Scenario Valuation of Universal Life Guarantees

Why Value Guarantees by Scenario?

A universal life policy may promise a death benefit or surrender amount that exceeds the policy account value. The account helps fund the benefit, but the insurer’s guarantee exposure is the shortfall between the guaranteed amount and the account available for that event. That shortfall depends on how the account develops and on which policies remain in force.

Scenario valuation makes these dependencies explicit. A scenario specifies assumptions such as the account’s credited rate, the valuation discount rate, mortality, and lapse rates. Project the account and guarantee shortfalls under each scenario, then weight the payments by the probability that the relevant event occurs while the policy is in force. The resulting present values can be compared across scenarios. They are scenario results, not automatically a probability-weighted average across scenarios.

Definition: For a scenario, the death-guarantee shortfall at the end of year $t$ is $D_t=\max(B_t^D-A_t,0)$, where $B_t^D$ is the guaranteed death benefit and $A_t$ is the projected policy account value immediately before the end-of-year event. The surrender-guarantee shortfall is $S_t=\max(B_t^S-A_t,0)$, where $B_t^S$ is the guaranteed surrender amount. These shortfalls are insurer-funded guarantee amounts; they are not the full death benefit, surrender amount, or policy account.

Specify the Timing and Scenario

Use the following annual timing convention throughout this tutorial. A policy is in force at the start of a year. During the year, death occurs with conditional probability $q_t$. Given survival, surrender occurs with conditional probability $l_t$. If neither event occurs, the policy is in force at the start of the next year. Death and surrender are therefore competing exits: a policy that dies during the year cannot also surrender in that year.

Let $a_t$ be the probability that a policy is in force at the start of year $t$, with $a_1=1$. The account value is projected under the scenario’s credited rate and cash-flow timing. For example, if a net deposit $C_t$ is made at the start of year $t$ and the annual credited rate is $g_t$, then $A_t=(A_{t-1}+C_t)(1+g_t)$, where $A_t$ is measured at the end of the year. Other premium or charge patterns require their corresponding account recursion.

The credited rate $g_t$ and the valuation interest rate $i$ have different roles. The credited rate determines the projected account and hence the guarantee shortfall. The valuation rate discounts the insurer’s future guarantee payments. They may be equal under a particular basis, but that equality must be an explicit assumption.

Model assumptions: The annual death and lapse probabilities are conditional on the status described above; lapse is conditional on survival. Benefits are paid at year-end when death or surrender occurs. The insurer’s valuation rate $i$ is effective annual interest and is constant in the formulas below. If benefits are paid at another time, or if probabilities and rates vary within a year, the cash-flow timing and valuation model must be adjusted.

Persistency-Weighted Present Value

The probability of death during year $t$ is $a_tq_t$. The probability of surrender during that year is $a_t(1-q_t)l_t$. The probability of remaining in force to the next year is $a_t(1-q_t)(1-l_t)$. These expressions account for competing exits and give the recursion for $a_t$.

a_1=1,\qquad a_{t+1}=a_t(1-q_t)(1-l_t)

Under the stated end-of-year payment convention, the present value of the expected guarantee shortfalls through year $n$ is

\operatorname{EPV} = \sum_{t=1}^{n}v^t \left[ a_tq_tD_t + a_t(1-q_t)l_tS_t \right], \qquad v=\frac{1}{1+i}.

If only one type of guarantee is present, set the other shortfall to zero. If a benefit is payable on death but is partly funded by the account, use the death shortfall $D_t$, rather than the full guaranteed death benefit, in this expression. The formula values the expected guarantee payments under the specified assumptions; it does not assert that the policy account itself is the insurer’s liability.

Why the Formula Holds

The formula follows by separating the possible end-of-year events. At the start of year $t$, the probability the policy is active is $a_t$. Conditional on being active, the probability of death is $q_t$, so the unconditional probability of a death payment is $a_tq_t$. Conditional on surviving, the probability of surrender is $l_t$; the unconditional probability of a surrender payment is therefore $a_t(1-q_t)l_t$. The two events are mutually exclusive under the timing convention.

The expected payment at the end of year $t$ is consequently $a_tq_tD_t+a_t(1-q_t)l_tS_t$. A payment at that time has present-value factor $v^t$. Summing the discounted expected payments over all years gives the stated expression. This argument also shows why applying lapse directly to the death probability, or treating death and surrender as simultaneous benefits, would be inconsistent with the model.

Set the scenario.
Record the credited rates, valuation rate, conditional mortality probabilities, conditional lapse probabilities, benefit guarantees, and account cash-flow timing.

Project account and shortfalls.
Calculate each year’s account value, then apply the maximum-with-zero definition separately to each guarantee.

Project in-force probabilities.
Start with $a_1=1$; multiply by the conditional survival-and-persistence probability to obtain each later year’s active probability.

Weight and discount payments.
Calculate death and surrender probabilities separately, multiply by their respective shortfalls, and discount at the valuation rate.

