Diagnosing Inconsistent Profits and Reserves

Why Reconciliation Matters

An annual profit test can appear plausible even when its parts disagree. A transition probability may be missing, a benefit may be placed at the wrong time, or a reserve may use a different valuation basis from the profit projection. In a multi-state product, these errors can be harder to spot because the cash flow and reserve depend on the state reached, not just on whether the policy remains in force.

A reliable diagnosis checks the supplied quantities in a fixed order: confirm that transition probabilities are valid, reconstruct the reserve recursion, calculate profit using the stated timing convention, and then check that the discounted profit signature agrees with the cash-flow and reserve identity. Continuous-time reserves provide a further cross-check through Thiele’s differential equation. A reconciliation only works when the model, timing, and valuation basis are stated explicitly.

Definition: For an annual multi-state model, let $V_t^j$ be the prospective reserve at the start of year $t$, immediately before the premium and expense at that time, conditional on being in state $j$. Let $p_{jk}$ be the probability of moving from state $j$ to state $k$ during the year, and let $B_{jk}$ be the benefit paid at the end of the year for that transition. If a premium $P_j$ and expense $E_j$ are paid at the start of the year, then, on a valuation basis with annual effective interest rate $i$, $$ V_t^j=-P_j+E_j+\frac{1}{1+i}\sum_k p_{jk}\bigl(B_{jk}+V_{t+1}^k\bigr). $$

This recursion assumes that the premium and expense occur at the start of the year, while the transition benefit and the next reserve are valued at year-end. The probabilities in each row must sum to one, including any absorbing state such as death. If a contract has mid-year cash flows, or the premium is paid at a different time, this equation must be adjusted rather than used unchanged.

Reconcile the Reserve and Annual Profit

Using the same timing convention, define the end-of-year profit conditional on starting the year in state $j$ as the accumulated starting resources less expected year-end benefits and reserves:

\Pi_t^j=(P_j-E_j+V_t^j)(1+i)-\sum_k p_{jk}(B_{jk}+V_{t+1}^k).

The vector with one entry for each starting state is the annual profit vector. If the reserve recursion and profit calculation use the same probabilities, cash flows, interest rate, and basis, expected profit is zero. In an experience profit test, actual experience may differ from the valuation basis; nonzero profit can then be appropriate. The diagnostic question is whether the difference is explained by the stated assumptions, not whether every profit entry must be zero.

Worked Example: Finding a Reserve Error in a Disability Test

A hypothetical one-year contract starts in the active state. At the start of the year it receives a premium of $50 and incurs an expense of $10. The annual effective valuation interest rate is 5%. At year-end, the transition probabilities are 0.80 to active, 0.15 to disabled, and 0.05 to death. The corresponding transition benefits are $0, $40, and $200. The next reserves are $50 in active, $80 in disabled, and $0 in death. A supplied table reports a starting active reserve of $24.00. Reconcile the reserve and annual profit.

Method. First check the probability row: $0.80+0.15+0.05=1.00$, so it is valid. Next calculate the expected year-end benefit plus reserve for each possible transition:

\begin{aligned} \text{Active: }&0+50=50, &0.80(50)&=40,\\ \text{Disabled: }&40+80=120, &0.15(120)&=18,\\ \text{Death: }&200+0=200, &0.05(200)&=10. \end{aligned}

The total expected year-end amount is $40+18+10=68$. Discounting at 5% gives $68/1.05=64.7619$. The correct starting reserve is therefore

V_0^{\text{active}}=-50+10+64.7619=24.7619.

The supplied reserve of $24.00 is too low by $0.7619. Using the corrected reserve, the annual profit is

\Pi_0^{\text{active}} =(50-10+24.7619)(1.05)-68 =64.7619(1.05)-68 =68-68=0.

With the supplied reserve, the reported profit would be $(50-10+24)(1.05)-68=67.20-68=-$0.80$. This is a reserve inconsistency, not an experience loss under the stated valuation assumptions. The common mistake is to round or copy the reserve before checking the recursion; retain unrounded intermediate values and compare the resulting profit with the stated basis.

