Valuation Assumptions and Emerging Surplus

Why the Assumptions Matter

A projected surplus figure depends on two distinct sets of assumptions. Experience assumptions describe what the insurer expects to happen during the projection period: for example, the earned investment return, deaths, lapses, and expenses. Valuation assumptions determine the reserves held for future obligations. Changing either set can change reported surplus, but the effects have different meanings.

This distinction matters when comparing projections. If projected experience changes while the valuation basis is held fixed, the difference is an expected gain or loss relative to that basis. If the valuation basis changes, the reserve itself changes; this may alter surplus immediately, even if the underlying expected experience has not changed. The framework below uses a one-year, discrete-time model to make those effects visible.

Definition: Consider one policy in force at the start of a year. Let $q$ be the probability of death during the year, $l$ the probability of lapse during the year, and $p=1-q-l$ the probability the policy remains in force at year-end. Death and lapse are mutually exclusive outcomes; the policy remains in force only if neither occurs. Let $B$ be the death benefit, $R$ the surrender payment on lapse, and $e$ the expense incurred per policy in force at the start of the year. The expense is incurred in every outcome. Let $V_1$ be the reserve per policy continuing at year-end.

A One-Year Reserve and Surplus Equation

For a valuation basis, write $q_v,l_v,p_v,e_v,i_v$ for its death probability, lapse probability, continuing probability, expense, and effective annual interest rate. The one-year prospective reserve is the present value of expected year-end outgo, including the reserve for a continuing policy:

V_0=\frac{q_vB+l_vR+e_v+p_vV_1}{1+i_v}, \qquad p_v=1-q_v-l_v.

This is a simplified annual model: benefits, surrender payments, expenses, and the closing reserve are treated as year-end amounts, and there are no premiums during the year. It is not a general formula for contracts with different payment timings, premiums, or more detailed states. Those cash flows must be included at their appropriate times in a model that matches the contract.

Now project the year using experience assumptions $q_e,l_e,p_e,e_e$ and an earned effective return $j$, while holding $V_0$ and $V_1$ on the chosen valuation basis. If the opening assets are $V_0$ per starting policy, expected year-end surplus per starting policy is

S=V_0(1+j)-q_eB-l_eR-e_e-p_eV_1, \qquad p_e=1-q_e-l_e.

The closing reserve is multiplied by the probability of continuing because it is held only for policies still in force at year-end. Death and lapse payments, by contrast, are weighted by their respective probabilities. The expense $e_e$ is not probability-weighted: it is incurred for every policy in force at the start.

Why the Valuation-Basis Projection Has Zero Expected Surplus

The reserve formula is constructed to balance the expected end-of-year value of assets against expected outgo and the reserve needed for continuing policies. To see this, use the valuation assumptions as the projected experience: $j=i_v$, $q_e=q_v$, $l_e=l_v$, $e_e=e_v$, and $p_e=p_v$. Multiplying the reserve formula by $1+i_v$ gives

$$ V_0(1+i_v)=q_vB+l_vR+e_v+p_vV_1. $$

Substitute the right-hand side into the surplus equation. The expected end-year assets exactly equal expected payments, expenses, and the reserve for continuing policies, so $S=0$. This is an expected-value balance, not a claim that actual surplus must be zero: realized deaths, lapses, expenses, and investment returns can differ from their expectations.

Fix the valuation basis.
Calculate $V_0$ and identify the closing reserve $V_1$ on that basis.

Specify projected experience.
State the return, death and lapse probabilities, and expense assumption for the projection. Confirm that $p_e=1-q_e-l_e$.

Calculate the expected end-year balance.
Accumulate opening assets, subtract expected payments and expenses, then subtract the reserve for continuing policies.

Interpret the result.
Compare it with the zero-surplus projection on the valuation assumptions, and identify which experience differences drive the change.

Worked Example: Separating Experience Variances on a Fixed Basis

A hypothetical policy has a \$1,000 death benefit, a \$50 surrender payment, a year-end expense of \$10, and a \$900 reserve for a policy continuing at year-end. On the valuation basis, $q_v=0.02$, $l_v=0.03$, and $i_v=0.04$. Projected experience instead has $q_e=0.03$, $l_e=0.02$, $e_e=\$12$, and an earned return $j=0.05$. Calculate the reserve on the valuation basis and the expected surplus per starting policy.

