Why an Endowment Profit Vector Needs Careful Timing
An endowment policy can produce different cash flows depending on whether the insured dies during the term or survives to maturity. In each policy year, a profit calculation must also account for the premium and expenses, investment earnings, and any reserve required for policies that remain in force. The key is to state when each item occurs and which policies it applies to.
This tutorial uses an annual, end-of-year profit convention. Each annual entry is calculated per policy in force at the start of that policy year. It is an expected amount under the stated mortality probabilities, not the profit on a particular individual policy. The reserve values are assumed to be supplied on a consistent valuation basis.
Model and Annual Profit Formulas
For policy year \(t\), let \(V_{t-1}\) be the reserve at the start of the year, before the year's premium and expense are paid. Let \(P_t\) be the premium and \(E_t\) the expense, both paid at the start of the year. The annual effective investment return is \(i\), assumed deterministic. Thus the amount available at the end of the year before end-of-year payments is \((V_{t-1}+P_t-E_t)(1+i)\).
Let \(q_t\) be the conditional probability of death during year \(t\), given that the policy is in force at its start, and let \(p_t=1-q_t\) be the corresponding probability of survival. A death benefit \(D_t\) is paid at the end of the year of death. In a nonfinal year, a closing reserve \(V_t\) is held at year-end for each survivor. In the final year \(n\), a maturity benefit \(M\) is paid at year-end if the insured survives; no reserve remains after the death or maturity payment.
The formula treats all policies in force at the start of the year as contributing their opening reserves and paying their premiums and expenses. It then subtracts the expected end-of-year outgo. In a nonfinal year, that outgo includes a death benefit for the fraction \(q_t\) expected to die and a reserve for the fraction \(p_t\) expected to survive. In the final year, the survival outgo is the maturity benefit instead of a continuing reserve.
Why the Formula Works
The profit entry is the expected amount left after accumulating the assets for the year and meeting the year-end obligations. The assets accumulated at year-end are known under the deterministic investment assumption:
For a nonfinal year, let the random year-end obligation be \(D_t\) if death occurs and \(V_t\) if survival occurs. These outcomes are mutually exclusive, with probabilities \(q_t\) and \(p_t\). Therefore, by the expectation rule,
Subtracting this expected obligation from the accumulated assets gives the nonfinal-year formula. In the final year, replace the survivor's closing reserve by the maturity benefit \(M\), so the expected obligation is \(q_nD_n+p_nM\). This derivation also explains why the reserve and benefit amounts must be weighted by the probability of the event that gives rise to them.
Worked Endowment Calculations
Worked Example: A Two-Year Endowment
A hypothetical two-year endowment has a 5% effective annual investment return. Premiums of $1,000 are paid at the start of each year. Expenses are $100 in year 1 and $50 in year 2, also paid at the start of the year. The death benefit is $1,500 in either year, and the maturity benefit is $2,000. The conditional death probabilities are \(q_1=0.01\) and \(q_2=0.02\). The reserves are \(V_0=0\) and \(V_1=\$600\); after the final benefit is paid, no reserve remains. Calculate the profit vector per policy in force at the start of each year.
Method. Accumulate the opening reserve plus premium less expense for each year. Then subtract the probability-weighted death benefit and either the survivor reserve or the survivor maturity benefit.
Year 1. The survival probability is \(p_1=1-0.01=0.99\). The accumulated amount before year-end outgo is
Expected death benefits are \(0.01(1{,}500)=15\). The closing reserve required for survivors is \(0.99(600)=594\). Hence
Year 2, the maturity year. The survival probability is \(p_2=1-0.02=0.98\). The accumulated amount before year-end outgo is
Expected death benefits are \(0.02(1{,}500)=30\), and expected maturity benefits are \(0.98(2{,}000)=1{,}960\). Therefore
The profit vector is \((\$336.00,-\$362.50)\). As a check, the final year's expected year-end outgo is \(30+1{,}960=\$1{,}990\), which exceeds its accumulated amount by $362.50. A negative final-year entry is possible: it means the year's assets are insufficient to meet expected final-year benefits, considered on this supplied reserve and cash-flow basis.
Worked Example: Building a Three-Year Profit Vector
A hypothetical three-year endowment earns 4% effective annually. A premium of $800 is paid at the start of each year. Expenses are $80 in year 1 and $40 in each of years 2 and 3. The death benefit is $1,000 in every year, and the maturity benefit is $2,500. Conditional death probabilities are \(q_1=0.01\), \(q_2=0.02\), and \(q_3=0.03\). The reserves are \(V_0=0\), \(V_1=\$500\), and \(V_2=\$700\); there is no reserve after maturity. Find all three annual entries.
