Worked Example Of A Cash Dividend Projection

Projecting Cash Dividends and Premium Offsets

A cash dividend can be paid to a policyholder, used to reduce a premium, or retained under an option that accumulates a balance. A projection must therefore track more than the declared dividend: it must also identify when the premium is due, how much of the available dividend value is applied, and what happens to any amount left over.

The calculations below use hypothetical policies and stated dividend amounts. They assume the policy remains in force through every projection date. A future dividend scale is not itself a guarantee: the projection treats each dividend shown as the amount assumed or declared for that year. It does not derive the dividend from the insurer's experience or surplus.

Definition: Under a premium-offset option, an available cash dividend value is applied toward a premium due. Any amount by which the premium exceeds the available value is paid by the policyholder. Any excess available value, if the option permits it, remains in a dividend balance. In this tutorial, that balance earns the stated effective annual interest rate.

Timing and Projection Equations

Use a consistent sequence at each policy anniversary. Let $B_{t-1}$ be the dividend balance immediately after the previous anniversary's premium offset, $j$ the effective annual interest rate credited to the balance, $D_t$ the cash dividend credited at anniversary $t$, and $P_t$ the premium due then. The convention here is: first credit interest on the opening balance for the elapsed policy year, then add the current dividend, and then apply value to the premium.

A_t=(1+j)B_{t-1}+D_t

Here $A_t$ is the total value available for the premium offset. The amount applied to the premium, the policyholder's out-of-pocket payment, and the closing balance are:

O_t=\min(P_t,A_t),\qquad C_t=\max(0,P_t-A_t),\qquad B_t=\max(0,A_t-P_t).

The option in these examples makes the full accumulated balance available, not just the current year's dividend. The equations also assume that no balance can be negative. If the contract instead pays a dividend in cash, restricts which balance amounts may offset premiums, or uses another crediting convention, the projection must follow those terms.

Timing convention: Interest is credited only on the balance carried into the year. The dividend credited at the current anniversary is not treated as if it had been in the account throughout the elapsed year. Premium offset occurs after both the interest credit and the current dividend credit.

Why the Offset Equations Work

The available value $A_t$ must be allocated between the premium offset and the closing balance. If $A_t$ is at least $P_t$, the entire premium can be covered, so $O_t=P_t$, $C_t=0$, and the remainder is $B_t=A_t-P_t$. If $A_t<P_t$, all available value is used, so $O_t=A_t$, $C_t=P_t-A_t$, and $B_t=0$. These two cases give the minimum and maximum expressions above.

In either case, the allocation conserves value:

A_t=O_t+B_t,\qquad P_t=O_t+C_t.

The first equality says that available dividend value is either used for the premium or carried forward. The second says that the premium is met by the offset plus any amount paid directly by the policyholder. These are useful checks on every annual row of a projection.

Worked Example: Four-Year Projection With a Carry-Forward Balance

A hypothetical policy has an annual premium of $500. At four successive anniversaries, its declared cash dividends are $300, $700, $400, and $650. The dividend balance earns an effective annual interest rate of 4%. The opening balance at the first anniversary is zero. Under the premium-offset option, calculate each year's offset, policyholder payment, and closing balance.

Method. For each year, calculate interest on the opening balance, add the current dividend, and then apply the available value to the $500 premium. This order avoids crediting interest prematurely on a new dividend.

Year 1. The opening balance is $B_0=0$, so interest is $0(0.04)=\$0$. Available value is $A_1=\$0+\$300=\$300$. The offset is $\min(\$500,\$300)=\$300$, the policyholder pays $\max(0,\$500-\$300)=\$200$, and the closing balance is $\max(0,\$300-\$500)=\$0$.

Year 2. Interest on the opening balance is $0(0.04)=\$0$. Available value is $A_2=\$0+\$700=\$700$. The offset is $500, the policyholder pays $0, and the closing balance is $\$700-\$500=\$200$.

Year 3. Interest is $\$200(0.04)=\$8$, giving an interest-credited opening balance of $\$200+\$8=\$208$. Available value is $A_3=\$208+\$400=\$608$. The offset is $500, the policyholder pays $0, and the closing balance is $\$608-\$500=\$108$.

Year 4. Interest is $\$108(0.04)=\$4.32$, giving $\$108+\$4.32=\$112.32$ before the new dividend. Available value is $A_4=\$112.32+\$650=\$762.32$. The offset is $500, the policyholder pays $0, and the closing balance is $\$762.32-\$500=\$262.32$.

Check. In year 4, $\$762.32=\$500+\$262.32$, so available value equals the offset plus the closing balance. The only out-of-pocket payment is $200 in year 1. All amounts are exact to the cent; no intermediate rounding was needed.

Reading a Projection Carefully

A dividend projection has two distinct kinds of input. The dividend schedule $D_t$ and premium schedule $P_t$ determine the cash flows at each anniversary; the interest rate $j$ determines how an existing balance grows between anniversaries. Keeping these inputs separate prevents a common conceptual error: treating the interest earned on retained dividends as another declared dividend. In the equations, interest increases the balance, while $D_t$ is the dividend credited for the current year.

A useful annual record includes the opening balance, interest credited, current dividend, total available value, premium, offset, policyholder payment, and closing balance. Each line should reconcile. If available value is below the premium, the balance is zero and the policyholder pays the shortfall. If available value is above the premium, the premium is fully offset and the excess remains in the account.

