Expense Charges and Expense Reserves

Why Separate Expenses from Charges?

A universal life policy can generate several different cash flows. The insurer may incur costs to administer an in-force policy and may collect expense charges under the contract. To understand the expense component of the insurer’s prospective liability, these items must be projected separately: the amount and timing of an expense need not match the amount and timing of a charge.

An expense charge is not itself a maintenance expense. It is a contractual amount deducted from the policy account and treated here as an inflow to the insurer. A maintenance expense is a cost incurred by the insurer. The calculation below isolates these two flows; it does not, by itself, calculate a total policy reserve that includes benefits, mortality margins, or other components. This expense-reserve method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: At a valuation date, the expense reserve for an in-force policy is the expected present value of future maintenance expenses minus the expected present value of future expense charges, calculated on the stated valuation assumptions and conditional on the policy being in force at the valuation date. A positive value means projected expenses exceed projected charges; a negative value means projected charges exceed projected expenses.

Set Up the Projection Before Calculating

For a discrete annual projection, let $e_k$ be the maintenance expense at the end of year $k$, and let $c_k$ be the expense charge at that same time. Let $r_k$ be the probability that the policy is still in force at the end of year $k$, given that it is in force at the valuation date. This probability may reflect the termination assumptions specified for the projection, including mortality and lapse where applicable.

If both the expense and charge are paid only when the policy remains in force at the end of the year, and the effective annual interest rate is $i$, then $v=1/(1+i)$, and the expense reserve is:

E_0=\sum_{k=1}^{n}v^k r_k(e_k-c_k).

This formula assumes the same end-of-year in-force condition for the expense and the charge. If a charge is collected at the beginning of a year, it has a different discount exponent and a different in-force probability. If an expense is incurred even when the policy terminates during the year, its probability condition must likewise be adjusted. Do not force differently timed cash flows into the same factor.

Identify each cash flow.
List projected maintenance expenses and expense charges separately, with amounts and payment dates.

Apply the correct condition.
For each item, identify the probability that the policy is in force, or that any other required event occurs, when that cash flow is due.

Discount to the valuation date.
Use the effective annual interest rate and the actual time of each payment.

Subtract charges from expenses.
Calculate the two present values separately, then take expenses less charges and check the result.

Why the Prospective Formula Holds

The formula follows from the definition of an expected present value. Consider one expense of amount $e_k$, payable at the end of year $k$ only if the policy is then in force. Its discounted value is $v^k e_k$ when payable and zero otherwise. The probability of payment, conditional on in-force status at the valuation date, is $r_k$. Therefore its expected present value is $v^k r_k e_k$. By the same reasoning, the expected present value of a charge due on the same condition is $v^k r_k c_k$. Subtracting the charge value from the expense value gives $v^k r_k(e_k-c_k)$; adding over all projection years gives the stated formula.

This argument also explains the limits of the formula. If the expense and charge have different payment dates or different conditions, they do not share a common $v^k r_k$ factor. Calculate each expected present value using its own timing and probability, then subtract.

Worked Applications

Worked Example: Annual Charges and Expenses Paid at Year-End

A hypothetical in-force policy is projected for two years. At each year-end, a maintenance expense is incurred and an expense charge is collected only if the policy remains in force. The conditional probability of remaining in force from one year-end to the next is 0.90. The effective annual interest rate is 5%. Expenses are $40 at the end of year 1 and $60 at the end of year 2; charges are $25 and $35, respectively. Find the expense reserve.

Method. Because both cash flows in each year are due only if the policy is in force at that year-end, use the same year-end persistence probability and discount factor for that year. The annual combined factor is $0.90v=0.90/1.05=6/7$.

The year 1 probability of being in force is $r_1=0.90$. The year 2 probability is $r_2=0.90^2=0.81$. The discount factors are $v=1/1.05=20/21$ and $v^2=400/441$. Thus $v r_1=6/7$ and $v^2r_2=36/49$.

The present value of expenses is:

40\left(\frac{6}{7}\right)+60\left(\frac{36}{49}\right) =34.285714+44.081633 =78.367347.

The present value of charges is:

25\left(\frac{6}{7}\right)+35\left(\frac{36}{49}\right) =21.428571+25.714286 =47.142857.

Therefore the expense reserve is $78.367347-47.142857=31.224490$, or $31.22 to the nearest cent. As a check, the net differences are $15 and $25, whose present values are $15(6/7)=12.857143$ and $25(36/49)=18.367347$; their sum is also $31.224490. The check confirms that the separate-value and net-flow calculations agree.

Common mistake avoided: Using a two-year survival probability for the year 1 cash flows understates their value. Year 1 payments require only $r_1=0.90$, not $r_2=0.81$.

Worked Example: An Immediate Charge and an Uncertain Expense

At the valuation date, a hypothetical policy is projected to remain in force through the end of the next year. The effective annual interest rate is 4.1667%, so use $v=0.96$. An expense charge of $10 has already been collected at the valuation date, and a further $30 charge is due at year-end. A $50 maintenance expense is due at year-end with probability 0.90; otherwise that expense is not incurred. Find the expense reserve, allowing for the expense’s stated uncertainty.

Method. Value each item at its own time and apply the expense occurrence probability only to the expense. The expected expense at year-end is $0.90(50)=45$, so its expected present value is $45(0.96)=43.20$. The $10 charge is immediate and is not discounted. The year-end charge has present value $30(0.96)=28.80$.