Compare scenario results.
Report each scenario’s assumptions and present value. Do not average scenarios unless scenario probabilities and an averaging purpose have been specified.

Worked Scenario Valuations

Worked Example: Death Guarantee Under a Two-Year Scenario

A hypothetical policy has an account value of $80 at the start of year 1 and a death benefit guarantee of $120. A net deposit of $10 is made at the start of each year. The credited rate is 5% each year; the valuation rate is 4% effective annually. Conditional death probabilities are $q_1=0.02$ and $q_2=0.03$. Conditional lapse rates after survival are $l_1=0.05$ and $l_2=0.04$. There is no surrender guarantee. Calculate the present value of the death-guarantee shortfall through year 2.

Method. Project the end-of-year account, calculate the shortfall from the guaranteed death benefit, find each year’s death probability while in force, and discount each expected payment at 4%.

Year 1 account and shortfall. The deposit is made before crediting, so $A_1=(80+10)(1.05)=\$94.50$. Thus $D_1=\max(120-94.50,0)=\$25.50$.

Year 2 account and shortfall. $A_2=(94.50+10)(1.05)=\$109.725$, so $D_2=\max(120-109.725,0)=\$10.275$.

Event probabilities. The year 1 death probability is $a_1q_1=1(0.02)=0.02$. The probability of reaching year 2 in force is $a_2=(1-0.02)(1-0.05)=0.931$. The year 2 death probability is therefore $a_2q_2=0.931(0.03)=0.02793$.

Present value. With $v=1/1.04$, the year 1 contribution is $0.02(25.50)/1.04=\$0.4903846$. The year 2 contribution is $0.02793(10.275)/(1.04)^2=\$0.2653298$. Their sum is $\$0.7557144$, or $0.76 to the nearest cent.

Check and common mistake. The year 2 active probability, 0.931, is less than the year 1 survival probability, 0.98, because it also allows for lapse. Using 0.98 alone would overstate year 2 exposure. The account value is not added to the result: only the guarantee shortfall is the insurer-funded payment being valued.

Worked Example: Comparing Two Scenarios With Death and Surrender Guarantees

A hypothetical policy begins with an account value of $70. A net deposit of $10 is made at the start of each year. The death benefit guarantee is $100 and the surrender guarantee is $85. Value both guarantees through year 2 under two alternative scenarios. In Scenario A, the credited rate is 0%, the valuation rate is 4%, $q_t=0.02$, and $l_t=0.10$ in each year. In Scenario B, the credited rate is 5%, the valuation rate is 3%, $q_t=0.04$, and $l_t=0.04$ in each year.

Method. In each scenario, project the account separately, calculate both shortfalls, then weight death and surrender shortfalls using the event probabilities for that scenario.

Scenario A account and shortfalls. $A_1=(70+10)(1.00)=\$80$ and $A_2=(80+10)(1.00)=\$90$. The death shortfalls are $D_1=\$20$ and $D_2=\$10$. The surrender shortfalls are $S_1=\$5$ and $S_2=\$0$.

The year 1 death probability is 0.02 and surrender probability is $(1-0.02)(0.10)=0.098$. The expected year 1 payment is $0.02(20)+0.098(5)=\$0.89$, giving present value $0.89/1.04=\$0.8557692$. The year 2 active probability is $(0.98)(0.90)=0.882$. Year 2 death probability is $0.882(0.02)=0.01764$; its expected payment is $0.01764(10)=\$0.1764$. The surrender shortfall is zero. The present value is $0.1764/(1.04)^2=\$0.1630096$. Scenario A total: $\$1.0187788$, or $1.02.

Scenario B account and shortfalls. $A_1=(70+10)(1.05)=\$84$ and $A_2=(84+10)(1.05)=\$98.70$. Thus $D_1=\$16$, $D_2=\$1.30$, $S_1=\$1$, and $S_2=\$0$.

The year 1 death probability is 0.04 and surrender probability is $(1-0.04)(0.04)=0.0384$. The expected year 1 payment is $0.04(16)+0.0384(1)=\$0.6784$, with present value $0.6784/1.03=\$0.6586408$. The year 2 active probability is $(0.96)(0.96)=0.9216$, so the year 2 death probability is $0.9216(0.04)=0.036864$. The year 2 expected payment is $0.036864(1.30)=\$0.0479232$, with present value $0.0479232/(1.03)^2=\$0.0451722$. Scenario B total: $\$0.7038130$, or $0.70.

Comparison and check. Scenario A produces the larger guarantee present value, $1.02 versus $0.70, despite its lower mortality assumption. Its account grows more slowly, leaving larger shortfalls. Scenario B has higher mortality but also faster account growth and a lower valuation rate. The result reflects all specified assumptions together; it is not a conclusion that higher mortality always reduces guarantee costs. Rounding the two scenario totals to cents gives the amounts reported above.

