Alternative Reserve Bases and Profit Comparability

Why Reserve Bases Can Produce Different Profit Signatures

Two profit tests can use the same hypothetical contract cash flows and still report different profits by year. The reason may be that they value the opening and closing reserves on different bases: one basis can hold more profit back in one period and release it in another. That difference does not, by itself, mean the contract has generated different total economic value.

A fair comparison therefore separates recognition timing from changed cash flows. The test must use the same projected premiums, benefits, expenses, decrements, and asset accumulation. It must also use consistent opening and terminal conditions, or explicitly allow for their difference. This tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A reserve basis is the set of valuation assumptions and methods used to determine the reserve at each duration. A profit signature is the sequence of profits emerging by duration under a specified profit-test convention. In a reserve-basis comparison, the projected contract cash flows are held fixed while the reserve values are recalculated on each basis.

Use an annual model with one policy state while the policy is in force. At the start of year $t$, the policy is in force; during the year it may terminate through a decrement, and only a policy surviving to the end of the year receives a closing reserve. Let $p_t$ be the probability of surviving year $t$, conditional on being in force at its start. Let $V_t^{(j)}$ be the reserve at the start of year $t$ on basis $j$. Let $C_t$ be the expected end-of-year value of all common non-reserve cash flows, conditional on being in force at the start of the year. It includes their accumulation at the profit-test rate $i$, but excludes the roll-forward of the opening reserve and the closing reserve. For example, with a premium $P_t$ and expense $e_t$ paid at the start of the year and a death benefit $B_t$ paid at year-end on death, $C_t=(P_t-e_t)(1+i)-(1-p_t)B_t$, provided there are no other cash flows.

Profit convention: The expected profit at the end of year $t$, conditional on the policy being in force at the start of that year, is $ \Pi_t^{(j)}=(1+i)V_t^{(j)}+C_t-p_tV_{t+1}^{(j)}. $ The term $p_tV_{t+1}^{(j)}$ is the expected closing reserve: it is held only for a policy that survives the year. All terms in this equation are measured at year-end. This convention assumes the opening reserve accumulates at the profit-test rate $i$. If a profit test uses a different accumulation convention, its formula must be adapted before comparing signatures.

How Reserve Differences Shift Profit

Write the difference between the reserves on two bases as $\Delta V_t=V_t^{(A)}-V_t^{(B)}$, and the difference between conditional profits as $\Delta\Pi_t=\Pi_t^{(A)}-\Pi_t^{(B)}$. When the cash flows and profit-test accumulation rate are the same, subtracting the profit equations gives

\Delta\Pi_t=(1+i)\Delta V_t-p_t\Delta V_{t+1}.

This equation gives the period-by-period effect directly. A higher opening reserve increases the current year's profit because more funds are available after accumulation. A higher closing reserve reduces the current year's profit because more funds must be retained for surviving policies. The sign of the change is not determined by either reserve alone; both reserve movements matter.

Result: If the opening reserves agree, the terminal reserves agree, and both bases are compared using the same cash flows, decrements, and profit-test accumulation rate, then the survival-weighted present value of the profit difference is zero over the comparison period. The signatures can differ by year even though their present values agree.

Why the Present Values Telescope

The result follows by valuing each year's conditional profit difference at the issue date and weighting it by the probability the policy is still in force when that year's profit is measured. For an $n$-year projection, the probability of being in force at the start of year $t$ is ${}_tp$, with ${}_0p=1$. The present value of the difference in year $t$ is therefore ${}_tp\,v^{t+1}\Delta\Pi_t$, where $v=1/(1+i)$.

Substitute the reserve-difference formula.
The contribution from year $t$ is ${}_tp\,v^{t+1}\big((1+i)\Delta V_t-p_t\Delta V_{t+1}\big)$.

Simplify the opening-reserve term.
Since $(1+i)v^{t+1}=v^t$, this term is ${}_tp\,v^t\Delta V_t$.

Recognize the next year's closing-reserve term.
Because ${}_{t+1}p={}_tp\,p_t$, the closing-reserve term is $-{}_{t+1}p\,v^{t+1}\Delta V_{t+1}$.

Sum and cancel intermediate terms.
Each intermediate reserve difference appears once as a closing reserve and once as the next year's opening reserve, with opposite signs. Only the boundaries remain.

