Reserves for Minimum Death and Maturity Guarantees

What a Guarantee Reserve Measures

A minimum death or maturity guarantee protects a policyholder when the policy fund is below a stated guaranteed amount. For the insurer, the guarantee creates a potential payment in addition to the fund value. Its reserve is therefore driven by the shortfall between the guaranteed benefit and the projected fund, not by the full benefit in every scenario.

This tutorial values the guarantee component of a hypothetical equity-linked policy. Assume the policy fund is separately identified and its value is available to meet the fund-linked part of the benefit. The reserve calculated here is the present value of the insurer’s additional guarantee payments. It is not, by itself, the reserve for every cash flow under the contract.

Definition: If the fund value at a benefit payment date is $F$ and the guaranteed amount is $G$, the insurer’s guarantee top-up is $(G-F)^+=\max(G-F,0)$. A death benefit of $\max(F,G)$, for example, consists of the fund value $F$ plus the top-up $(G-F)^+$.

The reserve is prospective: it is valued at a specified date for a policy then in force, using projected future benefits, decrement assumptions, investment scenarios, and discounting. A minimum maturity guarantee is generally payable only if the policy remains in force to maturity. A death guarantee is payable on death during the term. If lapse or another exit ends the guarantee without a payment, that exit removes the possibility of later death or maturity guarantee payments.

Model assumptions: A valuation specifies (1) the benefit amounts and dates, (2) how the fund is projected under each investment scenario, (3) the probabilities of death, lapse, and persistence, and (4) the discount basis. The formulas below assume benefit payments at the ends of policy years. If a contract pays at a different time, use that actual payment time for the discount factor and the relevant decrement probability.

Prospective Value of Future Top-Ups

Let time $t$ be the valuation date and $n$ the maturity date measured in years from issue. Write $F_k$ for the fund at the end of year $k$, $G_k^D$ for the death guarantee payable in year $k$, and $G_n^M$ for the maturity guarantee. Define $D_k$ as the event that death occurs during year $k$ while the policy is eligible for the guarantee, and $M_n$ as the event that the policy remains eligible and reaches maturity. With effective annual interest rate $i$, put $v=1/(1+i)$.

{}_tV^G = \sum_{k=t+1}^{n} v^{k-t} E_t\!\left[\mathbf{1}_{D_k}(G_k^D-F_k)^+\right] + v^{n-t}E_t\!\left[\mathbf{1}_{M_n}(G_n^M-F_n)^+\right].

Here $E_t$ is expectation conditional on the policy being in force at time $t$, and $\mathbf{1}_A$ equals 1 when event $A$ occurs and 0 otherwise. The expression allows the fund value and decrement events to be treated jointly. If investment outcomes and decrements are independent under the valuation assumptions, each expectation can be factored into an event probability multiplied by the conditional expected top-up. Do not make that factorization without the independence assumption.

For a simple annual competing-decrement model, let $q_k^D$ be the probability of death during year $k$, and $q_k^L$ the probability of lapse during that year, both conditional on being in force at its start. If death, lapse, and persistence are mutually exclusive outcomes, the persistence probability is $p_k=1-q_k^D-q_k^L$. The probability of reaching the start of year $k$ in force, conditional on being in force at $t$, is the product of persistence probabilities for the intervening years. Thus, the probability of death in year $k$ is that product multiplied by $q_k^D$, while the probability of reaching maturity in force is the product of persistence probabilities through the final year.

P_t(D_k)=\left(\prod_{j=t+1}^{k-1}p_j\right)q_k^D, \qquad P_t(M_n)=\prod_{j=t+1}^{n}p_j.

These probability formulas do not specify the guarantee amount or fund distribution; those still have to be projected. They also do not apply unchanged if the decrement assumptions are defined using a different convention, such as sequential rather than mutually exclusive decrement probabilities. Use probabilities that match the model and contract.

Why the Shortfall Formula Works

The insurer’s payment is the guaranteed benefit less the amount already provided by the fund, but it cannot be negative. For any nonnegative fund value $F$ and guarantee $G$, the total benefit $\max(F,G)$ can be split into the fund and a top-up:

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

If $F$ is at least $G$, the right-hand side is $F+0=F$. If $F<G$, it is $F+(G-F)=G$. This proves the identity in both cases. Since the fund portion is already represented by the fund, the incremental guarantee cash flow is $(G-F)^+$. Multiplying that payment by the indicator for the event on which it is due, discounting it to the valuation date, and taking expectation gives its prospective value. Adding the values for mutually exclusive death and maturity payments gives the guarantee reserve.

