Why Present Values Can Be Reserve-Basis Invariant

Why Reserve Basis Can Change Profit Timing Without Changing Present Value

A reserve basis can change the amount of profit reported in each policy year. A stronger closing reserve, for example, generally reduces current-year profit and may increase profit released in a later year. Yet if the reserve recursion and discounting use the same interest basis, those shifts can cancel in the present value. The cancellation is exact only after the initial and terminal reserve terms are accounted for.

The result is useful when comparing profit signatures calculated on different reserve bases. It does not say that annual profits are equal, or that present values must agree regardless of assumptions. It gives a precise test: the underlying cash flows and discount rate must be the same, and the boundary reserve terms must either agree or be included in the comparison.

Definition: Consider a deterministic annual model over years $1,\ldots,n$, with effective annual interest rate $i$ and discount factor $v=1/(1+i)$. Let $C_t$ be the net cash flow available at the end of year $t$, with inflows positive and outflows negative. Let $R_{t-1}$ be the reserve at the start of year $t$, and $R_t$ the reserve at its end. The annual reserve-adjusted profit is $$ \Pi_t=C_t+(1+i)R_{t-1}-R_t. $$

This convention treats the opening reserve as accumulating for one year and then combines it with the year-end net cash flow. The closing reserve is set aside; the remaining amount is profit. Equivalently, the reserve balance equation is $R_t=(1+i)R_{t-1}+C_t-\Pi_t$. This is a deterministic accounting identity, not a mortality or survival-weighted expected-profit formula. If cash flows occur at different times during a year, they must first be accumulated or discounted consistently with the model’s timing convention.

The Discounted-Profit Identity

Discount each year’s profit to time 0 using the same effective annual rate that accumulates the opening reserve. In year $t$, the discounted reserve contribution is especially simple: $v^t(1+i)R_{t-1}=v^{t-1}R_{t-1}$. It therefore cancels the previous year’s discounted closing reserve.

Write the annual profit.
For each year, use $\Pi_t=C_t+(1+i)R_{t-1}-R_t$.

Discount by the matching factor.
Multiply by $v^t$, where $v=1/(1+i)$, so the accumulated opening reserve becomes $v^{t-1}R_{t-1}$.

Sum over all years.
Adjacent reserve terms cancel, leaving only the initial reserve and discounted terminal reserve.

Theorem: Under the annual model and common discount and accumulation rate defined above, $$ \sum_{t=1}^{n}v^t\Pi_t = \sum_{t=1}^{n}v^tC_t+R_0-v^nR_n. $$ The present value of reserve-adjusted profits is the present value of the cash flows plus the initial reserve, less the discounted terminal reserve.

To justify the result, substitute the profit definition into the left-hand side. Since $v(1+i)=1$, each discounted opening-reserve term is $v^{t-1}R_{t-1}$. The reserve sum is therefore a telescoping sum:

\begin{aligned} \sum_{t=1}^{n}v^t\Pi_t &=\sum_{t=1}^{n}v^tC_t+ \sum_{t=1}^{n}\left(v^{t-1}R_{t-1}-v^tR_t\right)\\ &=\sum_{t=1}^{n}v^tC_t+ \left(R_0-vR_1\right)+\left(vR_1-v^2R_2\right)+\cdots+ \left(v^{n-1}R_{n-1}-v^nR_n\right)\\ &=\sum_{t=1}^{n}v^tC_t+R_0-v^nR_n. \end{aligned}

Every intermediate reserve $R_1,\ldots,R_{n-1}$ appears once with a positive sign and once with a negative sign at the same discounted value. Only the boundary terms remain. This cancellation is the reason a change in intermediate reserves can redistribute profit between years without changing its total present value.

Comparing Two Reserve Bases

Suppose two reserve bases, $R_t$ and $Q_t$, are applied to the same cash-flow sequence and use the same interest rate $i$. Let their annual profits be $\Pi_t^R$ and $\Pi_t^Q$. Applying the identity to each basis and subtracting gives:

\sum_{t=1}^{n}v^t\left(\Pi_t^R-\Pi_t^Q\right) = (R_0-Q_0)-v^n(R_n-Q_n).

Thus, present values of profit are equal if the initial reserves agree and the terminal reserves agree. A common case is a new policy with zero initial reserve and no reserve remaining at expiry under either basis. More generally, if the initial or terminal reserves differ, the discounted profit values differ by exactly the boundary adjustment shown above. Agreement of intermediate reserves is not required.

