Experience Deviations and Shareholder Profit

Why Separate Experience Deviations?

A change in mortality, lapse, expenses, or investment return can affect several parts of a product projection at once. For example, higher mortality increases death claims but can also reduce the number of policies remaining at the end of the period. Higher lapse rates increase surrender payments, while also reducing the reserve held for continuing policies. Looking at only one cash flow can therefore give the wrong impression of the change in shareholder profit or solvency.

The useful discipline is to calculate the revised cash flows and closing liability, then compare the resulting surplus with the original projection. This tutorial uses a deliberately simplified hypothetical portfolio to show that discipline. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and assumptions: At the start of a one-year period, $N$ identical policies are in force. Each policy pays a premium at the start of the year. Deaths occur during the year with probability $q$; among policies that survive the year, a proportion $l$ lapse. Death and lapse are mutually exclusive outcomes under this ordering. Premiums and opening assets earn the annual effective investment return $j$. Expenses, death benefits, and surrender benefits are paid at year-end. The closing liability is a fixed reserve $V_1$ for each policy still in force. All figures are expected-value projections, so fractional policy counts are permitted.

A One-Year Profit and Surplus Framework

Let $A_0$ be opening assets, $K_0$ opening shareholder capital, $P$ total premiums collected at the start of the year, $E$ year-end expenses, $B_D$ the benefit paid per death, and $B_L$ the surrender benefit paid per lapse. Opening assets support both the opening reserve and capital; in the example below, $A_0=NV_0+K_0$. The end-of-year asset amount is

$$ A_1=(A_0+P)(1+j)-E-NqB_D-N(1-q)lB_L. $$

The expected number of policies remaining is $N(1-q)(1-l)$, so the closing liability and shareholder profit for the period are

L_1=NV_1(1-q)(1-l), \qquad \Pi=A_1-L_1-K_0.

Here, $\Pi$ is the change in shareholder surplus over the period, measured from opening capital $K_0$. The solvency surplus at year-end is $A_1-L_1$. This simplified framework treats the closing reserve as the specified liability for solvency purposes; it does not recalculate the reserve basis when experience changes.

Derivation: At year-end, shareholder surplus is assets less the liability owed for policies still in force, or $A_1-L_1$. The period’s shareholder profit is the change from the opening surplus $K_0$. Therefore $\Pi=(A_1-L_1)-K_0$. In particular, a reduction in the closing liability raises surplus, all else equal: if assets fall by \$17,640 but the liability falls by \$26,460, surplus rises by $-17{,}640-(-26{,}460)=\$8{,}820$.

Worked Projection and Mortality Deviation

Worked Example: Isolating a Mortality Increase

A hypothetical portfolio has 1,000 policies. Each pays a \$100 premium at the start of the year. Opening assets are \$1,000,000, including \$900,000 of opening reserves and \$100,000 of shareholder capital. The annual effective investment return is 4%. Year-end expenses are \$20,000, the death benefit is \$5,000 per death, the surrender benefit is \$600 per lapse, and the closing reserve is \$900 per continuing policy. In the expected projection, $q=0.02$ and $l=0.05$, conditional on surviving. First calculate the original profit, then recalculate it for a mortality deviation to $q=0.025$, holding all other assumptions fixed.

Method. Project death claims, lapses, and continuing policies separately. Apply the investment return to opening assets plus premiums, then deduct year-end cash outgo and the closing liability.

Original projection. There are $1{,}000(0.02)=20$ expected deaths. The expected lapses are $1{,}000(1-0.02)(0.05)=49$, leaving $1{,}000(0.98)(0.95)=931$ continuing policies. Total premiums are $1{,}000(\$100)=\$100{,}000$. Accumulated assets before year-end outgo are $(\$1{,}000{,}000+\$100{,}000)(1.04)=\$1{,}144{,}000$. Death claims are $20(\$5{,}000)=\$100{,}000$; surrender payments are $49(\$600)=\$29{,}400$. Thus $A_1=\$1{,}144{,}000-\$20{,}000-\$100{,}000-\$29{,}400=\$994{,}600$. The closing liability is $931(\$900)=\$837{,}900$, and profit is $\Pi=\$994{,}600-\$837{,}900-\$100{,}000=\$56{,}700$.

