Valuation Interest Rates and Reserve Sensitivity

Why Separate the Two Interest Rates?

A profit test can involve two different interest rates. The actual investment return determines how assets grow during the projection period. The valuation interest rate is an assumption used to discount future cash flows when calculating a reserve. Changing one does not, by itself, change the other.

This distinction matters because a higher valuation rate will often reduce a reserve for future net outgo. If the actual return is held fixed, that reserve reduction changes the amount set aside for surviving policies and therefore changes expected annual profit. It does not create additional investment income. The discussion and calculations below are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $i$ be the effective annual valuation interest rate used to discount future cash flows, and let $j$ be the effective annual actual investment return earned on assets during a projection year. In this tutorial, $i$ determines the reserve and $j$ determines asset accumulation. They may be equal in a particular model, but changing $i$ does not imply a change in $j$.

A prospective reserve is the valuation-date value of specified future outgo less specified future income, conditional on the contract being in force at that date. Its calculation depends on the timing and probabilities of the cash flows, as well as on the valuation basis. Consequently, a reserve sensitivity calculation must state which cash flows and assumptions are held fixed.

Discounting and the Direction of Reserve Change

For a cash flow $C_t$ payable at the end of year $t$, its value at the valuation date is $C_t/(1+i)^t$. For a fixed set of future cash flows, the present value of net outgo can therefore be written as:

V(i)=\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}.

Here $C_t$ may represent expected net outgo at time $t$; premiums or other future income can be represented as negative amounts. This expression assumes annual effective discounting, fixed cash flows and probabilities, and no change in the timing or scope of the valuation. It is not a general formula for every reserve model: for example, a change in the valuation basis may also change decrements or other assumptions if the model specifies that they are linked.

Result: If every $C_t$ is nonnegative and fixed, increasing $i$ cannot increase $V(i)$. If net cash flows have both positive and negative amounts, the direction of the change is not guaranteed; it depends on their sizes and timing.

The result follows directly from discounting. For $t>0$, the factor $(1+i)^{-t}$ decreases as $i$ increases. Differentiating the fixed-cash-flow expression makes the condition explicit:

\frac{dV}{di} = -\sum_{t=1}^{n}\frac{tC_t}{(1+i)^{t+1}}.

When all $C_t\geq 0$, every term in the sum on the right before the minus sign is nonnegative, so $dV/di\leq 0$. If some $C_t$ are negative, some terms change sign, and the derivative need not be negative. This is why “a higher rate lowers the reserve” is a useful conditional rule, not a conclusion that should be applied without checking the cash-flow pattern.

Reserve Changes and Expected Annual Profit

For a simple annual profit-test step, suppose an opening reserve $V_0$, premium $P$, and expense $E$ are available at the start of the year. Assume the net amount $V_0+P-E$ earns the actual annual return $j$. At year-end, a death benefit $B$ is paid if death occurs, with probability $q$; otherwise a closing reserve $V_1(i)$ is held for the surviving policy, with probability $p=1-q$. Under these timing assumptions, expected year-end profit per policy in force at the start of the year is:

\Pi(i) = (V_0+P-E)(1+j)-qB-pV_1(i).

This formula treats $V_1(i)$ as a reserve per surviving policy and weights it by the probability of survival. It holds $V_0,P,E,j,q,$ and $B$ fixed when comparing valuation rates. If the opening reserve is also recalculated, or if cash flows occur at different times, those changes must be included rather than silently attributed to the closing reserve.

Key relationship: If only the closing reserve changes, then $\Delta\Pi=-p\,\Delta V_1$. A lower closing reserve increases expected profit by the survival-weighted reduction in the reserve. Actual investment income is unchanged when $j$ is held fixed.

Worked Reserve and Profit Comparisons

Worked Example: A Nested Future Liability

A hypothetical contract has a valuation-date outgo of $30. At the end of one year, it has certain outgo of $60. If the policyholder survives that year, an additional net amount is valued at that date as the present value of $120 payable one year later less a $30 premium payable at that date. The one-year survival probability is 0.90. Calculate the reserve at valuation rates of 4% and 6%. Then find the change in expected first-year profit when the closing reserve changes, assuming the actual return is fixed and the opening reserve and all other first-year amounts are unchanged.

