What an Actuarial Valuation Basis Must Specify
An equity-linked policy connects one or more benefits to an investment fund, but the fund projection is only one part of its valuation. A useful basis must also state what the contract pays, when it pays, how policies terminate, what expenses arise, and how future cash flows are discounted. Leaving any of these unspecified can change the result or make it impossible to interpret.
The benefit formula comes from the hypothetical contract; it is not itself an investment assumption. A projected fund value is calculated under a stated investment scenario. A guaranteed amount is a contractual minimum or other promised benefit. The projected fund may be below or above that guarantee, but the projection does not change the guarantee.
The basis should match the question being asked. For example, a deterministic scenario shows the result conditional on a particular sequence of returns. It does not, by itself, give the probability of that sequence or an expected value across possible investment outcomes. Likewise, a discount rate is not automatically the same as the rate earned by the policy fund. The two rates describe different things and should be stated separately unless the valuation framework explicitly relates them.
State the Contract Before Making Assumptions
Write the benefit rule in a way that can be applied to the fund value at the relevant payment date. For example, a hypothetical death benefit might be the greater of the fund and a fixed guarantee. A maturity benefit might instead be the fund plus a defined top-up, or the greater of the fund and a maturity guarantee. The wording must also specify which fund value applies—for example, the value just before a charge or after the final scheduled charge—and when payment is made.
For a floor guarantee \(G\) against a fund value \(F\), the benefit is \(\max(F,G)\). It can also be written as the fund plus a guarantee top-up:
This identity is useful for separating the projected fund component from the guarantee component. It does not mean that the top-up is always payable: it is positive only when \(F<G\). A basis should also say whether a surrender value, exit deduction, or other payment applies on lapse. A benefit payable on one event should not be silently treated as payable on another.
For investment assumptions, specify the return period, whether returns are gross or net of fund charges, and the order in which charges are deducted. For decrements, specify the covered population, the period, and whether death and lapse probabilities are mutually exclusive probabilities measured from the start of that period. For expenses, state the amount, the event that triggers it, and its payment timing.
How Decrements and Expenses Enter the Basis
Suppose each policy in force at the start of a year can experience death, lapse, or neither during that year. Let \(q_t^D\) be the probability of death and \(q_t^L\) the probability of lapse, each measured from the start of year \(t\). If these are mutually exclusive annual probabilities, then the probability of remaining in force through the year is \(1-q_t^D-q_t^L\). This condition requires \(q_t^D+q_t^L\leq 1\).
Here \(N_t\) is the number of policies in force at the start of year \(t\). The equation gives expected policy counts when the probabilities apply independently and consistently to each policy in that cohort; it does not assert that an actual group will contain fractional policies. If assumptions are instead expressed as cause-specific forces or in another competing-decrement framework, the corresponding conversion to annual probabilities must be used rather than simply adding incompatible rates.
Expenses need equally clear definitions. A maintenance expense may apply to every policy in force at the start of a year; a claim expense may apply only when a claim occurs; and a lapse expense may apply only on termination by lapse. The same expense should not be counted twice—for example, once as a deduction from the fund and again as an insurer expense—unless the valuation explicitly treats them as two distinct cash flows.
Worked Examples of Basis Construction
Worked Example: A One-Year Death Guarantee Basis
A hypothetical policy receives a premium of $1,000. The contract allocates 95% of premium to the fund. At the end of the first year, a death benefit is the greater of the fund and $1,200. On lapse at that date, the policyholder receives 98% of the fund. Specify and apply a one-year basis. Assume a gross fund return of 8%, a fund charge of 1% of the post-return fund, mutually exclusive annual death and lapse probabilities of 0.0200 and 0.0500, a $15 maintenance expense on each policy in force at the start of the year, a $100 claim expense on death, a $20 processing expense on lapse, and an annual effective discount rate of 4%. Death and lapse payments and event expenses occur at year-end; maintenance expense occurs at the start of the year.
Method. First separate the contractual benefits from the assumptions, then project the fund using the stated charge order, and finally check the event probabilities and timing. This avoids treating the projected fund as the guaranteed death benefit.
