Continuous and Discrete Benefit Payment Timing

Why Payment Timing Changes a Reserve

A benefit that accrues while a person is disabled and a benefit paid only on specified dates can have different prospective values, even when both are described using the same annual benefit amount. The continuous benefit pays in proportion to time spent in the qualifying state. A discrete benefit is instead triggered by the person’s state at a payment date. The probability of being disabled at that date—and the discount to that date—determines its value.

The distinction matters when calculating a state reserve: the reserve is conditional on the state occupied at the valuation time, and future payments depend on future state occupancy. This tutorial isolates the benefit component of the prospective reserve. If premiums, expenses, or transition lump sums are also present, value those cash flows using their own timing and add them consistently. The payment-timing methods here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A continuous state-occupancy benefit pays at a rate per unit time for as long as the insured occupies a specified state. A discrete state-contingent benefit pays a stated amount at a scheduled date if the insured occupies the qualifying state at that date. A prospective benefit value at time $t$, conditional on state $i$, is the expected present value at time $t$ of future benefits under the model.

State-Contingent Valuation Formulas

Consider a continuous-time Markov model with states $1,\ldots,m$. Let $p_{ij}(t,u)$ be the probability that the process is in state $j$ at time $u$, given that it is in state $i$ at time $t$. Use a constant force of interest $\delta$, so a payment at $u$ has value at $t$ multiplied by $e^{-\delta(u-t)}$. The formulas below assume that the benefit is payable only while the policy is in force, with no additional transition lump sums.

Suppose state $j$ pays a continuous benefit at rate $b_j(u)$ per unit time. The rate may vary by state and time. The benefit value conditional on state $i$ at time $t$, through a fixed horizon $T$, is

V_i^{\mathrm{cont}}(t) = \int_t^T e^{-\delta(u-t)} \sum_{j=1}^{m} p_{ij}(t,u)b_j(u)\,du.

For discrete payments at dates $t_1,\ldots,t_N$, suppose the payment at $t_k$ is $C_{jk}$ if the process is in state $j$ at that date, and zero otherwise. Then

V_i^{\mathrm{disc}}(t) = \sum_{k:\,t_k>t} e^{-\delta(t_k-t)} \sum_{j=1}^{m}p_{ij}(t,t_k)C_{jk}.

The condition $t_k>t$ excludes a payment already made at the valuation time. If the contract instead defines a payment at $t$, its treatment must follow the contract’s valuation-time convention. The continuous formula integrates future payment rates; the discrete formula sums payments at the actual scheduled dates. Neither is a substitute for the other.

Why the Continuous Formula Holds

Let $I_j(u)$ equal 1 when the process is in state $j$ at time $u$, and 0 otherwise. Over a short interval of length $du$, the benefit paid is approximately $b_j(u)I_j(u)\,du$. Its discounted value at $t$ is therefore integrated over the future. Taking expectations gives the formula because the expectation of the state indicator is the probability of occupying that state:

E[I_j(u)\mid X(t)=i]=p_{ij}(t,u).

Thus, summing across states and integrating the discounted expected payment rate gives $V_i^{\mathrm{cont}}(t)$. For a discrete benefit, the payment at $t_k$ is a lump sum, not a rate over an interval. Its expected amount is $\sum_j p_{ij}(t,t_k)C_{jk}$, which is discounted from $t_k$ and added to the other scheduled payments.

These expressions require the stated payment rules and transition probabilities to describe the same model. A benefit payable only at a scheduled date cannot be valued by integrating its annual amount as though it accrued continuously. Conversely, a continuous rate cannot generally be replaced by a full annual installment whenever the state is occupied at a year-end.

Worked Valuations

Worked Example: Disability Is Absorbing During the Benefit Term

A hypothetical two-year contract begins with an insured in the active state. The only transition during the term is from active to disabled, at constant force $\lambda=0.20$ per year; disability is absorbing through year 2. The force of interest is $\delta=0.05$ per year. Compare (a) a continuous benefit rate of $1,200 per year while disabled with (b) a discrete benefit of $1,200 at the end of each year if disabled on that date. No benefit is payable in the active state.

