Universal Life Reserve Projection Under Multiple Scenarios

Purpose and Projection Framework

A universal life projection can show a policy’s account value rising while the insurer’s guarantee liability also rises. These are different quantities: the account value tracks policy funds, while a guarantee reserve reflects the expected cost of benefits the insurer may have to pay beyond those funds. Comparing scenarios requires consistent assumptions about cash-flow timing, expenses, interest, mortality, and lapse.

The method here is an annual, discrete-time projection of a hypothetical policy. It isolates the insurer’s liability for a minimum death benefit guarantee. It does not include other prospective insurer cash flows, such as future maintenance expenses or premiums, in the reserve. Expenses and insurance charges do affect the account value, and therefore can affect the guarantee shortfall. A broader prospective net liability would require those additional cash flows to be included explicitly.

Definition: At duration $t$, let $A_t$ be the account value at the start of the year for a policy still in force. A premium $P_t$, expense deduction $e_t$, and insurance charge $c_t$ are applied at the start of the year. If the policy persists to year-end, its account value is $$ A_{t+1}=(A_t+P_t-e_t-c_t)(1+j_t), $$ where $j_t$ is the annual account crediting rate. For a guaranteed death benefit $G$, the insurer’s year-end guarantee shortfall is $H_{t+1}=\max(G-A_{t+1},0)$. This model assumes the account value available at death is the projected year-end value.

The timing convention matters. Mortality and lapse are treated as mutually exclusive events during each policy year, with annual probabilities $q_t$ and $w_t$, respectively, conditional on being in force at the start of that year. The probability of persisting to the next year is therefore $p_t=1-q_t-w_t$, which must be nonnegative. A lapsed policy receives its account value under this model, with no additional surrender guarantee. A policy that persists continues into the next projection year.

The valuation discount rate $i_t$ is separate from the account crediting rate $j_t$. The account crediting rate determines projected policy funds. The valuation rate discounts the insurer’s future guarantee payments. They need not be equal.

Definition: Let $V_t$ be the prospective reserve at the start of duration $t$, conditional on the policy being in force then. For a guarantee-only reserve over a finite projection ending at duration $n$, set $V_n=0$. Working backward, $$ V_t=\frac{q_tH_{t+1}+p_tV_{t+1}}{1+i_t}. $$ The reserve includes the expected discounted death-benefit shortfall in the coming year and, if the policy persists, the reserve at the next duration. Lapse terminates the guarantee liability.

This is a deliberately specified projection basis, not a complete description of every universal life contract. For example, a contract with a surrender guarantee, monthly charges, or a benefit that changes over time needs corresponding cash-flow and timing adjustments. State the model before applying the formulas.

Why the Reserve Recursion Holds

The recursion follows by conditioning on what happens during the next policy year. At duration $t$, death has probability $q_t$, and then the insurer pays the guarantee shortfall $H_{t+1}$ at year-end. Lapse has probability $w_t$, after which this guarantee-only liability is zero. Persistence has probability $p_t$, and leaves the insurer with the next-duration reserve $V_{t+1}$ at year-end. Thus the expected year-end value of these guarantee obligations is $q_tH_{t+1}+p_tV_{t+1}$. Discounting that amount for one year gives $V_t$. In particular, the lapse probability is not another claim probability; it removes the policy from future guarantee exposure.

V_t=(1+i_t)^{-1} \left[ q_t\max(G-A_{t+1},0)+(1-q_t-w_t)V_{t+1} \right].

Worked Projection Under One Scenario

Worked Example: Two-Year Account and Guarantee Projection

A hypothetical policy starts with $A_0=\$80$ and has a level guaranteed death benefit $G=\$110$. In each of the next two years, the premium is $\$10$, the expense deduction is $\$2$, and the insurance charge is $\$3$, all applied at the start of the year. The annual account crediting rate is $5\%$. In year 1, mortality is $q_0=0.02$ and lapse is $w_0=0.08$; in year 2, $q_1=0.03$ and $w_1=0.07$. The annual reserve discount rate is $4\%$ in both years. Calculate account values, guarantee shortfalls, and reserves.

Method. First project the account along the path on which the policy persists. Then calculate the shortfall at each year-end, and finally apply the reserve recursion backward from $V_2=0$.

Year 1 account value. The amount credited is $\$80+\$10-\$2-\$3=\$85$. Therefore,

A_1=85(1.05)=\$89.25.

The year-1 guarantee shortfall is $H_1=\max(110-89.25,0)=\$20.75$. The year-1 persistence probability is $p_0=1-0.02-0.08=0.90$.

Year 2 account value. The amount before crediting is $\$89.25+\$10-\$2-\$3=\$94.25$, so

A_2=94.25(1.05)=\$98.9625.

