Partial Withdrawals and Reserve Effects

Why a Partial Withdrawal Requires More Than One Calculation

A partial withdrawal gives a policyholder access to some of the value in a universal life policy while the contract remains in force. Its effects depend on the contract rules: the amount paid may differ from the amount removed from the account, and the death-benefit option determines how a lower account value affects the insurer’s exposure. A sound projection therefore keeps three quantities distinct: the cash paid, the post-withdrawal account value, and the benefit amount the insurer must fund beyond that account.

The account value and reserve are not interchangeable. The account is a contract value that changes with deposits, withdrawals, charges, and interest. A reserve is a liability measured on a stated valuation basis. The examples below use a deliberately specified one-year reserve model so that the effect of an account withdrawal can be isolated. Different contracts and valuation bases can produce different reserve amounts.

Definition: Let $A_t$ denote the policy account value immediately before a withdrawal at time $t$. Let $W$ be the cash amount paid to the policyholder, and let $C$ be any withdrawal charge deducted from the account. Under the convention used here, the post-withdrawal account value is $A_t^+=A_t-W-C$. If the account then earns an effective annual rate $i$ with no intervening cash flows or charges, its value one year later is $A_{t+1}=A_t^+(1+i)$.

The symbols $W$ and $C$ must follow the stated contract rule. A charge might be added to the amount deducted from the account, as in the examples here; another contract could define its charge differently. Do not assume that a quoted withdrawal amount is either the cash paid or the account reduction without checking the problem statement.

Account Value and Death-Benefit Exposure

For this tutorial, consider two explicitly specified death-benefit options. Under Option A, the death benefit is a fixed face amount $F$. The account is available to fund that benefit, so the insurer’s net amount at risk (NAR) at a specified time is the positive excess of the face amount over the account value. Under Option B, the death benefit is the face amount plus the account value; the account portion is funded by the account itself, leaving the face amount as the NAR. Actual contracts may include additional provisions, such as minimum benefits or charges, so these formulas apply only under the stated rules.

\text{Option A NAR at time }s=\max(F-A_s,0), \qquad \text{Option B NAR at time }s=F.

A withdrawal reduces the account, all else equal. Under Option A, that reduction generally raises the NAR because a smaller account remains to support the fixed benefit. Under Option B, the total death benefit also falls when the account falls, so the NAR remains the face amount under the specified definition. This difference is why a withdrawal cannot be translated into a reserve effect without first identifying the benefit option.

Worked Example: Projecting a Withdrawal Under Two Benefit Options

A hypothetical policy has an account value of $80,000 and a face amount of $150,000 immediately before a withdrawal. The policyholder receives $20,000 in cash. The contract deducts an additional charge equal to 2% of the cash paid. There are no premiums or other charges during the following year, and the account earns an effective annual rate of 4%. Compare the account value and NAR immediately after the withdrawal and one year later under Options A and B.

Method. First calculate the charge and account deduction, then project the account with interest. Apply each option’s benefit rule separately. This avoids treating the cash paid as the full account reduction or assuming both benefit options respond identically.

The withdrawal charge is $0.02(20{,}000)=\$400$. The account deduction is therefore $20{,}000+400=\$20{,}400$, and the immediate post-withdrawal account value is $80{,}000-20{,}400=\$59{,}600$.

After one year, the account value is $59{,}600(1.04)=\$61{,}984$. Under Option A, the immediate NAR is $\max(150{,}000-59{,}600,0)=\$90{,}400$. One year later it is $\max(150{,}000-61{,}984,0)=\$88{,}016$.

Under Option B, the immediate death benefit is $150{,}000+59{,}600=\$209{,}600$, of which $59,600 is the account portion; the NAR is $150,000. One year later the death benefit is $150{,}000+61{,}984=\$211{,}984$, and the NAR is still $150,000. The account and total benefit have increased with interest, but the face-amount exposure has not.

Check. Under Option A, account value plus NAR immediately after withdrawal is $59{,}600+90{,}400=\$150{,}000$, the face amount. Under Option B, subtracting the account portion from the total benefit leaves $150,000. These checks agree with the stipulated benefit rules.

