Asset Shares With Changing Annual Investment Returns

Why Changing Returns Require Careful Timing

An asset share is built up from the cash flows attributed to a policy and the investment income earned on funds held for that policy. When the effective investment return changes from year to year, it is not sufficient to apply one average return to the entire projection. The funds present in each year earn that year’s return; cash flows entering or leaving during the year earn a return only for the time they remain invested.

The key calculation is therefore a sequence of annual updates. Each update begins with the asset share at the start of that year, applies that year’s return to the funds held, and incorporates the cash flows at their specified times. The timing convention must be stated because a premium received at the start of a year and a benefit paid at the end of that year do not earn the same investment income.

Definition: Let $AS_t$ be the asset share immediately after all cash flows at the end of year $t$, and let $AS_0$ be the initial asset share. Let $i_t$ be the effective investment return earned during year $t$. A cash flow $C$ is positive if it increases the fund and negative if it reduces the fund. The return $i_t$ applies to funds invested during year $t$, not to funds that have not yet entered the fund or have already left it.

For a basic annual ledger, assume that all specified beginning-of-year cash flows occur at the start of the year, all specified end-of-year cash flows occur at the end, and the fund earns the year’s effective return between those times. If $B_t$ is the net beginning-of-year cash flow and $E_t$ is the net end-of-year cash flow, both signed as positive for an inflow, then

AS_t=(AS_{t-1}+B_t)(1+i_t)+E_t.

Thus a beginning-of-year premium is included before the year’s return is applied, while an end-of-year benefit is not reduced by that year’s return. A negative $B_t$ or $E_t$ represents an outflow. The return may be negative, provided $i_t>-1$, so the accumulation factor remains positive.

Cash Flows During the Year

An annual effective return determines the accumulation of funds invested for the whole year. To accumulate a cash flow occurring partway through the year, an additional within-year assumption is needed: the annual effective return by itself does not specify the actual path of investment performance between the start and end of the year.

For the calculations below, when a cash flow occurs at fraction $\tau$ of the year, assume that investment accumulates at a constant equivalent rate within that year. Under this convention, a full-year accumulation factor of $1+i_t$ corresponds to an accumulation factor of $(1+i_t)^s$ over a fraction $s$ of the year. A flow at time $\tau$ is invested for the remaining fraction $1-\tau$.

AS_t=AS_{t-1}(1+i_t)+\sum_k C_{t,k}(1+i_t)^{1-\tau_{t,k}}, \qquad 0\leq \tau_{t,k}\leq 1.

Here $C_{t,k}$ is the signed amount of the $k$th cash flow during year $t$, and $\tau_{t,k}$ is its time measured from the start of that year. An opening asset share is invested for the full year. A flow at the start of the year has $\tau=0$ and receives the full-year factor; a flow at the end has $\tau=1$ and receives no investment return for that year.

Why the Recurrence Holds

The recurrence follows by carrying each amount forward to the same valuation date: the end of year $t$. The opening asset share is present for the whole year, so it is multiplied by the year’s full accumulation factor. A cash flow occurring during the year is present only from its occurrence time to year-end.

Identify the time held.
The opening fund is held for one full year. A flow at time $\tau$ is held for $1-\tau$ years.

Accumulate each amount to year-end.
Under the stated within-year convention, the opening fund becomes $AS_{t-1}(1+i_t)$, and a flow $C_{t,k}$ becomes $C_{t,k}(1+i_t)^{1-\tau_{t,k}}$.

Add the year-end values.
All accumulated amounts are now measured at the same date, so their signed sum is $AS_t$.

This argument also shows why the return for year $t$ must be attached only to amounts invested during year $t$. It does not change the treatment of amounts that were already in the fund at the start of the year or those paid out before the relevant investment period.

Worked Example: Beginning Premiums and End-of-Year Benefits

A hypothetical policy has an initial asset share of $0. At the start of year 1, the fund receives a premium of $120 and incurs an expense of $20. At the end of year 1, it pays a benefit of $30. The effective return is 5% in year 1. At the start of year 2, a premium of $100 and an expense of $15 occur; an end-of-year benefit of $40 is paid. The effective return in year 2 is 2%. Find the asset share after each year.

Method. Combine the beginning-of-year flows, apply that year’s return to the opening share plus those flows, then include the end-of-year flow. This keeps the year’s benefit from incorrectly receiving investment income.

Year 1. The net beginning flow is $120-20=100$. The amount invested during the year is $0+100=100$, which accumulates to $100(1.05)=105$. The end-of-year benefit is an outflow of 30, so

$$ AS_1=(0+120-20)(1.05)-30=105-30=75. $$

Year 2. The net beginning flow is $100-15=85$. The amount invested for the year is $75+85=160$, which accumulates to $160(1.02)=163.20$. After the end-of-year benefit,

$$ AS_2=(75+100-15)(1.02)-40=163.20-40=123.20. $$

Check. The opening share plus beginning flows in year 2 is $160; a 2% return adds $3.20, and the $40 year-end payment reduces the fund to $123.20. The common mistake is to apply the 2% return to both years or to apply it to the year-2 benefit. The result correctly uses 5% only in year 1 and 2% only in year 2. Amounts are reported to the cent.

Worked Example: A Midyear Contribution

A hypothetical fund has an opening asset share of $500. During year 1, it receives a contribution of $60 exactly halfway through the year and pays an expense of $10 at year-end. The effective return for year 1 is 10%. In year 2, the fund receives a $25 premium at the start of the year and pays a $30 withdrawal at year-end; the effective return is $-4\%$. Under the constant equivalent within-year convention, find the asset share at the end of year 2.