Thus $H_2=\max(110-98.9625,0)=\$11.0375$, and $p_1=1-0.03-0.07=0.90$. The terminal reserve is $V_2=0$. At duration 1, only the year-2 death shortfall remains:

V_1=\frac{0.03(11.0375)+0.90(0)}{1.04} =\frac{0.331125}{1.04} =\$0.318389\ldots.

At duration 0, the year-1 death shortfall and the reserve if the policy persists both contribute:

V_0=\frac{0.02(20.75)+0.90(0.318389\ldots)}{1.04} =\frac{0.415+0.286550\ldots}{1.04} =\$0.674568\ldots.

The projected schedule is $A_0=\$80$, $A_1=\$89.25$, $A_2=\$98.9625$; the corresponding year-end shortfalls are $\$20.75$ and $\$11.0375$. The reserves at durations 0, 1, and 2 are $\$0.674568\ldots$, $\$0.318389\ldots$, and zero. Rounded to the nearest cent, the reserves are $\$0.67$, $\$0.32$, and $\$0.00$. The check is that both shortfalls are positive because both projected account values are below the guarantee, and the duration-0 reserve includes only the probability-weighted, discounted future guarantee costs—not the full face amount.

Common mistake avoided: using $1-q_t$ as the probability of reaching the next duration would ignore lapses. Here the correct year-1 persistence probability is $0.90$, not $0.98$.

Comparing Economic and Expense Scenarios

A useful scenario comparison changes stated assumptions while keeping the contract and projection timing clear. Lower credited interest or higher deductions can reduce account values and increase guarantee shortfalls. Changes in mortality and lapse affect the probability-weighted guarantee liability as well. The following example changes several assumptions together, so it compares complete scenarios rather than attributing the difference to one assumption in isolation.

Worked Example: Baseline and Adverse Assumption Sets

A hypothetical policy has $A_0=\$90$, $G=\$110$, and two years remaining. Each year it receives a $\$10$ premium and incurs a $\$3$ insurance charge. Under the baseline scenario, annual expenses are $\$2$, the account crediting rate is $5\%$, mortality is $2\%$, lapse is $8\%$, and the reserve discount rate is $4\%$ each year. Under the adverse scenario, annual expenses are $\$4$, the crediting rate is $2\%$, mortality is $3\%$, lapse is $5\%$, and the reserve discount rate is $3\%$ each year. Compare account values and duration-0 guarantee reserves. Assume the probabilities are constant within each scenario.

Baseline account projection. The net amount added before interest is $\$10-\$2-\$3=\$5$ per year. Therefore,

A_1=(90+5)(1.05)=\$99.75,\qquad A_2=(99.75+5)(1.05)=\$109.9875.

The shortfalls are $H_1=\$10.25$ and $H_2=\$0.0125$. Persistence is $p=1-0.02-0.08=0.90$. Backward calculation gives

V_1=\frac{0.02(0.0125)}{1.04} =\$0.000240\ldots, $$ $$ V_0=\frac{0.02(10.25)+0.90(0.000240\ldots)}{1.04} =\$0.197323\ldots.

Adverse account projection. The net amount added before interest is now $\$10-\$4-\$3=\$3$ per year:

A_1=(90+3)(1.02)=\$94.86,\qquad A_2=(94.86+3)(1.02)=\$99.8172.

The shortfalls are $H_1=\$15.14$ and $H_2=\$10.1828$. Persistence is $p=1-0.03-0.05=0.92$, so

V_1=\frac{0.03(10.1828)}{1.03} =\$0.296586\ldots, $$ $$ V_0=\frac{0.03(15.14)+0.92(0.296586\ldots)}{1.03} =\$0.705883\ldots.

The adverse assumptions produce a lower account value at both durations and a higher duration-0 guarantee reserve. Rounded to the nearest cent, the duration-0 reserves are $\$0.20$ under the baseline and $\$0.71$ under the adverse scenario. The calculation shows why reporting account values alone is insufficient: the reserve also depends on decrements and the valuation discount rate.

Common mistake avoided: discounting the reserve at the account crediting rate would use $5\%$ or $2\%$ instead of the stated valuation rates of $4\%$ or $3\%$. Crediting and valuation are separate assumptions in this model.

Isolating the Effect of Lapse

When comparing lapse assumptions, hold other assumptions fixed if the aim is to identify the direct lapse effect. The next example uses the same projected account path under both lapse scenarios. It therefore does not model selective lapse changing the characteristics of the policies that remain; it isolates how the persistence probability enters this guarantee-only reserve.

