Why Guarantee Projection Requires More Than an Account Value
A linked contract can provide a benefit based on an investment account while also promising a minimum amount in specified circumstances. The account projection alone does not determine the insurer’s payment: the contract must also be checked for the policy duration, the state in which the policy ends or pays a benefit, and the guarantee terms that apply in that state. A top-up is payable only when the contract’s trigger is met and the eligible account amount falls below the relevant guarantee.
The same account value can therefore produce different insurer payments at different durations or in different states. For example, a contract might have a minimum death benefit but no minimum surrender benefit, or a maturity minimum that differs from its death guarantee. The method here is to project the eligible account amount, apply the benefit rule for the relevant state and duration, and then weight the resulting payment by the probability of reaching that payment event.
Specify the Timing and the Trigger Before Calculating
A projection needs a clear timing convention. In the examples, \(A_t\) denotes the account value at the end of duration \(t\), after the investment return and any stated end-of-period charge. A premium paid at the start of a period is included before the return is applied. Unless a problem says otherwise, benefits triggered at the end of a period are calculated from that period’s end account value and paid immediately at that time.
For example, if \(A_t\) is the account value before a guarantee charge or benefit payment, the top-up should be calculated from that value unless the contract explicitly says the charge is deducted first. The order matters because changing the eligible balance changes the shortfall. Likewise, a guarantee applying only to death cannot be applied to a surrender or maturity simply because the account is below the guarantee amount.
A useful discrete-time account recursion, when a premium \(P_t\) is paid at the start of the year, the account earns annual return \(r_t\), and a charge \(C_{t+1}\) is deducted at year-end, is
This recursion assumes the specified premium, return and charge timing, and that the account remains nonnegative. If the contract has different charge timing, withdrawals, or account depletion rules, the recursion must be changed to match those terms; it is not a universal account formula.
Let \(E_{j,t}\) denote the event that a covered payment occurs in state \(j\) at duration \(t\). If \(p_{j,t}=P(E_{j,t})\), the expected present value of top-ups is obtained by weighting each state- and duration-specific top-up by its event probability and discounting to the valuation date. For effective annual valuation interest \(i\), a payment at time \(t\) is discounted by \(v^t\), where \(v=1/(1+i)\).
This expression assumes the event probabilities and top-up amounts refer to the same event definition and projection basis. If account values are uncertain, the top-up must be calculated separately for each relevant account scenario before taking the probability-weighted average. In general, averaging account values first and then applying the maximum function does not give the same result.
Why the Top-Up Formula Holds
For a covered event, the contract promises at least \(G_{j,t}\) in total and makes the eligible account amount \(B_{j,t}\) available toward that benefit. If \(B_{j,t}\) is already at least the guarantee, the account can fund the promised minimum, so no additional insurer payment is needed. If the account is below the guarantee, the shortfall \(G_{j,t}-B_{j,t}\) is precisely the amount needed to bring the total payment up to the guarantee. These two cases give the maximum expression.
Identify whether the state and duration satisfy the contract’s guarantee trigger. If not, the top-up is zero.
For a covered event, calculate \(G_{j,t}-B_{j,t}\).
A negative shortfall means the account already funds the guarantee, so the top-up is zero.
Use the probability of that particular state and duration, and discount from its actual payment time.
Worked Example: A Maturity Guarantee After Account Charges
A hypothetical linked contract has an initial account value of $1,000, no further premiums, a 6% annual account return, and a $40 charge deducted at each year-end. At the end of year 2, a surviving policyholder receives the greater of the account value and a guaranteed maturity benefit of $1,100. Assume annual effective valuation interest of 4%. Mortality is independent of account returns, with a 0.020 probability of death in each year conditional on being alive at the start of that year. The account path is deterministic. Calculate the conditional maturity top-up and its expected present value, allowing for survival to maturity.
Method. Project the account using the stated end-of-year charge timing, apply the maturity guarantee only to a policy surviving to the end of year 2, and then weight and discount the top-up.
At the end of year 1,
At the end of year 2,
The covered maturity top-up, conditional on surviving to maturity, is
Survival through both years has probability \(0.98^2=0.9604\). The expected present value of this maturity top-up is therefore
The conditional top-up is $58.80, and its expected present value is $52.21, rounded to the nearest cent. The calculation assumes the death benefit has no additional top-up; death does not trigger this maturity guarantee. A common mistake is to add the $58.80 top-up to the $1,100 guaranteed total and report a maturity payment of $1,158.80. That double-counts the shortfall: the account supplies $1,041.20 and the insurer supplies $58.80, for a total of exactly $1,100.
Worked Example: Different Top-Ups for Death and Maturity
A hypothetical contract has a projected eligible account amount of $900 if death occurs at the end of year 1. Its year-1 death guarantee is $1,050. If the policyholder survives year 1, the projected account amount is $980 at the end of year 2. At that duration, the death guarantee is $1,030 and the maturity guarantee is $1,000. There is no guarantee on surrender. The conditional mortality probabilities are 0.030 in year 1 and 0.040 in year 2. Assume mortality is independent of the account projection and annual effective valuation interest is 4%. Calculate the expected present value of the insurer’s top-ups for death and maturity events through year 2.
Method. Calculate each event’s shortfall using its own guarantee, then assign that shortfall the probability of the event occurring at that duration.
A year-1 death produces a top-up of
The probability of death in year 1 is 0.030, so the expected year-1 top-up is \(0.030(150)=4.50\), paid at time 1.