Method. Discount the year-end amount once to the valuation date, and include the survival-weighted net amount at year-end. The reserve is:

V(i)=30+\frac{60+0.90\left(\frac{120}{1+i}-30\right)}{1+i}.

At $i=0.04$, the inner one-year value is:

\frac{120}{1.04}=115.384615,\qquad 115.384615-30=85.384615,\qquad 0.90(85.384615)=76.846154.

The year-end amount to discount is $60+76.846154=136.846154$. Therefore:

V(0.04)=30+\frac{136.846154}{1.04} =30+131.582840 =161.582840.

At $i=0.06$:

\frac{120}{1.06}=113.207547,\quad 113.207547-30=83.207547,\quad 0.90(83.207547)=74.886792,

so the year-end amount is $60+74.886792=134.886792$, and:

V(0.06)=30+\frac{134.886792}{1.06} =30+127.251691 =157.251691.

The reserve change is $157.251691-161.582840=-4.331149$. With survival probability 0.90, the change in expected annual profit, holding the actual return and all other profit-test items fixed, is:

\Delta\Pi=-0.90(-4.331149)=3.898034.

Thus the valuation-rate increase raises expected first-year profit by $3.90 per policy in force at the start of the year, to the nearest cent, solely through the lower closing reserve. As a separate arithmetic check, a one-percentage-point change in earned return applied to these reserve amounts would be $0.01(161.582840)=1.615828$ at the 4% reserve and $-0.01(157.251691)=-1.572517$ for a one-point reduction at the 6% reserve. Those are investment-return calculations, not the reserve-basis profit change just computed.

Common mistake avoided: The immediate $30 is part of each full reserve. Omitting it would give incorrect one-point return checks of $1.315828 and $-$1.272517$. Also, changing the valuation rate does not permit changing the actual return in the profit formula.

Worked Example: A Different Survival-Weighted Liability

For a second hypothetical liability, the valuation-date outgo is $50. At the end of one year, certain outgo is $110. Conditional on surviving the year, a further net amount at that date is the present value of $90 payable one year later less a $50 premium due at that date. The one-year survival probability is 0.80. Compare reserves at 5% and 7%, then calculate the effect on expected annual profit if only the closing reserve changes.

The specified cash-flow pattern gives:

V(i)=50+\frac{110+0.80\left(\frac{90}{1+i}-50\right)}{1+i}.

At $i=0.05$:

\frac{90}{1.05}=85.714286,\quad 85.714286-50=35.714286,\quad 0.80(35.714286)=28.571429.

The year-end amount is $110+28.571429=138.571429$, so:

V(0.05)=50+\frac{138.571429}{1.05} =50+131.972789 =181.972789.

At $i=0.07$:

\frac{90}{1.07}=84.112150,\quad 84.112150-50=34.112150,\quad 0.80(34.112150)=27.289720.

The year-end amount is $110+27.289720=137.289720$, giving:

V(0.07)=50+\frac{137.289720}{1.07} =50+128.308149 =178.308149.

The reserve decreases by $178.308149-181.972789=-3.664640$. The corresponding change in expected profit is:

\Delta\Pi=-0.80(-3.664640)=2.931712.

The increase in valuation rate therefore raises expected annual profit by $2.93 per policy in force at the start of the year, to the nearest cent, when all other items are held fixed. Including the immediate $50, separate one-point earned-return checks are $0.01(181.972789)=1.819728$ and $-0.01(178.308149)=-1.783081$. This confirms that the amount used in such a return calculation is the full reserve, not only its discounted component.

Common mistake avoided: The survival probability weights the additional net amount at the one-year date; it does not weight the certain $110. Applying 0.80 to both amounts would change the stipulated cash-flow pattern and produce the wrong reserve.

