Reserving for Minimum Death Benefits

Why Separate the Guarantee from the Account?

A universal life policy may promise a minimum amount on death even when its policy account has fallen below that amount. The account helps fund the claim, but the insurer bears the additional cost needed to meet the minimum. A reserve calculation for this feature should therefore distinguish the account-funded benefit from the guarantee shortfall.

This tutorial values the shortfall for a hypothetical policy with a death benefit equal to the greater of the account value and a stated minimum. It uses a discrete annual model: account crediting and death claims are evaluated at year-end. The calculations omit premiums, expenses, lapses, and other guarantees so that the minimum death benefit contribution is visible on its own. They are claim-side guarantee contributions, not complete policy reserves.

Definition: Let $A_k$ be the policy account value at the end of year $k$, immediately before a death claim is paid, and let $G_k$ be the minimum death benefit at that time. For a benefit that pays the greater of account value and minimum, the claim is $\max(A_k,G_k)$. The guarantee shortfall is $(G_k-A_k)^+=\max(G_k-A_k,0)$. The account-funded portion is $A_k$; the guarantee-funded portion is the shortfall.

The comparison is made using the account value at the claim date, after any account crediting that applies through the year. It is not generally correct to compare the minimum with the account value at the beginning of the year. If a contract instead pays the account value plus an additional specified amount, its claim design is different and this shortfall formula does not apply unchanged.

One-Year Guarantee Contribution

Suppose the policy is in force at time $t$, with a known account value $A_t$. Assume it receives a deterministic annual credit rate $j$, so the account value at the end of the year is $A_{t+1}=A_t(1+j)$. Let $G_{t+1}$ be the minimum death benefit then, $q_t$ the probability of death during the year conditional on being in force at time $t$, and $i$ the annual effective valuation interest rate. Death claims are paid at year-end, and there are no other decrements.

V_t^{\text{guarantee}} = v q_t (G_{t+1}-A_{t+1})^+ = \frac{q_t}{1+i}(G_{t+1}-A_{t+1})^+.

This is a reserve contribution for the guarantee shortfall over the coming year, conditional on the policy being in force at time $t$. It is not the present value of the full death benefit: the account-funded portion is deliberately excluded. The crediting rate $j$ determines the account value used in the claim calculation; the valuation rate $i$ discounts that claim back to time $t$. They need not be equal.

Model condition: The one-year formula assumes a year-end claim, a specified one-year death probability, and an account value at claim that is known conditional on death. If the account value is random, replace the shortfall by its conditional expected value given death. If death timing or benefit amounts are modeled continuously, the claim timing and discounting must be adjusted to match that model.

Why the Shortfall Formula Works

The decomposition follows directly from the policy’s claim amount. If the account is at least the minimum, the account alone covers the claim. If the account is below the minimum, the insurer must add exactly the difference. Algebraically, for any nonnegative $A$ and $G$,

\max(A,G)=A+(G-A)^+.

To verify it, consider both possible cases. If $A\geq G$, then $(G-A)^+=0$, so the right-hand side is $A=\max(A,G)$. If $A<G$, then $(G-A)^+=G-A$, so the right-hand side is $A+(G-A)=G=\max(A,G)$. Thus the claim consists of the account amount plus the insurer-funded top-up, with no overlap.

Under the one-year model, the top-up is paid only if death occurs. Its expected value at year-end is therefore $q_t(G_{t+1}-A_{t+1})^+$. Discounting this expected payment by $v=1/(1+i)$ gives the reserve contribution. This derivation also explains why multiplying the whole minimum benefit by the death probability would value more than the guarantee: it would include the account-funded part.

Worked Example: A One-Year Minimum Benefit

A hypothetical policy has an account value of $80,000 at the start of the year and a minimum death benefit of $100,000. The account earns a guaranteed annual credit of 4%. The conditional probability of death during the year is 2%, claims are paid at year-end, and the valuation interest rate is 3% effective annually. Assume no other decrements or cash flows. Calculate the one-year guarantee reserve contribution.

Method. First project the account to the claim date, then value only the amount by which the minimum exceeds that projected account.

The year-end account value is

A_1=\$80{,}000(1.04)=\$83{,}200.

The full death benefit is $\max(\$83{,}200,\$100{,}000)=\$100{,}000$. The account funds $83,200, leaving a guarantee shortfall of

(\$100{,}000-\$83{,}200)^+=\$16{,}800.

The expected year-end shortfall payment is $0.02(\$16{,}800)=\$336$. Discounting one year at 3% gives

V_0^{\text{guarantee}} = \frac{\$336}{1.03} = \$326.2136\ldots = \boxed{\$326.21}.

The result is checked by noting that a death claim is $16,800 above the account only when death occurs; the one-year probability-weighted claim is $336, and its present value is less than $336. The calculation avoids treating the whole $100,000 benefit as insurer-funded.

Valuing the Guarantee Across Several Years

For a multi-year term, the guarantee shortfall may change as the account grows and the minimum benefit changes. Let $q_k$ be the probability of death in year $k$, conditional on surviving in force to the start of that year. Under a deterministic account path and a constant annual effective valuation rate $i$, the time-zero guarantee contribution over $n$ years is

V_0^{\text{guarantee}} = \sum_{k=1}^{n} v^k\,{}_{k-1}p\,q_k\,(G_k-A_k)^+, \qquad {}_{k-1}p=\prod_{r=1}^{k-1}(1-q_r).

