Single-State Prospective Reserve Calculation

Purpose and Valuation Convention

A prospective reserve measures the value, at a specified time, of the contract’s future obligations net of its future premiums. For a single-state contract, the policy is either in force or no longer in force; here it remains in force while the insured is alive and terminates on death or at its scheduled expiry. The reserve is calculated conditional on the policy being in force at the valuation time.

Cash-flow timing must be settled before the calculation. In this tutorial, premiums and ordinary expenses are paid at the beginning of a policy year if the policy is then in force. A death benefit is paid at the end of the policy year in which death occurs. A maturity benefit is paid at expiry, but only if the insured survives to that time. The reserve at time $k$ is measured immediately before any cash flow scheduled at time $k$. Thus, a premium due at time $k$ is included as a future premium.

Definition: A single-state prospective reserve at time $k$ is the conditional expected present value of future benefits and expenses minus the conditional expected present value of future premiums, given that the contract is in force at time $k$. A positive reserve represents a net future obligation on this valuation basis.

The calculation uses a stated interest basis and mortality basis. It is not automatically the same as an asset share, a profit-test reserve, or the amount of assets actually held for a policy. The focus here is the prospective valuation of specified future contract cash flows.

Formula for an Annual Contract

Consider a contract with scheduled expiry at time $N$. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let $P_j$ be the premium payable at time $j$, and let $e_j$ be an expense payable at that same time while the policy is in force. Let $B_j$ be the death benefit payable at time $j+1$ if death occurs during year $j$. Let $M_N$ be the maturity benefit payable at time $N$ if the insured survives to expiry.

Write ${}_tp_x$ for the probability that a life aged $x$ survives $t$ years, and ${}_tq_x=1-{}_tp_x$. The probability of surviving from time $k$ to the start of year $j$, conditional on being in force at $k$, is ${}_{j-k}p_{x+k}$. The probability of death during year $j$, on the same condition, is ${}_{j-k}p_{x+k}q_{x+j}$, where $q_{x+j}=1-p_{x+j}$.

{}_kV_x = \sum_{j=k}^{N-1} v^{j-k} {}_{j-k}p_{x+k}(e_j-P_j) + \sum_{j=k}^{N-1} v^{j+1-k} {}_{j-k}p_{x+k}q_{x+j}B_j + v^{N-k} {}_{N-k}p_{x+k}M_N

The first sum values beginning-of-year expenses and premiums: the expense is an outgo and the premium is an inflow, so their net contribution is $e_j-P_j$. The second sum values death benefits at year-end. The final term values a maturity benefit payable only on survival to expiry. If there is no maturity benefit, set $M_N=0$; if an expense is payable on death or at maturity, include its expected present value as an additional outgo at the appropriate time.

This expression assumes annual cash-flow times, deterministic interest at the stated effective rate, and the stated mortality probabilities. It also assumes that the listed premiums and expenses are payable only when the policy is in force at their scheduled times. If a contract uses different payment timing, the discount factors and event probabilities must be changed to match it.

Why the Prospective Formula Works

The formula follows by partitioning future cash flows according to when they occur and which event makes them payable. Conditional on being in force at time $k$, a beginning-of-year payment at time $j$ is payable only if the insured survives from $k$ to $j$. Its probability of payment is therefore ${}_{j-k}p_{x+k}$. Its value at time $k$ is its amount multiplied by that probability and by $v^{j-k}$.

A death benefit for year $j$ requires survival to the start of that year followed by death during it. Those events have conditional probability ${}_{j-k}p_{x+k}q_{x+j}$. Because payment is at time $j+1$, the benefit is discounted by $v^{j+1-k}$, not $v^{j-k}$. A maturity benefit requires survival all the way to time $N$, so its probability is ${}_{N-k}p_{x+k}$ and it is discounted by $v^{N-k}$. Adding the expected present values of all outgoes and subtracting those of all premiums gives the prospective reserve.

List future cash flows.
Separate premiums, ordinary expenses, death benefits, and any maturity benefit; record each amount and payment time.

Identify the event that triggers each flow.
For example, a start-of-year premium requires survival to that year’s start, while a death benefit requires survival to the year followed by death within it.

