Experience Assumptions Versus Valuation Assumptions

Why Two Sets of Assumptions Appear in One Projection

A profit projection and a reserve calculation answer different questions. The reserve is valued using a stated valuation basis; the projection measures how profit emerges under assumed experience. If the experience assumptions differ from the valuation assumptions, expected profit need not be zero, even when the reserve has been calculated consistently.

This distinction is important in multi-state contracts. A policy may move among active, disabled, and other in-force states before a claim or termination. Each transition can affect benefits, expenses, premiums, and the reserve carried into the next year. To calculate profit correctly, identify which basis determines each quantity and keep the cash-flow timing consistent.

Definition: Valuation assumptions are the interest and transition assumptions used to calculate reserves. Experience assumptions are the interest and transition assumptions used to project expected cash flows and profits. A reserve calculated on the valuation basis is then carried into a profit projection under the experience basis.

Annual Timing and the Reserve Equation

Use the following annual model. At the beginning of year $t$, the policy is in state $i$, the insurer holds reserve $V_t^i$, receives premium $P_t^i$, and pays expense $e_t^i$. At year-end, a transition from state $i$ to state $j$ produces benefit $b_{t,ij}$, and the policy carries reserve $V_{t+1}^j$ if state $j$ is still in force. A terminating state, such as death, has no subsequent reserve. Premiums and expenses occur at the start of the year; benefits and the next reserve are measured at year-end.

Let $q_{t,ij}^{\mathrm{val}}$ be the valuation-basis probability of transition $i$ to $j$, and let $i_t^{\mathrm{val}}$ be the valuation effective annual interest rate. The probabilities from a state sum to one, including transitions to terminating states. The prospective reserve is:

V_t^i=-P_t^i+e_t^i+ \frac{\displaystyle\sum_j q_{t,ij}^{\mathrm{val}} \left(b_{t,ij}+V_{t+1}^j\right)} {1+i_t^{\mathrm{val}}}.

This is a signed net reserve: it is the present value of future benefits and reserves, less future premiums, with current-year premium and expense represented explicitly in the equation. It can be negative when future premiums exceed the value of future obligations.

Timing convention: Define the amount available at the start of the year after the premium and expense as $C_t^i=V_t^i+P_t^i-e_t^i$. Under experience interest $i_t^{\mathrm{exp}}$, this amount accumulates to $C_t^i(1+i_t^{\mathrm{exp}})$ at year-end, before the transition benefit and next reserve are allowed for.

Transition-Contingent and Expected Profit

Let $p_{t,ij}^{\mathrm{exp}}$ denote the experience probability of transition $i$ to $j$. The profit conditional on that transition is the accumulated opening amount, less the transition benefit and the reserve required at year-end:

\Pi_{t,ij}= \left(V_t^i+P_t^i-e_t^i\right) \left(1+i_t^{\mathrm{exp}}\right) -b_{t,ij}-V_{t+1}^j.

The expected profit conditional on being in state $i$ at the start of the year is therefore:

E_{\mathrm{exp}}[\Pi_t\mid i] =\sum_j p_{t,ij}^{\mathrm{exp}}\Pi_{t,ij}.

The probabilities in this expectation are experience probabilities, not valuation probabilities. The $V$ values in the profit formula are the reserves established on the valuation basis. Thus the reserve basis and the profit-projection basis have distinct roles.

Why Matching Bases Give Zero Expected Profit

A short derivation makes the separation clear. By the reserve equation, $C_t^i=V_t^i+P_t^i-e_t^i$ equals the valuation present value of the year-end benefit plus next reserve. If the valuation and experience transition probabilities and interest rate are identical, multiplying the reserve equation by $1+i_t^{\mathrm{val}}$ gives:

C_t^i(1+i_t^{\mathrm{val}}) =\sum_j q_{t,ij}^{\mathrm{val}} \left(b_{t,ij}+V_{t+1}^j\right).

Taking expected profit under the same probabilities and interest rate gives:

\begin{aligned} E[\Pi_t\mid i] &=C_t^i(1+i_t^{\mathrm{val}}) -\sum_j q_{t,ij}^{\mathrm{val}} \left(b_{t,ij}+V_{t+1}^j\right)\\ &=0. \end{aligned}

The cancellation is conditional on the stated model and matching assumptions. If experience transition probabilities or interest differ, the two sides no longer match in general, and expected profit can be positive or negative.

Worked Example: One-Year Mortality Experience Differs

A hypothetical one-year policy starts in an active state. At the start of the year, it receives a premium of $30 and incurs an expense of $5. It pays $200 at year-end on death and nothing otherwise; there is no terminal reserve. The valuation death probability is 0.020 and valuation interest is 4% effective annually. The experience death probability is 0.030 and experience interest is 2% effective annually. Find the opening reserve and expected profit.

Method. First calculate the reserve using only valuation assumptions. Then use that reserve with the experience assumptions to calculate transition-contingent and expected profit.