Worked Example: Annual Profit with Actual Return Held Fixed

A hypothetical annual profit test starts with an opening reserve of $900. The premium is $120, the expense is $20, and the actual annual investment return is 5%. The probability of death during the year is 0.02, and the death benefit is $1,000. For a surviving policy, the only future liability is a benefit of $1,000 payable at the end of the following year if death occurs then; the next-year death probability is also 0.02. There is no future premium. Calculate annual expected profit using valuation rates of 4% and 6%.

First, the probability of surviving the projection year is $p=1-0.02=0.98$. At the end of the year, the prospective reserve for a survivor is the expected next-year death benefit discounted one year:

V_1(i)=\frac{0.02(1000)}{1+i}=\frac{20}{1+i}.

Thus $V_1(0.04)=20/1.04=19.230769$, while $V_1(0.06)=20/1.06=18.867925$. The net amount available to earn the actual return is $900+120-20=1{,}000$, so accumulated assets are $1{,}000(1.05)=1{,}050$ under both valuation rates. Expected current-year death claims are $0.02(1{,}000)=20$.

At 4%, expected closing reserves across the starting policy are $0.98(19.230769)=18.846154$. Therefore:

\Pi(0.04)=1050-20-18.846154=1011.153846.

At 6%, expected closing reserves are $0.98(18.867925)=18.490566$, and:

\Pi(0.06)=1050-20-18.490566=1011.509434.

The profit increase is $1011.509434-1011.153846=0.355588$, or $0.36 per starting policy to the nearest cent. This agrees with the reserve-only check:

-0.98\left(18.867925-19.230769\right)=0.355588.

Common mistake avoided: Replacing the actual 5% return with the 6% valuation rate would incorrectly change accumulated assets from $1,050 to $1,060. The comparison changes only the discount rate used for the closing reserve.

Interpretation and Exam Checks

The examples use positive future net outgo, so their reserves decrease as the valuation rate increases. The second example also includes future premium income, represented by a negative cash flow within a net amount; the direction still follows from that particular calculation, not from a universal rule. When a valuation model has substantial future premiums, refunds, or other negative net cash flows, compare the full reserve calculations rather than relying on a slogan about rate direction.

Before reporting a profit sensitivity, check the basis and timing:

Identify whether the changed rate is the valuation rate $i$, the actual return $j$, or both.
Recalculate only the reserve components that depend on $i$, unless the problem explicitly changes other assumptions.
Weight a closing reserve per survivor by the probability of survival when calculating expected profit per starting policy.
Keep immediate cash flows and discounted future amounts distinct; do not omit an immediate amount from the reserve total.
State whether the opening reserve is fixed or recalculated. A change in opening reserve can affect both opening assets and investment income.

These checks make the source of a profit change clear: reserve remeasurement affects the liability held at the end of the year, while an investment-return shock affects asset accumulation. Combining the two without being asked to do so obscures the separate effects.

Check Your Understanding

For each question, state any assumptions needed to follow the specified cash-flow timing.

A single $200 benefit is payable at the end of one year, with no other future cash flows. Calculate the reserve at 5% and 8%. (Answer: $190.48 at 5%; $185.19 at 8%.)
A valuation change reduces the reserve per surviving policy from $55 to $50. The probability of survival during the projection year is 0.97, and all other profit-test items are fixed. Find the change in expected profit per starting policy. (Answer: an increase of $4.85.)
Fixed net outgo is $100 at the end of year 1 and $200 at the end of year 2. Calculate the reserve at 4% and 5%, and give the change from 4% to 5%. (Answer: $281.07 at 4%; $276.64 at 5%; change $-$4.42$.)
A reserve is $V(i)=20+[40+0.80(100/(1+i)-25)]/(1+i)$. Calculate it at 4% and 6%. (Answer: $113.20 at 4%; $110.07 at 6%, to the nearest cent.)
An annual profit test has $V_0=$500$, $P=$60$, $E=$10$, $j=4\%$, death probability $q=0.01$, benefit $B=$1{,}000$, and closing reserve $V_1=$40$ per survivor. Use $\Pi=(V_0+P-E)(1+j)-qB-(1-q)V_1$. (Answer: $522.40.)

What You'll Learn