Here, $A_k$ and $G_k$ are the account value and minimum benefit at the end of year $k$. The factor ${}_{k-1}p q_k$ is the unconditional probability of surviving to the start of year $k$ and dying during it. Each year’s shortfall is discounted from its year-end payment date. If account values or benefit amounts are random, the shortfall must be averaged over their appropriate conditional distribution rather than treated as a known number.

Project the claim-date account.
Apply the assumed crediting basis through each year in which the policy remains in force.

Find each year’s guarantee shortfall.
For year $k$, calculate $(G_k-A_k)^+$; a negative difference means no top-up is due.

Weight by the death probability.
Use survival to the start of the year multiplied by the conditional probability of death in that year.

Discount and sum.
Discount each expected year-end shortfall to the valuation date, then add the contributions.

Worked Example: A Three-Year Guarantee Shortfall

A hypothetical policy starts with an account value of $70,000 and a constant minimum death benefit of $100,000. While the policy remains in force, the account earns 10% annually. The conditional probabilities of death in years 1, 2, and 3 are 2%, 3%, and 4%, respectively. Claims are paid at each year-end, and the valuation interest rate is 5% effective annually. Assume no lapse or other cash flows. Calculate the time-zero guarantee contribution for these three years.

Method. Project the account, determine each year’s top-up, then multiply by the probability of dying in that year and discount. The account values at the three year-ends are

A_1=\$70{,}000(1.10)=\$77{,}000,\qquad A_2=\$77{,}000(1.10)=\$84{,}700, $$ $$ A_3=\$84{,}700(1.10)=\$93{,}170.

The corresponding shortfalls are $23,000, $15,300, and $6,830. The unconditional probabilities of death in the three years are

P(D_1)=0.02, \qquad P(D_2)=0.98(0.03)=0.0294, $$ $$ P(D_3)=0.98(0.97)(0.04)=0.038024.

Thus the present value contribution from each year, in dollars, is

C_1=\frac{0.02(\$23{,}000)}{1.05}=\$438.0952\ldots, $$ $$ C_2=\frac{0.0294(\$15{,}300)}{1.05^2} =\frac{\$449.82}{1.1025} =\$408.00, $$ $$ C_3=\frac{0.038024(\$6{,}830)}{1.05^3} =\frac{\$259.70392}{1.157625} =\$224.3420\ldots.

Adding the discounted contributions gives

V_0^{\text{guarantee}} = \$438.0952\ldots+\$408.00+\$224.3420\ldots = \boxed{\$1{,}070.44}.

The contribution decreases across the years because the account grows and the shortfall narrows, although the death probabilities and survival weighting also affect each term. The check is that the year-3 death probability includes survival through both earlier years; using 4% alone would overstate that year’s unconditional probability.

Account Value and Reserve Contribution Are Not the Same

The account value is a policy-level amount used to determine how much of a claim is funded by the account. The guarantee contribution is the present value of the insurer’s expected top-ups under the stated assumptions. They answer different questions. In particular, the one-year guarantee contribution is not the policy account, not the full death benefit reserve, and not necessarily the complete reserve required under a broader valuation basis.

If the full claim-side expected present value is required under the same simplified assumptions, the decomposition can be used explicitly:

v q_t\max(A_{t+1},G_{t+1}) = v q_t A_{t+1} + v q_t(G_{t+1}-A_{t+1})^+.

The first term is the account-funded claim component under this model; the second is the minimum-benefit guarantee contribution. A complete policy reserve may also reflect future premiums, expenses, policy charges, and other cash flows. Do not infer those components from the guarantee-shortfall calculation alone.

Common Exam Trap: Comparing at the Wrong Time

Suppose a $100,000 minimum benefit applies at claim, the account is $90,000 at the start of the year, and it earns 2% before a year-end claim. The correct claim-date account is $91,800, so the guarantee shortfall is $8,200. Using the initial account instead gives a $10,000 shortfall and overstates the guarantee cost.

Key takeaway: For a minimum death benefit equal to the greater of account value and guarantee, value only $(G-A)^+$ as the insurer-funded claim top-up. Use the account and guarantee at the claim date, weight by the correct death probability, and discount according to the payment timing.

Check Your Understanding

Use the same discrete annual conventions as above: account crediting applies through the year, claims are paid at year-end, and the stated valuation rate discounts those claims. Each scenario is hypothetical.

A policy has a $48,000 account, a $55,000 minimum death benefit, a 3% annual account credit, a 1% conditional probability of death during the year, and a 4% valuation rate. What is the one-year guarantee contribution? (Answer: $53.46)
A policy has a $90,000 account, a $100,000 minimum benefit, a 2% annual account credit, a 2.5% conditional death probability, and a 5% valuation rate. Calculate the one-year guarantee contribution using the claim-date account. (Answer: $195.24)
A policy starts with a $60,000 account and a constant $70,000 minimum benefit. The account earns 5% annually; conditional death probabilities are 2% in year 1 and 3% in year 2; the valuation rate is 4%. Calculate the two-year time-zero guarantee contribution. (Answer: $239.27)
At the claim date, a policy account is $80,000 with probability 0.5 and $110,000 with probability 0.5. The minimum benefit is $100,000, the account outcome is independent of death, the one-year death probability is 2%, and the valuation rate is 4%. What is the guarantee contribution? (Answer: $192.31)
A policy pays the greater of its account and a $120,000 minimum. At the claim date the account is $125,000. State the full death benefit and the guarantee shortfall. (Answer: Death benefit $125,000; shortfall $0)

What You'll Learn