Check Signs with Thiele’s Equation

For a continuous-time model, let $V_j(t)$ be the prospective reserve while in state $j$. Suppose premiums are paid continuously at rate $\pi_j(t)$, expenses at rate $e_j(t)$, the force of interest is $\delta$, and transitions from $j$ to $k$ occur with intensity $\mu_{jk}(t)$. A transition pays benefit $b_{jk}(t)$. With reserve defined as the present value of future outgo less future premium income, Thiele’s equation is

\frac{dV_j}{dt} =\delta V_j+\pi_j-e_j -\sum_{k\ne j}\mu_{jk}\bigl(b_{jk}+V_k-V_j\bigr).

The transition term is the intensity multiplied by the insurer’s net cost of moving out of state $j$: pay $b_{jk}$, then hold reserve $V_k$, while releasing the reserve $V_j$ for the state left. Its sign is often the source of an apparent discrepancy.

Justification: Over a short interval of length $h$, the probability of a transition from $j$ to $k$ is $\mu_{jk}h+o(h)$, and the probability of no transition is $1-\sum_{k\ne j}\mu_{jk}h+o(h)$. The reserve earns interest at rate $\delta$; premium income increases the reserve and expenses reduce it. A transition replaces $V_j$ by $b_{jk}+V_k$, an incremental outgo of $b_{jk}+V_k-V_j$. Thus, to first order, the reserve change is $$ h\left[\delta V_j+\pi_j-e_j-\sum_{k\ne j}\mu_{jk}(b_{jk}+V_k-V_j)\right]+o(h). $$ Dividing by $h$ and taking $h$ to zero gives Thiele’s equation. This local equation assumes the stated continuous rates and intensities; annual transition probabilities cannot generally be substituted for intensities without a suitable model and time-step approximation.

Worked Example: Diagnosing a Thiele Sign Error

For a hypothetical long-term care contract, consider a time when the insured is active. The force of interest is $\delta=0.04$. Premiums and expenses are paid continuously at rates of $12 and $2 per year. The transition intensity from active to disabled is $0.20$ per year, the transition benefit is $30, and the reserves are $V_{\text{active}}=40$ and $V_{\text{disabled}}=70$. Find the instantaneous reserve derivative.

Method. The interest contribution is $0.04(40)=1.6$, and the net continuous premium and expense contribution is $12-2=10$. The net cost of transition is the benefit plus the destination reserve less the reserve released:

$$ 30+70-40=60. $$

The intensity-weighted transition cost is $0.20(60)=12$. Therefore,

\frac{dV_{\text{active}}}{dt}=1.6+10-12=-0.4.

The reserve is decreasing at an instantaneous rate of $0.40 per year at this point. For a short interval of $0.25$ years, a first-order Euler approximation gives $40+0.25(-0.4)=39.90$. This is an approximation, not an exact quarter-year reserve. A common error is to use $b+V_{\text{active}}-V_{\text{disabled}}$, which here equals $30+40-70=0$, omitting the reserve released in the wrong direction. The correct transition cost compares the destination amount $b+V_{\text{disabled}}$ with the current reserve $V_{\text{active}}$.

Check the Profit Signature and NPV

A profit vector is conditional on the state at the start of a period. To calculate expected profit for a policy beginning in a particular state, weight each entry by the probability of occupying that state in that period. Then discount the period’s profit to the valuation date. For end-of-year profits $\Pi_t$, the NPV over $n$ years is

\operatorname{NPV}=\sum_{t=0}^{n-1}v^{t+1}E[\Pi_t], \qquad v=\frac{1}{1+i}.

The exponent is $t+1$ because $\Pi_t$ is measured at the end of year $t$, when the first year is indexed $t=0$. If a profit is measured at the start of a period instead, its discount time changes. A second check follows by discounting the annual profit identity: intermediate reserves cancel between successive years. Under consistent timing, the discounted profit sum equals initial reserve plus the present value of net premiums and expenses less benefits and terminal reserve. A mismatch indicates that at least one cash flow, reserve, probability, or discount date has been treated differently.