The total present value of charges is $10+28.80=38.80$. Therefore:

E_0=43.20-38.80=\$4.40.

The result is positive: on these assumptions, expected expenses exceed charges by $4.40. The arithmetic can be checked directly: $10+28.80=38.80$, and $43.20-38.80=4.40$. In particular, this subtraction is not $5.20.

Common mistake avoided: Discounting the immediate $10 charge would give it the wrong timing. The valuation-date amount has discount factor 1; only the year-end amounts use $v=0.96$.

Worked Example: Charges at the Start of the Year

A hypothetical policy is valued at the start of year 1 and projected for three years. The effective annual interest rate is 5%, and the conditional probability of remaining in force for each year is 0.90. Maintenance expenses of $100, $120, and $140 are paid at the ends of years 1, 2, and 3, respectively, provided the policy is in force then. An expense charge of $40 is collected at the beginning of each year, provided the policy is in force at that time. Find the expense reserve.

Method. Expenses are paid at year-end, while charges are paid at the beginning of each year. The expense at the end of year $k$ has combined discount-and-persistence factor $(0.90/1.05)^k=(6/7)^k$. The beginning-of-year charges occur at times 0, 1, and 2. Their in-force probabilities are 1, 0.90, and 0.81, and their discount factors are 1, $v$, and $v^2$, respectively.

The expense present value is:

100\left(\frac{6}{7}\right) +120\left(\frac{36}{49}\right) +140\left(\frac{216}{343}\right) =85.714286+88.163265+88.163265 =262.040816.

For the charges, the time-0 factor is 1. At time 1, the combined factor is $0.90v=6/7$. At time 2, it is $0.81v^2=36/49$. Hence:

40\left(1+\frac{6}{7}+\frac{36}{49}\right) =40\left(\frac{127}{49}\right) =103.673469.

The expense reserve is $262.040816-103.673469=158.367347$, or $158.37 to the nearest cent. As a timing check, the first charge is not discounted because it is collected at the valuation date, while the first expense is discounted and weighted by the probability of remaining in force to the end of year 1.

Common mistake avoided: Treating all three charges as year-end payments incorrectly discounts the time-0 charge and applies the wrong persistence condition to charges at the start of later years.

Interpretation and Exam Traps

An expense reserve is sensitive to both the amounts projected and the cash-flow map. A charge collected earlier has a greater present value than the same charge collected later. A charge that is due only while the policy remains in force must be weighted by the probability of that condition; a charge already collected at the valuation date has no future persistence risk. Similar care is required for expenses. State whether each cost is fixed, expected, or conditional on a specified event.

Several errors are especially easy to make in a multi-year projection:

Mixing policy account activity with insurer expenses. A charge deducted from the account is a projected charge receipt in this calculation, not automatically an expense of the same amount.
Using the wrong payment date. A beginning-of-year charge is not discounted like an end-of-year charge.
Applying persistence twice or not at all. Check whether a supplied amount or factor already incorporates the probability that the policy remains in force.
Subtracting in the wrong direction. The expense reserve defined here is expenses less charges. Reversing the order changes the sign and the interpretation.
Stopping after a plausible-looking total. Reconcile the separately calculated expense and charge present values, or recompute using net flows when they share the same timing and conditions.

A negative expense reserve is possible under this definition when projected charges exceed projected expenses. It is not, by itself, a conclusion about the total adequacy of the policy reserve: other projected cash flows and any applicable valuation requirements would need separate consideration.

Key takeaway: Project each maintenance expense and expense charge with its own amount, payment time, and applicable probability. Then calculate expected present values on a consistent basis and subtract charges from expenses. The timing and condition attached to a cash flow are part of its value, not bookkeeping details.

Check Your Understanding

For each problem, identify the payment timing and applicable probability before calculating. Answers are rounded to the nearest cent where appropriate.

A policy has a 0.90 probability of remaining in force to each year-end, conditional on being in force at the start of that year. At year-end, expenses are $20 and $30 in years 1 and 2, while charges are $12 and $18. The effective annual interest rate is 0%. Find the expense reserve. (Answer: $16.92)
A policy is assumed to remain in force through year-end. At the valuation date a $5 charge is collected. At year-end a $20 charge is due. A $40 expense is incurred at year-end with probability 0.75. If the effective annual interest rate is 4%, find the expense reserve. (Answer: $4.62)
The effective annual interest rate is 0%, and the probability of remaining in force each year is 0.80. Expenses of $60 and $70 are due at the ends of years 1 and 2 if the policy remains in force to those dates. Charges of $25 are due at the beginnings of years 1 and 2 if the policy is in force at those times. Find the expense reserve at the start of year 1. (Answer: $47.80)
For three years, a policy has a 0.90 conditional probability of remaining in force each year. The effective annual interest rate is 0%. Each year-end expense is $15 if the policy remains in force to that year-end. A $20 charge is collected at the beginning of each year if the policy is in force then. Find the expense reserve at time 0. (Answer: -$17.62)
For a two-year projection, expenses of $50 and $70 and a level charge $C$ are each paid at year-end if the policy is in force at that year-end. The conditional annual probability of remaining in force is 0.95, and the effective annual interest rate is 5%. Find $C$ that makes the expense reserve zero. (Answer: $59.50 per year)

What You'll Learn