Worked Valuations

Worked Example: One-Year Minimum Death Guarantee

A hypothetical one-year policy has a death guarantee of $900, payable at the end of the year if death occurs during the year. The fund at that date is $600 with probability 0.40 and $1,000 with probability 0.60. Assume fund outcomes are independent of mortality, the probability of death is 0.0200, there is no lapse, and the effective annual interest rate is 5%. Calculate the issue-date guarantee reserve.

Method. Calculate the top-up in each fund scenario, average those top-ups, multiply by the death probability, and discount one year. This avoids reserving for the full guarantee when the fund already exceeds it.

If the fund is $600, the top-up is $(900-600)^+=\$300$. If the fund is $1,000, the top-up is $(900-1{,}000)^+=\$0$. The expected top-up conditional on death is therefore

0.40(300)+0.60(0)=\$120.

The expected payment at the end of the year is $0.0200(120)=\$2.40$. The one-year discount factor is $v=1/1.05$, so the issue reserve is

{}_0V^G=\frac{2.40}{1.05}=\$2.285714\ldots=\boxed{\$2.29}.

Check. Discounting the death probability and the conditional expected top-up together gives $0.0200(120)/1.05=\$2.285714\ldots$, the same result. The reserve is small relative to the $900 guarantee because the fund usually covers the guarantee and death is not certain.

Worked Example: Death and Maturity Guarantees with Lapse

A hypothetical two-year policy has a $1,000 death guarantee payable at the end of either year and a $1,200 maturity guarantee payable at the end of year 2 if the policy remains in force. There are two equally likely investment scenarios, independent of decrements: scenario A has fund values $800 at year 1 and $900 at year 2; scenario B has fund values $1,100 at year 1 and $1,300 at year 2. At the start of year 1, the probabilities of death and lapse are 0.0200 and 0.0800. Conditional on reaching year 2 in force, the corresponding probabilities in year 2 are 0.0300 and 0.0700. The effective annual interest rate is 5%. Calculate the issue-date reserve for the guarantee top-ups.

Method. First average the top-up across the two investment scenarios at each payment date. Then weight each date’s amount by the probability of the relevant decrement or persistence, and discount it to issue. This avoids counting a maturity payment after death or lapse.

At year 1, the death top-up is $200 in scenario A and $0 in scenario B. Its expected amount conditional on death is $0.50(200)+0.50(0)=\$100$. The probability of death in year 1 is 0.0200, so the expected year-1 payment is $0.0200(100)=\$2.00$.

The probability of persisting through year 1 is $1-0.0200-0.0800=0.9000$. At year 2, the death top-up is $100 in scenario A and $0 in scenario B, giving a conditional expected top-up of $0.50(100)+0.50(0)=\$50$. The probability of death in year 2 from issue is $0.9000(0.0300)=0.0270$, so the expected year-2 death payment is $0.0270(50)=\$1.35$.

For maturity, scenario A has a top-up of $1{,}200-900=\$300$, while scenario B has no top-up. The conditional expected maturity top-up is $0.50(300)+0.50(0)=\$150$. The probability of reaching maturity in force is $0.9000(1-0.0300-0.0700)=0.9000(0.9000)=0.8100$. The expected maturity payment is $0.8100(150)=\$121.50$.

The year-1 payment is discounted one year; the year-2 death and maturity payments are each discounted two years. The reserve is

{}_0V^G = \frac{2.00}{1.05} + \frac{1.35+121.50}{1.05^2} = 1.9047619\ldots+111.4285714\ldots = \boxed{\$113.33}.

Check. The year-2 probabilities used in the calculation include persistence through year 1, and the maturity probability includes persistence through both years. The year-2 payments are discounted by $1.05^2=1.1025$, not by 1.05.