A One-Year Reserve Model and Its Justification

To quantify a reserve effect, use the following simplified valuation basis. At the valuation date, the policy is in force. During the next year, death occurs with probability $q$; otherwise the policy survives. A death claim is valued as paid at the end of the year, and the account grows deterministically at effective annual rate $i$. There are no premiums, expenses, charges, or other cash flows during the year. The insurer’s supplementary reserve covers only the expected present value of the death benefit in excess of the account available at the end of the year. It does not include the account value itself.

Under these assumptions, if the face amount is $F$ and the account at the valuation date is $A$, the year-end account is $A(1+i)$. Under Option A, the year-end NAR is $\max(F-A(1+i),0)$. The reserve is the probability of death times this claim amount, discounted one year at the valuation rate $i$.

V=\frac{q}{1+i}\max\!\bigl(F-A(1+i),0\bigr)

This formula follows directly from the model rather than from a general reserve identity. If death occurs, the insurer pays the year-end NAR; if the policy survives, there is no supplementary death claim in this model. The expected year-end claim is therefore $q\max(F-A(1+i),0)+(1-q)(0)$. Discounting that expected payment one year gives the displayed expression. The formula does not apply unchanged if claims are paid at another time, future premiums or expenses are included, account growth is uncertain, or the policy has different benefit rules.

Apply the withdrawal rule.
Calculate the cash paid and every account deduction, including any specified charge.

Project the account to the claim time.
Apply the stated crediting rate and include only the cash flows the model specifies.

Calculate the benefit exposure.
Use the specified death-benefit option to determine the NAR; apply a zero floor where the rule requires one.

Value the reserve on its stated basis.
Weight the claim by its probability and discount it using the stated timing and rate assumptions.

Worked Example: Measuring the Reserve Increase

A hypothetical Option A policy has a $160,000 face amount and an account value of $100,000 before a withdrawal. The policyholder receives $29,000, and a specified $1,000 charge is also deducted from the account. For the coming year, the account earns 5% effective annual interest, the probability of death is $q=0.02$, and the claim is valued at year-end. Use the one-year reserve model just defined. Compare the reserve before and after the withdrawal.

Method. Project each account value to year-end, calculate the Option A NAR at that time, and discount the expected claim. The reserve is not calculated by applying the mortality probability to the account reduction alone.

Before withdrawal, the year-end account is $100{,}000(1.05)=\$105{,}000$, giving a year-end NAR of $160{,}000-105{,}000=\$55{,}000$. The reserve before withdrawal is $0.02(55{,}000)/1.05=\$1{,}047.6190\ldots$, or $1,047.62.

After withdrawal, the account is $100{,}000-29{,}000-1{,}000=\$70{,}000$. Its year-end value is $70{,}000(1.05)=\$73{,}500$, so the year-end NAR is $160{,}000-73{,}500=\$86{,}500$. The reserve after withdrawal is $0.02(86{,}500)/1.05=\$1{,}647.6190\ldots$, or $1,647.62.

The reserve increases by $1{,}647.62-1{,}047.62=\$600.00$. Before rounding, the change is $(86{,}500-55{,}000)(0.02)/1.05=31{,}500(0.02)/1.05=\$600$. The increase is driven by the higher year-end NAR, not by treating the withdrawal as an immediate claim.

Check. The withdrawal reduces the starting account by $30,000, and after 5% interest that difference is $31,500. The NAR rises by exactly $31,500, consistent with a fixed face amount. Multiplying that NAR increase by $q/(1+i)=0.02/1.05$ gives the $600 reserve increase.

When the Reserve Effect Is Zero or Capped

Option A’s NAR cannot be negative under the definition used here. If the projected account at the claim time is at least the face amount, the excess benefit exposure is zero. A withdrawal may move a policy from this region into positive NAR, but an account decrease does not always translate dollar-for-dollar into a positive reserve increase: the floor must be applied after projecting the account to the relevant claim time.