Worked Example: Lapse Sensitivity With the Account Path Held Fixed

A hypothetical policy starts with $A_0=\$90$ and has a guaranteed death benefit of $\$110$. There is no account interest in either year. In year 1, the premium less expense and insurance charge adds a net $\$8$ to the account; in year 2 it adds a net $\$6$. Thus the projected account values are the same in both lapse scenarios. Mortality is $4\%$ in each year and the valuation discount rate is $5\%$. Compare duration-0 reserves if lapse is $5\%$ per year versus $20\%$ per year.

The account values are $A_1=90+8=\$98$ and $A_2=98+6=\$104$, giving $H_1=\$12$ and $H_2=\$6$. At duration 1, the reserve is the same under both scenarios because the year-2 mortality, shortfall, and discount rate are fixed:

V_1=\frac{0.04(6)}{1.05} =\frac{0.24}{1.05} =\$0.228571\ldots.

With $5\%$ lapse, year-1 persistence is $p=1-0.04-0.05=0.91$. Therefore,

V_0=\frac{0.04(12)+0.91(0.228571\ldots)}{1.05} =\frac{0.688}{1.05} =\$0.655238\ldots.

With $20\%$ lapse, persistence is $p=1-0.04-0.20=0.76$, and

V_0=\frac{0.04(12)+0.76(0.228571\ldots)}{1.05} =\frac{0.653714\ldots}{1.05} =\$0.622585\ldots.

Rounded to the nearest cent, the reserves are $\$0.66$ and $\$0.62$, respectively. The higher lapse assumption lowers the reserve here because fewer policies are expected to persist into the second year. This conclusion is specific to the fixed account path and decrement assumptions in the example; a scenario that also changes the remaining policy population or its mortality would require those effects to be projected explicitly.

Scenario Controls and Common Traps

A reliable policy-level scenario projection follows a disciplined order: define the timing and contract cash flows, project the account on the in-force path, calculate the guarantee shortfalls, set the terminal reserve, and work backward using each scenario’s mortality, lapse, and discount assumptions. Comparing scenarios at identical durations prevents a further common error: treating a start-of-year reserve as though it were measured at year-end.

For the first worked example, a wrong duration-0 calculation that ignores lapse would use $1-q_0=0.98$ as the persistence probability:

V_0^{\text{wrong}} =\frac{0.02(20.75)+0.98(0.318389\ldots)}{1.04} =\$0.699\ldots.

The corrected calculation uses $p_0=1-q_0-w_0=0.90$, giving $V_0=\$0.674568\ldots$. The wrong calculation overstates the reserve because it carries the future reserve forward for policies that actually lapse. Other checks include confirming that $q_t+w_t\leq 1$, that guarantee shortfalls cannot be negative, and that the reserve uses the valuation discount rate rather than the account crediting rate.

Key takeaway: A universal life scenario projection links three calculations: account accumulation, guarantee shortfall, and decrement-weighted reserve. Keep their timing and assumptions distinct, and state precisely which future liabilities the reserve includes.

Check Your Understanding

Each problem uses the annual timing convention defined above. Amounts are in dollars unless stated otherwise.

A hypothetical policy has $A_0=70$, a premium of $\$8$, expense of $\$2$, and insurance charge of $\$1$ at the start of the year. The account credits $4\%$, and the guaranteed death benefit is $\$90$. Find the year-end account value and guarantee shortfall. (Answer: account value $\$78.00$; shortfall $\$12.00$)
At a duration with one year remaining, the year-end guarantee shortfall is $\$20$, mortality is $3\%$, lapse is $7\%$, and the reserve discount rate is $5\%$. Find the conditional reserve. (Answer: $\$0.5714$)
A two-year guarantee has year-end shortfalls $H_1=\$18$ and $H_2=\$8$. In both years, mortality is $2\%$, lapse is $8\%$, and the reserve discount rate is $4\%$. With $V_2=0$, calculate $V_0$. (Answer: $\$0.4793$)
A policy has $A_0=\$100$, a net addition of $\$4$ before interest in each of two years, and $G=\$110$. Compare the two-year account paths and duration-0 reserves under (a) a $0\%$ account crediting rate and (b) a $3\%$ account crediting rate. In both scenarios, mortality is $5\%$, lapse is $5\%$, and the reserve discount rate is $4\%$ each year. (Answer: scenario (a) account values $\$104,\ \$108$, reserve $\$0.3717$; scenario (b) account values $\$107.12,\ \$114.4536$, reserve $\$0.1385$)
For a one-year guarantee reserve, the shortfall is $\$15$, mortality is $4\%$, lapse is $6\%$, the account crediting rate is $6\%$, and the reserve discount rate is $3.5\%$. Find the reserve and state which rate discounts it. (Answer: $\$0.5797$; discount at the $3.5\%$ valuation rate)

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