Negative Reserves and Reserve Floors

Why a Prospective Reserve Can Be Negative

A prospective reserve compares the present value of future outgo with the present value of future income, conditional on the policy being in force at the valuation date. For a hypothetical universal life policy, future outgo may include death benefits and expenses, while future income may include charges deducted from the policy. If expected future charge income exceeds expected future outgo, the unfloored prospective reserve can be negative.

A negative result is a feature of the stated cash-flow model and valuation assumptions; it is not automatically an error. A valuation rule may require a zero floor, but that floor is a separate step. The underlying unfloored result should still be calculated and understood.

Definition: At time $t$, the unfloored prospective reserve is the expected present value of future benefits and expenses less the expected present value of future premiums, charges, and other income, conditional on the policy being in force at time $t$. Under a specified zero-floor rule, the floored reserve is $V_t^{\mathrm{floor}}=\max(0,V_t)$, where $V_t$ is the unfloored reserve. The floor is applied to the valuation result; it does not change the sign convention used to calculate that result.

A Discrete-Time Model for Future Margins

Use an annual model with valuation dates at integer times. For a policy in force at the start of year $j$, let $q_j$ be the probability of death during that year and $p_j=1-q_j$ the probability of surviving to the next valuation date. Assume death and survival are the only possible outcomes in the model. Let $B_j$ be the death benefit payable at year-end if death occurs, $e_j$ an expense payable at year-end regardless of the outcome, and $C_j$ a charge collected at year-end only if the policy survives. Let $i$ be the effective annual interest rate and $v=1/(1+i)$.

The expected net outgo for year $j$, conditional on being in force at the start of that year, is

$$ g_j=q_jB_j+e_j-p_jC_j. $$

Positive $g_j$ means expected outgo exceeds expected income in that year; negative $g_j$ means expected income exceeds expected outgo. If $V_j$ is the unfloored reserve at the start of year $j$, and $V_{j+1}$ is the unfloored reserve at the next valuation date, the recursion is

V_j=v\left(g_j+p_jV_{j+1}\right).

This recursion assumes all listed year-$j$ cash flows occur at year-end, the charge is collected only on survival, and the reserve at the next date is required only if the policy survives. Other cash-flow timing, additional decrements, or a different charge collection condition would require a different projection.

Why the Recursion Holds

At the start of year $j$, the year-end expected net outgo is $g_j$. If the policy survives, which has probability $p_j$, the insurer also carries the reserve $V_{j+1}$ at the next valuation date. The expected amount due at that date is therefore $g_j+p_jV_{j+1}$. Discounting one year gives $V_j$.

Repeated substitution gives a useful direct expression. For a finite projection ending at time $n$ with $V_n=0$, define the probability of surviving from time $t$ to the start of year $j$ by $\prod_{r=t}^{j-1}p_r$, with an empty product equal to 1. Then

V_t=\sum_{j=t}^{n-1}v^{j-t+1} \left(\prod_{r=t}^{j-1}p_r\right)g_j.

Each year’s expected net outgo is discounted to time $t$ and weighted by the probability that the policy remains in force to the start of that year. This form explains how several years of negative expected net outgo can produce a negative reserve, even when the policy has a positive death benefit.

Worked Calculations

Worked Example: A Small Positive Reserve After Two Years

A hypothetical two-year policy has an effective annual interest rate of 5%. In each year, the conditional probability of death is 0.01 and the probability of survival is 0.99. A death benefit of $8,000 is payable at year-end on death. An expense of $20 is payable at year-end whether the policyholder dies or survives. A charge of $100 is collected at year-end only on survival. There are no cash flows after year 2. Calculate the unfloored reserve at the start of year 1 and apply a zero floor.

Method. Calculate each year’s expected net outgo, work backward from the terminal date, and then apply the floor. This avoids treating the survival charge as certain or omitting the second year’s contribution.

For either year, the expected death benefit is $0.01(8{,}000)=\$80.00$. Adding the expense gives expected outgo of $80+20=\$100.00$. Expected charge income is $0.99(100)=\$99.00$. Thus $g_1=g_2=100-99=\$1.00$.

At the start of year 2, there is one year remaining, so

V_2=\frac{1.00}{1.05}=\$0.95238095\ldots

At the start of year 1, include the current year’s expected net outgo and the next reserve only if the policy survives:

V_1=\frac{1.00+0.99(0.95238095\ldots)}{1.05} =\frac{1.94285714\ldots}{1.05} =\$1.85034014\ldots

Rounded to the nearest cent, the unfloored reserve is $1.85. Since it is positive, the floored reserve is also $1.85. As a check, the present value is $1/1.05+0.99/1.05^2=\$1.85034014\ldots$, agreeing with the recursion. A frequent mistake is to report $1.84; the unrounded result rounds to $1.85.

Worked Example: Two Years of Negative Expected Net Outgo

A hypothetical two-year policy has an effective annual interest rate of 4%. In each year, the probability of death is 0.02 and the probability of survival is 0.98. The year-end death benefit is $1,000, the year-end expense is $10 regardless of outcome, and a $80 charge is collected at year-end only on survival. There are no later cash flows. Find the unfloored reserve at the start of year 1 and the reserve under a specified zero floor.

Method. Calculate conditional expected net outgo, then use the reserve recursion. This makes the sign of each year’s margin explicit before discounting.

In either year, expected death benefit is $0.02(1{,}000)=\$20.00$. Total expected outgo is $20+10=\$30.00$, while expected charge income is $0.98(80)=\$78.40$. Therefore,

g_1=g_2=30.00-78.40=-\$48.40.

