Tracking Equity-Linked Units Through Charges and Claims

Why Track Units Separately from Claims?

In an equity-linked contract, the policyholder’s account may be represented by units in a fund. Premiums purchase units, unit prices change with investment performance, and charges may be collected by cancelling units. A claim, however, is determined by the contract’s benefit rule. The account value and the claim amount are therefore related, but they are not always the same.

The central discipline is to keep three quantities distinct: units held, the value of those units, and the insurer’s contractual payment. Confusing them can misstate both the policyholder account and the insurer’s exposure. The methods below use hypothetical contracts with explicit transaction timing; they are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: If a policy has $U$ units at unit price $p$, its linked account value is $F=Up$. If an allocated premium of $A$ buys units at price $p$, it purchases $A/p$ units. If a monetary charge $C$ is deducted by cancelling units at price $p$, the number of units cancelled is $C/p$. These statements assume the contract uses the stated price and timing for each transaction.

A projection must state whether a premium is allocated before or after a unit-price movement, which price applies to each charge, and when a claim is valued. The examples use annual periods and deterministic unit-price changes. A unit-price return is an investment return, not an assumed effective interest rate $i$; no mortality probability or probability distribution is needed when an example specifies that the policy survives or that a claim occurs.

A Transaction Sequence for a Linked Account

A practical projection records transactions in chronological order. In the simple convention used here, the allocated premium first buys units at the opening price, the unit price then changes, a monetary charge is deducted at the resulting price, and any specified claim is valued after that charge. A different contract may use a different order, so the contractual rules must take precedence.

Allocate the premium.
Apply any allocation percentage or deduction, then divide the amount invested by the applicable unit price to find units purchased.

Move the unit price.
Apply the stated investment return to the price, or use the given new price. The number of units does not change solely because the price moves.

Deduct charges.
For a monetary charge taken from the linked account, divide the charge by the price used for the deduction and cancel that many units.

Value the account and any claim.
Multiply remaining units by the applicable price. Apply the contractual claim rule separately to determine the insurer’s payment.

Worked Example: Premium Allocation and a Unit-Deducted Charge

A hypothetical policy begins with 500 units priced at $20.00. A $1,200 premium is received, of which 95% is allocated to the linked account. The policyholder survives the year. The unit price rises by 4%, and a $60 administrative charge is deducted at the year-end price. Find the closing units and account value. The price movement is deterministic; no interest-rate assumption or mortality probability is used.

Method. Apply the allocation rule, reprice the units, then convert the charge to units at the closing price. This avoids deducting a dollar charge as though it were a fixed number of units.

The amount allocated is $1{,}200(0.95)=\$1{,}140$. The remaining $60 of the premium is not invested under this allocation rule. At the opening price, the premium purchases

\frac{1{,}140}{20}=57 \text{ units}.

There are $500+57=557$ units before the price movement. The closing unit price is $20(1.04)=\$20.80$, so the account before the administrative charge is $557(20.80)=\$11{,}585.60$. The charge cancels

\frac{60}{20.80}=2.8846153846\ldots \text{ units}.

Using unrounded values, closing units are $557-60/20.80=554.1153846154\ldots$, or 554.115385 units to six decimal places. The closing account value is $11{,}585.60-60=\$11{,}525.60$. As a check, the exact closing units multiplied by $20.80 also give $11,525.60. A common mistake is to subtract $60 directly from the number of units; the correction is to divide the monetary charge by the applicable unit price.

Account Value and the Contractual Claim

The account value is a fund measure. The claim is a benefit specified by the contract. For example, a hypothetical death benefit might equal the post-charge account value plus a stated additional amount. In that case, the account value is one component of the claim, not the total claim.

Key distinction: If a claim rule specifies $B=F+G$, where $F$ is the account value at the claim valuation time and $G$ is an additional contractual benefit, the insurer’s gross claim payment is $B$. The account-linked component is $F$, and the amount above that component is $G$. This decomposition assumes the contract actually defines the benefit that way; it is not a general formula for every linked product.

