Surrender Guarantees and Their Reserve Effects

Why a Surrender Guarantee Can Create an Additional Liability

A universal life policy's account value is not always the amount the policyholder receives on surrender. A contract may specify a guaranteed surrender amount that exceeds the account value in some circumstances. If the policyholder exercises the surrender option then, the insurer must fund the difference. The guarantee therefore creates an incremental liability whose value depends on both the size of any shortfall and whether surrender occurs while the guarantee applies.

The account value alone does not determine the reserve for this guarantee. The valuation must reflect the account-value outcomes at eligible surrender dates, the timing of those dates, and the relationship between account performance and surrender behavior. This tutorial values that incremental guarantee cost; it does not revalue the policy's other benefits or teach scenario valuation across alternative economic paths.

Definition: At an eligible surrender date $j$, let $A_j$ be the policy account value immediately before surrender and $G_j$ the guaranteed surrender amount. The incremental guarantee payment is the shortfall $$ (G_j-A_j)^+=\max(G_j-A_j,0). $$ It is zero when the account value meets or exceeds the guarantee. If the contract pays the greater of the account value and the guaranteed amount, the total surrender payment is $\max(A_j,G_j)$; the expression above is the part in excess of the account value.

A shortfall is not, by itself, a payment. The policy must be in force and eligible for the guarantee, and the policyholder must surrender at that date. For clarity, use a discrete annual model: surrender can occur at anniversaries $j=1,\ldots,n$, and all benefit amounts are paid at the anniversary when surrender occurs. The valuation interest rate is an effective annual rate $i$, with $v=1/(1+i)$. Assume the guarantee is payable only on surrender, not on death or another termination.

Valuation setup: Let $I_j$ equal 1 if the policy is in force and eligible immediately before the surrender opportunity at time $j$, and 0 otherwise. Let $S_j$ equal 1 if surrender occurs at time $j$, and 0 otherwise. The time-0 reserve for the incremental guarantee is $$ V^{\text{guar}}_0 = \sum_{j=1}^{n} v^j E\!\left[I_jS_j(G_j-A_j)^+\right]. $$

The product $I_jS_j$ ensures that a guarantee payment is counted only when the policy is eligible and actually surrendered. Surrender opportunities at different dates are mutually exclusive for an individual policy: surrender at one date prevents that same policy from surrendering again later. If surrender probabilities are conditional on the account history, their dependence on the shortfall must remain inside the expectation. A single marginal surrender probability cannot generally be multiplied by an unconditional expected shortfall.

Why the Expected-Present-Value Formula Holds

At each eligible date $j$, the incremental cash flow is $I_jS_j(G_j-A_j)^+$. Discounting that payment to time 0 multiplies it by $v^j$. The total present value for a policy is the sum of these discounted cash flows over all possible surrender dates. Taking expectations and using linearity of expectation gives the reserve formula. This argument does not require account values or surrender events at different dates to be independent.

When account outcomes are represented by states $h$ with probabilities $P(h)$, the expectation can be evaluated state by state. If the conditional probability of surrender at date $j$, given the relevant history $h$ and eligibility, is $q^s_j(h)$, then

E\!\left[I_jS_j(G_j-A_j)^+\right] = E\!\left[I_jq^s_j(h)(G_j-A_j)^+\right].

This follows by conditioning on the history: given that history, the shortfall is known, and the conditional expected value of $S_j$ is $q^s_j(h)$. Only if that conditional probability is constant across relevant account histories can it be taken outside the expectation. In that special case, surrender is independent of the shortfall conditional on eligibility.

Worked Example: Independent Surrender After a One-Year Account Return

A hypothetical policy has an account value of $15,000 at the start of the year. After one year, the account earns a return of $-20\%$ with probability 0.25 or $+10\%$ with probability 0.75. The guaranteed surrender amount is $15,000. At the anniversary, surrender occurs with probability 0.40, independently of the account return. The valuation interest rate is 5%. Find the probability of a shortfall, the probability of an actual top-up, and the time-0 reserve for the guarantee.

Method and account outcomes. Calculate the account value in each return state using starting account value times one plus the return:

A_1= \begin{cases} \$15{,}000(0.80)=\$12{,}000, & \text{with probability }0.25,\\ \$15{,}000(1.10)=\$16{,}500, & \text{with probability }0.75. \end{cases}

In the low-return state, the shortfall is $\max(\$15{,}000-\$12{,}000,0)=\$3{,}000$. In the high-return state, it is $\max(\$15{,}000-\$16{,}500,0)=\$0$. Thus the probability of a shortfall is 0.25.