Year 1. Here \(p_1=0.99\). The accumulated amount is \((0+800-80)(1.04)=720(1.04)=748.80\). Expected death benefits are \(0.01(1{,}000)=10\), and the survivor reserve is \(0.99(500)=495\). Thus
Year 2. Here \(p_2=0.98\). The accumulated amount is \((500+800-40)(1.04)=1{,}260(1.04)=1{,}310.40\). Expected death benefits are \(0.02(1{,}000)=20\), and the survivor reserve is \(0.98(700)=686\). Therefore
Year 3, the maturity year. Here \(p_3=0.97\). The accumulated amount is \((700+800-40)(1.04)=1{,}460(1.04)=1{,}518.40\). Expected death benefits are \(0.03(1{,}000)=30\), while expected maturity benefits are \(0.97(2{,}500)=2{,}425\). Hence
The resulting vector is \((\$243.80,\$604.40,-\$936.60)\). The calculation uses each reserve only for survivors in a nonfinal year. In the final year it uses the maturity benefit for survivors instead; it does not carry forward the year-2 reserve as a separate final-year outgo.
Worked Example: Solving for a Closing Reserve
A hypothetical nonfinal endowment year has a 5% effective annual investment return, opening reserve \(V_0=\$400\), premium \(P=\$600\), and beginning-of-year expense \(E=\$40\). The conditional death probability is \(q=0.02\), and the death benefit is $1,000. The insurer wants the annual profit entry to be $100 per policy in force at the start of the year. Find the required closing reserve \(V_1\) for survivors.
Step 1: Accumulate the available amount.
Step 2: Set the profit equal to the target. The expected death benefit is \(0.02(1{,}000)=20\), and the survival probability is \(p=0.98\). The nonfinal-year equation is
Rearranging gives \(0.98V_1=888\), so
Rounded to the nearest cent, the required reserve is $906.12. Checking before rounding, \(1{,}008-20-0.98(906.122448\ldots)=100\), as required.
Common mistake. Omitting survival weighting would give \(V_1=888\), from incorrectly setting \(100=1{,}008-20-V_1\). That treats the closing reserve as payable for every policy in force at the start, including policies that have died. Under the stated model it is held only for survivors, so the correct reserve term is \(0.98V_1\).
Common Traps and Scope of the Calculation
The most reliable way to avoid timing and event errors is to classify each item before calculating. The table summarizes the convention used here.
| Item | Timing or event | Amount in the annual entry |
|---|---|---|
| Opening reserve | At the start of the year | Included in assets accumulated over the year |
| Premium and expense | At the start of the year | Added or subtracted before investment accumulation |
| Death benefit | Paid at year-end if death occurs during the year | Expected amount \(q_tD_t\) |
| Closing reserve | Held at year-end for survivors in a nonfinal year | Expected amount \(p_tV_t\) |
| Maturity benefit | Paid at year-end if the insured survives the final year | Expected amount \(p_nM\) |
A frequent error is to apply the maturity benefit to everyone in force at the start of the final year. Under this model, only survivors receive it, so its expected amount is \(p_nM\). Another error is to add investment income on the opening reserve separately after using the accumulated-asset term: the factor \((1+i)\) already includes that investment income. Adding it again double counts the return.
The calculation also depends on its stated assumptions. If premiums stop before the final year, expenses occur at year-end, benefits are paid immediately on death, or investment returns are stochastic, the formula must reflect those features. Likewise, if there are lapses or other termination events, their cash flows and probabilities must be included rather than silently treated as survival. The entries here are not yet weighted by the probability that a policy remains in force from issue to a given duration; they are conditional annual entries per policy in force at the start of each year.
Check Your Understanding
Use the same timing convention as in the tutorial. Round money to the nearest cent where needed.
- A nonfinal year has \(i=4\%\), \(V_0=\$300\), \(P=\$500\), \(E=\$20\), \(q=0.02\), \(D=\$1{,}000\), and \(V_1=\$600\). Find the annual profit entry. (Answer: $203.20)
- In a final year, \(i=3\%\), \(V_{n-1}=\$900\), \(P_n=\$700\), \(E_n=\$50\), \(q_n=0.04\), \(D_n=\$1{,}500\), and \(M=\$2{,}200\). Find the profit entry. (Answer: -$575.50)
- A nonfinal year has \(i=5\%\), \(V_0=\$200\), \(P=\$900\), \(E=\$100\), \(q=0.01\), \(D=\$1{,}200\), and \(V_1=\$700\). Calculate the profit and identify the probability applied to the closing reserve. (Answer: $345.00; survival probability \(0.99\))
- A nonfinal year has \(i=4\%\), \(V_0=\$500\), \(E=\$30\), \(q=0.02\), \(D=\$1{,}000\), and \(V_1=\$800\). What start-of-year premium produces a profit of $100? (Answer: $399.23)
- In a final year, \(i=6\%\), \(V_{n-1}=\$400\), \(P_n=\$500\), \(E_n=\$25\), \(q_n=0.02\), \(D_n=\$1{,}000\), and \(M=\$1{,}800\). Find the profit entry, explicitly using the survival probability for maturity. (Answer: -$856.50)