Modified Reserves And Profit Emergence

Why a Modified Reserve Changes Profit Emergence

A reserve is a liability measured at a particular duration; it is not an additional premium or benefit paid under the contract. Changing the reserve basis can therefore change the profit reported in a year even when the policyholder’s contractual cash flows are unchanged. The key is to account consistently for both the reserve at the start of the year and the reserve established for survivors at the end.

This tutorial studies annual profit emergence in a simple discrete-time model. It separates the assumptions used to measure actual experience from the assumptions used to calculate the modified reserves. The resulting formula shows directly how a reserve change can defer or accelerate profit across policy years.

Model and timing convention: Consider a policy in force at the start of a year. A premium $P_t$ and expense $E_t$ are paid at the start of the year. Death occurs during the year with actual probability $q_t$, and the death benefit $B_t$ is paid at year-end. A surviving policy has probability $p_t=1-q_t$ and is assigned an end-of-year reserve. The accumulated funds earn the actual annual effective rate $i$. There are no lapses, taxes, or other cash flows in this model.

Let $M_{t-1}$ be the modified reserve at the start of policy year $t$, and let $M_t$ be the modified reserve at year-end for a policy that survives that year. Profit is measured per policy in force at the start of the year. Thus the year-end reserve is required only for the proportion $p_t$ expected to survive. The quantities $P_t$, $E_t$, and $B_t$ are contractual or expense cash flows; changing the reserve basis does not change them.

Definition: Under the stated timing convention, expected annual profit per policy in force at the start of year $t$, using modified reserves, is $$ \pi_t^{M}=(M_{t-1}+P_t-E_t)(1+i)-q_tB_t-p_tM_t. $$ The first term is the accumulated opening reserve and net start-of-year cash flow. The second term is the expected death benefit, and the last term is the expected reserve held for survivors.

This is an expected, per-in-force-policy calculation: it is not the actual profit on every individual policy. It uses actual experience assumptions for the year’s interest and mortality, while using the stated modified reserve schedule for the liability held. If premiums, expenses, or benefits are paid at different times, or if lapses or other transitions occur, the corresponding cash flows and probabilities must be included with their actual timing.

Deriving the Reserve-Movement Formula

The formula follows from a year-end funds balance. At the start, the insurer holds $M_{t-1}$ and receives the premium while paying the expense, leaving $M_{t-1}+P_t-E_t$. After interest, expected funds are $(M_{t-1}+P_t-E_t)(1+i)$. The insurer pays the death benefit to the fraction $q_t$ that dies and retains $M_t$ for the fraction $p_t$ that survives. The remaining amount is profit. Subtracting those expected outflows gives the definition above.

To isolate the effect of the reserve basis, compare a modified schedule $M_t$ with a reference schedule $V_t$, while holding actual assumptions and all contractual cash flows fixed. Define the reserve difference $\Delta_t=M_t-V_t$. Subtracting the reference profit formula from the modified-basis formula cancels the premium, expense, and benefit terms:

\Delta\pi_t =\pi_t^M-\pi_t^V =(1+i)\Delta_{t-1}-p_t\Delta_t.

This equation is the central result. A higher opening reserve increases current profit because more funds are available to accumulate during the year. A higher closing reserve for survivors decreases current profit because more funds are retained as a liability. The closing difference is multiplied by $p_t$, not by one, because only survivors receive that reserve.

Theorem: If the actual experience assumptions and contractual cash flows are identical under two reserve schedules, their annual profit difference is determined entirely by the opening and closing reserve differences: $$ \Delta\pi_t=(1+i)\Delta_{t-1}-p_t\Delta_t. $$ In particular, a reserve change can redistribute profit between years. Under the conditions derived below, it does not change the discounted expected total profit over the full contract term.

For a cohort, the probability of being in force at the start of year $t$ is the product of the preceding survival probabilities. Discounting each year’s expected profit difference to the outset makes the reserve movements telescope. Write $s_{t-1}=\prod_{j=1}^{t-1}p_j$, with $s_0=1$, and $v=1/(1+i)$. Then:

\sum_{t=1}^{n}s_{t-1}v^t\Delta\pi_t = \sum_{t=1}^{n}s_{t-1}v^{t-1}\Delta_{t-1} - \sum_{t=1}^{n}s_tv^t\Delta_t = \Delta_0-s_nv^n\Delta_n.

