Reserves Between Policy Anniversaries

Why the Exact Valuation Time Matters

A prospective reserve at an anniversary is based on future benefits and premiums measured from that anniversary. Between anniversaries, the policy has already been in force for part of a policy year. The reserve must therefore account for the remaining part of that year, as well as the time until each future premium or benefit is paid. In particular, it is conditional on the insured being alive at the exact valuation time.

This is not just a matter of changing an age in an anniversary reserve formula. A death benefit may be payable at the end of the current policy year, less than one year from the valuation date. The next annual premium may be payable on that same anniversary, but only if the insured is then alive. Both the probabilities and the discount periods must match these payment times.

Definition: A prospective reserve at time $t$, conditional on the policy being in force and the insured being alive then, is the expected present value at time $t$ of future benefits minus the expected present value at time $t$ of future premiums. Future cash flows are valued using probabilities conditional on survival to $t$.

Suppose a policy was issued at age $x$, has annual policy anniversaries, and is valued at duration $t=k+s$, where $k$ is a whole number and $0<s<1$. The insured has reached age $x+k+s$, and the next anniversary is $1-s$ years away. Here, ${}_tp_x$ is the probability of survival for $t$ years from age $x$, and ${}_tq_x=1-{}_tp_x$. Fractional-duration probabilities require a stated mortality assumption; they should not be inferred from an annual mortality rate without one.

Cash-Flow Dates and Conditional Probabilities

For an annual-premium policy with premiums due at durations $0,1,\ldots,n-1$, and death benefits paid at the end of the policy year of death, valuation at $t=k+s$ has a clear timeline:

The next possible death benefit is paid at duration $k+1$, after $1-s$ years.
The next premium is due at duration $k+1$, if the insured survives to that anniversary and the premium is then payable under the contract.
Later death benefits are paid at subsequent policy-year ends. Later premiums are paid at subsequent anniversaries, subject to survival to each payment date.

For example, the probability of death before the next anniversary, given survival to $t=k+s$, is ${}_{1-s}q_{x+k+s}$. It is not generally the annual probability ${}_1q_{x+k}$. The probability of reaching the next anniversary is ${}_{1-s}p_{x+k+s}$. These probabilities are conditional on being alive at the valuation time.

The following general setup makes the timing explicit. Assume an $n$-year term insurance with benefit $B$ paid at the end of the policy year of death, and level annual premiums $P$ due at durations $0,\ldots,n-1$ while the insured is alive. Let $v=(1+i)^{-1}$, where $i$ is the effective annual interest rate. At $t=k+s$, where $0\leq k<n$ and $0<s<1$, the reserve is:

{}_tV = Bv^{1-s}{}_{1-s}q_{x+t} + B\sum_{r=1}^{n-k-1}v^{r+1-s}{}_{r-s}p_{x+t}\,{}_1q_{x+k+r} - P\sum_{r=1}^{n-k-1}v^{r-s}{}_{r-s}p_{x+t}.

The first benefit term covers death between $t$ and the next anniversary. In each term of the sum, $r$ counts the number of anniversaries after that next anniversary: the insured must first survive $r-s$ years from $t$, then die during the following policy year. Each premium term is discounted to its anniversary and weighted by the probability of being alive to pay it. If there are no future premium dates in the contract, the premium sum is empty. Other benefit timings or premium schedules require corresponding changes to the cash-flow times and probabilities.

Why Survival Conditioning Is Required

The probability of an event after time $t$, given survival to $t$, is obtained by dividing the joint probability of survival to $t$ and the event by the probability of survival to $t$. For instance, survival from age $x$ to $x+t$ followed by death in the next $1-s$ years has probability ${}_tp_x\,{}_{1-s}q_{x+t}$. Dividing by ${}_tp_x$ gives ${}_{1-s}q_{x+t}$, the appropriate probability for a reserve conditional on being alive at $t$.

This also explains why an annual death probability cannot simply be assigned to the remaining fraction of a year. Under the uniform distribution of deaths (UDD) assumption, deaths are spread uniformly across each policy year when measured from its start. If the annual death probability is $q$, then, for $0\leq s<1$, survival to fraction $s$ of the year has probability $1-sq$. Conditional on surviving to that point, the probability of death before the year ends is:

{}_{1-s}q_{x+k+s} = \frac{sq}{1-sq}, \qquad {}_{1-s}p_{x+k+s} = \frac{1-q}{1-sq}.

The denominator matters: the fraction $sq$ is the probability of death during the elapsed portion of the year measured from the start, not the conditional probability of death during the remaining portion for someone who has survived that far.

Locate the valuation date.
Write $t=k+s$, and identify the time $1-s$ remaining until the next policy anniversary.

List future cash-flow dates.
Write down when each possible benefit is paid and when each premium is due. Do not include cash flows that occurred before $t$.

