Interest Assumption Changes in Profit and Reserve Projections

Why an Interest Change Has Two Effects

Changing an interest assumption can affect a profit projection in two distinct ways. First, it changes the investment earnings on funds held during the projection. Second, it changes the discount factors used to express future profits or liabilities at an earlier date. These effects are related, but they are not interchangeable: a profit amount may itself change because of earned interest, while its present value also changes because of discounting.

The distinction matters when comparing projections. If the projected cash flows are held fixed, a rate change affects their present values but not their amounts. If investment earnings are part of the projected profit calculation, changing the rate can change both the profit amounts and the factors applied to those amounts. A sound comparison identifies which quantities are being recalculated and which are being held constant.

The methods below use annual effective interest rates and annual cash-flow timing. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $i$ be the effective annual interest rate and $v=1/(1+i)$. A prospective reserve at time $t$ is the present value, at time $t$, of expected future benefits and expenses less expected future premiums, using the specified future cash-flow assumptions and interest rate. A profit signature is a sequence of projected profits by time period. Its net present value is the sum of those profits discounted to a common valuation date.

Revaluing a Prospective Reserve

For an annual model, let $C_k$ denote expected net outgo at the end of the $k$-th year after the valuation date. Net outgo means benefits and expenses less premiums, with their expected amounts already reflecting the applicable probability assumptions. If the interest rate is $i$, the prospective reserve at the valuation date is

V_0(i)=\sum_{k=1}^{n} C_k v^k =\sum_{k=1}^{n}\frac{C_k}{(1+i)^k}.

When comparing rates, hold the cash-flow amounts and probability assumptions fixed if the intended question is the isolated effect of interest. In a full profit projection, however, investment earnings may depend on the rate and must be recalculated. For a stream of positive future net outgo, a lower rate increases the reserve. If net cash flows include both outgo and income, that direction is not guaranteed; the amounts and timing determine the result.

Worked Example: Revaluing a Two-Year Liability

A hypothetical contract has expected net outgo of $105 at the end of year 1 and $110.25 at the end of year 2. There are no other future cash flows. Calculate the reserve at effective annual rates of 5% and 10%, and find the change when the rate rises. The expected net outgo amounts are fixed in this comparison.

Method. Discount each dated amount to time 0 using the rate being tested, then compare the two reserves. This avoids applying one discount factor to cash flows at different durations.

At 5%, the discount factors are $v=1/1.05$ and $v^2=1/(1.05)^2=1/1.1025$. Thus

V_0(5\%)=\frac{105}{1.05}+\frac{110.25}{1.1025} =100+100=200.

At 10%, the factors are $1/1.10$ and $1/(1.10)^2=1/1.21$. Therefore

V_0(10\%)=\frac{105}{1.10}+\frac{110.25}{1.21} =95.454545\ldots+91.115702\ldots =186.570247\ldots.

The reserve falls by $200-186.570247\ldots=13.429752\ldots$. To the nearest cent, it is $200.00 at 5% and $186.57 at 10%, a decrease of $13.43. As a check, each amount at 5% discounts to exactly $100, while each amount at 10% discounts to less than its 5% value.

Common mistake. Discounting both payments by one year would give the wrong answer because the second payment is due at the end of year 2. Its discount factor is $v^2$, not $v$.

Investment Earnings and Discounting a Profit Signature

Write $\Pi_t(i)$ for profit at the end of year $t$, when that profit is projected using rate $i$. The present value at time 0 is

\operatorname{NPV}(i)=\sum_{t=1}^{n}\Pi_t(i)v^t.

The notation $\Pi_t(i)$ highlights a crucial issue: profit amounts may depend on the interest rate. To compare two rates, first recalculate each profit amount whose investment earnings depend on the rate. Then discount the resulting profit signature using the corresponding rate. If the profit amounts are explicitly fixed independently of interest, only the discount factors change.

A simple annual profit identity makes the earnings effect visible. Suppose an amount $A$ is held for a full year, earns the effective annual rate $i$, and an outgo $E$ is paid at year-end. The year-end amount before any closing reserve is $A(1+i)-E$. The interest contribution is $Ai$; it is not a discounting adjustment.

Worked Example: Separating Earnings from NPV Effects

A hypothetical projection starts year 1 with $1,000 invested for a full year. A $1,080 outgo is paid at the end of that year. The year-1 profit is the investment amount with its earnings, less the outgo. An independently specified profit of $200 occurs at the end of year 2 and is unchanged by the rate assumption. Compare the profit signatures and NPVs at 5% and 8%.