Method. First find the valuation reserve, then use that reserve in the surplus projection with experience assumptions. This keeps the change in experience separate from a change in valuation basis.

On the valuation basis, $p_v=1-0.02-0.03=0.95$. Expected year-end outgo and continuing reserve are

0.02(1000)+0.03(50)+10+0.95(900) =20+1.50+10+855 =\$886.50.

Therefore,

V_0=\frac{886.50}{1.04}=\$852.403846\ldots

For projected experience, $p_e=1-0.03-0.02=0.95$. Expected death payments are $0.03(1000)=\$30$; expected surrender payments are $0.02(50)=\$1$; the expense is $\$12$; and the expected closing reserve is $0.95(900)=\$855$. The accumulated opening reserve is

852.403846\ldots(1.05)=\$895.024038\ldots.

Thus,

S=895.024038\ldots-30-1-12-855 =-\$2.975962\ldots.

Rounded to the nearest cent, expected surplus is -\$2.98 per starting policy. As a check, the valuation-basis projection gives zero. Relative to that projection, the 1 percentage-point higher return adds $0.01V_0=\$8.52$; the increased death probability adds \$10 of expected claims; the lower lapse probability reduces expected surrender payments by \$0.50; and the higher expense adds \$2. The net change is $8.5240-10+0.50-2=-\$2.9760$, consistent with the direct calculation. The continuing probability is unchanged, so the closing-reserve term does not change.

Changing the Valuation Basis

A valuation-basis change is not simply another experience variance. It changes the reserve required at the valuation date. If assets are fixed and the reserve rises by $\Delta V_0$, initial surplus, defined here as assets less reserve, falls by the same amount. A changed basis can also change the closing reserve and therefore later surplus emergence.

To compare bases cleanly, state what is held fixed. A useful comparison first calculates the reserve under each basis for the same hypothetical contract. If projecting one-year surplus as well, specify whether opening assets are held fixed or reset to equal each basis’s opening reserve. Those are different comparisons and can produce different surplus amounts.

Worked Example: Comparing Two Reserve Bases

A hypothetical one-year projection has a \$1,000 death benefit, a \$50 surrender payment, a \$10 year-end expense, and a \$900 closing reserve. Under the old basis, $q_v=0.02$, $l_v=0.03$, and $i_v=0.04$. Under the new basis, $q_v=0.03$, $l_v=0.02$, and $i_v=0.02$. Calculate both opening reserves and the increase.

Method. Recalculate the expected outgo and continuing-reserve amount under each basis, then discount using that basis’s interest rate. Death and lapse are mutually exclusive; the expense is included in full under both bases.

On the old basis, $p_v=1-0.02-0.03=0.95$. The numerator is

0.02(1000)+0.03(50)+10+0.95(900) =20+1.50+10+855=\$886.50.

Therefore, $V_{0,\mathrm{old}}=886.50/1.04=\$852.403846\ldots$, or \$852.40.

On the new basis, $p_v=1-0.03-0.02=0.95$. The numerator is

0.03(1000)+0.02(50)+10+0.95(900) =30+1+10+855=\$896.

Therefore, $V_{0,\mathrm{new}}=896/1.02=\$878.431373\ldots$, or \$878.43. The reserve increase is $878.431373\ldots-852.403846\ldots=\$26.027527\ldots$, or \$26.03 per policy to the nearest cent. If assets were unchanged at the old reserve, this strengthening would reduce initial surplus by \$26.03. The calculation alone does not say how future surplus changes; that also depends on future experience and the closing-reserve basis.

From Per-Policy Results to a Cohort

For a homogeneous cohort of $N$ starting policies, multiply an expected per-policy amount by $N$ to obtain expected cohort surplus. For an actual cohort, use actual counts and actual cash flows, not expected counts, and apply the closing reserve only to policies that remain in force. This distinction is especially important when expected counts are fractional or when actual deaths and lapses differ from expectation.

Worked Example: Actual Cohort Surplus and the Closing Reserve

A hypothetical cohort has 100 policies. Opening assets equal the aggregate valuation reserve, with $V_0=\$849.038462$ per policy. Assets earn 5% during the year. There are 3 deaths, 2 lapses, and 95 policies continue. The death benefit is \$1,000, the surrender payment is \$100, expenses are \$6 per starting policy, and the closing reserve is \$900 per continuing policy. Calculate actual year-end surplus.