Mortality deviation. At $q=0.025$, there are $1{,}000(0.025)=25$ expected deaths. Lapses are $1{,}000(0.975)(0.05)=48.75$, and continuing policies are $1{,}000(0.975)(0.95)=926.25$. Death claims are $25(\$5{,}000)=\$125{,}000$, and surrender payments are $48.75(\$600)=\$29{,}250$. Assets are $\$1{,}144{,}000-\$20{,}000-\$125{,}000-\$29{,}250=\$969{,}750$. The closing liability is $926.25(\$900)=\$833{,}625$, so profit is $\$969{,}750-\$833{,}625-\$100{,}000=\$36{,}125$.

The profit change is $\$36{,}125-\$56{,}700=-\$20{,}575$. Assets fall by $\$24{,}850$, while the liability falls by $\$4{,}275$; hence the solvency-surplus change is $-\$24{,}850-(-\$4{,}275)=-\$20{,}575$, as required. Death-benefit payments rise by \$25,000, while surrender payments fall by \$150 because fewer policies survive to lapse. This check accounts for both policyholder cash flows and the continuing-policy liability.

Common mistake avoided: Applying the higher death rate to the lapse rate without respecting the conditional definition of $l$. Lapses are calculated from survivors, $N(1-q)l$, not from all policies, $Nl$.

Lapse Deviations and the Reserve Release

Worked Example: When More Lapses Increase Projected Profit

Use the same hypothetical portfolio and original assumptions as in the preceding example, but change only the conditional lapse rate from $0.05$ to $0.08$. Calculate the revised profit and explain how it can increase even though more surrender benefits are paid.

There are still 20 expected deaths and \$100,000 of death claims. Expected lapses are $1{,}000(0.98)(0.08)=78.4$, giving surrender payments of $78.4(\$600)=\$47{,}040$. Continuing policies are $1{,}000(0.98)(0.92)=901.6$, so the closing liability is $901.6(\$900)=\$811{,}440$. Accumulated assets before outgo remain \$1,144,000. After expenses, death claims, and surrender payments, $A_1=\$1{,}144{,}000-\$20{,}000-\$100{,}000-\$47{,}040=\$976{,}960$. The solvency surplus is $\$976{,}960-\$811{,}440=\$165{,}520$, and shareholder profit is $\$165{,}520-\$100{,}000=\$65{,}520$.

Profit increases by $\$65{,}520-\$56{,}700=\$8{,}820$. Surrender payments increase by $\$47{,}040-\$29{,}400=\$17{,}640$, reducing assets by that amount. But the closing liability decreases by $\$837{,}900-\$811{,}440=\$26{,}460$. The net change in surplus is $-\$17{,}640+\$26{,}460=\$8{,}820$.

This result is a consequence of the stated reserve and benefit amounts, not a general rule that lapses are profitable. It also does not mean policyholders as a group are better off: more surrendering policyholders receive cash, while fewer policies remain with a reserve attached to them. The calculation keeps these effects distinct rather than treating a liability release as a policyholder payment.

Common mistake avoided: Counting the additional \$17,640 of surrender payments as the entire lapse effect and reporting profit of $\$56{,}700-\$17{,}640=\$39{,}060$. That omits the \$26,460 reduction in the closing liability. The corrected profit is \$65,520.

Expense and Investment Deviations

In this model, a change in year-end expense reduces assets and profit dollar for dollar, with no change in policyholder cash flows or the stipulated closing reserve. A change in investment return applies to the specified invested amount $A_0+P$. It can therefore be calculated directly as the change in return on that amount. These conclusions depend on the assumptions: if investment returns determine policyholder account values or the closing liability, the liability and possibly benefits must also be recalculated.

Worked Example: Combining Four Experience Deviations

Starting from the original portfolio, suppose mortality changes to $q=0.025$, the conditional lapse rate changes to $l=0.08$, expenses rise to \$24,000, and the annual effective investment return falls to 3%. Calculate the combined profit, then compare it with the sum of the isolated effects.

The expected deaths are $1{,}000(0.025)=25$, with death claims of \$125,000. Expected lapses are $1{,}000(0.975)(0.08)=78$, with surrender payments of $78(\$600)=\$46{,}800$. Continuing policies are $1{,}000(0.975)(0.92)=897$, giving a closing liability of $897(\$900)=\$807{,}300$.

Accumulated assets before year-end outgo are $(\$1{,}000{,}000+\$100{,}000)(1.03)=\$1{,}133{,}000$. Thus $A_1=\$1{,}133{,}000-\$24{,}000-\$125{,}000-\$46{,}800=\$937{,}200$. The solvency surplus is $\$937{,}200-\$807{,}300=\$129{,}900$, and profit is $\$129{,}900-\$100{,}000=\$29{,}900$. Compared with original profit of \$56,700, the combined change is $-\$26{,}800$.

