Expected Present Value of Discrete State Benefits

Valuing Benefits Payable at Discrete Dates

A contract may pay a benefit at the end of each policy year depending on the insured’s state on that date. For example, a hypothetical disability contract might pay one amount if the insured is active and a different amount if disabled. To value the contract, first determine the probability of each state at each payment date; then combine those probabilities with the amounts payable and discount each expected payment to the valuation date.

This tutorial considers benefits determined by the state occupied at specified discrete payment dates. It does not value payments made continuously during state occupancy or benefits triggered by a transition between states. Those cash flows require different timing calculations. The result here is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $Z(t)$ denote the insured’s state at time $t$, and let the contract make a payment at each of the dates $t_1,\ldots,t_n$. If the payment at $t_k$ is $b_{k,j}$ when $Z(t_k)=j$, then the discrete state benefit at that date is determined by the state occupied at the payment date. A state with no benefit has $b_{k,j}=0$.

Assume the state probabilities are specified by a Markov model, the state set is exhaustive for the model, the payment amounts are deterministic once the date and state are known, and the effective annual interest rate $i$ is deterministic and constant. The Markov assumption means that, conditional on the current state, the distribution of the next state does not depend on the earlier path. If $t_k$ is measured in years, the discount factor to time 0 is $v^{t_k}$, where $v=1/(1+i)$.

Theorem: If the insured begins in state $r$, the expected present value of the discrete state benefits is $$ \operatorname{EPV} =\sum_{k=1}^{n} v^{t_k} \sum_j P\bigl(Z(t_k)=j\mid Z(0)=r\bigr)b_{k,j}. $$ The inner sum is the expected payment at date $t_k$, before discounting. If payments occur at the ends of years $1,\ldots,n$, then $t_k=k$.

For calculations using transition matrices, let $\boldsymbol{\alpha}_k$ be a row vector whose $j$th entry is the probability of being in state $j$ at time $t_k$. If $P_k$ is the transition probability matrix from $t_{k-1}$ to $t_k$, then $\boldsymbol{\alpha}_k=\boldsymbol{\alpha}_{k-1}P_k$. Let $\boldsymbol{b}_k$ be the column vector of benefits at date $t_k$, one amount for each state. The expected payment at that date is $\boldsymbol{\alpha}_k\boldsymbol{b}_k$. Thus the matrix form of the valuation is $\sum_{k=1}^{n}v^{t_k}\boldsymbol{\alpha}_k\boldsymbol{b}_k$.

Why the Expected Present Value Formula Holds

At date $t_k$, define $I_{k,j}$ to equal 1 if the insured is in state $j$ and 0 otherwise. The payment at that date can then be written as a sum of state-specific amounts multiplied by indicators. Discounting and taking expectations gives the following derivation.

Write the payment using indicators.
The payment at $t_k$ is $B_k=\sum_j b_{k,j}I_{k,j}$. Exactly one indicator is 1 when the state set is exhaustive.

Take the expectation.
By linearity, $E[B_k]=\sum_j b_{k,j}E[I_{k,j}]=\sum_j b_{k,j}P(Z(t_k)=j)$, conditional on the specified initial state.

Discount and add across dates.
With deterministic interest, the present value is $\sum_k v^{t_k}B_k$. Taking expectations and applying linearity again gives $\sum_kv^{t_k}E[B_k]$, which is the stated formula.

No independence assumption between payments at different dates is needed for this expected value calculation: linearity of expectation applies even when the payments are dependent. The formula does rely on using the correct state probabilities at each payment date. It also does not automatically cover stochastic interest, a benefit triggered by a transition during an interval, or an amount depending on the duration already spent in a state.

Worked Applications

Worked Example: Two Annual Benefits in a Three-State Model

A hypothetical two-year contract begins with the insured in state A. The possible states are A (active), D (disabled), and C (a more severe disabled state). The annual transition matrix, with rows and columns ordered A, D, C, is

P= \begin{pmatrix} 0.50&0.50&0\\ 0&0.50&0.50\\ 0&0&1 \end{pmatrix}.

At the end of year 1, the contract pays $400 in A and $1,080 in D; it pays $0 in C. At the end of year 2, it pays $500 in A, $1,200 in D, and $2,596 in C. The effective annual interest rate is $i=5\%$. Mortality is ignored in this illustration: the stated states and transition matrix fully specify the modeled state movement over the two-year term.