Method. First find the probability of disability by time $u$, then apply the continuous integral or the scheduled-date sum. The annual discrete installment is paid in full if the insured is disabled at the payment date; it is not prorated for the fraction of the year spent disabled.

Starting active, the probability of still being active at time $u$ is $e^{-0.20u}$. Therefore

p_{AD}(0,u)=1-e^{-0.20u}.

For the continuous benefit, integrate the discounted probability of being disabled over the two years:

\begin{aligned} V_A^{\mathrm{cont}}(0) &=1200\int_0^2 e^{-0.05u}(1-e^{-0.20u})\,du\\ &=1200\left[ \frac{1-e^{-0.10}}{0.05} - \frac{1-e^{-0.50}}{0.25} \right]\\ &=1200(1.903252-1.573877)\\ &=1200(0.329374)\\ &=\$395.25. \end{aligned}

For the discrete benefit, disability probabilities at the two payment dates are $p_{AD}(0,1)=0.1813$ and $p_{AD}(0,2)=0.3297$, rounded to four decimals. Using the unrounded exponential expressions in the valuation,

\begin{aligned} V_A^{\mathrm{disc}}(0) &=1200\left[e^{-0.05}(1-e^{-0.20}) +e^{-0.10}(1-e^{-0.40})\right]\\ &=1200\left[(e^{-0.05}-e^{-0.25}) +(e^{-0.10}-e^{-0.50})\right]\\ &=1200(0.172429+0.298307)\\ &=\$564.88. \end{aligned}

The calculations check: the discrete value includes two possible full installments, each conditional on disability at its payment date; the continuous value accumulates only while disabled. The two benefit designs are not equivalent merely because both use the number $1,200. The common mistake is to treat the discrete payment as a continuous rate, or to prorate the year-end installment without a contract rule requiring proration.

Worked Example: Reserve Conditional on Disability Before a Scheduled Date

At time $t=0.25$, a hypothetical insured is disabled. The remaining coverage ends at time 1, and recovery to the active state occurs at constant force $\mu=0.20$ per year; there is no return to disability after recovery. The force of interest is $\delta=0.04$. Compare a continuous disability benefit of $500 per year with a $500 discrete payment at time 1 if the insured is still disabled then.

Method. Value future payments from time $t=0.25$, not from issue. Let $s$ be time measured from the valuation time, so the remaining term is $0\le s\le0.75$. The probability of still being disabled after $s$ years is $e^{-0.20s}$.

For continuous payment, discount from $t$ and multiply by the probability of remaining disabled:

\begin{aligned} V_D^{\mathrm{cont}}(0.25) &=500\int_0^{0.75}e^{-0.04s}e^{-0.20s}\,ds\\ &=500\frac{1-e^{-0.24(0.75)}}{0.24}\\ &=500\frac{1-e^{-0.18}}{0.24}\\ &=500\left(\frac{0.164730}{0.24}\right)\\ &=\$343.19. \end{aligned}

The discrete payment is 0.75 years after the valuation time. Its probability of being payable is $e^{-0.20(0.75)}=0.8607$, rounded to four decimals. Using the combined discount and recovery expression,

\begin{aligned} V_D^{\mathrm{disc}}(0.25) &=500e^{-0.04(0.75)}e^{-0.20(0.75)}\\ &=500e^{-0.18}\\ &=\$417.64. \end{aligned}

The calculations use the same remaining term and the same valuation date. The discrete payment is made only if disability persists to time 1, but if it is payable it is the full $500. A common timing error is to discount it for one year from issue; the correct discount period from the reserve date is 0.75 years.

Worked Example: Recovery Changes Occupancy Probabilities

A hypothetical insured begins active in a one-year contract. The process has two states: active $A$ and disabled $D$. Transitions occur from $A$ to $D$ at force $\lambda=0.50$ and from $D$ to $A$ at force $\mu=0.50$, both per year. The force of interest is $\delta=0.10$. Compare a continuous rate of $1,000 per year while disabled with one $1,000 payment at time 1 if disabled at that date.

Method. Use the two-state transition probability, which allows both entry into and recovery from disability. The total transition rate out of either state is $1.00$, and the long-run disabled proportion is $\lambda/(\lambda+\mu)=0.5$. Starting active,

p_{AD}(0,u)=\frac{\lambda}{\lambda+\mu} \left(1-e^{-(\lambda+\mu)u}\right) =\frac12(1-e^{-u}).