Common mistake. Applying the surrender probability to all policies in force would give $a_tl_t$, but this model defines lapse conditional on surviving death. The correct surrender probability is $a_t(1-q_t)l_t$, as used in both scenarios.

Worked Example: Separating Account Growth From Discounting

A hypothetical policy starts with an account value of $60 and has no further deposits. Its death benefit guarantee is $100. Compare the two-year death-guarantee present value under two scenarios. Scenario C has credited rate 0%, valuation rate 4%, $q_t=0.02$, and $l_t=0.05$ in each year. Scenario D has credited rate 5%, valuation rate 3%, $q_t=0.03$, and $l_t=0.02$ in each year. Benefits are paid at year-end.

Method. Keep the account projection and discount calculation distinct: credited interest changes the shortfalls, while valuation interest discounts the expected payments.

Scenario C. The account remains $60 in both years, so each death shortfall is $40. The year 1 expected payment is $0.02(40)=\$0.80$; its present value is $0.80/1.04=\$0.7692308$. The year 2 active probability is $(0.98)(0.95)=0.931$, and the year 2 death probability is $0.931(0.02)=0.01862$. The year 2 expected payment is $0.01862(40)=\$0.7448$, with present value $0.7448/(1.04)^2=\$0.6886095$. Total: $\$1.4578403$, or $1.46.

Scenario D. The account values are $60(1.05)=\$63$ and $63(1.05)=\$66.15$. The death shortfalls are therefore $37 and $33.85. The year 1 expected payment is $0.03(37)=\$1.11$, with present value $1.11/1.03=\$1.0776699$. The year 2 active probability is $(0.97)(0.98)=0.9506$, giving death probability $0.9506(0.03)=0.028518$. The year 2 expected payment is $0.028518(33.85)=\$0.9653343$, with present value $0.9653343/(1.03)^2=\$0.9099200$. Total: $\$1.9875899$, or $1.99.

Check and common mistake. Scenario D’s credited rate reduces the shortfall relative to a constant $60 account, but its higher mortality increases the probability of a death payment. Its valuation rate is also lower, so future payments receive less discount. The total is consequently higher in this example. Using the credited rate as the discount rate would conflate two different assumptions and would not reproduce either scenario’s stated valuation.

Interpretation and Exam Traps

Scenario results are useful because they expose which assumptions drive a guarantee’s value. Lower credited rates may increase shortfalls by slowing account growth. Higher mortality may increase the chance of a death shortfall being paid, while higher lapse can reduce later in-force exposure. However, lapse can also trigger a surrender guarantee. The direction of the total effect depends on the guarantee design, account projection, event probabilities, and discount rate; it should be calculated rather than assumed.

Do not value the account as the guarantee liability. The account helps fund the contractual benefit. For a minimum-benefit guarantee, the relevant additional amount is the positive excess of the guarantee over the account.
Do not confuse credited and valuation interest. Project the account at the credited rate, then discount the insurer’s expected payments at the valuation rate specified for that scenario.
Apply competing-event probabilities in the stated order. With lapse conditional on survival, the surrender probability is $a_t(1-q_t)l_t$, not $a_tl_t$.
Respect payment timing. An end-of-year payment is discounted by $v^t$. Moving it to the start of the year changes the present value.
Keep scenarios separate unless weights are supplied. A collection of alternative assumption sets produces a collection of present values. Averaging them requires explicit scenario probabilities and a stated reason for doing so.

Key takeaway: For each scenario, project account values, calculate insurer-funded guarantee shortfalls, weight them by the appropriate death or surrender probabilities while in force, and discount at the scenario’s valuation rate. The policy account is part of the benefit funding, not the guarantee shortfall itself.

Check Your Understanding

Use the end-of-year timing convention and show the event probabilities separately before calculating each present value.

A policy has account value $75 and a one-year death guarantee of $100. The conditional death probability is 0.04 and the valuation rate is 5%. There is no surrender guarantee. What is the present value of the death-guarantee shortfall? (Answer: $0.95)
At the start of a year, account value is $90 and the guaranteed surrender amount is $95. Conditional mortality is 0.02, lapse conditional on survival is 0.08, and the valuation rate is 4%. What is the present value of the surrender-guarantee shortfall for that year? (Answer: $0.38)
A policy starts with account value $80, has no deposits, and has a $100 death guarantee. The credited rate is 5% in each of two years; $q_t=0.01$ and $l_t=0.10$ in each year; the valuation rate is 4%. What is the two-year present value of the death-guarantee shortfall? (Answer: $0.25)
A one-year policy has account value $70, death guarantee $90, surrender guarantee $75, conditional mortality 0.03, conditional lapse after survival 0.10, and valuation rate 5%. What is the present value of both guarantee shortfalls for the year? (Answer: $1.03)
Conditional mortality and lapse rates are $q_1=0.02,l_1=0.10$ in year 1 and $q_2=0.03,l_2=0.05$ in year 2. What is the probability the policy remains in force at the start of year 3? (Answer: 0.8128)

What You'll Learn