Method. Recalculate the year-1 profit from investment earnings, retain the stated fixed year-2 profit, and discount each end-of-year profit to time 0. The common mistake this avoids is changing discount factors while incorrectly leaving an interest-sensitive profit amount unchanged.

At 5%, the year-1 investment earns $1000(0.05)=50$. Hence

\Pi_1(5\%)=1000(1.05)-1080=1050-1080=-30, \qquad \Pi_2(5\%)=200.

The year-1 profit is discounted one year and the year-2 profit two years:

\operatorname{NPV}(5\%) =\frac{-30}{1.05}+\frac{200}{(1.05)^2} =-28.571428\ldots+181.405895\ldots =152.834467\ldots.

At 8%, the year-1 investment earns $1000(0.08)=80$, so

\Pi_1(8\%)=1000(1.08)-1080=1080-1080=0, \qquad \Pi_2(8\%)=200.

The new NPV is

\operatorname{NPV}(8\%) =\frac{0}{1.08}+\frac{200}{(1.08)^2} =\frac{200}{1.1664} =171.467764\ldots.

Rounded to the nearest cent, the NPVs are $152.83 and $171.47. The change is an increase of $171.467764\ldots-152.834467\ldots=18.633296\ldots$, or $18.63. It can also be separated into two effects: the year-1 profit change contributes $0-(-30/1.05)=28.571428\ldots$, while discounting the fixed year-2 profit at 8% rather than 5% changes its present value by $171.467764\ldots-181.405895\ldots=-9.938131\ldots$. Their sum is $18.633296\ldots$, agreeing with the direct comparison.

Check. The higher rate raises the year-1 profit by $30, and the year-2 profit has a lower present value. The earnings effect is larger here, so total NPV rises.

Why the Reserve Recursion Holds

A reserve projection and a profit projection must use compatible timing and interest assumptions. Consider a discrete annual model in which the opening reserve $V_{t-1}$ and premium $P_t$ are available at the start of year $t$, earn interest for the full year, and end-of-year outgo $E_t$ is paid at the end of that year. Let $V_t$ be the reserve immediately after that outgo. The profit for the year is the amount accumulated from opening funds and premium, less outgo and the closing reserve.

\Pi_t=(V_{t-1}+P_t)(1+i)-E_t-V_t.

This is an accounting identity under the stated timing, rather than a formula that applies unchanged to every cash-flow convention. If premiums are paid later, claims occur throughout the year, or a different interest-crediting convention is used, the accumulation terms must reflect that timing.

The identity also shows why reserve changes matter to profit. A higher closing reserve leaves less of the accumulated amount as current profit; a lower closing reserve leaves more. When the valuation rate changes, both the opening and closing reserves may change, as well as the investment earnings. It is therefore unsafe to change only the rate in the investment term while carrying forward reserves calculated on the old basis without stating that this is a transition comparison.

The prospective reserve recurrence follows directly from separating the first future year from the remaining cash flows. If $C_1$ is end-of-year net outgo and $V_1$ is the value at time 1 of all later net outgo, then the value at time 0 is the present value of $C_1+V_1$:

V_0=\frac{C_1+V_1}{1+i} \quad\Longleftrightarrow\quad V_0(1+i)-C_1=V_1.

Thus, if the reserve is calculated prospectively on the same assumptions as the projection, the accumulated opening reserve less the first-year outgo equals the closing reserve, when there are no premiums or other profit items. The resulting profit is zero in this deliberately self-consistent valuation projection. Actual profit testing can produce nonzero profits when experience, pricing, expenses, or the valuation basis differ from the projection basis.

Specify what changes.
Decide whether the exercise changes only the valuation rate, the projected investment return, or both. State which cash flows and probability assumptions remain fixed.

Recalculate reserves.
Discount each future net cash flow using its own time-specific factor under the new rate. Recalculate reserves at later projection dates on that same basis if needed.

Recalculate profits.
Update interest earnings and reserve movements using the specified cash-flow timing. Do not assume a previously calculated profit vector remains fixed if it includes interest-sensitive amounts.

Discount the new profit signature.
Apply the new rate's discount factors to the recalculated profits, then compare with the old NPV. Keep this discounting effect distinct from the change in profit amounts.

Worked Example: A Consistent Rate Change and a Reserve-Basis Transition

A hypothetical contract has no premiums, and its expected net outgo is $105 at the end of year 1 and $110.25 at the end of year 2. First, calculate the reserves at times 0 and 1 using 5%. Then recalculate them using 10%. Finally, suppose the time-0 reserve remains at its old 5% value when the projection earns 10%; calculate the year-1 profit using a time-1 closing reserve valued at 10%.