Method. Propagate the initial state distribution to each payment date, calculate the expected payment at that date, and discount it using that date’s power of $v$. The initial distribution is $\boldsymbol{\alpha}_0=(1,0,0)$.

At the end of year 1,

\boldsymbol{\alpha}_1 =(1,0,0)P =(0.50,0.50,0).

The expected year-1 payment is

E[B_1]=0.50(400)+0.50(1{,}080)+0(0) =200+540 =\$740.

At the end of year 2, multiply the year-1 distribution by the transition matrix:

\boldsymbol{\alpha}_2 =(0.50,0.50,0)P =(0.25,0.50,0.25).

The expected year-2 payment is

E[B_2]=0.25(500)+0.50(1{,}200)+0.25(2{,}596) =125+600+649 =\$1{,}374.

Since $v=1/1.05$, the expected present value is

\operatorname{EPV} =\frac{740}{1.05}+\frac{1{,}374}{1.05^2} =704.761904\ldots+1{,}246.258503\ldots =\$1{,}951.02.

The final amount is rounded to the nearest cent. As a check, the year-2 amount is larger than the year-1 amount, but it is also discounted for an additional year; its present value, about $1,246.26, is therefore below $1,374. The common mistake to avoid is using the year-1 state distribution for the year-2 benefit, rather than propagating it through $P$.

Worked Example: A Disability Benefit Over Three Years

A hypothetical contract has two states, A (active) and D (disabled), and begins in A. Each year, the transition matrix is

P= \begin{pmatrix} 0.80&0.20\\ 0.30&0.70 \end{pmatrix},

where the rows and columns are ordered A, D. At the ends of years 1, 2, and 3, respectively, the contract pays $1,000, $1,200, and $1,400 if the insured is in D; it pays nothing in A. The effective annual interest rate is $i=5\%$. Mortality is ignored; these two states and the given matrix define the entire model over the three-year term.

Method. Find the disabled-state probability at each date, multiply it by that date’s benefit, and discount. Starting from $\boldsymbol{\alpha}_0=(1,0)$, the distribution after one year is

\boldsymbol{\alpha}_1=(1,0)P=(0.80,0.20).

Thus the probability of disability at the first payment date is 0.20, and the expected payment is $0.20(1{,}000)=\$200$. The next distribution is

\boldsymbol{\alpha}_2 =(0.80,0.20)P =(0.80(0.80)+0.20(0.30),\;0.80(0.20)+0.20(0.70)) =(0.70,0.30).

The year-2 expected payment is $0.30(1{,}200)=\$360$. Propagating once more gives

\boldsymbol{\alpha}_3 =(0.70,0.30)P =(0.70(0.80)+0.30(0.30),\;0.70(0.20)+0.30(0.70)) =(0.65,0.35).

The year-3 expected payment is $0.35(1{,}400)=\$490$. Therefore,

\begin{aligned} \operatorname{EPV} &=\frac{200}{1.05}+\frac{360}{1.05^2}+\frac{490}{1.05^3}\\ &=190.476190\ldots+326.530612\ldots+423.280423\ldots\\ &=\$940.29. \end{aligned}

The final amount is rounded to the nearest cent. The probabilities at the three dates are 0.20, 0.30, and 0.35, so they are not interchangeable. A common error is to use the year-1 probability of disability for all three payments; the Markov transition model must be applied repeatedly before valuing each year’s benefit.

Worked Example: Benefits Payable in Either State

A hypothetical three-year contract begins in state A and has states A and D. Its annual transition matrix is

P= \begin{pmatrix} 0.80&0.20\\ 0.50&0.50 \end{pmatrix},

with rows and columns ordered A, D. At the ends of years 1, 2, and 3, the benefit vectors $(b_{k,A},b_{k,D})$ are $(300,900)$, $(400,1{,}000)$, and $(500,1{,}500)$, respectively. The effective annual interest rate is $i=10\%$. Mortality is ignored, and the stated model specifies all possible state transitions during the term.

