Projecting Dividend Options Across Policy Years

Projecting Dividends One Policy Year at a Time

A dividend scale specifies the dividends projected for an eligible participating policy; it does not, by itself, specify what happens to those dividends after declaration. A policyholder may take each dividend in cash, use it to reduce a premium, leave it on deposit to accumulate, or use it to buy paid-up additions. The dividend amount and the resulting cash flow or benefit therefore need to be projected as separate steps.

The calculations below use hypothetical policies and scales. Unless a problem states otherwise, assume the policy remains in force through every projected dividend date, dividends are declared at policy anniversaries, and taxes, expenses, and transaction charges are ignored. A projected dividend under a scale is not automatically guaranteed.

Definition: Let $D_t$ be the dividend declared at the end of policy year $t$. A fixed-dollar scale specifies $D_t$ directly for each year. A scale stated as a rate $b_t$ on an eligible benefit base $B_{t-1}$ instead gives $D_t=b_tB_{t-1}$. The scale and its eligible base must be identified before projecting any dividend option.

These two ways of stating a scale are not interchangeable. If the scale lists fixed dollar dividends, buying additions does not change those listed amounts unless the problem says that the scale is recalculated. If the scale gives a rate on the current eligible benefit, additions may increase the base used for later dividends. State which interpretation applies and use it consistently.

Cash and Premium-Offset Options

With the cash option, the policyholder receives $D_t$ at the dividend date. This is a dividend cash inflow; it does not reduce a premium unless the policyholder has selected the offset option. With a premium offset, apply the dividend to the premium due at the stated date. If the dividend is less than the premium, the policyholder pays the difference. If it exceeds the premium, specify how the excess is handled; in the examples here, the excess is paid to the policyholder in cash.

Worked Example: Cash Dividends and Premium Offsets

A hypothetical policy has a fixed-dollar scale of $180, $620, and $240 at the ends of policy years 1, 2, and 3. A level premium of $500 is due at each of those same anniversaries, immediately after the dividend is declared. Under the offset option, apply the dividend first and pay any excess in cash. Project both the cash receipts under the cash option and the premium payments or excess receipts under the offset option.

Method. Under the cash option, record the full premium and the dividend as separate cash flows. Under the offset option, compare the dividend with the premium at each anniversary.

Year 1. The cash option gives a $180 dividend receipt and requires a $500 premium payment. With an offset, the remaining premium is $500-180=\$320$.

Year 2. The cash option gives a $620 dividend receipt and requires a $500 premium payment. With an offset, the dividend exceeds the premium by $620-500=\$120$, so the premium is fully covered and $120 is paid in cash.

Year 3. The cash option gives a $240 dividend receipt and requires a $500 premium payment. With an offset, the remaining premium is $500-240=\$260$.

Thus the offset-option payments by the policyholder are $320 and $260 in years 1 and 3, and the policyholder receives $120 in year 2. The common mistake is to treat the year-2 dividend as a negative $120 premium payment without saying where the excess goes. Under the stated assumption, it is a cash receipt. The check is that each dividend is fully accounted for: year 1, $180+320=500$; year 2, $620=500+120$; year 3, $240+260=500$.

Accumulating Dividends on Deposit

Under an accumulation option, each dividend is credited to a balance that earns interest. Let $A_t$ be the balance immediately after the dividend at the end of policy year $t$ has been credited. If the effective annual accumulation rate is $j$, the dividend is credited at each year-end, and interest is credited annually, then

A_t=A_{t-1}(1+j)+D_t,\qquad A_0=0.

The opening balance earns a year of interest before the next year-end dividend is added. Consequently, a dividend credited at the end of year $t$ earns interest only in later years, not during the year that has just ended.

Derivation: At the start of policy year $t$, the accumulated balance is $A_{t-1}$. During that year it grows to $A_{t-1}(1+j)$. At the end of the year, the newly declared dividend $D_t$ is credited, giving $A_t=A_{t-1}(1+j)+D_t$. Repeated substitution gives $A_n=\sum_{t=1}^{n}D_t(1+j)^{n-t}$, because the year-$t$ deposit earns interest for exactly $n-t$ full years before the end of year $n$.

Worked Example: Accumulating Three Year-End Dividends

A hypothetical policy pays fixed dividends of $100, $150, and $200 at the ends of years 1, 2, and 3. Each credited balance earns an effective annual interest rate of 4%, with interest applied annually. Find the balance immediately after the year-3 dividend is credited.

Method. Use the accumulation recurrence, or accumulate each deposit to the end of year 3. The year-1 dividend earns two full years of interest, the year-2 dividend earns one, and the year-3 dividend earns none.

Starting from $A_0=0$, the year-1 balance is $A_1=0(1.04)+100=\$100$. The year-2 balance is $A_2=100(1.04)+150=\$254$. The year-3 balance is $A_3=254(1.04)+200=\$464.16$.

Equivalently, $100(1.04)^2+150(1.04)+200=108.16+156+200=\$464.16$. The answer is $464.16. A common timing error is to give every dividend one extra year of interest, producing $100(1.04)^3+150(1.04)^2+200(1.04)=\$482.73$ to the nearest cent. That would be appropriate only if the deposits were made one year earlier than stated. The recurrence confirms that the end-of-year-3 deposit receives no interest before the end-of-year-3 valuation point.

Using Dividends to Buy Paid-Up Additions

A paid-up addition is an increment of insurance benefit that requires no future premium. To project additions, the problem must provide a purchase basis, such as the cost $c_t$ at the dividend date for each $1 of additional benefit. If the full dividend is used to buy additions, the amount of additional benefit is $D_t/c_t$. This calculation assumes $c_t>0$, that fractional amounts of benefit may be purchased, and that the dividend is entirely applied to the purchase.