Worked Example: Applying the Zero Floor After Projection

A hypothetical Option A policy has a $120,000 face amount and a current account value of $130,000. The policyholder receives $25,000, and a $1,000 charge is deducted in addition. Assume 3% effective annual interest, a one-year death probability of $q=0.015$, and the same year-end claim convention and reserve model. Calculate the reserve before and after the withdrawal.

Method. Project the account first, then apply the NAR floor. Applying the floor to the current account difference and stopping there would miss the effect of interest and the withdrawal.

Before withdrawal, the year-end account is $130{,}000(1.03)=\$133{,}900$. Since this exceeds the $120,000 face amount, the NAR is $\max(120{,}000-133{,}900,0)=\$0$, and the reserve is $0.

The withdrawal reduces the account by $25{,}000+1{,}000=\$26{,}000$, leaving $130{,}000-26{,}000=\$104{,}000$. At year-end, the account is $104{,}000(1.03)=\$107{,}120$. The year-end NAR is $120{,}000-107{,}120=\$12{,}880$. Thus the reserve is $0.015(12{,}880)/1.03=\$187.5728\ldots$, or $187.57.

The reserve therefore rises from $0 to $187.57. The zero floor is important: the pre-withdrawal NAR is not $-\$13,900$, and the pre-withdrawal reserve is not negative under this model. The calculated amount is rounded to the nearest cent.

Common Mistakes and Practical Checks

Using the cash paid as the account deduction. In the first example, the policyholder receives $20,000, but the account falls by $20,400 because of the charge. Use the contract’s stated rule to calculate both amounts.
Applying interest before the withdrawal when the withdrawal occurs first. If the policy account is reduced at the start of the projection period, calculate interest on the post-withdrawal account. Reversing that order changes the projected account and the resulting NAR.
Ignoring the benefit option. A lower account increases the NAR for a fixed-face Option A benefit, but under the specified Option B rule the account portion of the benefit falls along with the account. Do not transfer one option’s NAR formula to another.
Forgetting the NAR floor. When the face amount is below the projected account, use the stipulated maximum with zero rather than a negative NAR.
Calling a simplified reserve a universal contract reserve. The one-year formula omits premiums, expenses, future charges, and other benefits by assumption. State the valuation model before using it; add any required cash flows if the problem includes them.

Key takeaway: A partial withdrawal changes the account by the contract-defined deduction, not necessarily by the cash paid. Project the resulting account under the specified timing, calculate NAR using the actual benefit option, and then value the reserve on an explicit basis. Each step has its own assumptions and should be checked separately.

These distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus, where a projection may require the candidate to connect contract cash flows, benefit exposure, and reserve assumptions without confusing them.

Check Your Understanding

For each problem, use the benefit and valuation rules stated in that problem. Round money answers to the nearest cent when needed.

A hypothetical Option A policy has a $140,000 face amount and a $90,000 account value. The policyholder receives $12,000; the charge is 3% of the cash paid and is deducted in addition. What is the immediate post-withdrawal account value and immediate NAR? (Answer: Account value $77,640; NAR $62,360.)
A hypothetical Option A policy has a $200,000 face amount and a $150,000 account. A $28,000 cash withdrawal and a $2,000 charge are deducted at the start of the year. The account earns 4% for the year. What is the year-end account and NAR? (Answer: Account value $124,800; NAR $75,200.)
In the one-year reserve model, an Option A policy has face amount $250,000, post-withdrawal account value $190,000, annual interest $i=0.03$, and death probability $q=0.012$. What is the supplementary reserve? (Answer: $632.62.)
A hypothetical Option B policy has a $175,000 face amount and a $90,000 account. The policyholder receives $15,000, and a $500 charge is also deducted. The remaining account earns 5% for one year. Under Option B, what are the year-end account, total death benefit, and NAR? (Answer: Account value $78,225; death benefit $253,225; NAR $175,000.)
An Option A policy has a $180,000 face amount and a $160,000 account before a withdrawal. A $20,000 cash payment and a $1,000 charge are deducted at the start of the year. The account earns 2%, the death probability is $0.01$, and the year-end one-year reserve model applies. What is the reserve after withdrawal? (Answer: Year-end account $141,780; NAR $38,220; reserve $374.71.)

What You'll Learn