Condition on the valuation state.
Use probabilities from the valuation time onward, given that the policy is in force at that time.

Discount to the valuation time and apply the sign.
Include benefits and expenses as outgoes; subtract premiums as inflows.

Worked Calculations

Worked Example: A One-Year Term Reserve

A hypothetical one-year term policy is valued at time 0, immediately before its premium is due. The annual effective interest rate is $i=5\%$. The insured’s probability of surviving the year is $p_x=0.90$, so $q_x=0.10$. The premium is \$70 and a beginning-of-year expense is \$5, both payable only while the policy is in force. A \$1,000 death benefit is paid at the end of the year if death occurs during the year. There is no maturity benefit. Calculate the reserve.

Method. Value the beginning-of-year premium and expense at time 0, then add the expected discounted death benefit. This avoids discounting a cash flow that occurs at the valuation time.

The beginning-of-year net amount is an outgo of \$5 less a premium of \$70:

$$ e_0-P_0=5-70=-65 $$

The death benefit is payable with probability $q_x=0.10$, one year from valuation. Since $v=1/1.05$, its expected present value is:

v q_x B_0 = \frac{1}{1.05}(0.10)(1000) = \$95.238095\ldots

Therefore:

{}_0V_x=-65+95.238095\ldots=\$30.238095\ldots

Rounded to the nearest cent, the reserve is \$30.24. As a check, the expected present value of outgoes is \$5 plus \$95.238095, or \$100.238095; subtracting the \$70 premium gives \$30.238095, the same result.

Worked Example: A Two-Year Endowment Reserve

A hypothetical two-year endowment policy is valued at time 1, immediately before the premium due at that time. The annual effective interest rate is $i=25\%$, so $v=0.80$. The insured’s survival probability is $0.80$ in each year, independently of the other year; thus the annual death probability is $0.20$. A premium of \$100 and an expense of \$10 are payable at the beginning of each remaining year while the policy is in force. A \$500 death benefit is paid at the end of either policy year in which death occurs. A \$500 maturity benefit is paid at time 3 if the insured survives to expiry. Calculate the reserve at time 1.

Method. Condition on being in force at time 1. Value the two remaining years separately, then value the maturity benefit. This avoids including any survival probability for years before the valuation date.

In the first remaining year, at time 1, the net beginning-of-year cash flow is $10-100=-\$90$. The death benefit at time 2 has expected present value $v(0.20)(\$500)=\$80$. The first year’s contribution is therefore $-\$90+\$80=-\$10$.

For the second remaining year, the insured must survive the first remaining year, with probability $0.80$, for the time-2 premium and expense to be payable. Their expected present value at time 1 is:

v(0.80)(10-100)=0.80(0.80)(-90)=-\$57.60

The second-year death benefit requires survival to time 2 and death during the following year. Its probability is $0.80(0.20)=0.16$, and it is paid at time 3:

v^2(0.80)(0.20)(500) = (0.80)^2(0.16)(500) = \$51.20

The maturity benefit is payable with probability $0.80^2=0.64$ at time 3. Its expected present value is:

v^2(0.64)(500)=(0.80)^2(0.64)(500)=\$204.80

Adding all contributions gives:

{}_1V_{x+1}=-10-57.60+51.20+204.80=\$188.40

The result is \$188.40. As a check, the two death-benefit values total \$131.20 and the maturity value is \$204.80; after subtracting the net future premium contributions, the reserve is positive \$188.40.

Worked Example: Nonlevel Cash Flows and Conditional Survival

A hypothetical contract expires at time 4 and is valued at time 2, just before the premium due then. The annual effective interest rate is $i=4\%$, so $v=1/1.04$. Conditional on being in force at time 2, the probability of surviving to time 3 is $0.90$; conditional on being in force at time 3, the probability of surviving to time 4 is $0.80$. Premiums at times 2 and 3 are \$150 and \$120, respectively. Expenses at those times are \$12 and \$18. The death benefits for deaths in years 2 and 3 are \$600 and \$800, payable at times 3 and 4, respectively. A \$1,000 maturity benefit is payable at time 4 if the insured survives. Calculate the reserve at time 2.

Method. Use the conditional probabilities from time 2 and separate the two death-benefit events from survival to maturity. This avoids multiplying by the probability of surviving before time 2, which is already incorporated by the condition that the policy is in force then.