When the scale is a rate on the benefit base, the base must be updated after each purchase if later dividends use that increased base. Let $B_0$ be the initial eligible benefit and $B_t$ the benefit base after the year-$t$ addition. Under the stated purchase assumptions,

D_t=b_tB_{t-1},\qquad \text{addition}_t=\frac{D_t}{c_t},\qquad B_t=B_{t-1}+\frac{D_t}{c_t}.

The addition increases the base because it adds paid-up benefit to the policy. The recurrence follows by first calculating the dividend from the base that applies before the year-$t$ purchase, then converting that dividend to benefit, and finally adding the new benefit to the base. Reversing those steps would use a benefit amount that has not yet been purchased to calculate the current dividend.

Worked Example: Projecting Additions Under a Benefit-Based Scale

A hypothetical policy starts with an eligible benefit base of $20,000. Its annual dividend rates are 0.60% in years 1 and 2, and 0.50% in year 3. Each full dividend buys paid-up benefit at a cost of $0.12 per $1 of addition in year 1 and $0.15 per $1 in years 2 and 3. The dividend is calculated on the base at the start of the year, before that year’s addition. Find each dividend and addition, and the benefit base after year 3.

Method. For each year, calculate the dividend from the opening base, divide by the purchase cost per dollar of benefit, and add the resulting benefit to the base. No rounding is needed in these calculations.

Year 1. The dividend is $0.006(20{,}000)=\$120$. The addition is $120/0.12=\$1{,}000$, so $B_1=20{,}000+1{,}000=\$21{,}000$.

Year 2. The dividend is $0.006(21{,}000)=\$126$. The addition is $126/0.15=\$840$, so $B_2=21{,}000+840=\$21{,}840$.

Year 3. The dividend is $0.005(21{,}840)=\$109.20$. The addition is $109.20/0.15=\$728$, so $B_3=21{,}840+728=\$22{,}568$.

The projected dividends are $120, $126, and $109.20; the additions are $1,000, $840, and $728; and the year-3 benefit base is $22,568. The check is that the additions total $1{,}000+840+728=\$2{,}568$, and $20{,}000+2{,}568=\$22{,}568$. A common mistake is to apply the year-2 rate to the year-2 closing base, which would include the year-2 addition before it is purchased. The stated timing requires the opening base of $21,000 instead.

Keep the Scale, Option, and Timing Distinct

A reliable projection keeps three decisions separate: how the dividend is determined, what the policyholder does with it, and when the relevant cash flow or benefit change occurs. This order matters because the option generally changes the use of a dividend, not the scale’s declared amount. A benefit-based scale is an important exception to the idea that future dividends are unaffected: additions can change the later eligible base when the scale explicitly uses that base.

Determine the dividend.
Use the fixed dollar amount or apply the specified rate to the correct eligible base. Confirm whether the base is measured before or after the current year’s transaction.

Apply the selected option.
Record a cash receipt, reduce the premium due, add the dividend to the deposit balance, or convert it to paid-up benefit using the stated purchase basis.

Update the next year.
Carry forward the accumulation balance or benefit base as appropriate. Do not alter a fixed-dollar dividend schedule unless the problem says the scale depends on the changed base.

These projections are conditional on the policy being in force and eligible for the stated dividend at each declaration date. If death, lapse, or another event occurs first, later projected dividends may not be payable; the event’s effect must be specified before continuing the projection. Similarly, an accumulation rate or paid-up addition purchase basis must not be inferred from the dividend scale. They are separate assumptions.

The procedure is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In an exam calculation, state the scale basis, the dividend date, the option, and any interest or purchase assumptions before computing. Doing so makes the timing and benefit-base choices visible and helps prevent double counting.

Check Your Understanding

For each question, use the stated timing and assumptions. Amounts are hypothetical; round money to the nearest cent only if needed.

A fixed scale pays $75, $105, and $130 at the ends of years 1, 2, and 3. Dividends accumulate at an effective annual rate of 3%, with each dividend credited at year-end. Find the balance just after the year-3 credit. (Answer: $317.72)
A $400 premium is due at each anniversary immediately after the dividend. Fixed dividends are $350, $430, and $280. Under the premium-offset option, any excess is paid in cash. State the policyholder’s premium payment or excess receipt for each year. (Answer: Year 1 pays $50; year 2 receives $30; year 3 pays $120.)
A policy has an opening eligible base of $15,000. The year-1 dividend rate is 0.80% and the purchase cost is $0.10 per $1 of addition. The year-2 rate is 0.75% and the purchase cost is $0.135 per $1. Each dividend uses the opening base and buys additions in full. Find both dividends and the base after year 2. (Answer: Dividends $120 and $121.50; additions $1,200 and $900; year-2 base $17,100.)
Fixed dividends of $90 and $110 are credited at the ends of years 1 and 2. The annual effective accumulation rate is 5%. Find the balance immediately after the year-2 credit, and state how many years the first dividend earns interest. (Answer: $204.50; the first dividend earns interest for one year.)
A policy starts with a $12,000 eligible base. Its dividend rates are 1.00%, 0.80%, and 0.75% in years 1–3; purchase costs are $0.12, $0.13, and $0.15 per $1 of addition. Each dividend is based on the opening base and is used entirely to buy additions. Find the three additions and the year-3 closing base. (Answer: Additions $1,000, $800, and $690; year-3 closing base $14,490.)

What You'll Learn