This conclusion depends on more than the labels attached to the bases. The cash flows must be the same, the reserves must satisfy the stated recursion, and the profits must be discounted at the rate used to accumulate opening reserves. If the underlying cash flows also change, the difference in their present values must be included. If the discount rate does not match the reserve accumulation rate, the reserve terms no longer cancel.

Worked Example: Different Intermediate Reserves, Equal Present Values

A hypothetical two-year contract has certain end-of-year net cash flows $C_1=\$100$ and $C_2=-\$30$. There is no mortality, lapse, or uncertainty in these stipulated amounts. The annual effective interest rate is $10\%$, and all cash flows are treated as year-end amounts. Under basis A, $R_0=0$, $R_1=\$40$, and $R_2=0$. Under basis B, $R_0=0$, $R_1=\$20$, and $R_2=0$. Calculate the annual profits and their present values.

Method. Apply the reserve-adjusted profit definition separately under each basis, then check both present values against the discounted cash flows. This avoids mistaking a change in annual profit timing for a change in total present value.

The discount factor is $v=1/1.10=10/11$. Under basis A:

Year 1 profit is $\Pi_1^A=100+1.10(0)-40=\$60$. Year 2 profit is $\Pi_2^A=-30+1.10(40)-0=\$14$. Discounting gives $60/1.10=\$54.5455$ and $14/(1.10)^2=\$11.5702$, so the total is $\$66.1157$.

Under basis B, year 1 profit is $\Pi_1^B=100+1.10(0)-20=\$80$. Year 2 profit is $\Pi_2^B=-30+1.10(20)-0=-\$8$. The discounted amounts are $80/1.10=\$72.7273$ and $-8/(1.10)^2=-\$6.6116$, for a total of $\$66.1157$.

As a direct check, the present value of cash flows is $100/1.10-30/(1.10)^2=\$90.9091-\$24.7934=\$66.1157$. Both bases have $R_0=R_2=0$, so the boundary adjustment is zero. Rounded to the nearest cent, each profit present value is \$66.12, even though the annual profit patterns differ.

Boundary Reserves Are Part of the Answer

A terminal reserve should not be silently discarded simply because the calculation horizon has ended. If the contract still has a liability at time $n$, the terminal reserve is a boundary value in the identity. Likewise, an initial reserve is not automatically zero: a portfolio transferred into the model may begin with an established reserve. The next example shows how both terms affect the result and how different boundary values change the comparison between bases.

Worked Example: Initial and Terminal Reserve Adjustments

A hypothetical two-year contract has certain year-end net cash flows $C_1=\$50$ and $C_2=\$60$, no mortality or lapse, and effective annual interest $i=5\%$. All cash flows occur at year-end. Under basis A, reserves are $R_0=\$10$, $R_1=\$30$, and $R_2=\$5$. Under basis B, they are $Q_0=\$8$, $Q_1=\$25$, and $Q_2=\$4$. Calculate each basis’s profit present value and verify the difference.

Method. Use the annual recursion to calculate profits and the boundary identity to check them. The common cash-flow present value is $50/1.05+60/(1.05)^2=\$47.6190+\$54.4218=\$102.0408$.

For basis A, year 1 profit is $50+1.05(10)-30=\$30.50$. Year 2 profit is $60+1.05(30)-5=\$86.50$. Its discounted profit value is $30.50/1.05+86.50/(1.05)^2=\$29.0476+\$78.4580=\$107.5057$.

The identity gives the same result: basis A’s boundary adjustment is $10-5/(1.05)^2=\$5.4649$. Adding this to the cash-flow present value gives $\$102.0408+\$5.4649=\$107.5057$.

For basis B, year 1 profit is $50+1.05(8)-25=\$33.40$. Year 2 profit is $60+1.05(25)-4=\$82.25$. Its discounted profit value is $33.40/1.05+82.25/(1.05)^2=\$31.8095+\$74.6032=\$106.4127$.

The difference between the two profit present values is $\$107.5057-\$106.4127=\$1.0930$. The boundary formula gives the same value: $(10-8)-(5-4)/(1.05)^2=\$2-\$0.9070=\$1.0930$. Rounded to the nearest cent, the present values are \$107.51 under basis A and \$106.41 under basis B. They differ because their boundary reserves differ, not because of their intermediate reserve values.