The fund at issue is \(0.95(1{,}000)=\$950\). It earns the gross return first, giving \(950(1.08)=\$1{,}026\). The charge is then \(0.01(1{,}026)=\$10.26\), so the year-end fund is \(1{,}026-10.26=\$1{,}015.74\). Equivalently, the net accumulation factor is \(1.08(0.99)=1.0692\), and \(950(1.0692)=\$1{,}015.74\).
The death benefit is \(\max(1{,}015.74,1{,}200)=\$1{,}200\). Its fund component is $1,015.74 and its guarantee top-up is \(1{,}200-1{,}015.74=\$184.26\). The lapse payment is \(0.98(1{,}015.74)=\$995.4252\), or $995.43 to the nearest cent.
The in-force probability at year-end is \(1-0.0200-0.0500=0.9300\), which is nonnegative. Per policy in force at issue, the stated basis therefore assigns death probability 0.0200, lapse probability 0.0500, and continuation probability 0.9300. The maintenance expense is due at issue under the stated timing; the claim and lapse expenses apply only on their respective events. The 4% discount rate is specified for discounting future insurer cash flows; it was not used to project the fund.
Check. The death and lapse probabilities sum to 0.0700, leaving 0.9300 in force, and the projected fund is below the contractual death guarantee. The contract formula and the fund projection have not been conflated.
Worked Example: A Three-Year Deterministic Investment Scenario
A hypothetical policy has a fund of $2,000 at the start of year 1. Consider the deterministic gross annual returns \(10\%\), \(-5\%\), and \(6\%\), in that order. At each year-end, deduct a fund charge equal to 1% of the fund after that year's return. Assume the policy remains in force throughout the three years. Its maturity benefit is the greater of the year-3 fund and $2,100. Project the fund and identify whether the guarantee binds under this scenario.
Method. Apply each gross return and then the charge, year by year. This makes the result conditional on the specified return path; it is not an expected fund value across different paths.
For year 1, the fund after gross return is \(2{,}000(1.10)=\$2{,}200\). The charge is \(0.01(2{,}200)=\$22\), leaving \(2{,}200-22=\$2{,}178\).
For year 2, the fund after gross return is \(2{,}178(0.95)=\$2{,}069.10\). The charge is \(0.01(2{,}069.10)=\$20.691\), leaving \(2{,}069.10-20.691=\$2{,}048.409\).
For year 3, the fund after gross return is \(2{,}048.409(1.06)=\$2{,}171.31354\). The charge is \(0.01(2{,}171.31354)=\$21.7131354\), leaving \(2{,}171.31354-21.7131354=\$2{,}149.6004046\). Rounded to cents, the projected fund is $2,149.60.
The maturity benefit under this scenario is \(\max(2{,}149.6004046,2{,}100)=\$2{,}149.60\); the guarantee does not bind. The guaranteed floor remains $2,100, not $2,149.60. The latter is the scenario-dependent fund projection.
Check. Each year uses the specified return followed by the specified charge, and the final fund exceeds the floor by $49.6004046 before rounding.
Worked Example: Decrement and Expense Assumptions with Discounting
A hypothetical block has 1,000 policies in force at time 0. In year 1, death and lapse probabilities are 0.0100 and 0.0800. Among year-1 survivors, the corresponding year-2 probabilities are 0.0200 and 0.0500. There is a $60 acquisition expense per policy at time 0, an $18 maintenance expense per policy in force at each year-start, a $150 expense per death, and a $25 expense per lapse. Year-end death and lapse expenses are paid at the end of the year. Use an annual effective discount rate of 5%. Calculate the expected present value of these expenses through the end of year 2.
Method. Project expected in-force and decrement counts year by year, apply each expense only to its triggering count, and discount each year's expected expenses to time 0. This avoids applying year-2 probabilities to the original cohort instead of to year-1 survivors.
At time 0, acquisition expenses are \(1{,}000(60)=\$60{,}000\). In year 1, expected deaths are \(1{,}000(0.0100)=10\), expected lapses are \(1{,}000(0.0800)=80\), and expected survivors are \(1{,}000(1-0.0100-0.0800)=910\). Year-1 maintenance expense is \(1{,}000(18)=\$18{,}000\); death expenses are \(10(150)=\$1{,}500\); and lapse expenses are \(80(25)=\$2{,}000\). Total expected year-1 expense is \(18{,}000+1{,}500+2{,}000=\$21{,}500\).