\sum_{t=0}^{n-1} {}_tp\,v^{t+1}\Delta\Pi_t =\Delta V_0-{}_np\,v^n\Delta V_n.

Thus, equal opening reserves and equal terminal reserves make the right side zero. If the opening reserves differ, the present-value difference includes that opening amount. If terminal reserves differ, the result includes the survival-weighted present value of that terminal difference. A comparison over a finite projection should not quietly treat unequal terminal reserves as if they were equal.

Worked Example: Same Cash Flows, Different One-Year Reserve

Consider a hypothetical two-year contract with certain survival, an annual profit-test accumulation rate of $i=5\%$, and common end-of-year operating contributions $C_0=\$100$ and $C_1=\$80$. On basis A, reserves are $V_0^{(A)}=\$0$, $V_1^{(A)}=\$40$, and $V_2^{(A)}=\$0$. On basis B, they are $V_0^{(B)}=\$0$, $V_1^{(B)}=\$20$, and $V_2^{(B)}=\$0$. Find both profit signatures and compare their present values. The method is to calculate conditional profits on each basis, then discount the year-end profits.

Year 0, basis A: $\Pi_0^{(A)}=1.05(0)+100-1(40)=\$60.00$.

Year 0, basis B: $\Pi_0^{(B)}=1.05(0)+100-1(20)=\$80.00$. Basis A recognizes $\$20.00$ less in this year.

Year 1, basis A: $\Pi_1^{(A)}=1.05(40)+80-1(0)=\$122.00$.

Year 1, basis B: $\Pi_1^{(B)}=1.05(20)+80-1(0)=\$101.00$. Basis A recognizes $\$21.00$ more in this year, including the one-year accumulation of the $\$20.00$ reserve difference.

The present value on basis A is $60+122/1.05=\$176.19$. The present value on basis B is $80+101/1.05=\$176.19$. The opening and terminal reserves agree, so the reserve basis changes the timing but not the present value. This avoids the common mistake of comparing only one year's profit and treating the difference as a change in total value.

Worked Example: Survival-Weighted Comparison

A hypothetical two-year policy has $i=5\%$, survival probabilities $p_0=0.80$ and $p_1=0.75$, and common conditional operating contributions $C_0=\$100$ and $C_1=\$70$. Basis A has reserves $(V_0,V_1,V_2)=(\$0,\$80,\$0)$; basis B has $(\$0,\$30,\$0)$. Calculate the conditional profit signatures and the issue-date present value on each basis. The key method is to weight year 1 profit by the probability of reaching year 1.

Year 0, basis A: $\Pi_0^{(A)}=100-0.80(80)=\$36.00$. Basis B: $\Pi_0^{(B)}=100-0.80(30)=\$76.00$.

Year 1, basis A: $\Pi_1^{(A)}=70+1.05(80)-0.75(0)=\$154.00$. Basis B: $\Pi_1^{(B)}=70+1.05(30)-0.75(0)=\$101.50$.

The year 0 profit is already measured per policy in force at issue, so its issue-date present value is its amount divided by $1.05$. For year 1, only $0.80$ of the issued policies are expected to be in force at its start, and the year-end profit is discounted two years. Hence basis A has present value $36/1.05+(0.80)(154)/(1.05)^2=\$146.03$. Basis B has present value $76/1.05+(0.80)(101.50)/(1.05)^2=\$146.03$.

The conditional profit differences are $-\$40.00$ in year 0 and $+\$52.50$ in year 1. Their correctly weighted present values are $-40/1.05=-\$38.10$ and $(0.80)(52.50)/(1.05)^2=+\$38.10$, which cancel. The common mistake is to add conditional differences without survival weighting or discounting: $-40+52.50=\$12.50$ is not a present-value difference.

Worked Example: A Terminal-Reserve Difference

A hypothetical two-year contract has certain survival, $i=10\%$, and common operating contributions $C_0=\$60$ and $C_1=\$70$. Basis A has reserves $(\$100,\$80,\$20)$, while basis B has reserves $(\$100,\$50,\$10)$ at times 0, 1, and 2. Calculate the profit differences and their issue-date present value. The method is to use the reserve-difference equation and retain the terminal boundary rather than assume it is zero.

