From Annual State Results to a Profit Signature
A profit test for a multi-state product can produce a different expected annual result at each duration. The result depends both on the profit associated with each state and on the proportion of the original policy cohort occupying that state at the beginning of the year. A profit signature collects these cohort-level expected results, one value for each year.
This distinction matters for disability income insurance. A policy may begin with an active life, then move into disability, recover, or terminate on death. The annual profit for an active policy is not necessarily the same as the annual profit for a disabled policy. In addition, the proportion of the cohort still active or disabled generally changes over time. This tutorial assembles those elements without discounting the annual results; discounting a signature is a separate calculation.
The conditional result \(\phi_{t,j}\) is a per-policy amount for a policy starting the year in state \(j\). It can include cash flows triggered by transitions during that year, if those cash flows are part of the stated annual profit definition. The cohort result \(g_t\), in contrast, is averaged over all policies in the original cohort, including policies that have already terminated and therefore contribute no later profit.
For a discrete-time Markov model, let \(P_t\) be the transition matrix from the start of year \(t\) to the start of year \(t+1\). Use row vectors for state probabilities. Then $$ \boldsymbol{\pi}_{t+1}=\boldsymbol{\pi}_tP_t, \qquad g_t=\boldsymbol{\pi}_t\boldsymbol{\phi}_t, $$ where \(\boldsymbol{\phi}_t\) is a column vector of conditional annual profits. If transition assumptions do not vary by duration, the same matrix \(P\) is used each year. The probability vector is not usually renormalized after termination: it continues to describe the original cohort.
Why the Weighting Formula Holds
The cohort profit in a year is a random amount determined partly by the state at the beginning of that year. Let \(Z_t\) denote the annual profit for one policy. Conditioning on its starting state gives the law of total expectation: $$ E[Z_t]=\sum_j P(\text{state }j\text{ at start of year }t)\,E[Z_t\mid \text{state }j\text{ at start of year }t]. $$ By definition, those two terms are \(\pi_{t,j}\) and \(\phi_{t,j}\). Therefore \(E[Z_t]=\sum_j\pi_{t,j}\phi_{t,j}=g_t\). This derivation does not require profits to be independent of transitions: any transition-related cash flows must simply be included consistently in the conditional expectation \(\phi_{t,j}\).
The result applies when the state probabilities and conditional profit amounts refer to the same policy basis, starting time, and annual period. It does not by itself give a discounted value, a reserve, or a profit conditional on remaining in force. Those quantities require additional definitions and calculations.
Worked Disability-Income Model
The examples use a hypothetical annual model with four states: active \(A\), disabled \(D\), recovered and terminated \(R\), and dead \(M\). Recovery and death are absorbing states: once entered, the policy produces no further annual profit in this projection. All dollar results are undiscounted expected profit during the stated year. The transition matrix is $$ P= \begin{pmatrix} 0.70&0.20&0.05&0.05\\ 0.10&0.70&0.10&0.10\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, $$ with rows and columns ordered \(A,D,R,M\). Each row sums to 1. The matrix is an assumption for the examples, not an industry estimate.
Worked Example: Building the First Two Signature Values
A hypothetical policy begins active. During a year beginning in \(A\), the insurer collects a premium of $8 and incurs an expense of $5. At year-end it pays $10 if the next state is \(D\), and $100 if the next state is \(M\). There is no transition payment on recovery. A year beginning in \(D\) has a continuing premium of $4, a disability payment of $6, and an expense of $1; it has no additional payment on death or recovery during that year. Find the first two cohort-level expected annual profits.
Method. First calculate the conditional annual profit for each starting state, then weight those amounts by the beginning-of-year state probabilities. The premiums, expenses, and payments described are the complete annual profit cash flows for this example.