Year 1. The opening $500 is invested for the full year, producing $500(1.10)=550$. The midyear contribution is invested for half a year, so its accumulation factor is $(1.10)^{1/2}=\sqrt{1.10}$, approximately 1.04880885. Its year-end value is $60(1.04880885)=62.928531$. The expense is paid at year-end and receives no year-1 return. Therefore,

AS_1=500(1.10)+60(1.10)^{1/2}-10 =550+62.928531-10 =602.928531.

Year 2. The beginning premium is invested for the full year. The amount exposed to the year-2 return is $602.928531+25=627.928531$. A return of $-4\%$ means the accumulation factor is $1-0.04=0.96$, so this amount becomes $627.928531(0.96)=602.811390$. The year-end withdrawal is then deducted:

$$ AS_2=(602.928531+25)(0.96)-30 =602.811390-30 =572.811390. $$

Check. The negative return reduced the year-2 invested amount, and the withdrawal was not exposed to that return because it occurred at year-end. Rounded to the nearest cent, $AS_2=$$572.81. Treating the midyear contribution as if it had been invested for a full year would overstate the asset share.

Using a Different Return in Every Projection Year

When returns change, repeat the update with the return attached to its own year. A return sequence such as $10\%$, $-10\%$, and $5\%$ means accumulation factors of 1.10, 0.90, and 1.05, respectively. It does not mean that the final return applies retrospectively to earlier years. Nor does it mean that each cash flow should be multiplied by every return in the projection: a cash flow is affected only by returns for years after it enters the fund and before it leaves.

Worked Example: Three Successive Returns, Including a Negative Year

A hypothetical policy has an opening asset share of $100. At the start of year 1, it receives $20; at year-end it pays $12. In year 2, it receives $30 at the start and pays $5 at year-end. In year 3, it receives $10 at the start and pays $7 at year-end. The effective returns are 10%, $-10\%$, and 5%, respectively. Find the ending share after each year.

Year 1. The start-of-year amount is $100+20=120$. It grows by 10% to $120(1.10)=132$. After the end-of-year payment, $AS_1=132-12=120$.

Year 2. The start-of-year amount is $120+30=150$. The 10% loss gives an accumulation factor of 0.90, so the amount becomes $150(0.90)=135$. After the year-end payment, $AS_2=135-5=130$.

Year 3. The start-of-year amount is $130+10=140$. It grows by 5% to $140(1.05)=147$. After the year-end payment, $AS_3=147-7=140$.

AS_1=120(1.10)-12=120,\qquad AS_2=150(0.90)-5=130,\qquad AS_3=140(1.05)-7=140.

Check. Each year starts with the previous year’s ending share, and its own return is used once. The sequence of ending shares is $120, $130, and $140. Applying 5% to all three years would not represent the stated return history. Amounts are exact in this example.

Common Mistakes and Practical Checks

A year-by-year ledger is easiest to audit when each row records the opening share, beginning flows, return factor, investment gain or loss, end flows, and closing share. Before accepting a result, check that the return is applied to the correct balance and that every flow is included at its stated time.

Using one return for the whole projection. Use $i_t$ in year $t$, rather than an average or the final year’s return, unless the problem explicitly specifies such a simplification.
Giving a year-end payment a full year of return. A payment made at year-end is deducted after that year’s accumulation under the stated timing convention.
Giving a midyear flow a full-year return. Under the constant equivalent convention, a flow at midyear receives the factor $(1+i_t)^{1/2}$, not $1+i_t$.
Assuming an annual return specifies the within-year path. It does not. For an intrayear cash flow, use a stated convention or actual subperiod returns if supplied.
Applying a negative return as a positive addition. A return of $-4\%$ has accumulation factor $0.96$, not $1.04$.

These projections are deterministic cash-flow calculations: any premiums, expenses, benefits, or other specified amounts are treated as the cash flows in the ledger. No mortality or lapse assumption is needed unless the problem asks for expected or pooled cash flows. This topic is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: For each year, first determine which funds are invested and for how long; then apply that year’s effective return only to those amounts. Carry the resulting closing asset share forward as the next year’s opening share.

Check Your Understanding

Use the stated timing conventions and round monetary answers to the nearest cent where needed.

A fund starts with $200, earns 5% during the year, and has no cash flows. What is its ending asset share? (Answer: $210.00)
A policy has an opening share of $0. At the start of the year it receives a $120 premium and incurs a $20 expense; at year-end it pays a $15 benefit. The effective return is 4%. Find the ending share. (Answer: $89.00)
A fund starts year 1 with $300. It receives $40 at the start of year 1 and pays $25 at year-end; the year-1 return is 10%. In year 2 it receives $20 at the start, pays $10 at year-end, and earns $-5\%$. Find the year-2 ending share. (Answer: $340.55)
A fund starts with $1,000, earns 8%, receives $100 exactly halfway through the year, and pays $50 at year-end. Use the constant equivalent within-year convention. Find the ending share. (Answer: $1,133.92)
A fund starts with $250. In year 1, it receives $50 at the start, pays $10 at year-end, and earns 4%. In year 2, it pays $12 at the start, receives $30 at year-end, and earns $-2\%$. In year 3, it receives $20 at the start, pays $15 at year-end, and earns 6%. Find the ending share after year 3. (Answer: $339.25)

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