Condition on survival to $t$.
Use probabilities from the attained age and exact duration, including fractional-year probabilities where needed.

Discount to $t$ and take the difference.
Discount each expected future benefit and premium for the time from $t$ to its payment date; subtract the premium value from the benefit value.

Worked Examples

Worked Example: A Two-Year Term Policy at Half-Year Duration

A hypothetical two-year term policy pays $B=\$10{,}000$ at the end of the policy year of death. Annual premiums are due at durations 0 and 1 while the insured is alive. The annual death probability is $q=0.20$ in each policy year, and UDD applies within each year. The effective annual interest rate is $i=23.46\%$, so use $v=0.81$. First find the level net premium $P$, then find the reserve at $t=0.5$.

Method. First apply the equivalence principle at issue: expected present value of premiums equals expected present value of benefits. Then revalue the remaining cash flows conditional on survival to duration 0.5.

At issue, the benefit is paid at duration 1 if death occurs in year 1, or duration 2 if the insured survives year 1 and dies in year 2. Thus:

\operatorname{EPV}(\text{benefits}) = 10{,}000\left(0.20(0.81)+(0.80)(0.20)(0.81^2)\right) = 10{,}000(0.162+0.104976) = \$2{,}669.76.

The premium at duration 0 is certain, and the premium at duration 1 is payable with probability $0.80$. Therefore:

\operatorname{EPV}(\text{premiums}) = P\left(1+(0.80)(0.81)\right) = 1.648P.

Equating the two present values gives $P=\$2{,}669.76/1.648=\$1{,}620.00$.

At duration 0.5, UDD gives survival to the anniversary of $1-(0.5)(0.20)=0.90$. Conditional on survival to 0.5, the probability of death before duration 1 is $(0.5)(0.20)/0.90=1/9$, or 0.1111. The first benefit is paid at duration 1, so its expected present value is:

\$10{,}000\left(\frac{1}{9}\right)(0.81^{0.5}) = \$10{,}000\left(\frac{1}{9}\right)(0.90) = \$1{,}000.00.

For death in policy year 2, the conditional probability is $(8/9)(0.20)=8/45$, or 0.1778. The payment at duration 2 is 1.5 years from the valuation date:

\$10{,}000\left(\frac{8}{45}\right)(0.81^{1.5}) = \$10{,}000(0.177777\ldots)(0.729) = \$1{,}296.00.

The premium at duration 1 is payable only if the insured survives to that date. Its present value is $\$1{,}620(8/9)(0.90)=\$1{,}296.00$. The reserve is therefore $\$1{,}000+\$1{,}296-\$1{,}296=\mathbf{\$1{,}000.00}$. The calculation is checked against the cash-flow timeline: the two-year benefit is one and a half years away, while the only remaining premium is half a year away.

Worked Example: A Three-Year Endowment at Midyear

A hypothetical three-year endowment policy pays $\$10{,}000$ at the end of the policy year of death, or at duration 3 if the insured survives to maturity. Annual premiums of $\$1{,}000$ are due at durations 0, 1, and 2 while the insured is alive. Assume $q=0.20$ each year, UDD within each year, and $v=0.81$, corresponding to an effective annual interest rate of 23.46%. Calculate the reserve at $t=1.5$.

Method. Separate the possible benefits into death before the next anniversary, death during the final policy year, and survival to maturity. Then value the one remaining premium, which is due at duration 2.

From duration 1.5, UDD gives a conditional probability of death before duration 2 of $1/9$, and a probability of survival to duration 2 of $8/9$. The first benefit is paid after 0.5 years:

\text{First benefit EPV} = \$10{,}000\left(\frac{1}{9}\right)(0.90) = \$1{,}000.00.

Conditional on survival to duration 2, the probability of death during the final policy year is $0.20$. The joint conditional probability of surviving to duration 2 and then dying in that year is $(8/9)(0.20)=8/45$. This benefit is paid at duration 3, 1.5 years from valuation:

\text{Final-year death benefit EPV} = \$10{,}000\left(\frac{8}{45}\right)(0.81^{1.5}) = \$1{,}296.00.

The conditional probability of surviving to maturity is $(8/9)(0.80)=32/45$. The maturity benefit is also paid at duration 3:

\text{Maturity benefit EPV} = \$10{,}000\left(\frac{32}{45}\right)(0.81^{1.5}) = \$5{,}184.00.

The total expected present value of benefits is $\$1{,}000+\$1{,}296+\$5{,}184=\$7{,}480.00$. The remaining premium at duration 2 is payable with probability $8/9$ and is discounted for 0.5 years:

\text{Premium EPV} = \$1{,}000\left(\frac{8}{9}\right)(0.90) = \$800.00.