For a policy starting the first year in \(A\), the transition probabilities to \(D\) and \(M\) are 0.20 and 0.05. Thus $$ \phi_{1,A}=8-5-(0.20)(10)-(0.05)(100) =3-2-5=-4. $$ For a policy starting the year in \(D\), $$ \phi_{1,D}=4-6-1=-3. $$ A policy starting in \(R\) or \(M\) is terminated, so its future annual profit is 0. The first-year starting distribution is \(\boldsymbol{\pi}_1=(1,0,0,0)\). Therefore $$ g_1=(1)(-4)+(0)(-3)+(0)(0)+(0)(0)=-4. $$
The year-two starting distribution is the first row of \(P\), because the policy started in \(A\): $$ \boldsymbol{\pi}_2=(0.70,0.20,0.05,0.05). $$ The active-state contribution is \(0.70(-4)=-2.80\), not \(-4\): only 70% of the original cohort is active at the beginning of year two. The disabled-state contribution is \(0.20(-3)=-0.60\), and the two terminated-state contributions are zero. Hence $$ g_2=-2.80-0.60+0+0=-3.40. $$ As a check, the year-two probabilities sum to \(0.70+0.20+0.05+0.05=1\). The result is an expected profit per original policy, not a profit per policy still in force.
Common mistake avoided: applying the active-state result to the entire cohort at year two would give \(-4\), incorrectly ignoring both the reduced active probability and the disabled-state result. Weighting each starting state separately gives \(-3.40\).
Propagating Probabilities and Keeping Terminations
A profit signature requires a starting distribution for each duration. When annual transitions are homogeneous, multiply the distribution at the start of one year by the transition matrix to obtain the distribution at the start of the next. A terminated state remains in the probability vector, but its future profit is zero. Retaining that probability mass makes the vector a complete description of the original cohort.
Specify the probabilities across all states at the beginning of the first projected year. They must sum to 1 for a single-policy cohort.
For each starting state, determine expected profit during the year, including any transition-contingent payments specified by the model.
Take the dot product of the starting probability vector and the conditional profit vector for that duration.
Multiply the probability row vector by the transition matrix, check that probabilities still sum to 1, and repeat using the next year's profit vector.
Worked Example: Finding Year-Three Profit With Termination States
Continue the hypothetical model in the preceding example, with the policy starting active and the same transition matrix and conditional annual profits \(\boldsymbol{\phi}=(-4,-3,0,0)^{\mathsf T}\). Calculate the year-three starting distribution and the year-three cohort profit. Also find the expected profit conditional on being in force at the start of year three.
The year-two distribution is \((0.70,0.20,0.05,0.05)\). Multiply it by \(P\). The probability of being active at the start of year three is $$ \pi_{3,A}=(0.70)(0.70)+(0.20)(0.10)=0.49+0.02=0.51. $$ The probability of being disabled is $$ \pi_{3,D}=(0.70)(0.20)+(0.20)(0.70)=0.14+0.14=0.28. $$ For recovery, include both new recoveries and policies already recovered: $$ \pi_{3,R}=(0.70)(0.05)+(0.20)(0.10)+(0.05)(1) =0.035+0.020+0.050=0.105. $$ For death, $$ \pi_{3,M}=(0.70)(0.05)+(0.20)(0.10)+(0.05)(1) =0.035+0.020+0.050=0.105. $$ Thus \(\boldsymbol{\pi}_3=(0.51,0.28,0.105,0.105)\), and its entries sum to \(0.51+0.28+0.105+0.105=1\).
The year-three cohort profit is $$ g_3=(0.51)(-4)+(0.28)(-3)+(0.105)(0)+(0.105)(0) =-2.04-0.84=-2.88. $$ The in-force probability is \(0.51+0.28=0.79\). Conditional on being in force, the expected profit is therefore $$ \frac{-2.88}{0.79}=-3.645569\ldots $$ or \(-\$3.65\) per in-force policy, rounded to the nearest cent. This conditional figure is different from the cohort signature value of \(-\$2.88\) per original policy.
Check. The in-force conditional result, multiplied by the in-force probability, returns the cohort result: \((-2.88/0.79)(0.79)=-2.88\), apart from displayed rounding. Terminated policies contribute zero future profit but remain part of the original cohort.
Duration-Dependent Results
The transition probabilities and the annual conditional profits need not be constant over time. For example, a premium might change at renewal, or expenses may differ by duration. In that case, use the appropriate vector \(\boldsymbol{\phi}_t\) for each year. Do not carry a first-year result forward just because the state labels are unchanged.
Worked Example: Assembling a Three-Year Signature
Use the same initial active state and transition matrix. Suppose the conditional annual profit vectors, in the order \(A,D,R,M\), are \(\boldsymbol{\phi}_1=(-4,-3,0,0)^{\mathsf T}\), \(\boldsymbol{\phi}_2=(-2,-5,0,0)^{\mathsf T}\), and \(\boldsymbol{\phi}_3=(-1,-4,0,0)^{\mathsf T}\). These are hypothetical duration-specific results, already calculated on a consistent annual basis. Find the three signature values.
