Equivalence of State and Decrement Valuations

Why Two Valuation Methods Can Agree

A benefit payable when a person leaves an active state can be valued either by following transitions in a state model or by using a multiple-decrement calculation. These are not automatically equivalent merely because both describe exits. They agree when they use the same causes of exit, the same probabilities or rates, the same period of exposure, and the same benefit amounts and payment times.

The simplest setting is one active state with several mutually exclusive exits. Once an exit occurs, the person enters an absorbing state: no further transition affects the benefit being valued. Examples might include a hypothetical pension member leaving active service through retirement, withdrawal, or death. The model must specify which exits are covered and what benefit, if any, is paid for each.

Definition: An absorbing-exit model has one initial active state and a set of mutually exclusive exit causes. While active, the individual can leave through cause $j$; after leaving, the individual remains in the corresponding absorbing state. A multiple-decrement model describes the same population by its probability of leaving through each cause during a specified period, while accounting for the other competing causes.

Matching Continuous-Time Rates and Decrement Probabilities

Suppose an individual is active at time 0. Let $\lambda_j(t)$ be the transition intensity from active status to absorbing exit state $j$ at time $t$, conditional on being active immediately before $t$. Assume these are deterministic, nonnegative rates, there are no re-entries, and the only transitions out of the active state are the listed exits. Write $\Lambda(t)=\sum_j\lambda_j(t)$ for the total exit intensity.

The probability of remaining active to time $t$ is the state occupancy probability $p_0(t)$. Under these assumptions, the probability of an exit of cause $j$ in the short interval from $t$ to $t+dt$ is approximately $p_0(t)\lambda_j(t)dt$. Thus the probability of cause $j$ being the first exit by time $T$, and the probability of still being active at $T$, are:

p_0(t)=\exp\left(-\int_0^t\Lambda(s)\,ds\right),\qquad q_j(T)=\int_0^T p_0(t)\lambda_j(t)\,dt,\qquad p_0(T)+\sum_j q_j(T)=1.

Here $q_j(T)$ is the multiple-decrement probability for exit cause $j$ over the interval, allowing for all other causes competing during that interval. It is not generally equal to the rate $\lambda_j$ multiplied by the interval length. The formula applies to a single spell in the active state with the specified transition intensities; recurrent transitions require a state model that represents re-entry.

Theorem: For an exit benefit $B_j(t)$ payable immediately at an exit of cause $j$, the expected present value through time $T$ is $$ \sum_j\int_0^T e^{-\delta t}p_0(t)\lambda_j(t)B_j(t)\,dt. $$ A multiple-decrement calculation gives the same value if it uses the same cause-specific exit probabilities and payment times, with the payment amount and discount factor applied to each exit at its actual time. A calculation using only period-end probabilities matches only when its payment-timing assumption also matches.

Why the Exit-Valuation Formula Holds

The state approach values an exit at its time of occurrence. Conditional on being active at $t$, the chance of cause $j$ in a sufficiently short interval $dt$ is $\lambda_j(t)dt$, ignoring terms whose order is smaller than $dt$. The probability of being active at that time is $p_0(t)$. The expected discounted payment in that interval is therefore, to first order, $e^{-\delta t}B_j(t)p_0(t)\lambda_j(t)dt$. Adding these expected payments over time and over causes gives the integral.

The multiple-decrement view partitions the same possible outcomes by exit cause and time. Its cause-$j$ exit density is $p_0(t)\lambda_j(t)$, exactly the density used in the state calculation. Integrating the same discounted benefit against that density must give the same value. This argument also explains the conditions: if one method pays at the end of the year while the other pays immediately on exit, they do not discount the cash flows at the same times.

Specify the exits.
List mutually exclusive causes and identify which state is occupied before each exit.

Match probabilities or rates.
Use cause-specific intensities in both continuous-time calculations, or use the same conditional decrement probabilities in both discrete-time calculations.

Match cash flows and dates.
For each cause, state the benefit amount and whether it is paid immediately, at a period boundary, or on another specified date.

Value and reconcile.
Discount each benefit to the same valuation date, then check that the state and decrement approaches give the same total.