When the Discount Rate Does Not Match

The cancellation in the derivation relies on $v(1+i)=1$. If reserves accumulate at rate $i$ but profits are discounted at a different rate, the opening reserve term is not the previous discounted closing reserve. In that situation, the boundary-only identity does not apply using the different discount rate. This is a common source of incorrect claims that present values must be reserve-basis invariant.

Worked Example: A Mismatched Discount Rate

Use the cash flows and basis A reserves from the first example: $C_1=\$100$, $C_2=-\$30$, $R_0=0$, $R_1=\$40$, and $R_2=0$. Reserves accumulate at $10\%$, and there is no mortality or lapse. The resulting profits are $\Pi_1=\$60$ and $\Pi_2=\$14$. Now, incorrectly for the identity, discount profits and cash flows at $5\%$. Quantify the discrepancy.

Method. Calculate the two present values directly and isolate the reserve contribution. This makes clear why the cancellation fails when the discount factor does not match reserve accumulation.

At $5\%$, the discounted profit value is $60/1.05+14/(1.05)^2=\$57.1429+\$12.6984=\$69.8413$. The discounted cash-flow value is $100/1.05-30/(1.05)^2=\$95.2381-\$27.2109=\$68.0272$. The difference is $\$1.8141$.

That difference is exactly the discounted reserve contribution at the mismatched rate. In year 1, the reserve contribution is $-\$40$; in year 2, it is $1.10(40)=\$44$. Their $5\%$-discounted sum is $-40/1.05+44/(1.05)^2=-\$38.0952+\$39.9093=\$1.8141$. Rounded to the nearest cent, the profit present value is \$69.84, the cash-flow present value is \$68.03, and the discrepancy is \$1.81.

Practical Interpretation and Common Trap

The identity is an algebraic consistency check for annual profit tests. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. Use it to separate two questions: whether profit is emerging in the same pattern each year, and whether its discounted total is the same. A reserve basis can materially change the first answer while leaving the second unchanged, provided the required conditions hold.

A frequent incorrect argument is: “Intermediate reserves cancel, so all reserve bases have the same profit present value.” That statement omits the initial and terminal reserves. For the second example, incorrectly treating the bases as invariant would report no difference. The correct difference is $R_0-Q_0-v^n(R_n-Q_n)=\$1.09$. The other frequent error is to discount at a rate chosen separately from the reserve accumulation rate and still invoke the telescoping identity. The third example shows that this leaves a nonzero reserve contribution.

Key takeaway: With common cash flows and a consistent annual interest basis, discounted reserve-adjusted profits equal discounted cash flows plus $R_0-v^nR_n$. Two bases have equal profit present values when their boundary adjustment is the same; matching intermediate reserves is unnecessary.

Check Your Understanding

For each problem, use the stated cash-flow timing and interest basis. Assume no mortality or lapse unless otherwise specified.

A two-year contract has year-end cash flows $C_1=\$40$ and $C_2=\$70$, effective annual interest $4\%$, and reserves $R_0=0$, $R_1=\$20$, $R_2=0$. Find both annual profits and their present value. (Answer: profits $\$20.00$ and $\$90.80$; present value $\$103.18$.)
For a three-year model at $6\%$, two bases have the same cash flows, $R_0=Q_0=\$12$, $R_3=\$3$, and $Q_3=\$3$. What is the difference in profit present values, basis $R$ minus basis $Q$, regardless of their intermediate reserves? (Answer: $\$0.00$.)
At $5\%$, basis A and basis B use the same cash flows over three years. Their reserve differences are $R_0-Q_0=0$ and $R_3-Q_3=\$6$. Find the present-value difference, basis A minus basis B, to the nearest cent. (Answer: $-\$5.18$.)
In a one-year model, $C_1=\$100$, $R_0=0$, and $R_1=\$20$. Reserves accumulate at $8\%$, but profits and cash flows are discounted at $5\%$. Find the difference between discounted profit and discounted cash flow. (Answer: $-\$19.05$.)
State the boundary condition on two reserve bases that makes their profit present values equal when cash flows and the interest basis are common. (Answer: $R_0-Q_0=v^n(R_n-Q_n)$.)

What You'll Learn