The first-year distribution is \((1,0,0,0)\), so $$ g_1=(1)(-4)=-4. $$ After one transition the year-two distribution is \((0.70,0.20,0.05,0.05)\). Apply the year-two profit vector, not the year-one vector: $$ g_2=(0.70)(-2)+(0.20)(-5)+(0.05)(0)+(0.05)(0) =-1.40-1.00=-2.40. $$ The year-three distribution, calculated from the transition matrix, is \((0.51,0.28,0.105,0.105)\). Therefore $$ g_3=(0.51)(-1)+(0.28)(-4)+(0.105)(0)+(0.105)(0) =-0.51-1.12=-1.63. $$ The profit signature is \((-4,-2.40,-1.63)\) dollars per original policy for years one through three. Each entry uses the matching duration's state distribution and conditional profit vector. The check is that the terminal-state contributions are zero in all three years, while their probability remains included in each distribution.
Common mistake avoided: reusing the year-one profit vector would produce different values and would ignore the stated duration changes. The signature is formed duration by duration, with both the distribution and profit vector matched to the same year.
Interpreting and Checking the Signature
A signature is a sequence of expected annual profits on a specified basis. Its entries need not have the same sign, and a later entry can change because the state mix changes, because conditional state profits change, or because both change. The signature alone does not reveal which cause dominates; inspect the probability and profit vectors together.
For a disability product with recovery and death, several checks are especially useful:
- Confirm that every transition-matrix row is a probability distribution: its entries are nonnegative and sum to 1.
- Confirm that the starting-state probabilities sum to 1 and that each propagated distribution continues to sum to 1.
- Make the termination convention explicit. If recovered and dead policies have no further projected cash flows, their later conditional profits are zero.
- Apply the same annual profit definition in every state. If a transition payment is included, include it in the conditional result for the state from which the transition occurs.
- Label whether a result is per original policy or conditional on remaining in force. These are different averages whenever termination probability is positive.
A frequent error is to drop recovered and dead probabilities and then treat the remaining probabilities as if they still referred to the original cohort. Renormalizing can be appropriate when a question explicitly asks for a result conditional on being in force, but it is not the cohort profit signature. For the signature, retain the original-cohort probabilities and weight every state result, including zeros for terminated states.
Check Your Understanding
For each problem, treat profits as undiscounted annual amounts and use the state order stated in the question. Give cohort results per original policy unless a question asks for an in-force conditional result.
- A beginning-of-year distribution is \((0.60,0.30,0.10,0)\), and the conditional annual profit vector is \((-2,-5,0,0)^{\mathsf T}\). Find the cohort-level annual profit. (Answer: \(-\$2.70\) per original policy)
- A policy starts active. The transition probabilities from \(A\) to \((A,D,R,M)\) are \((0.80,0.10,0.05,0.05)\). The first-year profit vector is \((-3,-7,0,0)^{\mathsf T}\), and the second-year vector is \((-2,-4,0,0)^{\mathsf T}\). Find the two-year profit signature. (Answer: \((-\$3.00,-\$2.00)\) per original policy)
- At a given duration, the state distribution is \((0.50,0.25,0.15,0.10)\) and the profit vector is \((4,-8,0,0)^{\mathsf T}\). Find the cohort profit and the expected profit conditional on being in force, rounded to the nearest cent. (Answer: \(\$0.00\) per original policy and \(\$0.00\) per in-force policy)
- A policy starts active. The transitions from \(A\) are \((0.60,0.25,0.10,0.05)\), and the first-year conditional profit vector is \((-5,-2,0,0)^{\mathsf T}\). The second-year vector is \((-3,-4,0,0)^{\mathsf T}\). Find the two-year signature. (Answer: \((-\$5.00,-\$2.80)\) per original policy)
- At the start of a year, the distribution is \((0.40,0.30,0.20,0.10)\), with conditional profits \((5,-6,0,0)^{\mathsf T}\). Find the cohort profit and the in-force conditional profit, rounded to the nearest cent. (Answer: \(\$0.20\) per original policy and \(\$0.29\) per in-force policy)