Worked Example: Continuous Exit Benefits With Interest

A hypothetical contract starts with an active member at time 0 and ends at time 2. Two absorbing exits have constant intensities $\lambda_1=0.10$ and $\lambda_2=0.20$ per year. The force of interest is constant at $\delta=0.10$. A benefit of $10,000 is paid immediately on exit 1 and $4,000 immediately on exit 2. Find the expected present value by the state method and reconcile it with the multiple-decrement method.

Method. The total exit intensity is $\Lambda=0.10+0.20=0.30$, so the probability of remaining active at time $t$ is $p_0(t)=e^{-0.30t}$. The discounted cause-$j$ exit density is $e^{-0.10t}e^{-0.30t}\lambda_j=e^{-0.40t}\lambda_j$.

For exit 1, the expected present value is

10{,}000(0.10)\int_0^2 e^{-0.40t}\,dt =1{,}000\frac{1-e^{-0.80}}{0.40} =1{,}000\frac{0.550671}{0.40} =\$1{,}376.68.

For exit 2, it is

4{,}000(0.20)\int_0^2 e^{-0.40t}\,dt =800\frac{0.550671}{0.40} =\$1{,}101.34.

The total state-model value is $1,376.68 + $1,101.34 = $2,478.02. The multiple-decrement method uses the same cause-specific exit densities. Its discounted value for each cause is therefore the same integral, giving the same total, $2,478.02. This check confirms that both the exit rates and the immediate-payment rule were matched.

Discrete-Time Equivalence

A discrete annual model uses conditional probabilities rather than continuous intensities. If an individual is active at the start of a year, let $q_j$ be the probability of exit $j$ during that year and let $q_0=1-\sum_jq_j$ be the probability of remaining active to year-end. The exits must be mutually exclusive outcomes for that year, so their probabilities sum to no more than 1.

When benefits are all paid at year-end, the expected present value for an individual active at the start of the year is $v\sum_j q_jB_j$, where $v=1/(1+i)$ and $B_j$ is the year-end benefit for cause $j$. This is also the value from a discrete state-transition model with the same transition probabilities and end-of-year payment rule. For several years, apply the same conditional probabilities to the individuals still active at each year’s start, and discount each year’s payments to time 0.

Worked Example: Annual Decrements With Year-End Payments

A hypothetical one-year contract begins with 1,000 active members. Conditional on being active at the start of the year, each member has probability $q_1=0.08$ of withdrawal, probability $q_2=0.12$ of retirement, and probability $q_0=0.80$ of remaining active. At year-end, the contract pays $9,000 for each withdrawal and $3,000 for each retirement. The effective annual interest rate is $i=0.05$. Find the expected present value of benefits per initial member and show the matching state calculation.

Method. The expected year-end payment per initial member is $0.08(9{,}000)+0.12(3{,}000)=720+360=\$1{,}080$. Discounting by $v=1/1.05=0.95238095$ gives $1{,}080/1.05=\mathbf{\$1{,}028.57}$, rounded to the nearest cent.

In the state model, the probability of reaching the withdrawal state is 0.08 and the probability of reaching the retirement state is 0.12. The corresponding expected year-end payments are $720 and $360 per initial member. Discounting both at the same year-end gives $(720+360)/1.05=\$1,028.57$, as required. The remaining probability, 0.80, generates no exit benefit in this example. The method avoids discounting exits at a time other than the stated year-end payment date.

Matching Continuous Benefits and Exit Payments

The same equivalence applies when a contract pays a continuous benefit while the individual occupies the active state as well as lump sums on exit. If the payment rate while active is $c(t)$, its expected present value is the integral of the payment rate multiplied by the discounted probability of active occupancy. Add the exit-benefit value using the cause-specific exit densities. The multiple-decrement calculation must include the same active exposure as well as the same exit benefits.

Worked Example: Active-State Payments Plus Exit Benefits

A hypothetical benefit pays $600 per year continuously while a member is active, for at most one year. The member begins active and can leave through two absorbing exits, with constant intensities $\lambda_1=0.20$ and $\lambda_2=0.10$ per year. There is no interest. Exit 1 pays $1,000 immediately and exit 2 pays $2,000 immediately. Find the total expected value using both descriptions.