Method. Accumulate cohort opening assets, subtract actual payments and expenses, and subtract the reserve for actual continuing policies. Do not apply the closing reserve to all 100 starting policies.

Opening assets are $100(849.038462\ldots)=\$84{,}903.8462\ldots$. After investment earnings, assets are

84{,}903.8462\ldots(1.05)=\$89{,}149.0385\ldots.

Actual death benefits total $3(1000)=\$3{,}000$; surrender payments total $2(100)=\$200$; and expenses total $100(6)=\$600$. Assets after these outgo amounts are

89{,}149.0385\ldots-3{,}000-200-600 =\$85{,}349.0385\ldots.

The closing reserve is $95(900)=\$85{,}500$. Therefore actual surplus is $85{,}349.0385\ldots-85{,}500=-\$150.9615\ldots$, or -\$150.96 to the nearest cent. The calculation correctly reserves for all 95 continuing policies; using \$855 instead of \$85,500 would be a factor-of-100 error.

Common Traps and Interpretation

When attributing a surplus change, change one assumption at a time where possible, or calculate each contribution explicitly. A combined projection can show the net result but conceal offsetting effects: for example, a higher investment return can partly offset higher death claims. Also distinguish an expected projection from a realized cohort result. Expected death and lapse amounts are probability-weighted; actual payments depend on actual outcomes.

Treating expense as a mutually exclusive outcome. Death and lapse are mutually exclusive events, but the stated expense is incurred in every outcome. For example, expected year-end outgo is $B I_{\text{death}}+R I_{\text{lapse}}+e$, so its expectation is $qB+lR+e$, not a formula that multiplies $e$ by a probability.
Using the wrong continuing probability. The reserve term is $pV_1$, with $p=1-q-l$ for the assumptions used in that projection. If $q$ or $l$ changes, the continuing probability may also change.
Confusing experience changes with reserve changes. A lower expected surplus under one experience scenario is not automatically a reserve strengthening. State whether the valuation basis, opening reserve, and closing reserve are fixed.
Applying a per-policy reserve to the wrong count. For a cohort, multiply $V_1$ by the number continuing, not the number initially in force.
Calling zero expected surplus a guarantee. The reserve recurrence balances expected values on its own basis; actual experience can produce gains or losses.

Key takeaway: First calculate reserves on a clearly stated valuation basis. Then project expected surplus using explicitly stated experience assumptions and the correct continuing-policy probability. Compare scenarios with the basis held fixed before interpreting reserve-basis changes separately.

Check Your Understanding

For each problem, use the one-year model stated above. Round money to the nearest cent where applicable.

A hypothetical policy has $B=\$800$, $R=\$120$, $e_v=\$8$, $V_1=\$500$, $q_v=0.01$, $l_v=0.04$, and $i_v=0.05$. Find $V_0$. (Answer: \$472.19)
A policy has $B=\$600$, $R=\$80$, $V_1=\$400$. Its valuation basis is $q_v=0.02$, $l_v=0.01$, $e_v=\$7$, and $i_v=0.03$. Projected experience is $q_e=0.03$, $l_e=0.01$, $e_e=\$9$, and $j=0.04$. Find expected surplus per starting policy using the valuation reserve. (Answer: -\$0.04)
For a hypothetical contract, $B=\$1{,}200$, $R=\$100$, $e_v=\$12$, and $V_1=\$700$, with $q_v=0.015$ and $l_v=0.025$. Find the reserve at $i_v=0.04$ and at $i_v=0.02$, holding the other assumptions fixed, and give the increase. (Answer: \$677.40 and \$690.69; increase \$13.28)
A cohort starts with 80 policies. During the year, 2 die and 4 lapse. If the closing reserve is \$1,250 per continuing policy, find the aggregate closing reserve. (Answer: \$92,500)
A hypothetical projection starts with opening reserve and assets of \$500 per policy. The earned return is $j=0.04$; experience has $q_e=0.02$, $l_e=0.03$, $B=\$1{,}000$, $R=\$60$, and $e_e=\$8$; the closing reserve is $V_1=\$450$. Find expected year-end surplus per starting policy. (Answer: \$62.70)

What You'll Learn