The reserve differences are $\Delta V_0=0$, $\Delta V_1=\$30$, and $\Delta V_2=\$10$. Therefore, year 0 has $\Delta\Pi_0=1.10(0)-30=-\$30.00$. Year 1 has $\Delta\Pi_1=1.10(30)-10=\$23.00$.

The issue-date present value of the difference is $-30/1.10+23/(1.10)^2=-\$8.26$. The boundary formula gives the same result: $0-(1)(10)/(1.10)^2=-\$8.26$. The signatures do not have equal present values over this horizon because the terminal reserves differ by $\$10$. The difference is a boundary adjustment, not evidence that the common projected cash flows changed.

When a Comparison Is Not Purely About Timing

A reserve-basis comparison is informative only after the items being held fixed have been made explicit. Reserve valuation assumptions may differ, but if the projected benefits, premiums, expenses, decrement probabilities, or profit-test accumulation rate also differ, the resulting signatures do not isolate reserve timing. Likewise, a different opening reserve or a different terminal reserve creates a boundary effect in the present-value comparison.

For an actuarial review, first label each profit as conditional on being in force at the start of the year or already survival-weighted per issued policy. Then use the same timing convention, accumulation rate, and cash-flow projection for both bases. Finally, compare survival-weighted discounted profits over the same horizon and state any opening or terminal reserve adjustment. These checks help distinguish a genuine change in projected economics from a redistribution of profit among durations.

Do not compare raw conditional profits across years. Later-year conditional profits apply only to policies reaching those years.
Do not omit discounting. Profits paid or measured at different year-ends are not directly additive as present values.
Do not assume a zero terminal adjustment. It is valid only when the relevant terminal reserves agree or when the comparison explicitly includes their difference.
Do not attribute every signature change to reserve timing. Confirm that projected cash flows, decrements, and the profit-test accumulation convention are unchanged.

Key takeaway: Under a consistent annual profit convention, alternative reserve bases change conditional profit by $(1+i)\Delta V_t-p_t\Delta V_{t+1}$. Survival-weighted discounting makes these reserve movements telescope. Equal opening and terminal reserves therefore imply equal present values, even when the annual profit signatures differ.

Check Your Understanding

For each question, distinguish conditional profit differences from survival-weighted present-value differences.

A certain-survival two-year projection uses $i=4\%$, with reserve differences $\Delta V_0=0$, $\Delta V_1=\$25$, and $\Delta V_2=0$. Find the profit differences in the two years and their issue-date present-value difference. (Answer: $-\$25.00$ and $+\$26.00$; present-value difference $\$0.00$.)
A two-year projection has $p_0=0.90$, $p_1=0.80$, and $i=5\%$, with $\Delta V_0=0$, $\Delta V_1=\$40$, and $\Delta V_2=0$. Find both conditional profit differences and verify their survival-weighted present-value difference. (Answer: $-\$36.00$, $+\$42.00$; present-value difference $\$0.00$.)
In a certain-survival two-year projection at $i=8\%$, reserve differences are $\Delta V_0=\$5$, $\Delta V_1=\$20$, and $\Delta V_2=0$. Find the two conditional profit differences and their issue-date present-value difference. (Answer: $-\$14.60$, $+\$21.60$; present-value difference $+\$5.00$.)
A two-year projection has $p_0=0.75$, $p_1=0.60$, and $i=4\%$, with $\Delta V_0=0$, $\Delta V_1=\$50$, and $\Delta V_2=\$10$. Find the conditional profit differences and the issue-date present-value difference, rounding the latter to cents. (Answer: $-\$37.50$, $+\$46.00$; present-value difference $-\$4.16$, the discounted survival-weighted terminal reserve difference.)
Two bases have identical projected cash flows except that one projects an additional $\$5$ death benefit paid at the end of year 0 on death. The year-0 death probability is $0.10$, $i=5\%$, and reserve differences are $\Delta V_0=0$, $\Delta V_1=\$20$, and $\Delta V_2=0$. What is the issue-date present-value difference between the bases, including both the reserve-timing effect and the changed benefit? (Answer: $-\$0.48$ per issued policy; the reserve-timing effect cancels, and the additional expected benefit costs $0.10(\$5)/1.05$.)

What You'll Learn