For a simple immediate funding comparison, one may compare the contractual payment with assets backing the account. If those assets exactly match the account value and are available to meet the claim, the additional amount $G$ is the payment beyond that account backing. This is only a limited cash-flow comparison: expenses, asset mismatch, other liabilities, and the applicable valuation basis can all affect the insurer’s actual financial position.

Worked Example: Claim After a Premium, Price Rise, and Charge

A hypothetical policy has 400 units at $15.00. A $900 premium is received, with 98% allocated to units at that price. The price then rises by 8%. At the resulting price, a $45 charge is deducted. The policyholder dies immediately afterward, and the contract pays the post-charge account value plus $1,500. Find the claim and identify its account-linked and additional portions. The specified death occurs; no mortality probability is required. The unit-price path is deterministic.

Method. Project the account first, then apply the stated claim formula. This prevents the common error of treating the additional benefit as the entire claim or omitting the account value from it.

The allocated premium is $900(0.98)=\$882$, so it buys $882/15=58.8$ units. Total units before the price movement are $400+58.8=458.8$. The new price is $15(1.08)=\$16.20$, giving a pre-charge account value of

458.8(16.20)=\$7{,}432.56.

The $45 charge cancels $45/16.20=2.7777777778\ldots$ units. Thus the remaining units are $458.8-45/16.20=456.0222222222\ldots$. The post-charge account value is exactly $7{,}432.56-45=\$7{,}387.56$, which is also the remaining units multiplied by $16.20 using unrounded units.

The contractual claim is

B=7{,}387.56+1{,}500=\$8{,}887.56.

The account-linked portion is $7,387.56; the additional portion is $1,500. The check is that these components sum to the $8,887.56 claim. If the insurer pays the entire benefit, its gross payment obligation is $8,887.56, even though the account-linked amount may be backed by linked assets. The $1,500 is the amount above that account value under this particular benefit rule, not a complete solvency result.

Why Cancelling Units Deducts the Correct Amount

The unit-cancellation rule follows directly from the account-value definition. Before a charge, $U$ units at price $p$ have value $Up$. Cancelling $C/p$ units leaves $U-C/p$ units. Their value is the original value less the charge:

\left(U-\frac{C}{p}\right)p=Up-C.

Thus the rule preserves the intended monetary deduction, provided the units are cancelled at the same price $p$ used in the calculation and enough units are available. If $C>Up$, the formula would require cancelling more units than the policy holds. The contract must specify what happens in that situation; the simple unit-cancellation calculation cannot by itself determine the resulting benefit, lapse, or insurer payment.

The same logic gives a useful check on any transaction projection: calculate the account value both as units times price and as the prior value adjusted for premiums, investment movement, and monetary deductions. Differences often reveal a transaction applied at the wrong price or in the wrong order.

Worked Example: Two Price Movements Before a Claim

A hypothetical policy starts with 300 units at $12.50 and receives a fully allocated $600 premium at that price. The policyholder survives the first year. The price rises by 4%, after which a $30 charge is deducted at the year-end price. In the second year the price falls by 10%, then a $25 charge is deducted. The policyholder dies immediately after that second charge, and the contract pays the post-charge account value plus $800. Find the claim and its two components. Both price movements are deterministic; the claim event is specified rather than probabilistically modelled.

Method. Carry the unrounded unit count into the second year and apply each charge at its own stated price. This avoids rounding drift and prevents the second charge being valued at the first-year price.

The premium buys $600/12.50=48$ units, giving $300+48=348$ units before the first price movement. The first-year price becomes $12.50(1.04)=\$13.00$, and the account value before the first charge is $348(13)=\$4{,}524$. After the $30 charge, the account is $4,494, represented by

348-\frac{30}{13}=345.6923076923\ldots \text{ units}.