An actual top-up requires both a shortfall and surrender. Because surrender is independent of the account return, its probability in the low-return state is 0.40. Therefore the probability of an actual top-up is $0.25(0.40)=0.10$.

The expected payment at the anniversary is

E[\text{top-up}] =0.25(0.40)(\$3{,}000)+0.75(0.40)(\$0) =\$300.

Discount one year at 5%: $v=1/1.05$, so the reserve is $\$300/1.05=\$285.714\ldots$, or $285.71 to the nearest cent. Check: the payment occurs with probability 0.10 and is $3,000 when it occurs, giving an expected payment of $300 before discounting. This method avoids confusing the probability of a shortfall with the probability of a guarantee payment.

When Surrender Depends on the Account Value

Policyholders may be more likely to surrender when account performance is poor, or less likely to surrender when the guarantee is valuable. In either case, the event that creates the shortfall may be associated with the event that triggers payment. The correct calculation uses the surrender probability within each account state, rather than applying a marginal surrender probability to an average shortfall.

Worked Example: Account-Dependent Surrender Probabilities

At the end of one year, a hypothetical policy's account value is $8,000 with probability 0.40 and $14,000 with probability 0.60. Its guaranteed surrender amount is $12,000. Conditional on the account value, the probability of surrender is 0.75 in the $8,000 state and 0.10 in the $14,000 state. The valuation interest rate is 5%. Calculate the probability of a shortfall, the probability of an actual top-up, and the time-0 guarantee reserve.

Method. First calculate the shortfall within each account state. At $8,000, the shortfall is $\$12{,}000-\$8{,}000=\$4{,}000$. At $14,000, the shortfall is zero. The shortfall probability is therefore 0.40.

The probability of an actual top-up is the probability of each state multiplied by its conditional surrender probability, summed over states. Only the low-account state has a positive shortfall:

P(\text{top-up}) =0.40(0.75)+0.60(0.10)(0) =0.30.

The expected top-up payment at the anniversary is $0.40(0.75)(\$4{,}000)=\$1{,}200$. Discounting at 5% gives

V^{\text{guar}}_0 =\frac{\$1{,}200}{1.05} =\$1{,}142.857\ldots =\boxed{\$1{,}142.86}.

Check. A top-up occurs with probability 0.30, and whenever it occurs the amount is $4,000, so the expected payment is $0.30(\$4{,}000)=\$1{,}200$. The calculation correctly uses the 0.75 surrender probability in the low-account state. Using the marginal surrender probability $0.40(0.75)+0.60(0.10)=0.36$ as though it applied equally to every account state would not preserve the joint relationship between surrender and shortfall.

Valuing the Guarantee Across Multiple Durations

For a multi-year policy, eligibility at a later surrender date depends on the policy having remained in force until then. A practical calculation proceeds date by date: determine the probability the policy is eligible, determine the shortfall in each relevant account state, apply the conditional surrender probability to that state and date, and discount the resulting expected payment.

Describe each eligible date.
Specify the account value, guarantee, and policy-in-force condition immediately before surrender can occur.

Calculate state-specific shortfalls.
For every relevant account outcome, calculate $(G_j-A_j)^+$; outcomes with no shortfall contribute zero.

Apply joint eligibility and surrender probabilities.
Use the probability of reaching the date and the surrender probability conditional on the relevant account history.

Discount and sum.
Discount each date's expected incremental payment by $v^j$, then add the date-specific values.

Worked Example: Shortfall Exposure at Two Surrender Dates

A hypothetical policy is eligible to surrender at the end of years 1 and 2. Its guaranteed surrender amount is $10,000 at each date. The account value is $9,000 at the end of year 1 and, if the policy remains in force, $9,900 at the end of year 2. At year 1, the policy surrenders with probability 0.20. If it does not surrender, it reaches year 2 and surrenders then with conditional probability 0.25. These surrender probabilities are specified for the policy's account path. The valuation interest rate is 5%. Find the probability of a guarantee payment at each date, the probability of any guarantee payment, the expected number of eligible anniversary dates with a shortfall, and the reserve.

