Asset Shares at Maturity, Surrender, and Other Exit Dates

Why the Exit Date Needs Its Own Calculation

An asset share accumulates premiums and other receipts, investment earnings, and allocated outgo according to their timing. At maturity, surrender, or another termination date, the fund is used to meet a terminal payment. The relevant amount for that comparison is the asset share immediately before that payment. The balance immediately after payment is a different quantity.

The distinction matters especially when another cash flow is due on the exit date. A final premium, expense, or other routine transaction may occur just before the terminal payment. It belongs in the pre-payment asset share if the stated event order places it first. Treating every cash flow dated at the exit time as either automatically included or automatically excluded can produce the wrong result.

Definition: The pre-exit asset share is the accumulated asset share immediately before the terminal payment is made. The post-exit balance is the pre-exit asset share less that payment. If the terminal payment is $B_T$, then $A(T^+)=A(T^-)-B_T$. These are accounting balances on the stated asset-share basis; a negative post-exit balance indicates a shortfall on that basis, not a change in the contractual amount owed.

Cash-Flow Timing and Accumulation

Use a consistent event order at every date. For example, the fund might first receive a scheduled premium, then earn investment income over the next period, and finally make a maturity payment. A different contract or projection may place the premium after the investment period. The calculation must follow the order specified in the problem rather than relying on a default convention.

Let $A_0$ be the asset share at the starting time, and let $C_j$ be a signed cash flow: receipts are positive and outgo is negative. Let $G(s,t)$ be the accumulation factor from time $s$, immediately after the cash flow at $s$, to time $t$, immediately before the event at $t$. For annual periods with effective annual returns $i_k$, the factor over complete periods is the product of the applicable $1+i_k$. A fractional-period factor should be used only when it is supplied or justified by the assumed investment accumulation model.

Calculation rule: Accumulate the opening asset share and every nonterminal cash flow that occurs before the terminal payment. Include a cash flow at time $T$ if it occurs before that payment, using an accumulation factor of 1 from the cash-flow time to $T$. Exclude the terminal payment from this pre-payment accumulation, then subtract it once to obtain the post-payment balance.

In notation, let $\mathcal{C}_T$ be the set of nonterminal cash flows occurring before the terminal payment. This set can include flows strictly before $T$ and flows at $T$ that precede the terminal payment. Then

A(T^-)=A_0G(0,T)+\sum_{j\in\mathcal{C}_T} C_jG(t_j,T), \qquad A(T^+)=A(T^-)-B_T.

Here $B_T$ is the terminal payment and $t_j$ is the time of cash flow $C_j$. If a routine expense at $T$ occurs before the terminal payment, it can be represented as a negative $C_j$ in the sum. If the expense occurs after the terminal payment, it is not part of $A(T^-)$; it belongs to a later balance, if one is being calculated. When multiple events share a date, the problem's ordering determines which side of the terminal payment they belong on.

Why the Accumulation Formula Holds

The formula follows by applying the asset-share ledger one interval at a time. During an interval, the existing balance is multiplied by that interval's accumulation factor. A cash flow made at the end of the interval is then added or subtracted at that date, and only subsequent investment factors apply to it.

Accumulate the opening balance.
The opening amount $A_0$ grows from time 0 to $T$, giving $A_0G(0,T)$.

Accumulate each signed cash flow.
A flow $C_j$ at $t_j$ contributes $C_jG(t_j,T)$. It earns investment return only after it enters or leaves the fund.

Respect the terminal event order.
Sum only flows occurring before the terminal payment. A flow at $T$ that is earlier in the event order has $G(T,T)=1$, so it is included without further accumulation.

Make the terminal payment.
Subtract $B_T$ from the pre-payment balance to obtain the post-payment balance.

This is also why the terminal payment should not be accumulated as though it had been made before the investment period ending at $T$. It is paid at the exit event, after the pre-exit balance has been determined.

Worked Calculations at Exit

Worked Example: Maturity With a Final Premium Due First

A hypothetical policy has an opening asset share of $500 at time 0. The fund earns 4% effective annually during year 1. At time 1, a net renewal contribution of $200 is added before the next year's investment period. The fund then earns 3% effective annually during year 2. At time 2, a final premium of $50 is received immediately before the maturity payment of $780. Find the pre-maturity asset share and the post-payment balance.

Method. Apply each investment factor to the balance held during that year, add the time-1 contribution before year 2, and include the time-2 premium because it precedes the maturity payment.

After year 1, the asset share is $500(1.04)=520$. Add the renewal contribution at time 1: $520+200=720$. After year 2, just before the time-2 events, the balance is $720(1.03)=741.60$. Add the final premium due before maturity: $741.60+50=791.60$.

A(2^-)=791.60, \qquad A(2^+)=791.60-780=11.60.

The pre-maturity asset share is $791.60, and the post-payment balance is $11.60. As a check, the premium adds exactly $50 at time 2 and the terminal payment removes $780, leaving $11.60 after the accumulated pre-payment balance is established. The common mistake is to omit the final premium just because it is dated at maturity; the specified order puts it before the terminal payment.