Thus the reserve is $\$7{,}480-\$800=\mathbf{\$6{,}680.00}$. As a check, the death and maturity probabilities after valuation sum to one: $1/9+8/45+32/45=1$.

Worked Example: A Whole-Life Policy Between Anniversaries

A hypothetical whole-life policy pays $\$10{,}000$ at the end of the policy year of death. Annual premiums of $\$1{,}000$ are due at each policy anniversary while the insured is alive. Assume a constant annual death probability $q=0.20$, UDD within each policy year, and $v=0.81$, corresponding to an effective annual interest rate of 23.46%. Find the reserve at duration $t=1.5$.

Method. Value the benefit for death before duration 2 separately. From duration 2 onward, use the geometric series for the annual whole-life insurance and premium values. The geometric series converges because $pv=(0.80)(0.81)=0.648<1$.

As before, the conditional probability of death before duration 2 is $1/9$, and the discount factor to duration 2 is 0.90. The expected present value of that benefit is $\$10{,}000(1/9)(0.90)=\$1{,}000.00$.

At duration 2, the whole-life insurance value per dollar of benefit is the sum of the discounted probabilities of death in each future policy year:

A = \sum_{j=0}^{\infty}v^{j+1}(0.80)^j(0.20) = \frac{0.20(0.81)}{1-(0.80)(0.81)} = \frac{81}{176}.

The probability of reaching duration 2 from the valuation time is $8/9$, and the discount factor to duration 2 is 0.90. Their product is 0.80. Therefore, the expected present value of death benefits after duration 2 is:

\$10{,}000(0.80)\left(\frac{81}{176}\right) = \$3{,}681.82.

At duration 2, the expected present value of premiums starting immediately at that anniversary is $\$1{,}000\sum_{j=0}^{\infty}(0.80v)^j=\$1{,}000/(1-0.648)$. Adjusting for survival and discounting from duration 1.5 to duration 2 gives:

\$1{,}000(0.80)\left(\frac{1}{1-0.648}\right) = \$2{,}272.73.

The reserve is $\$1{,}000+\$3{,}681.82-\$2{,}272.73=\mathbf{\$2{,}409.09}$. This includes the benefit for death during the remaining half-year as well as all subsequent cash flows; the premium due at duration 2 is included because the insured must survive to pay it.

Common Exam Trap: Using an Annual Probability for a Partial Year

In the first worked example, a tempting but incorrect calculation is to treat the annual $q=0.20$ as the conditional probability of death between durations 0.5 and 1. That gives a first benefit value of $\$10{,}000(0.20)(0.90)=\$1{,}800$, rather than the correct $\$1{,}000$. It also incorrectly uses 0.80 as the probability of surviving from duration 0.5 to 1, instead of $8/9=0.8889$.

Applying those incorrect probabilities to the full example gives a later death-benefit value of $\$10{,}000(0.80)(0.20)(0.729)=\$1{,}166.40$, and a premium value of $\$1{,}620(0.80)(0.90)=\$1{,}166.40$. The resulting reserve would be $\$1{,}800.00$, not $\$1{,}000.00$. The correction is to calculate the conditional probability from the exact valuation time, and then use it consistently for both the remaining-year benefit and survival to the next premium date.

Key takeaway: At a between-anniversary valuation, first identify the next benefit and premium dates. Then condition every future cash flow on survival to the exact valuation time, use the appropriate fractional-duration mortality probabilities, and discount for the time from valuation to payment.

Check Your Understanding

For each problem, use the stated mortality and interest assumptions and condition on survival to the valuation time.

A one-year term policy pays $\$6{,}000$ at duration 1 if death occurs during the year. At duration 0.5, the annual death probability is 0.20 under UDD, the effective annual interest rate is zero, and a premium of $\$500$ is due at duration 1 if the insured survives. Find the reserve. (Answer: $\$222.22$)
A two-year term policy pays $\$9{,}000$ at the end of the policy year of death. At duration 0.25, annual $q=0.20$ in each year, UDD applies, and the effective annual interest rate is zero. A premium of $\$1{,}000$ is due at duration 1 if the insured survives. Find the reserve. (Answer: $\$2{,}094.74$)
A two-year endowment pays $\$5{,}000$ at the end of the policy year of death or at duration 2 if the insured survives. At duration 0.5, annual $q=0.20$, UDD applies, and the effective annual interest rate is zero. A $\$400$ premium is due at duration 1 if the insured survives. Find the reserve. (Answer: $\$4{,}644.44$)
A whole-life policy pays $\$10{,}000$ at the end of the policy year of death. At duration 0.5, annual $q=0.20$ is constant, UDD applies, and the effective annual interest rate is zero. A $\$100$ premium is due at each anniversary while the insured is alive, beginning at duration 1. Find the reserve. (Answer: $\$9{,}555.56$)

What You'll Learn