The second-year price is $13(0.90)=\$11.70$. Before the second charge, the account value is $345.6923076923\ldots(11.70)=\$4{,}044.60$. Deducting $25 leaves an account value of $4{,}044.60-25=\$4{,}019.60$. The corresponding unrounded units are

345.6923076923\ldots-\frac{25}{11.70} =343.5555555556\ldots.

The claim is $4{,}019.60+800=\$4{,}819.60$. Its account-linked portion is $4,019.60, and its additional portion is $800. As a check, $343.5555555556\ldots(11.70)=\$4{,}019.60$ using the unrounded unit count. The total of the two benefit components is $4,819.60.

Reading the Projection for Solvency

A linked account is not, by itself, a complete measure of the insurer’s liability or solvency position. It records the value attributable to the policyholder’s units under the chosen price and transaction rules. The insurer must separately identify the benefits promised by the contract and the assets available to support them. A claim may exceed the account, as in the examples, or a contract may specify a different relationship between account value and benefit.

In an integrated projection, report at least the following items separately:

Unit balance: units after each premium purchase and unit cancellation.
Account value: remaining units multiplied by the appropriate unit price at the valuation time.
Gross claim: the amount payable under the stated contractual benefit rule.
Account-linked component and excess: the portion represented by the account value and any further payment specified by the contract.
Solvency interpretation: the comparison of obligations with supporting assets under the relevant valuation and capital framework.

Only the first four can be determined from a basic unit-and-claim specification. A solvency conclusion also needs information about assets, valuation assumptions, other cash flows, and required capital. In particular, it is unsafe to call the excess over the account “the insurer’s solvency requirement”: it is only an identifiable component of the claim under the specified benefit rule.

Common Errors to Avoid

Subtracting a dollar charge from units. Convert the charge to units by dividing by the price at which the deduction is made.
Using the wrong price for a transaction. A premium or charge may use a price set at a particular time. Follow the stated timing rather than applying every transaction at the opening or closing price by habit.
Rounding units too early. Keep full precision during the projection and round only the reported result. Otherwise, multiplying rounded units by the price may not reproduce the account value.
Equating account value with claim amount. Apply the contractual benefit rule after calculating the account. An account-plus-additional-benefit claim includes both components.
Equating claim excess with total solvency exposure. The excess above the account is not a complete solvency measure without the wider asset, liability, and capital information.

Key takeaway: Track units and money on a stated timeline. Use unit prices to convert premiums and charges, calculate the account value from the remaining units, and apply the claim rule separately. Then interpret the resulting cash flows in the context of the insurer’s wider assets and obligations.

Check Your Understanding

For each problem, follow the stated transaction order and retain unrounded values until the final answer.

A policy has 120 units at $10.00. A $500 premium is 96% allocated at that price. The unit price then rises to $10.50, and a $21 charge is deducted at that price. Find the closing account value and units. (Answer: $1,743.00; 166 units)
A policy has 600 units at $8.50 and receives no premium. The unit price rises by 6%, then a $51 charge is deducted at the new price. Find the closing account value and units, rounding units to six decimal places. (Answer: $5,355.00; 594.339623 units)
A policy has 180 units at $25.00. A $1,000 premium is 90% allocated at that price. The price rises by 3%, then a $40 charge is deducted. A death claim immediately afterward is the post-charge account plus $600. Find the claim and account-linked portion. (Answer: $6,122.00 claim; $5,522.00 account-linked portion)
Ten identical policies each have 250 units at $12.00. There are no premiums. The price falls by 5%, and a $18 charge is deducted from each account at the new price. Three policyholders then die, with each death benefit equal to that policy’s post-charge account plus $400. Find aggregate claims for the three deaths and separate the account-linked and additional portions. (Answer: $9,696.00 aggregate claims; $8,496.00 account-linked and $1,200.00 additional)
For the situation in Question 4, explain why the $1,200 aggregate additional benefit is not, by itself, a complete measure of the insurer’s solvency exposure. (Answer: It is only the claim amount above the specified account-linked portions; a solvency assessment also needs the supporting assets, valuation basis, other obligations, and applicable capital requirements.)

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