Worked Example: Surrender at a Fractional-Year Date

A hypothetical policy has an asset share of $300 at time 0. It earns 5% effective annually during year 1. At time 1, a $20 expense is paid, followed immediately by a net contribution of $100. The fund earns 4% effective annually during year 2. At time 2, a further $10 expense is paid. The fund then earns a specified half-year accumulation factor of 1.02, and the policy is surrendered at time 2.5 for $390. Find both balances at surrender.

Method. Follow the dated ledger in order, including the time-2 expense before the final half-year accumulation. The supplied factor 1.02 applies only to that last half-year.

Year 1 accumulation gives $300(1.05)=315$. At time 1, pay the expense and then add the contribution: $315-20+100=395$. Year 2 accumulation gives $395(1.04)=410.80$. At time 2, pay the expense: $410.80-10=400.80$. Accumulate for the final half-year: $400.80(1.02)=408.816$.

A(2.5^-)=408.816, \qquad A(2.5^+)=408.816-390=18.816.

Rounding only the final dollar amounts to cents, the pre-surrender asset share is $408.82 and the post-payment balance is $18.82. The event-by-event calculation avoids applying the half-year factor to the $390 surrender payment: that payment occurs at the end of the accumulation period.

Worked Example: Another Termination Date With Same-Date Outgo

A hypothetical policy has an asset share of $1,000 at time 0. It earns 6% effective annually for one year. At time 1, an expense of $25 is paid, followed by a net contribution of $100. The fund then earns a specified half-year accumulation factor of 1.03. At time 1.5, an exit payment of $1,100 is made. A separate $15 administration expense is also due at time 1.5, and the event order is that this expense is paid immediately before the exit payment. Calculate the pre-payment asset share and the post-payment balance.

Method. Accumulate the opening balance through year 1, apply the time-1 cash flows in order, and use the half-year factor. Include the time-1.5 expense in the pre-payment balance because it is paid first.

After the first year, the balance is $1{,}000(1.06)=1{,}060$. At time 1, apply the expense and contribution: $1{,}060-25+100=1{,}135$. After the half-year accumulation, the balance is $1{,}135(1.03)=1{,}169.05$. Deduct the expense due at the exit date: $1{,}169.05-15=1{,}154.05$.

A(1.5^-)=1{,}154.05, \qquad A(1.5^+)=1{,}154.05-1{,}100=54.05.

The pre-payment asset share is $1,154.05, and the balance after the exit payment is $54.05. If the $15 expense were ordered after the exit payment instead, it would not belong in the pre-payment balance. Here, event order—not merely the date label—settles the issue.

Common Timing Trap

A frequent error is to use a strict “cash-flow time less than $T$” rule and therefore exclude every routine cash flow dated at $T$. That rule fails when a same-date flow occurs before the terminal payment. The opposite error is to include all time-$T$ flows without checking whether they occur before or after the terminal event.

In the maturity example, the incorrect calculation that excludes the final premium gives a pre-payment balance of $720(1.03)=741.60$, and a post-payment balance of $741.60-780=-38.40$. But the stated event order requires the $50 premium to be received first. The correct balances are $741.60+50=791.60$ before payment and $791.60-780=11.60$ after payment. Thus, the mistake changes both the pre-payment amount and the sign of the post-payment balance.

Key takeaway: Identify the exact instant at which the requested asset share is measured. Accumulate each receipt or outgo only over the periods during which it is in the fund, include same-date flows that precede the terminal payment, and subtract the terminal payment once.

Check Your Understanding

For each problem, follow the stated event order. Report dollar amounts to the nearest cent where needed.

A fund has an asset share of $250 at time 0 and earns 4% effective annually for one year. At time 1, a premium of $40 is received immediately before a maturity payment of $300. Find the pre-payment and post-payment balances. (Answer: $300.00 before payment; $0.00 after payment)
A fund starts at $500 and earns 3% effective annually for one year. At time 1, an expense of $15 is paid. The fund then earns a specified half-year factor of 1.02 before a time-1.5 exit payment of $520. Find both balances. (Answer: $510.00 before payment; −$10.00 after payment)
A fund starts at $700 and earns 5% effective annually for one year. At time 1, a contribution of $60 is received. It earns 4% effective annually for the second year. At time 2, an expense of $12 is paid immediately before an exit payment of $850. Find both balances. (Answer: $814.80 before payment; −$35.20 after payment)
A fund starts at $1,000, earns 2% effective annually for one year, pays an expense of $30 at time 1, and earns 3% effective annually for year 2. At time 2, a contribution of $25 is received before a payment of $1,050. Find the balances. (Answer: $1,044.70 before payment; −$5.30 after payment)
A fund starts at $400 and earns 5% effective annually for one year. A contribution of $50 is then received, and the fund earns 4% effective annually for the next year. At time 2, an exit payment of $500 is made. Find the pre-payment asset share and the shortfall or excess after payment. (Answer: $488.80 before payment; $11.20 shortfall after payment)

What You'll Learn