Worked Example: Weighting a Two-State Profit Signature

A hypothetical contract begins year 1 in state H. Its year-1 profit vector, conditional on the starting state, is $12 in H and \(-$4 in D. The year-1 transition probabilities from H are 0.80 to H and 0.20 to D. In year 2, the conditional profit vector is $10 in H and $6 in D. The annual effective interest rate is 5%. Calculate the two-year NPV.

Method. The policy starts in H, so the year-1 expected profit is $12. At the start of year 2, the state probabilities are 0.80 in H and 0.20 in D. Thus year-2 expected profit is

$$ 0.80(10)+0.20(6)=8+1.2=9.2. $$

Discount each expected profit from its year-end date:

\begin{aligned} \operatorname{NPV} &=\frac{12}{1.05}+\frac{9.2}{1.05^2}\\ &=11.428571+8.344671\\ &=19.773242. \end{aligned}

Rounded to the nearest cent, the NPV is $19.77. The common mistake is to use the year-2 H profit alone, as if the policy were certain to remain in H. That gives $12/1.05+10/1.05^2=$20.50$, which overstates NPV because it ignores the 0.20 probability of being in D. The profit vector is not itself an expected profit until it is weighted by state occupancy.

A Practical Reconciliation Sequence

When supplied results disagree, do not adjust a reserve or profit entry until the source of the discrepancy is identified. A compact audit can follow these steps:

Validate the model.
Check that each transition row sums to one, each transition has a defined destination state, and absorbing states are treated consistently.

Write the timing.
Mark when premiums, expenses, benefits, interest, and reserves are measured. Do not discount an end-of-year amount as if it occurred at the start.

Rebuild the reserve.
Use the stated valuation probabilities, interest rate, benefits, and next reserves in the prospective recursion. Keep separate any experience assumptions used for the profit projection.

Recompute the profit vector.
Calculate profit conditional on each starting state. Then weight it by the appropriate state probabilities to obtain expected profit for the policy cohort.

Reconcile the NPV.
Discount profits at their actual measurement dates and compare the result with the discounted cash-flow and reserve identity, including any terminal reserve.

A reserve and profit test can legitimately use different bases, such as a valuation basis for reserves and an experience basis for projected cash flows. That difference should be explicit: a profit discrepancy is not automatically an error. Conversely, agreement in total NPV does not prove every annual entry is correct; positive and negative errors can offset. Check signs, timing, state weights, and basis period by period before relying on an aggregate result. This reconciliation discipline is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, state the key consistency check before interpreting the result.

A one-year disability model has transitions from active of 0.65 to active, 0.25 to disabled, and 0.15 to death. Can this row be used as a probability distribution without adjustment? (Answer: No; the probabilities sum to 1.05, not 1.)
A hypothetical annual model has $P=80$, $E=12$, $i=0.04$, transition probabilities 0.70 to active, 0.20 to care, and 0.10 to death. Transition benefits are $0, $30, and $150; next reserves are $40, $60, and $0. Using the stated start-of-year premium and expense convention, calculate the starting reserve. (Answer: $-$9.35$.)
At a point in a continuous-time model, $\delta=0.03$, $V_H=50$, $\pi_H=8$, $e_H=1$, and the only transition from H has intensity 0.10, benefit $25, and destination reserve $V_D=45$. Find $dV_H/dt$. (Answer: $6.50$ dollars per year.)
A policy starts in H. Its year-1 profit is $5. At the start of year 2, the state probabilities are 0.60 in H and 0.40 in D; year-2 profits are $8 in H and $-$2 in D. If \(i=0.04$ and profits are paid at year-end, find the two-year NPV. (Answer: $8.51.)
For one year, $P=30$, $E=5$, $V_t=20$, and $i=0.05$. The probability-weighted year-end total of benefits plus next reserves is $47. Calculate end-of-year profit. (Answer: $0.25.)

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