The equality uses $s_{t-1}p_t=s_t$ and $v^t(1+i)=v^{t-1}$. Every intermediate reserve difference cancels: the end-of-year term for one policy year is the opening-reserve term for the next. If the two schedules agree at inception and at the terminal duration, so that $\Delta_0=\Delta_n=0$, the discounted expected total profit difference is zero. This result requires the same actual interest and survival assumptions for both profit calculations, and a consistent terminal boundary. If the terminal reserves differ, the final boundary term remains.

Worked Examples

In each example, reserve values are stated as the outputs of the valuation schedules being compared. The calculations use actual experience assumptions for profit emergence; they do not assume that the modified valuation basis is the actual experience basis.

Worked Example: One-Year Profit Under a Modified Reserve

A hypothetical one-year term policy has a premium of $200 and an expense of $20, both paid at the start of the year. The death benefit of $1,000 is paid at year-end if death occurs. Actual annual interest is $4\%$, and actual mortality is $q_1=0.10$, so $p_1=0.90$. The modified reserve is $40 at the start and $120 at year-end for a survivor. Calculate the expected annual profit per policy in force at the start of the year. Then compare it with a reference schedule having the same opening reserve and a $100 closing reserve.

Method. Use the annual funds-balance formula with actual interest and mortality. The opening funds after premium and expense are $40+200-20=220$. After interest, they are $220(1.04)=228.80$. The expected death benefit is $0.10(1{,}000)=100$, and the expected modified reserve for survivors is $0.90(120)=108$.

\pi_1^M=228.80-100-108=\$20.80.

For the reference schedule, the opening funds are unchanged. Its expected closing reserve is $0.90(100)=90$, so $$ \pi_1^V=228.80-100-90=\$38.80. $$ The modified schedule therefore reduces first-year profit by $20.80-38.80=-\$18.00$. The reserve difference is $\Delta_0=0$, $\Delta_1=20$; the reserve-movement formula gives $(1.04)(0)-(0.90)(20)=-\$18.00$, matching the direct calculation.

Check and common mistake. Only survivors receive the year-end reserve, which is why the $20 closing reserve difference is multiplied by $0.90$. Subtracting the full $20 would incorrectly give a profit difference of $-\$20$.

Worked Example: Profit Shifts Across Two Policy Years

A hypothetical two-year term policy pays a $500 death benefit at the end of a year in which death occurs. It charges a $150 premium and incurs a $20 expense at the start of each year in force. Actual annual interest is $5\%$. Mortality is $q_1=0.10$ in year 1 and $q_2=0.20$ in year 2, with conditional survival probabilities $p_1=0.90$ and $p_2=0.80$. Compare the modified reserve schedule $M_0=0, M_1=80, M_2=0$ with the reference schedule $V_0=0, V_1=50, V_2=0$. Calculate the annual profit difference and the discounted expected total difference at issue.

Year 1. The common net start-of-year cash flow is $150-20=130$, which accumulates to $130(1.05)=136.50$. Expected death benefits are $0.10(500)=50$. Modified expected closing reserves are $0.90(80)=72$; reference expected closing reserves are $0.90(50)=45$. Therefore:

\pi_1^M=136.50-50-72=14.50,\qquad \pi_1^V=136.50-50-45=41.50.

The year 1 profit difference is $\Delta\pi_1=14.50-41.50=-\$27.00$.

Year 2. The modified opening reserve is $80. The accumulated modified funds are $(80+130)(1.05)=220.50$. Expected death benefits are $0.20(500)=100$, and the terminal reserve is zero. Thus $\pi_2^M=220.50-100=\$120.50$. For the reference schedule, accumulated funds are $(50+130)(1.05)=189.00$, so $\pi_2^V=189.00-100=\$89.00$. The year 2 profit difference is $\Delta\pi_2=120.50-89.00=\$31.50$.

At issue, year 1 is certain to be reached, but year 2 is reached only if the policy survives year 1. Discounting at the actual $5\%$ rate gives:

-\frac{27.00}{1.05} +\frac{0.90(31.50)}{(1.05)^2} =-25.7143+25.7143 =\$0.00.

The schedules agree at inception and at the terminal duration, so the zero discounted difference also follows from the telescoping result. The modified basis moves profit out of year 1 and into year 2 without changing the discounted expected total under these assumptions.