Worked Example: Valuing a Guarantee Reserve Mid-Contract

At the end of year 1, a hypothetical policy is in force with one year remaining. Its death guarantee is $1,000, and its maturity guarantee is $1,050. At the end of year 2, the projected fund is $900 with probability 0.25, $1,000 with probability 0.50, and $1,200 with probability 0.25. These outcomes are independent of whether death occurs. The probability of death during the year is 0.0400, there is no lapse, the policy matures if the policyholder survives, and the effective annual interest rate is 4%. Find the guarantee reserve at the end of year 1.

Method. From the valuation date, calculate separate expected top-ups for death and maturity, apply the mutually exclusive event probabilities, and discount each end-of-year payment once.

For death, the top-ups under the three fund outcomes are $100, $0, and $0. Thus the expected death top-up conditional on death is

0.25(100)+0.50(0)+0.25(0)=\$25.

For maturity, the top-ups are $150, $50, and $0. The expected maturity top-up conditional on maturity is

0.25(150)+0.50(50)+0.25(0)=\$62.50.

The probability of maturity is $1-0.0400=0.9600$. The expected payment at the end of the year is $0.0400(25)+0.9600(62.50)=1.00+60.00=\$61.00$. Therefore,

{}_1V^G=\frac{61.00}{1.04} =\$58.653846\ldots =\boxed{\$58.65}.

Check. The death and maturity probabilities sum to 1.0000, and both payments are at the end of the remaining year, so each uses the same one-year discount factor.

Common Trap: Applying the Guarantee to an Average Fund

A frequent error is to calculate the expected fund first and then take the shortfall against that average. In general, this does not give the expected guarantee payment. The shortfall function has a floor at zero, so outcomes below the guarantee can generate top-ups even when the average fund is above the guarantee.

For the maturity outcomes in the previous example, the expected fund is $0.25(900)+0.50(1{,}000)+0.25(1{,}200)=\$1{,}025$. Taking a shortfall against that average gives $(1{,}050-1{,}025)^+=\$25$. But the correct expected maturity top-up is $0.25(150)+0.50(50)+0.25(0)=\$62.50$. Applying the guarantee to the average fund understates the expected top-up by $37.50 before allowing for survival and discounting.

Other checks help prevent related errors:

Calculate $(G-F)^+$ separately for each scenario before averaging.
Apply death probabilities only to death guarantees and persistence probabilities to maturity guarantees.
Include survival or persistence to the start of a year before applying that year’s decrement probability.
Discount from the actual payment date, not automatically from the start of the policy year.
Distinguish the incremental top-up reserve from a reserve that also includes fund-related amounts, premiums, expenses, or other benefits.

Key takeaway: A minimum guarantee reserve is the expected present value of future shortfall payments. Project each fund scenario, calculate its top-up, apply the probability of the event that triggers payment, and discount from that payment date using the stated valuation assumptions.

Check Your Understanding

For each problem, calculate only the reserve for guarantee top-ups, using the assumptions stated.

A one-year death guarantee is $800. At year-end the fund is $500 with probability 0.30 and $900 with probability 0.70. The death probability is 0.0500, the outcomes are independent of death, there is no lapse, and $i=0.03$. Find the issue reserve. (Answer: $4.37)
A two-year policy pays a $1,000 maturity guarantee only if it remains in force to maturity. The year-2 fund is $700 with probability 0.40 and $1,100 with probability 0.60. In each year, conditional on being in force at the start, death probability is 0.0200 in year 1 and 0.0300 in year 2; lapse probability is 0.0800 in year 1 and 0.0700 in year 2. The fund outcome is independent of decrements and $i=0.05$. Find the issue reserve. (Answer: $88.16)
A two-year death guarantee is $900, with no maturity guarantee. In year 1, the conditional expected death top-up is $50 and the death probability is 0.0200. In year 2, the conditional expected death top-up is $80 and the death probability is 0.0400. The lapse probability is 0.0800 in year 1; in year 2, use the death probability stated and a lapse probability of 0.0600. Decrements are mutually exclusive, investment outcomes are independent of decrements, and $i=0.04$. Find the issue reserve. (Answer: $3.62)
At a valuation date one year before maturity, a policy has a $1,000 death guarantee and a $1,100 maturity guarantee. The year-end fund is $800 with probability 0.20, $1,000 with probability 0.30, and $1,200 with probability 0.50. The death probability is 0.0500, there is no lapse, and $i=0.02$. Find the reserve at the valuation date. (Answer: $85.78)

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