Method. Calculate each prospective reserve under its stated rate, then apply the annual accumulation identity to the transition case. This separates a consistent new-basis projection from the effect of carrying an old opening reserve into a projection using a new rate.

At 5%, the time-0 reserve is

V_0(5\%)=\frac{105}{1.05}+\frac{110.25}{(1.05)^2}=100+100=200.

At time 1, only the year-2 outgo remains. Its time-1 reserve is

V_1(5\%)=\frac{110.25}{1.05}=105.

At 10%, the time-0 reserve is

V_0(10\%)=\frac{105}{1.10}+\frac{110.25}{(1.10)^2} =95.454545\ldots+91.115702\ldots =186.570247\ldots.

The time-1 reserve on the 10% basis is

V_1(10\%)=\frac{110.25}{1.10}=100.227272\ldots.

For a projection that starts with the new-basis reserve, accumulated funds after the year-1 outgo are

186.570247\ldots(1.10)-105 =205.227272\ldots-105 =100.227272\ldots.

That equals $V_1(10\%)$, so the year-1 profit is zero, as expected for a projection matching the reserve basis and cash-flow assumptions. In the transition case, the opening reserve instead remains at $200. It accumulates to $200(1.10)=220$; after the $105 outgo, $115 remains. Subtracting the new-basis closing reserve gives

\Pi_1=200(1.10)-105-100.227272\ldots =14.772727\ldots.

The year-1 transition profit is $14.77 to the nearest cent. This is not the profit of a fully consistent 10% reserve projection: it results from combining an old-basis opening reserve with a new-basis closing reserve and a 10% earned rate. The calculation is a useful check on the timing and size of the basis transition, but the treatment of a reserve change in reported profit depends on the projection and accounting convention being used.

Common mistake. Using the new 10% closing reserve while also silently replacing the opening $200 with the 10% opening reserve would erase the specified transition. Conversely, using the old closing reserve would not measure the stated new-basis transition. Label each reserve by its valuation basis before applying the profit identity.

Common Traps in Rate Comparisons

Trap	Why it is wrong	Correction
Changing the discount rate but not rate-sensitive profit amounts	Interest earnings may change the profit signature itself.	Recalculate investment earnings and any resulting profits before discounting them.
Discounting a later cash flow for too few years	The discount exponent must match the time from the valuation date to payment.	Use $v^t$ for a payment at the end of year $t$.
Changing the earned rate but retaining old reserves without explanation	The resulting profit includes the effect of a mixed-basis transition.	Label opening and closing reserve bases and identify the transition being measured.
Assuming every reserve falls when interest rises	This is assured for positive future net outgo, but mixed-sign cash flows can behave differently.	Revalue the actual dated net cash flows rather than relying on a direction rule.
Adding individually rounded present values to report an exact rounded NPV	Intermediate rounding can change the final digit.	Calculate from unrounded factors and amounts, then round only the final result.

Key takeaway: An interest-rate comparison has two separate tasks: reproject any interest-sensitive profit amounts, and discount the resulting dated amounts using the appropriate rate. Revalue reserves consistently with their cash-flow timing and clearly identify any opening or closing reserve carried over from another basis.

Check Your Understanding

For each problem, state the relevant timing and use unrounded intermediate values when calculating present values.

A contract has expected net outgo of $210 at the end of year 1 and $242 at the end of year 2. Calculate its time-0 reserve at 10% and at 5%, and give the increase when the rate falls to 5%. (Answer: $390.91 at 10%; $419.50 at 5%; increase $28.59.)
An account of $500 earns interest for one year, after which a $530 outgo is paid. Calculate the year-end profit at 6% and at 4%. (Answer: $0 at 6%; -$10.00 at 4%.)
A profit signature consists of $100 at the end of year 1 and $121 at the end of year 2; both amounts are fixed independently of interest. Calculate its NPV at 10% and 5%, and the increase when the rate falls. (Answer: $190.91 at 10%; $204.99 at 5%; increase $14.08.)
Expected net outgo is $60 at the end of year 1 and $121 at the end of year 2. Calculate the time-0 reserve at 10% and 5%, then state whether it rises or falls when the rate decreases. (Answer: $154.55 at 10%; $166.89 at 5%; it rises by $12.35.)
A fixed profit of -$20 occurs at the end of year 1 and a fixed profit of $60 at the end of year 2. Calculate the NPV at 5% and at 10%, then give the change from the 5% NPV to the 10% NPV. (Answer: $35.37 at 5%; $31.40 at 10%; decrease $3.97.)

What You'll Learn