Method. Since a positive amount is payable in either state, use the full state distribution at each date rather than only the probability of disability. Starting with $\boldsymbol{\alpha}_0=(1,0)$, the first two distributions are

\boldsymbol{\alpha}_1=(1,0)P=(0.80,0.20), \qquad \boldsymbol{\alpha}_2=(0.80,0.20)P=(0.74,0.26).

The third distribution is

\boldsymbol{\alpha}_3 =(0.74,0.26)P =(0.74(0.80)+0.26(0.50),\;0.74(0.20)+0.26(0.50)) =(0.722,0.278).

The expected payments are

\begin{aligned} E[B_1]&=0.80(300)+0.20(900)=240+180=\$420,\\ E[B_2]&=0.74(400)+0.26(1{,}000)=296+260=\$556,\\ E[B_3]&=0.722(500)+0.278(1{,}500)=361+417=\$778. \end{aligned}

Discount each expected payment from its own year:

\begin{aligned} \operatorname{EPV} &=\frac{420}{1.10}+\frac{556}{1.10^2}+\frac{778}{1.10^3}\\ &=381.818182\ldots+459.504132\ldots+584.522915\ldots\\ &=\$1{,}425.85. \end{aligned}

The final amount is rounded to the nearest cent. The year-3 state probabilities sum to $0.722+0.278=1$, and the expected year-3 payment is between $500 and $1,500, as it must be for a weighted average of those two amounts. The mistake to avoid is omitting a state merely because it is not the state associated with the largest benefit.

Timing, Scope, and Exam Traps

A state-contingent benefit is valued using the probability of occupying the relevant state at the payment date. This timing condition matters. If a contract instead pays when the insured enters a state, the event is a transition, not simply occupancy at a scheduled date. Similarly, a continuous benefit during disability depends on the time spent in that state, rather than only on the state observed at a finite set of dates.

The model also needs an unambiguous convention for a payment date that coincides with a possible state change. The examples above define the payment by the state at the end of the year, after the year’s transition has been reflected in the distribution. A contract using the state immediately before a transition could require a different state probability.

Step	Question to check	Common error
1. Identify dates	At which dates is a benefit actually paid?	Discounting from the wrong date or adding a payment at time 0 when none is specified.
2. Find state probabilities	What is the distribution at each payment date?	Reusing an earlier distribution without applying the intervening transition matrix.
3. Calculate expected payments	What amount is payable in each possible state?	Using only one state’s probability when multiple states carry benefits.
4. Discount and sum	What is the correct discount factor for each date?	Discounting every payment by the same power of $v$.

For a finite-horizon contract, the calculation is complete when the expected payment at every specified date has been included exactly once and discounted to the same valuation date. Transition probabilities at each step should be valid probabilities: entries are nonnegative and each row sums to 1. These checks do not establish that a model is appropriate for a particular product, but they can reveal calculation and transcription errors.

Key takeaway: At each discrete payment date, multiply each state-contingent benefit by the probability of being in that state on that date. Sum those state-weighted amounts, discount the result from its payment date, and then add across dates.

Check Your Understanding

For each hypothetical contract, use the stated discrete-time Markov model and deterministic effective annual interest rate. Unless specified otherwise, mortality is ignored and the stated states exhaust the model. Answers are rounded to the nearest cent.

A contract starts in A. At the end of year 1, the probability of being in D is 0.30. It pays $900 in D and $0 in A at that date. If $i=5\%$, what is the EPV? (Answer: $257.14)
A contract starts in A and has transition matrix $\begin{pmatrix}0.70&0.30\\0.20&0.80\end{pmatrix}$, with states ordered A, D. It pays $1,000 in D at year 1 and $1,500 in D at year 2, and nothing in A. If $i=10\%$, find the EPV. (Answer: $830.58)
A contract starts in A and has transition matrix $\begin{pmatrix}0.60&0.40\\0.25&0.75\end{pmatrix}$, with states ordered A, D. At each of years 1 and 2 it pays $400 in A and $1,000 in D. If $i=4\%$, find the EPV. (Answer: $1,284.76)
A contract starts in A. From year 0 to 1, the probability of moving from A to D is 0.20. From year 1 to 2, the probabilities of moving to D are 0.40 from A and 0.70 from D. It pays $500 in D at year 1 and $800 in D at year 2, with no payment in A. If $i=5\%$, find the EPV. (Answer: $429.02)

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