Method. The total intensity is 0.30, and $p_0(t)=e^{-0.30t}$. The expected value of the continuous active-state payments is

600\int_0^1 e^{-0.30t}\,dt =600\frac{1-e^{-0.30}}{0.30} =600\frac{0.259182}{0.30} =\$518.36.

The cause-specific exit probabilities by time 1 are

q_1(1)=\frac{0.20}{0.30}(1-e^{-0.30})=0.172788,\qquad q_2(1)=\frac{0.10}{0.30}(1-e^{-0.30})=0.086394.

The exit benefits have expected value $0.172788(1{,}000)+0.086394(2{,}000)=172.788+172.788=\$345.58$. Adding the active-state payments gives $518.36 + $345.58 = $863.94, with probabilities rounded only for display.

The state calculation gives the same result directly: the total expected payment rate at time $t$, conditional on being active up to that time, is $600+0.20(1{,}000)+0.10(2{,}000)=\$1{,}000$ per year. Its value is $1{,}000\int_0^1e^{-0.30t}dt=\$863.94$. The active payments stop at exit in both methods.

Common Mistake: Treating an Intensity as a Probability

An intensity is a rate per unit of time, not an exit probability over an interval. For example, with constant rates $\lambda_1=0.10$ and $\lambda_2=0.20$ over two years, multiplying each rate by 2 gives 0.20 and 0.40. Those figures are not the competing-exit probabilities: they ignore the fact that a member who exits early cannot experience another exit from the active state.

With no interest, the correct two-year probabilities are

q_1(2)=\frac{0.10}{0.30}(1-e^{-0.60})=0.150396,\qquad q_2(2)=\frac{0.20}{0.30}(1-e^{-0.60})=0.300792.

They sum to $1-e^{-0.60}=0.451188$, leaving probability $e^{-0.60}=0.548812$ active at two years. If the exit benefits are $10,000 and $4,000, the incorrect rate-times-time calculation gives $0.20(10{,}000)+0.40(4{,}000)=\$3{,}600$. The correct expected benefit is $0.150396(10{,}000)+0.300792(4{,}000)=\$2{,}707.13$. The error is not a rounding issue; it results from using quantities that are not probabilities.

A second common error is to use the correct exit probabilities but discount an immediate benefit as if it were paid at year-end, or vice versa. State and decrement calculations are equivalent only after payment dates are aligned as well as transition assumptions. This is especially important when a problem distinguishes payment at exit from payment at the end of a valuation period.

Key takeaway: An absorbing-exit state model and a multiple-decrement valuation agree when their cause-specific exit distributions, exposure period, benefit amounts, and payment dates agree. In continuous time, the cause-$j$ exit density is $p_0(t)\lambda_j(t)$; in discrete time, use the same conditional decrement probabilities in both models.

Check Your Understanding

For each problem, align the state or decrement probabilities with the stated payment timing before calculating.

A one-year discrete contract has exit probabilities 0.06 and 0.14, with all benefits paid at year-end. The corresponding benefits are $5,000 and $2,000, and $i=0.04$. Find the expected present value per initial member. (Answer: $557.69)
In a one-year continuous model with no interest, constant exit intensities are 0.10 and 0.10. Immediate benefits are $1,000 and $3,000. Find the expected benefit value. (Answer: $362.54)
In a one-year continuous model, the constant exit intensities are 0.20 and 0.30, the force of interest is 0.10, and immediate exit benefits are $4,000 and $1,000. Find the expected present value. (Answer: $827.18)
A continuous benefit pays $120 per year while an individual remains active for at most two years. The total exit intensity is constant at 0.40, and there is no interest. Find the expected value of the active-state payments only. (Answer: $165.20)
In a one-year discrete model, exit probabilities are 0.10 and 0.20, and the remaining 0.70 stay active. Exit benefits of $10,000 and $5,000 are paid at year-end; $i=0.05$. Find the expected present value per initial member and state the probability of remaining active. (Answer: $1,904.76; probability 0.70)

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