For comparison, the isolated profit changes, each measured against the original projection, are: mortality, $-\$20{,}575$; lapse, $+\$8{,}820$; expenses, $-\$4{,}000$; and investment return, $-\$11{,}000$. The expense-only profit is $\$56{,}700-\$4{,}000=\$52{,}700$. The investment-only profit is $\$45{,}700$, because the return falls by $(\$1{,}100{,}000)(0.04-0.03)=\$11{,}000$. The sum of isolated changes is $-\$20{,}575+\$8{,}820-\$4{,}000-\$11{,}000=-\$26{,}755$, which is \$45 greater than the combined change.

The \$45 difference arises because the lapse count depends on the number surviving mortality. For a fully reconciled attribution, change assumptions in a stated order. Apply mortality first, giving profit \$36,125. Then increase lapses at the stressed mortality rate: with $q=0.025$, $l=0.08$, but the original expenses and return, assets are $\$1{,}144{,}000-\$20{,}000-\$125{,}000-\$46{,}800=\$952{,}200$; the reserve is \$807,300, so profit is \$44,900. This lapse step changes profit by $+\$8,775$. Raising expenses then reduces profit by \$4,000 to \$40,900. Reducing the return then reduces it by $(\$1{,}100{,}000)(0.01)=\$11{,}000$ to \$29,900. The sequential changes sum to $-\$20{,}575+\$8{,}775-\$4{,}000-\$11{,}000=-\$26{,}800$, exactly reconciling the combined projection.

Common mistake avoided: Assuming isolated effects always add exactly. Because lapses are conditional on survival, changing mortality also changes the population exposed to the lapse assumption. The combined projection, rather than an unadjusted sum of isolated effects, determines the actual result.

Checks for a Reliable Profit Attribution

An experience analysis is easier to audit when each deviation is traced through assets, policyholder payments, and liabilities. Use the following sequence:

State the basis and timing.
Identify which probabilities are conditional, when premiums and outgo occur, what earns investment return, and how the closing liability is defined.

Recalculate exposure counts.
For the model here, deaths are $Nq$, lapses are $N(1-q)l$, and continuing policies are $N(1-q)(1-l)$.

Recalculate assets and liability.
Apply the return to the stipulated invested amount, deduct year-end outgo, and value the remaining policies using the stated closing reserve.

Reconcile profit and policyholder effects.
Calculate $\Pi=A_1-L_1-K_0$, then separately report changes in death claims, surrender payments, continuing reserves, and solvency surplus.

A change in shareholder profit is not itself a complete description of a deviation. Higher death claims may be accompanied by fewer continuing policies; more lapses may release reserves while increasing surrender payments; and investment returns may affect liabilities when product terms link benefits to investment performance. Keeping the cash-flow and liability effects visible makes the result interpretable and helps distinguish a true change in economic surplus from a change in timing or exposure.

Key takeaway: Recalculate the full projected balance sheet under each deviation. Shareholder profit changes by the change in assets less the change in liabilities; policyholder payments and remaining-policy values should be shown separately. If several assumptions change together, calculate the combined scenario and identify any interaction rather than assuming isolated effects add exactly.

Check Your Understanding

Use the hypothetical portfolio and conventions defined above. Unless a question says otherwise, hold all assumptions at their original values: $q=0.02$, $l=0.05$, $j=0.04$, and expenses of \$20,000.

Change mortality only to $q=0.03$. What are the expected death claims, closing liability, and shareholder profit? (Answer: death claims \$150,000; closing liability \$829,350; profit \$15,550.)
Change the conditional lapse rate only to $l=0.06$. Calculate the shareholder profit and its change from the original projection. (Answer: profit \$59,640; increase of \$2,940.)
Change expenses only to \$23,500. What is the revised profit and the change in year-end solvency surplus? (Answer: profit \$53,200; solvency-surplus change $-\$3,500$.)
Change the annual effective investment return only to 3.5%. Calculate the revised profit and the investment-income change. (Answer: profit \$51,200; investment-income change $-\$5,500$.)
Now use $q=0.02$, $l=0.06$, expenses of \$23,500, and $j=0.035$ together. Calculate expected lapses, the closing liability, and shareholder profit. (Answer: 58.8 expected lapses; closing liability \$829,080; profit \$50,640.)

What You'll Learn