Check and common mistake. The year 2 profit difference is weighted by the probability of surviving year 1 when it is valued at issue. Omitting the factor $0.90$ would incorrectly produce a nonzero discounted total.

Worked Example: Finding a Closing Reserve for a Target Profit

A hypothetical policy is in force at the start of a year with a modified opening reserve of $20. The premium is $100 and the start-of-year expense is $10. The death benefit is $400 at year-end. Actual interest is $4\%$, and actual mortality is $q=0.10$, so $p=0.90$. Find the closing reserve for survivors that produces expected annual profit of $14.40 per policy in force at the start of the year.

Method. First accumulate the opening reserve and net start-of-year cash flow: $$ (20+100-10)(1.04)=110(1.04)=114.40. $$ The expected death benefit is $0.10(400)=40$. Let the required closing reserve be $M_1$. The target profit equation is $$ 114.40-40-0.90M_1=14.40. $$ Therefore $0.90M_1=60.00$, and $$ M_1=\frac{60.00}{0.90}=\$66.666\ldots. $$ To the nearest cent, the closing reserve is **$66.67**.

Check and common mistake. Using the unrounded reserve gives profit $114.40-40-0.90(66.666\ldots)=\$14.40$. Using the rounded reserve gives $114.40-40-0.90(66.67)=\$14.397$, which rounds to $14.40. Dividing the required reserve outflow by one instead of by the survivor probability would fail to account for the fact that the reserve is held only for survivors.

Interpreting Profit Emergence and Avoiding Traps

A reserve basis is a measurement choice, while premiums and benefits are contractual cash flows. A modified reserve can make reported profit lower in one duration and higher in another. That movement is not, by itself, evidence that the contract has become more or less profitable overall. To assess a total, use consistent survival weighting, discounting, and terminal reserves.

The formula is also useful in reverse. If a target profit is specified, rearrange it to solve for the required closing reserve: $$ M_t=\frac{(M_{t-1}+P_t-E_t)(1+i)-q_tB_t-\pi_t^M}{p_t}, $$ provided $p_t>0$. If no policy can survive the year, a survivor reserve is not defined by this calculation. In practical valuation work, a reserve must also satisfy the applicable valuation method and boundary conditions; solving this equation alone does not establish that a reserve schedule is an appropriate statutory or actuarial reserve.

The most common exam errors are timing and weighting errors. A start-of-year premium earns a full year of interest in this model, while a year-end death benefit does not. The closing reserve is multiplied by the actual probability of survival, and a later year’s profit must be weighted by the probability of reaching that year when calculating an inception value. Using reserve-basis mortality in the profit calculation when the question specifies actual mortality also confuses the liability measurement basis with actual experience.

Key takeaway: With contractual cash flows and actual experience assumptions fixed, annual profit changes only through reserve movements: the opening reserve difference accumulates with interest, while the closing reserve difference reduces profit for survivors. If initial and terminal reserve differences are both zero, these changes cancel in discounted expected total profit.

Check Your Understanding

Use the timing convention and formulas in this tutorial. Answers are rounded to the nearest cent where needed.

A one-year policy has opening modified reserve $15, premium $90, expense $8, death benefit $300, actual interest $2\%$, and mortality $q=0.04$. Its closing reserve for survivors is $40. Find expected profit per policy in force at the start of the year. (Answer: $48.54)
Actual interest is $3\%$, survival probability is $0.80$, and the reserve differences are $\Delta_0=12$ and $\Delta_1=30$. Find the modified-minus-reference annual profit difference. (Answer: $-\$11.64$)
In a two-year model, actual interest is $5\%$, $p_1=0.90$, and $p_2=0.80$. The reserve differences are $\Delta_0=0,\Delta_1=40,\Delta_2=0$. Find each year’s per-in-force profit difference and the discounted expected total difference at issue. (Answer: Year 1 $-\$36.00$; year 2 $+\$42.00$; discounted expected total $\$0.00$)
A policy has opening modified reserve $50, premium $130, expense $10, death benefit $250, actual interest $4\%$, and mortality $q=0.20$. What closing reserve for survivors produces expected annual profit of $20? (Answer: $133.50)
Actual interest is $5\%$ and survival probability is $0.90$. A modified schedule has an opening reserve $10 higher than the reference schedule and a closing reserve $20 higher. Find the annual profit difference and state whether current profit increases or decreases. (Answer: $-\$7.50$; current profit decreases)

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