\\ &=282.9931973. \end{aligned}

The benefits use the disabled probabilities at those same dates:

\begin{aligned} EPV(\text{benefits}) &=\frac{2{,}400(0.70)}{1.05}+\frac{2{,}700(0.51)}{(1.05)^2}\\ &=1{,}600+1{,}248.9795918\\ &=2{,}848.9795918. \end{aligned}

The net benefit outgo is $2{,}848.9795918-282.9931973=2{,}565.9863945$. Rounded to the nearest cent, the premium present value is $282.99, the benefit present value is $2,848.98, and net benefit outgo is $2,565.99. The calculation allows recovery to generate a premium at a later date, while retaining the possibility of subsequent disability.

Timing, Interpretation, and Exam Checks

The state at a payment time and the path taken to reach it are different kinds of information. If a contract pays at time $k$ whenever the insured is disabled at time $k$, then $P(J_k=D)$ is the relevant probability. If it instead pays for a new disability onset during the preceding year, the endpoint probability $P(J_k=D)$ is not enough: some insureds may have been disabled earlier, while others may have entered disability during the year. The payment rule determines which probability is needed.

Similarly, identify whether cash flows occur at the beginning or end of a period. A payment at time 0 has discount factor 1; a payment at time 1 has factor $v$; and a payment at time 2 has factor $v^2$. A payment amount should not be discounted merely because its probability was calculated using transitions over a period. Discounting follows the payment date, not the number of transitions used to calculate the probability.

Write the payment rule.
For each date, state the amount and the state or event that makes it payable.

Find the state distribution.
Start from the initial distribution and propagate it to every payment date using the applicable transition probabilities.

Calculate expected payments.
Multiply each state-contingent amount by the probability of the required state at that date; sum across states if more than one state pays.

Discount and reconcile.
Discount each expected payment from its own date, total premiums and benefits separately, then subtract premiums from benefits if net benefit outgo is requested.

A frequent exam error is to use the right amount and probability with the wrong date, or to use a probability of ever becoming disabled when the benefit is payable only to those disabled at a scheduled date. Before calculating, check that the state definition, event condition, transition period, payment time, and discount exponent all refer to the same contract rule.

Check Your Understanding

For each problem, use the specified state probabilities at the payment dates and the stated effective annual interest rate.

An active insured has probability 0.90 of being active at time 1. A $500 premium is payable then only if active. At $i=4\%$, find its expected present value. (Answer: $432.69)
A hypothetical policy pays $8,000 at time 1 only if the insured is disabled then. The probability of being disabled at time 1 is 0.12. At $i=5\%$, find the benefit’s expected present value. (Answer: $914.29)
An insured begins active. At time 1 the probabilities of active, disabled, and dead are 0.70, 0.20, and 0.10. From active, the next-year probabilities of active and disabled are 0.80 and 0.15; from disabled, the next-year probability of disabled is 0.60. Benefits of $3,000 at time 1 and $4,000 at time 2 are payable only if disabled at the respective dates. At $i=4\%$, find the benefit present value. (Answer: $1,409.02)
A cohort has initial probabilities 0.60 active and 0.40 disabled. Over one year, active transitions to active, disabled, and dead with probabilities 0.80, 0.15, and 0.05; disabled transitions to those states with probabilities 0.30, 0.60, and 0.10. A $1,000 premium is due at time 0 if active and an $800 premium at time 1 if active. A $5,000 benefit is due at time 1 if disabled. At $i=5\%$, find expected net benefit outgo. (Answer: $514.29)
A contract pays a benefit only when disability first begins during a year. Explain why the probability of being disabled at the end of that year may not by itself give the probability needed to value the benefit. (Answer: The endpoint disabled probability can include insureds disabled before the year and does not isolate new onsets; the onset probability or appropriate transition-event information is required.)

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