Worked Example: Starting With an Existing Dividend Balance

A hypothetical policy begins the first projected year with a dividend balance of $240. The balance earns 5% effective annual interest. The premium due at each of the next three anniversaries is $900, $900, and $400, respectively; the cash dividends credited at those anniversaries are $500, $1,100, and $250. Calculate the offset, any direct policyholder payment, and the closing balance for each year.

Year 1. The opening balance earns $\$240(0.05)=\$12$, so it becomes $\$252$. Add the $500 dividend to obtain $A_1=\$752$. The premium offset is $\min(\$900,\$752)=\$752$; the policyholder pays $\$900-\$752=\$148$; and the closing balance is $0.

Year 2. The opening balance is zero, so interest is $0. Available value is $A_2=\$0+\$1{,}100=\$1{,}100$. The $900 premium is fully offset, the policyholder pays $0, and the closing balance is $\$1{,}100-\$900=\$200$.

Year 3. Interest on the $200 opening balance is $\$200(0.05)=\$10$, so the interest-credited balance is $210. After adding the $250 dividend, available value is $A_3=\$210+\$250=\$460$. The $400 premium is fully offset, the policyholder pays $0, and the closing balance is $\$460-\$400=\$60$.

Check. In year 1, the available $752 plus the direct payment of $148 meets the $900 premium. In year 3, the $460 available is divided into a $400 offset and a $60 balance. The projection is consistent with the stated anniversary timing.

When Dividends Are Expressed as a Percentage

A dividend schedule may be stated as a percentage of a specified base rather than as dollar amounts. The base must be identified before the projection begins. In the next example, the stated percentage is applied to the annual premium, and each resulting dollar amount is treated as that anniversary's dividend. It is not applied to the dividend balance.

Worked Example: Converting a Dividend Scale to Dollar Credits

A hypothetical policy has an annual premium of $500. Its assumed cash-dividend rates, each expressed as a percentage of that premium, are 40%, 80%, 120%, and 60% for policy years 1 through 4. The balance earns 3% effective annual interest and starts at zero. Calculate the dollar dividend, premium offset, direct payment, and closing balance in each year.

Convert the scale. The four dividends are $0.40(\$500)=\$200$, $0.80(\$500)=\$400$, $1.20(\$500)=\$600$, and $0.60(\$500)=\$300$.

Year 1. Available value is $200. The offset is $200, the policyholder pays $\$500-\$200=\$300$, and the closing balance is $0.

Year 2. The opening balance and its interest are both zero. Available value is $400. The offset is $400, the policyholder pays $100, and the closing balance is $0.

Year 3. Available value is $600. The offset is $500, the policyholder pays $0, and the closing balance is $\$600-\$500=\$100$.

Year 4. Interest on the $100 opening balance is $\$100(0.03)=\$3$, making $103 before the dividend. Available value is $\$103+\$300=\$403$. The offset is $403, the policyholder pays $\$500-\$403=\$97$, and the closing balance is $0.

Check. In year 4, $\$403+\$97=\$500$, the premium due. The year 3 excess of $100 earns $3 before the year 4 dividend is credited; it does not change the stated 60% dividend rate.

Common Timing Trap

A frequent error is to apply the annual interest rate to the sum of the opening balance and the current dividend. That calculation assumes the current dividend was in the balance for the entire policy year, contrary to the timing convention used here. For example, in year 4 of the first example, the incorrect calculation would be $(\$108+\$650)(1.04)=\$788.32$ before premium offset. The correct available value is $\$108(1.04)+\$650=\$762.32$. The incorrect method overstates available value by $\$788.32-\$762.32=\$26.00$, which is $0.04$ times the $650 current dividend.

The correct timing depends on the contract and the projection specification. If a dividend were credited before an entire interest-earning period, interest might apply to it for that period. Do not infer such timing silently: identify the credit date, interest period, and premium due date, then use the stated sequence consistently.

Key takeaway: At each anniversary, reconcile the dividend account before moving to the next year: grow the opening balance for the elapsed period, add the current declared or assumed dividend, apply no more than the premium due, and carry forward only the excess. Treat the dividend schedule and balance interest as separate components.

Check Your Understanding

Use the anniversary order and premium-offset equations stated above. Assume each hypothetical policy remains in force for every year described.

A policy has a $700 premium, a $500 dividend at the first anniversary, a zero opening balance, and a 4% effective annual balance rate. Find the first-year offset, direct payment, and closing balance. (Answer: offset $500, direct payment $200, closing balance $0.)
A policy opens the year with a $100 balance, earns 5% effective annual interest, receives a $350 dividend, and has a $400 premium due. Find the available value and closing balance. (Answer: available value $455; offset $400; closing balance $55; direct payment $0.)
A policy has a $600 annual premium, a zero opening balance, 4% effective annual interest, and dividends of $800, $250, and $900 in three successive years. Find the direct payment and closing balance in each year. (Answer: year 1 payment $0 and balance $200; year 2 payment $142 and balance $0; year 3 payment $0 and balance $300.)
A $1,200 annual premium has assumed dividend rates of 35%, 80%, and 110% of premium in three successive years. The balance starts at zero. Find the dollar dividend in each year and the closing balance after year 3, assuming no interest is credited. (Answer: dividends $420, $960, and $1,320; closing balance after year 3 is $120.)
A policy opens with a $150 balance, earns 4% effective annual interest, receives a $300 dividend at the anniversary, and has a $400 premium due. Calculate the correct available value and balance. Then identify the amount by which available value would be overstated if interest were incorrectly applied to both the opening balance and current dividend. (Answer: correct available value $456 and closing balance $56; incorrect method overstates available value by $12.)

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