A year-2 death requires survival through year 1 followed by death in year 2. Its probability is \(0.970(0.040)=0.0388\). Its top-up is
The probability of surviving to maturity is \(0.970(0.960)=0.9312\). The maturity top-up is
The expected present value is
Thus the expected present value of top-ups is $23.34, rounded to the nearest cent. The death and maturity guarantees are applied to mutually exclusive year-2 outcomes. A common error is to use 0.040 as the unconditional probability of year-2 death. That omits the required year-1 survival factor: the correct probability is \(0.970(0.040)=0.0388\). The surrender state creates no top-up because the contract explicitly has no surrender guarantee.
Combining Market Scenarios and Insured States
When account values vary by investment scenario, each scenario must be carried through the guarantee calculation. The market scenario determines the account amount; mortality or other state-transition assumptions determine whether and when a covered benefit is paid. If those sources of uncertainty are independent, their probabilities can be multiplied. If they are dependent, the joint probabilities must be used instead.
The following example illustrates the required order: project the account along each market path, calculate the top-up separately for each state and path, and then apply the relevant event probabilities. The expected payment is not obtained by applying the guarantee to a single average account value.
Worked Example: Two-Year Projection Under Two Investment Returns
A hypothetical contract begins with an account value of $1,000 and has no premiums. At each year-end, the account earns either a return of -10% with probability 0.25 or +10% with probability 0.75, independently from year to year, and then incurs a $20 charge. A death at the end of year 1 has a guaranteed total benefit of $1,100. At the end of year 2, the death guarantee is $1,150 and the maturity guarantee is $1,200. There is no surrender guarantee. Conditional mortality probabilities are 0.020 in year 1 and 0.030 in year 2; mortality is independent of investment returns. Use annual effective valuation interest of 4%. Find the expected present value of top-ups through year 2.
Method. Enumerate the investment paths, find each path’s account value and state-specific top-up, and combine those amounts with the applicable mortality, survival, and market probabilities.
At the end of year 1, the low-return and high-return account values are, respectively,
The corresponding year-1 death top-ups are \(1{,}100-880=220\) and \(1{,}100-1{,}080=20\). The expected top-up conditional on a year-1 death is \(0.25(220)+0.75(20)=70\). Since the death probability is 0.020, the expected year-1 payment is \(0.020(70)=1.40\).
For year 2, project each of the four equally specified return paths. Their probabilities and account values are
| Path | Probability | End-year-2 account |
|---|---|---|
| Low, then low | 0.25(0.25) = 0.0625 | 880(0.90) - 20 = 772 |
| Low, then high | 0.25(0.75) = 0.1875 | 880(1.10) - 20 = 948 |
| High, then low | 0.75(0.25) = 0.1875 | 1,080(0.90) - 20 = 952 |
| High, then high | 0.75(0.75) = 0.5625 | 1,080(1.10) - 20 = 1,168 |
For a year-2 death, the top-ups along these paths are $378, $202, $198, and $0. The expected top-up conditional on a year-2 death is
For maturity, the corresponding top-ups are $428, $252, $248, and $32. The expected maturity top-up conditional on reaching maturity is
The probabilities of year-2 death and maturity are \(0.980(0.030)=0.0294\) and \(0.980(0.970)=0.9506\), respectively. Thus the expected year-2 top-up is
Discounting the year-1 and year-2 expected payments gives
The expected present value of top-ups is $125.75, rounded to the nearest cent. The result checks against the event logic: the year-2 death and maturity probabilities sum to \(0.0294+0.9506=0.9800\), the probability of surviving year 1. A common mistake is to average the year-2 account values first and then apply a guarantee. That loses the path-specific shortfalls: a low account path can generate a substantial top-up even when the average account value appears relatively high.
Projection Checks and Common Traps
A guarantee projection is most reliable when the account recursion, covered states, event probabilities, and payment times are checked separately. In particular, confirm whether the guarantee is a minimum total benefit or an additional amount payable on top of the account. The examples use a minimum total benefit, so the account amount is credited toward the guarantee and only the shortfall is an insurer top-up.
- Applying a guarantee to the wrong state. A death guarantee does not automatically apply to surrender or maturity. Set the top-up to zero when the trigger is not met.
- Double-counting the linked account. For a guaranteed total amount, pay the account amount plus only the shortfall, not the account amount plus the full guarantee.
- Using the wrong duration probability. A death probability conditional on being alive at the start of a year must be multiplied by the probability of surviving to that year.
- Ignoring the account timing convention. Returns, charges, premiums, and benefits must be applied in the order specified by the contract.
- Averaging before applying the guarantee. When account values vary by scenario, calculate the nonnegative shortfall in each scenario before probability-weighting.
The general projection principle is simple but strict: define the guarantee trigger, project the account under the stated timing, calculate the state- and duration-specific shortfall, and then weight and discount the resulting payments. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus and makes the source of each projected insurer payment explicit.
Check Your Understanding
For each question, apply the guarantee only in the stated covered state and use the payment timing given.
- A covered maturity event has an eligible account value of $870 and a guaranteed total benefit of $950. What are the top-up and total benefit? (Answer: top-up $80; total benefit $950.)
- A policy has conditional mortality probabilities of 0.050 in year 1 and 0.100 in year 2. What is the unconditional probability of death in year 2? (Answer: 0.0950.)
- A linked account starts at $1,000, earns 5% for one year, and has a $20 charge deducted at year-end. A covered maturity guarantee is $1,000. Calculate the end-year account value and top-up. (Answer: account value $1,030; top-up $0.)
- A two-year maturity top-up is $40 if the policyholder survives both years. Conditional mortality probabilities are 0.020 and 0.030. What is the undiscounted expected maturity top-up? (Answer: $38.02.)
- At a covered event, the account is $800 with probability 0.40 and $1,100 with probability 0.60. The guaranteed total benefit is $1,000 in either scenario. Find the expected top-up conditional on the event. (Answer: $80.)