The valuation expected year-end payment is $0.020(200)=\$4.00$. Thus:

V_0=-30+5+\frac{4}{1.04} =-25+\frac{4}{1.04} =-21.153846\ldots

The opening amount after premium and expense is $C_0=V_0+30-5=3.846153\ldots$. Accumulated at the 2% experience rate, this becomes $3.846153\ldots(1.02)=3.923076\ldots$. The profit on survival is therefore $3.923076..., while the profit on death is $3.923076\ldots-200=-\$196.076923\ldots$.

Using experience probabilities, expected profit is:

0.970(3.923076\ldots)+0.030(-196.076923\ldots) = -2.076923\ldots

To the nearest cent, the opening reserve is −$21.15 and expected profit is −$2.08. The negative expected profit reflects the difference between the experience and valuation assumptions; it is not a reason to substitute experience assumptions into the reserve.

Common mistake avoided: The valuation reserve is not calculated using the 3% experience death probability. Using that probability in the reserve would give $-30+5+(0.030)(200)/1.04=-\$19.23$, which is a reserve on a mixed basis, not the specified valuation basis.

Worked Example: A One-Year Policy With Two In-Force States

A hypothetical policy starts the year in healthy state $H$. It receives $10 and incurs $2 at the start of the year. At year-end, it pays $40 on transition to disabled state $D$, $100 on death, and zero if the insured remains healthy. There are no terminal reserves. Valuation probabilities from $H$ are 0.90 to $H$, 0.08 to $D$, and 0.02 to death; valuation interest is 0%. Experience probabilities are 0.85, 0.10, and 0.05, respectively, and experience interest is also 0%. Calculate the reserve and expected profit.

Reserve. The valuation expected year-end payment is $0.90(0)+0.08(40)+0.02(100)=3.20+2.00=\$5.20$. With zero interest:

V_0^H=-10+2+5.20=-\$2.80.

The amount available before year-end transitions is $V_0^H+10-2=-2.80+8=\$5.20$. Since interest is zero, this amount does not change before the transition.

Conditional profits. Remaining healthy gives $5.20-0=\$5.20$. Becoming disabled gives $5.20-40=-\$34.80$. Death gives $5.20-100=-\$94.80$.

Expected profit. Weight those profits by experience probabilities:

0.85(5.20)+0.10(-34.80)+0.05(-94.80) =4.42-3.48-4.74 =-\$3.80.

The reserve is −$2.80 and expected profit is −$3.80. As a check, the experience expected year-end payment is $0.10(40)+0.05(100)=\$9.00$; the opening amount of $5.20 less this payment is −$3.80.

Common mistake avoided: The 0.90, 0.08, and 0.02 probabilities determine the reserve, but the expected profit uses 0.85, 0.10, and 0.05. Using the valuation probabilities to average the conditional profits would answer a different question.

Worked Example: Following Profit Through Two States and Two Years

Consider a hypothetical two-year contract with states $H$, $D$, and death. Interest is 0% on both bases. At the start of a year in $H$, the premium is $10 and expense is $2. In $D$, there is no premium and expense is $1. A transition from $H$ to $D$ pays $5; death from $H$ pays $100; death from $D$ pays $30. Other transitions pay nothing. At the end of year 2 all remaining reserves are zero.

At time 1, valuation probabilities from $H$ are 0.90 to $H$ and 0.10 to death. From $D$, they are 0.80 to $D$ and 0.20 to death. At time 0, valuation probabilities from $H$ are 0.80 to $H$, 0.15 to $D$, and 0.05 to death. Experience probabilities at time 0 from $H$ are 0.70 to $H$, 0.20 to $D$, and 0.10 to death. At time 1, experience probabilities are 0.95 to $H$ and 0.05 to death from $H$, and 0.70 to $D$ and 0.30 to death from $D$. Find the valuation reserves and expected profits.

Step 1: Reserves at time 1. In $H$, the valuation expected payment is $0.10(100)=\$10$, so $V_1^H=-10+2+10=\$2$. In $D$, it is $0.20(30)=\$6$, so $V_1^D=0+1+6=\$7$.

Step 2: Reserve at time 0. The valuation expected year-end amount from $H$ is $0.80(0+2)+0.15(5+7)+0.05(100+0)=1.60+1.80+5.00=\$8.40$. Therefore $V_0^H=-10+2+8.40=\$0.40$.

Step 3: Year 0 experience profits. The opening amount after premium and expense is $0.40+10-2=\$8.40$. The profits for transitions to $H$, $D$, and death are respectively $8.40-2=\$6.40$, $8.40-5-7=-\$3.60$, and $8.40-100=-\$91.60$. The $H$-transition includes the $2 reserve at time 1; the $D$-transition includes both its $5 transition benefit and its $7 reserve. Expected year 0 profit is:

0.70(6.40)+0.20(-3.60)+0.10(-91.60) =4.48-0.72-9.16 =-\$5.40.