With no reserve after year 2, the reserve at the start of year 2 is

V_2=\frac{-48.40}{1.04}=-\$46.5384615\ldots

At the start of year 1,

V_1=\frac{-48.40+0.98(-46.5384615\ldots)}{1.04} =\frac{-94.0076923\ldots}{1.04} =-\$90.392012\ldots

Rounded to the nearest cent, the unfloored reserve is −$90.39, and the floored reserve is $0.00. The direct present-value check is $ -48.40/1.04+0.98(-48.40)/1.04^2=-\$90.392012\ldots $. The negative result is consistent with two years of expected charge income exceeding expected benefits and expenses; the floor changes the reported liability, not that underlying comparison.

Worked Example: Changing Margins by Year

A hypothetical two-year policy has an effective annual interest rate of 5%. In year 1, the death probability is 0.01, the death benefit is $5,000, the expense is $15 regardless of outcome, and a $60 charge is collected only on survival. In year 2, those values are respectively 0.02, $3,000, $15, and $100. Each year’s benefit and expense are payable at year-end, and the charge is collected at year-end only on survival. Find the reserve at the start of year 1 and at the start of year 2, both before and after applying a zero floor.

Method. Calculate the two annual expected net outgo amounts separately, then value the later year first. This avoids carrying year 1’s mortality or charge assumptions into year 2.

For year 1, $p_1=0.99$, so expected death benefit is $0.01(5{,}000)=\$50.00$, expected expense is $15.00, and expected charge income is $0.99(60)=\$59.40$. Hence $g_1=50+15-59.40=\$5.60$.

For year 2, $p_2=0.98$, so expected death benefit is $0.02(3{,}000)=\$60.00$, expected expense is $15.00, and expected charge income is $0.98(100)=\$98.00$. Hence $g_2=60+15-98=-\$23.00$. With no reserve after year 2,

V_2=\frac{-23.00}{1.05}=-\$21.9047619\ldots

At the start of year 1,

V_1=\frac{5.60+0.99(-21.9047619\ldots)}{1.05} =\frac{-16.0857143\ldots}{1.05} =-\$15.3197279\ldots

Rounded to the nearest cent, the unfloored reserves are $V_1=-\$15.32$ and $V_2=-\$21.90$. Applying the zero floor gives $0.00 at both dates. As a check, the direct value at time 0 is $5.60/1.05+0.99(-23)/1.05^2=-\$15.3197279\ldots$. The current-year positive margin does not make the total reserve positive because the survival-weighted present value of the later negative margin is larger.

Applying and Interpreting the Floor

A zero floor is a specified valuation constraint, not part of the definition of the unfloored prospective reserve. At each date for which the rule applies, first calculate the signed prospective value and then set the reported floored value to zero if that result is negative. If the result is positive, the floor leaves it unchanged.

Keep the two figures distinct in analysis. The unfloored reserve shows the balance of projected future outgo and income under the assumptions. The floored reserve shows the liability after applying the stated constraint. A negative unfloored reserve does not by itself establish that a policy has no economic value, nor does a zero floored reserve mean the future charges or benefits disappear.

In the model used here, future reserves inside the recursion are unfloored prospective values. Replacing a negative continuation value with zero while calculating an earlier unfloored reserve would change the cash-flow valuation being computed. If a different valuation specification explicitly defines a different recursive basis, follow that specification rather than silently substituting floored values.

Common Exam Traps

Reversing the sign. With the stated convention, reserve is outgo less income. Expected charges reduce $g_j$; they are not added to benefits and expenses.
Applying survival incorrectly. A charge collected only on survival contributes $p_jC_j$ to expected income, not $C_j$. A benefit payable only on death contributes $q_jB_j$.
Ignoring cash-flow timing. The recursion above discounts the year-end amount by one year. A cash flow paid at the start of the year would not receive the same discount factor.
Flooring too early. Calculate and retain the signed prospective value before applying the specified floor. Do not report only a zero if the task asks for both results.
Rounding prematurely. Keep unrounded intermediate reserves through the recursion, then round the final money amounts to cents.

Key takeaway: A negative prospective reserve can arise when the present value of future charge income exceeds the present value of future benefits and expenses. Calculate that unfloored value using the stated cash flows, probabilities, and timing; apply a zero floor only as a separate specified valuation step.

Check Your Understanding

For each question, use the same convention: reserve equals expected present value of outgo less expected present value of income. Unless stated otherwise, charges are collected only on survival, expenses are paid regardless of the outcome, and all cash flows occur at year-end.

A one-year policy has $i=4\%$, death probability 0.02, a $2,000 death benefit, a $8 expense, and a $50 survival charge. Find the unfloored and zero-floored reserves at the start of the year. (Answer: Unfloored −$0.96; floored $0.00.)
A two-year policy has $i=5\%$. Each year, death probability is 0.01, the death benefit is $4,000, the expense is $10, and the survival charge is $55. Find the start-of-year-1 unfloored and floored reserves. (Answer: Unfloored −$8.23; floored $0.00.)
At a valuation date, a policy has two years remaining and $i=5\%$. The first year’s expected net outgo is $10 and its survival probability is 0.90. The second year’s expected net outgo is −$20. There are no later cash flows. Find the start-date unfloored reserve and its zero-floored value. (Answer: Unfloored −$6.80; floored $0.00.)
A one-year policy has $i=4\%$, death probability 0.05, a $1,000 death benefit, a $10 expense, and a $20 charge collected only on survival. Find the unfloored and floored reserves. (Answer: Unfloored $39.42; floored $39.42.)
A one-year policy has $i=5\%$, death probability 0.01, a $5,000 death benefit, and a $20 expense. A charge $C$ is collected only on survival. What is the smallest charge, to the nearest cent, that makes the unfloored reserve nonpositive? (Answer: $70.71.)

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