The continuous value is

\begin{aligned} V_A^{\mathrm{cont}}(0) &=1000\int_0^1 e^{-0.10u}\frac12(1-e^{-u})\,du\\ &=500\left[ \frac{1-e^{-0.10}}{0.10} -\frac{1-e^{-1.10}}{1.10} \right]\\ &=500(0.951626-0.606481)\\ &=\$172.57. \end{aligned}

At time 1, the disability probability is $p_{AD}(0,1)=0.3161$, rounded to four decimals. The discrete payment value, using the unrounded probability expression, is

\begin{aligned} V_A^{\mathrm{disc}}(0) &=1000e^{-0.10}\frac12(1-e^{-1})\\ &=1000(0.285983)\\ &=\$285.98. \end{aligned}

The probability at time 1 is not the probability of ever having been disabled during the year: an insured who recovered is active at the payment date. The two-state model accounts for that recovery. The calculation checks by conditioning the discrete benefit on the state at time 1 and applying the one-year discount factor.

Comparison and Exam Traps

The values in the examples need not move in the same direction in every contract comparison. In particular, a full installment payable at a date may be larger than the benefit accrued continuously during the preceding period, but this depends on the benefit amounts and state probabilities. For a valid comparison, specify the cash-flow rule, the state required for payment, the payment dates, and the valuation date before calculating.

Feature	Continuous occupancy benefit	Discrete state-contingent benefit
Payment rule	Rate per unit time while in the qualifying state	Specified amount if in the qualifying state at a scheduled date
Expected cash flow	State probability multiplied by the payment rate, integrated over time	State probability multiplied by the scheduled amount, summed over dates
Discount period	Each infinitesimal payment is discounted from its payment time	Each installment is discounted from its scheduled date
Typical trap	Replacing the integral by a year-end payment	Using “ever entered the state” instead of occupancy at the payment date

For the absorbing-disability example, one incorrect calculation would use the probability of disability by year 2 for both annual installments:

1200\left[e^{-0.05}p_{AD}(0,2)+e^{-0.10}p_{AD}(0,2)\right].

That is wrong because the year-1 installment depends on disability at year 1, not year 2. The corrected expression uses $p_{AD}(0,1)$ for the first payment and $p_{AD}(0,2)$ for the second. In a recoverable model, the relevant probability at each date is likewise occupancy at that date, not the probability of ever entering disability.

Key takeaway: For a continuous benefit, integrate the discounted state-occupancy probability multiplied by the payment rate. For a discrete benefit, sum the discounted state-occupancy probability multiplied by the scheduled amount. Keep the model, state condition, payment date, and valuation date explicit.

Check Your Understanding

For each problem, compare the specified benefit values using the stated force of interest and transition assumptions. Report monetary answers to the nearest cent.

A hypothetical insured starts active. Disability is absorbing during a one-year term and occurs at force $\lambda=0.10$. With $\delta=0.05$, calculate the value of a continuous benefit rate of $900 per year and the value of a $900 payment at year-end if disabled then. (Answer: continuous $42.12; discrete $81.47.)
An insured is disabled at time 0.25 and recovers at force $\mu=0.20$, with no subsequent disability. The term ends at time 1 and $\delta=0.04$. Calculate the value at time 0.25 of a continuous benefit rate of $500 per year and of a $500 payment at time 1 if still disabled. (Answer: continuous $343.19; discrete $417.64.)
A hypothetical insured starts active in a one-year, zero-interest model. The transition forces are $\lambda=0.40$ from active to disabled and $\mu=0.60$ from disabled to active. Find the value of a continuous benefit rate of $1,000 per year while disabled and a $1,000 payment at year-end if disabled then. (Answer: continuous $147.15; discrete $252.85.)
A hypothetical insured starts active and can transition to disabled at force $\lambda=0.20$, after which disability is absorbing for two years. The force of interest is zero. Compare a continuous benefit rate of $700 per year with a $700 payment at each year-end if disabled on that date. (Answer: continuous $246.12; discrete $357.66.)

What You'll Learn