Step 4: Year 1 experience profits. In $H$, the opening amount is $V_1^H+10-2=2+8=\$10$. Persistence in $H$ gives profit $10, and death gives $10-100=-\$90$. Expected profit conditional on reaching $H$ is $0.95(10)+0.05(-90)=\$5.00$. In $D$, the opening amount is $V_1^D-1=7-1=\$6$. Persistence gives $6, and death gives $6-30=-\$24$. Expected profit conditional on reaching $D$ is $0.70(6)+0.30(-24)=-\$3.00$.

The experience probabilities of reaching $H$ and $D$ at time 1 are 0.70 and 0.20. Thus expected year 1 profit, measured from the initial $H$ state, is $0.70(5.00)+0.20(-3.00)=\$2.90$. The remaining 0.10 is already in the terminating death state and produces no year 1 profit. Total expected profit over both years is $-5.40+2.90=-\$2.50$.

Check. The time 0 reserve includes the expected value of the time 1 reserves and the time 0 transition benefits on the valuation basis. The profit projection instead uses the experience probabilities at each year. The reserves are $V_1^H=\$2$, $V_1^D=\$7$, and $V_0^H=\$0.40$; the expected total profit is −$2.50.

Common mistake avoided: For the $H$-to-$D$ transition at time 0, subtract both the $5 transition benefit and the $7 closing reserve. Omitting either component misstates conditional profit and its expected value.

Interpreting the Profit Pattern

A change in assumptions affects expected profit through the expected accumulated opening amount and the expected value of transition benefits and closing reserves. Writing $W_{t,ij}=b_{t,ij}+V_{t+1}^j$, the difference from zero expected profit can be expressed as:

E_{\mathrm{exp}}[\Pi_t\mid i] =C_t^i(1+i_t^{\mathrm{exp}}) -\sum_j p_{t,ij}^{\mathrm{exp}}W_{t,ij}.

The reserve equation separately sets $C_t^i$ equal to the valuation-basis present value of those year-end amounts. Consequently, a deviation in transition probabilities changes the experience-weighted value of $W$, while a deviation in interest changes the accumulated value of $C$. In a multi-year projection, transitions also alter which states are reached, and therefore the future premiums, expenses, benefits, and reserves in the profit pattern.

State the timing and basis.
Identify when premiums, expenses, benefits, and reserves apply. Record valuation and experience probabilities and interest separately.

Calculate reserves.
Use valuation assumptions in the prospective reserve equation, including all relevant next-state reserves and transition benefits.

Calculate conditional profits.
Accumulate the opening reserve plus premium less expense at the experience rate, then subtract the transition benefit and closing reserve for each transition.

Take the experience expectation.
Weight each conditional profit by the experience transition probability. In later years, also account for the experience probability of reaching each state.

Common Exam Traps

Mixing bases in the reserve. A reserve specified on a valuation basis uses valuation probabilities and valuation interest, even if the profit projection uses different experience assumptions.
Using valuation probabilities to average projected profit. Conditional profits may be calculated for every transition, but their expected value under projected experience uses experience probabilities.
Omitting the next-state reserve. A transition to an in-force state generally requires subtracting that state’s closing reserve as well as any transition benefit.
Forgetting the terminating-state boundary condition. A terminating state has no future reserve unless the problem explicitly defines a further liability.
Applying interest to the wrong amount or period. Under the timing used here, the beginning reserve plus beginning premium less expense accumulates for one year before year-end benefits and reserves are allowed for.
Assuming expected profit must be zero. Zero conditional expected profit follows when the valuation and experience transition assumptions and interest rate match, under the stated timing and reserve model. It is not a general result when the bases differ.

Key takeaway: Calculate the reserve on its valuation basis, then project profit using experience assumptions while carrying that reserve into each transition. The difference between those bases explains why expected profit can emerge and how it varies by state and year.

Check Your Understanding

For each problem, keep the reserve calculation and the experience profit calculation on their stated bases.

A one-year policy receives $24 and incurs $3 at the start of the year. It pays $150 on death and nothing on survival, with no terminal reserve. Valuation death probability is 0.040 and valuation interest is 5%. Find the opening reserve, to the nearest cent. (Answer: −$15.29)
For the policy in question 1, experience death probability is 0.060 and experience interest is 2%. Find expected profit, using the unrounded valuation reserve and rounding only the final answer to the nearest cent. (Answer: −$3.17)
A one-year contract starts in state $A$, receives $8, and incurs $1 at the start. At year-end, a transition to $B$ pays $12, death pays $60, and remaining in $A$ pays zero; terminal reserves are zero. Valuation transition probabilities are 0.70 to $A$, 0.20 to $B$, and 0.10 to death, with 0% interest. Find the opening reserve. (Answer: $1.40)
For the contract in question 3, experience probabilities are 0.60 to $A$, 0.30 to $B$, and 0.10 to death, with 0% interest. Find the transition-contingent profits and expected profit. (Answer: $A$: $8.40; $B$: −$3.60; death: −$51.60; expected profit: −$1.20)
A one-year policy has valuation death probability 0.050 and valuation interest 0%. Its premium is $10, expense is $2, death benefit is $100, and there is no terminal reserve. Experience death probability is also 0.050, but experience interest is 4%. Find expected profit. (Answer: $0.20)

What You'll Learn