Shortfalls and eligibility. At year 1, the shortfall is $\$10{,}000-\$9{,}000=\$1{,}000$. At year 2, it is $\$10{,}000-\$9{,}900=\$100$. The policy is eligible at year 1 with probability 1. It reaches year 2 only if it did not surrender at year 1, which has probability $1-0.20=0.80$.

The probability of a guarantee payment at year 1 is 0.20. The probability of a guarantee payment at year 2 is the probability of reaching year 2 times the conditional surrender probability then: $0.80(0.25)=0.20$. Since surrender at the two dates cannot both occur for the same policy, the probability of any guarantee payment is $0.20+0.20=0.40$.

The policy is eligible at both dates with a shortfall along this specified account path. The expected number of eligible shortfall anniversaries is $1+0.80=1.80$: every policy is eligible at year 1, and 80% remain eligible at year 2. This exposure measure is not the same as the probability of a payment; the latter also requires surrender.

The expected incremental payment at year 1 is $0.20(\$1{,}000)=\$200$. At year 2, it is $0.80(0.25)(\$100)=\$20$. Hence

V^{\text{guar}}_0 =\frac{\$200}{1.05}+\frac{\$20}{1.05^2} =\$190.4762+\$18.1406 =\boxed{\$208.62}.

Check. The discounted year-1 and year-2 amounts are based on distinct surrender dates; their expected present values add. This avoids treating the year-2 surrender probability as unconditional: only 80% of policies reach that opportunity.

Common Mistakes and Interpretation

Multiplying marginal probabilities when surrender depends on account value. The incorrect calculation $P(\text{surrender})E[(G-A)^+]$ generally does not equal $E[S(G-A)^+]$. Calculate the joint expectation by account state or use a conditional surrender probability inside the expectation.
Confusing a shortfall with a guarantee payment. A shortfall can exist when no surrender occurs. The payment requires both eligibility and surrender.
Forgetting persistence to later dates. A later-date expected payment must include the probability of remaining in force and eligible up to that date.
Discounting an expected payment incorrectly. If the payment occurs at the end of year $j$, discount by $v^j$, not by $v$ for every duration.
Including the account value twice. The formula here is for the incremental guarantee above the account value. A valuation of the full surrender cash flow must distinguish the account-value component from the additional guaranteed amount.

A surrender guarantee matters because its cost can be concentrated in account states where the policyholder is particularly likely to exercise the option. Its reserve is therefore not determined by the average account value or a stand-alone surrender rate. The valuation must preserve the joint timing of account shortfalls, policy persistence, and surrender behavior. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: Value the incremental surrender guarantee as the expected present value of $(G_j-A_j)^+$ only when the policy is both eligible and surrendered. Preserve dependence between account history and surrender, and include the probability of reaching each future surrender date.

Check Your Understanding

For each problem, distinguish the probability of a shortfall from the probability and amount of an actual guarantee payment.

A policy has an account value of $20,000. At the end of one year it earns $-15\%$ with probability 0.30 or $+5\%$ with probability 0.70. The guarantee is $19,000, surrender is independent of return with probability 0.50, and the valuation interest rate is 4%. Find the shortfall probability, top-up probability, and reserve. (Answer: shortfall probability 0.3000; top-up probability 0.1500; reserve $288.46.)
At one year, an account is $7,000 with probability 0.25 and $11,000 with probability 0.75. The guarantee is $9,000. Conditional surrender probabilities are 0.80 and 0.20, respectively. At a 5% valuation rate, find the top-up probability and reserve. (Answer: top-up probability 0.2000; reserve $380.95.)
A policy has shortfalls of $500 at year 1 and zero at year 2. It surrenders at year 1 with probability 0.10; if it persists, it is eligible at year 2. The valuation interest rate is 4%. If surrender at year 2 would create no top-up, find the probability of any guarantee payment and the reserve. (Answer: probability 0.1000; reserve $48.08.)
A policy has a $10,000 guarantee and account values of $9,400 at year 1 and $9,700 at year 2 if it persists. It surrenders at year 1 with probability 0.25; conditional on persistence, it surrenders at year 2 with probability 0.20. At 5%, find the probability of any guarantee payment and the reserve. (Answer: probability 0.4000; reserve $183.67.)
A one-year account value is $9,000 with probability 0.40 and $14,000 with probability 0.60; the guarantee is $12,000. Surrender occurs with probability 1 in the low-account state and 0 in the high-account state. At a 6% valuation rate, find the probability of a guarantee payment and the reserve. (Answer: probability 0.4000; reserve $1,132.08.)

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