In year 2, the 910 expected survivors are the starting in-force count. Expected deaths are \(910(0.0200)=18.2\), expected lapses are \(910(0.0500)=45.5\), and expected survivors at year-end are \(910(1-0.0200-0.0500)=846.3\). Year-2 maintenance expense is \(910(18)=\$16{,}380\); death expenses are \(18.2(150)=\$2{,}730\); and lapse expenses are \(45.5(25)=\$1{,}137.50\). Total expected year-2 expense is \(16{,}380+2{,}730+1{,}137.50=\$20{,}247.50\).
Discounting gives
The acquisition expense is not discounted because it occurs at time 0. The answer is rounded only at the final step to the nearest cent. Fractional expected counts are appropriate for expected expenses, even though actual policies occur in whole numbers.
Check. Year-1 probabilities leave 910 expected survivors, and year-2 probabilities are applied to 910. The discount factors are \(1.05\) for year 1 and \(1.05^2=1.1025\) for year 2.
Why the Expected-Expense Calculation Works
For a group of \(N_t\) policies with the same annual decrement probabilities, let \(I_j\) equal 1 if policy \(j\) remains in force through the year and 0 otherwise. Then the number continuing is \(\sum_{j=1}^{N_t} I_j\). If each policy has continuation probability \(1-q_t^D-q_t^L\), linearity of expectation gives
This justification does not require the continuation indicators to be independent for the expected-count formula, provided each has the stated probability. Independence or other dependence assumptions matter for measures involving the distribution or variability of total claims. For a valuation based on expected cash flows, the expected count can be multiplied by the applicable per-policy expense when the expense amount is fixed and the trigger is correctly identified.
Common Basis Errors
A common mistake is to treat a gross investment return as if it were already net of a fund charge. In the three-year example, the year-1 factor is \(1.10(0.99)=1.089\), not \(1+0.10-0.01=1.09\). The incorrect calculation would give \(2{,}000(1.09)=\$2{,}180\), rather than the correct $2,178. The specified charge is 1% of the post-return fund, so the correct factor is multiplicative. If contract terms instead define the charge as a percentage of the opening fund, the calculation must follow that definition.
Other errors arise from failing to state timing or mixing distinct assumptions. In particular:
- Using the fund projection as the guarantee. Apply the contract's benefit formula to the fund and guaranteed amount; do not label a scenario result as guaranteed.
- Adding probabilities that are not on the same basis. Death and lapse probabilities can be added in the annual decrement formula only when they are mutually exclusive probabilities measured over the same period from the same in-force group.
- Applying decrements to the wrong starting count. Later-year probabilities apply to the policies surviving into that year, not automatically to the original cohort.
- Discounting at the fund return without justification. The investment return projects the policy fund; the discount basis values future insurer cash flows. They need not be equal.
- Double-counting charges or expenses. Check whether a charge already reduces the fund and whether an expense is a separate insurer cash flow.
Check Your Understanding
For each problem, state the relevant basis assumptions before calculating.
- A $1,600 premium has 94% allocated to a fund. The fund earns 5% gross for one year, then a 1.5% charge is deducted from the post-return value. Find the year-end fund. (Answer: $1,555.51)
- At the start of a year, 800 policies face mutually exclusive death and lapse probabilities of 0.0150 and 0.0450. Find the expected deaths, lapses, and policies still in force at year-end. (Answer: 12 deaths, 36 lapses, and 752 in force)
- A block of 500 policies has annual death and lapse probabilities of 0.0200 and 0.0600. A $10 maintenance expense applies per policy in force at the year-start, a $200 expense applies per death, and a $15 expense applies per lapse. Find the expected total expense for the year. (Answer: $7,450)
- A fixed $12,100 insurer payment is due at the end of year 2. Using an annual effective discount rate of 10%, find its present value at time 0. (Answer: $10,000.00)
- A hypothetical maturity benefit is the greater of the fund and $2,500. Under a stated deterministic scenario, the projected fund at maturity is $2,420. Identify the benefit, guarantee top-up, and whether the guarantee binds. (Answer: benefit $2,500; top-up $80; guarantee binds)