At time 2, the net beginning-of-year amount is $12-150=-\$138$. The death benefit for the next year is payable with probability $0.10$, at time 3, so its expected present value is:

v(0.10)(600)=\frac{60}{1.04}=\$57.692308

The probability of surviving from time 2 to time 3 is $0.90$. Thus the expected present value of the time-3 net beginning-of-year amount is:

v(0.90)(18-120) = \frac{0.90(-102)}{1.04} = -\$88.269231

Death in the second remaining year requires survival to time 3 and death before time 4. Its probability is $0.90(0.20)=0.18$. The benefit is paid at time 4, two years after valuation:

v^2(0.18)(800) = \frac{144}{1.04^2} = \$133.136095

Survival to maturity has probability $0.90(0.80)=0.72$. The maturity benefit’s expected present value is:

v^2(0.72)(1000) = \frac{720}{1.04^2} = \$665.680473

The reserve is the sum of the four contributions:

{}_2V_{x+2} = -138+57.692308-88.269231+133.136095+665.680473 = \$630.239645

Rounded to the nearest cent, the reserve is \$630.24. The calculation is consistent with the stated timing: the time-2 premium and expense are undiscounted, the time-3 net amount and first death benefit are discounted one year, and the time-4 benefits are discounted two years.

Common Timing and Conditioning Traps

A frequent error is to apply the wrong probability or discount period to a cash flow. For the death benefit in the first year of the endowment example, an incorrect calculation might use $0.20(500)=\$100$ as its value at time 1. This is wrong because the benefit is paid at time 2, one year after valuation, not at time 1. The correct value is $v(0.20)(500)=\$80$. Similarly, in a reserve at time 2, it is wrong to multiply every future probability by the chance of surviving from issue to time 2: the reserve is already conditional on the policy being in force at time 2.

Check payment time. A beginning-of-year cash flow at time $j$ is discounted $j-k$ years from time $k$; a year-end death benefit at $j+1$ is discounted $j+1-k$ years.
Check the triggering event. A death benefit needs survival to the start of the year and death during that year; a maturity benefit needs survival to expiry.
Check signs. Expenses and benefits are outgoes; premiums reduce the reserve.
Check the condition. A reserve at time $k$ is conditional on being in force then, so use future conditional survival probabilities rather than unconditional probabilities from issue.

Key takeaway: A single-state prospective reserve is built cash flow by cash flow: identify its time, its payment condition, its conditional probability, its discount factor, and its sign. Consistent use of these five details is more reliable than trying to recall a formula without checking the contract timing.

Check Your Understanding

For each problem, use the stated annual timing conventions and round monetary answers to the nearest cent.

A one-year term policy is valued at time 0 before its premium. The effective annual interest rate is $5\%$; $p_x=0.90$. A \$80 premium and \$5 expense are due immediately, and a \$500 death benefit is paid at year-end on death. There is no maturity benefit. Calculate the reserve. (Answer: $-\$27.38$)
A two-year endowment is valued at time 0 before its premium. The effective annual interest rate is $10\%$; survival probability is $0.90$ in each year. A \$50 premium and \$5 expense are due at the start of each year while in force. A \$300 death benefit is paid at the end of the year of death, and a \$300 maturity benefit is paid at time 2 on survival. Calculate the reserve. (Answer: \$168.60)
A two-year contract is valued at time 2 before the premium due then. The effective annual interest rate is $5\%$. Conditional survival probabilities for years 2 and 3 are $0.95$ and $0.90$. Premiums at times 2 and 3 are \$40 and \$30; expenses are \$4 and \$6. Death benefits for those years are \$200 and \$300, payable at their respective year-ends. A \$500 maturity benefit is paid at time 4 on survival. Calculate the reserve. (Answer: \$365.41)
A two-year contract is valued at time 0 before the first premium. The effective annual interest rate is $0\%$, and survival probability is $0.90$ in each year. A \$200 premium is payable at the start of each year while in force; there are no expenses. A \$100 death benefit is payable at each year-end on death, and a \$1,000 maturity benefit is payable at time 2 on survival. Calculate the reserve. (Answer: \$449.00)

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