Reserves at Fractional Durations

Why the Exact Valuation Time Matters

A retrospective reserve is based on past cash flows accumulated to a valuation date. If that date falls between policy anniversaries, it is not enough to calculate a reserve at the preceding anniversary and label it a fractional-duration reserve. The premiums and benefits included, the number of policies still in force, and the interest accumulated on each cash flow must all match the actual valuation time.

The key practical step is to set a clear timeline. A premium paid at duration 0 has accumulated for more time than one paid at duration 1; a benefit paid at the end of year 1 has accumulated for less time than that initial premium. Fractional exponents handle these different time spans when the effective annual interest rate is applied consistently.

Model and convention: The formulas and examples below use an annual discrete mortality model. At the start of year $j$, $n_j$ policies are in force. Premiums are paid at the start of each year, and death benefits are paid at the end of the year in which death occurs. Thus $m_j=n_j-n_{j+1}$ claims are paid at duration $j+1$. There are no other cash flows between anniversaries, and deaths are modeled at year-end. If the valuation time is $t=k+s$, where $k$ is an integer and $0<s<1$, the policies in force during that partial year are $n_k$. Interest accumulates for $u$ years by the factor $(1+i)^u$, where $i$ is the effective annual rate.

This is a specific, internally consistent timing model. It does not directly describe a contract with benefits paid immediately on death, premiums paid monthly, or a fractional-mortality assumption that allows deaths within the policy year. In those cases, the actual cash-flow times and the appropriate survival counts must be used instead. State the model before applying a formula; the fractional-duration calculation depends on it.

The Cohort Formula at a Fractional Duration

Let the valuation time be $t=k+s$, with $0<s<1$. Premiums at durations $0,1,\ldots,k$ have been paid by time $t$, because the premium at duration $k$ is paid at the start of the current year. Benefits for deaths during years $0$ through $k-1$ have also been paid, at durations $1$ through $k$. The benefit for a death during the current year is not paid until duration $k+1$.

{}_tV = \frac{ \displaystyle \sum_{j=0}^{k} n_j P_j(1+i)^{t-j} - \displaystyle \sum_{j=0}^{k-1} m_j B_j(1+i)^{t-(j+1)} }{ n_k }, \qquad t=k+s,\quad 0<s<1.

Here $P_j$ is the premium per policy paid at duration $j$, and $B_j$ is the benefit per claim for deaths during year $j$. For level premiums and benefits, set $P_j=P$ and $B_j=B$. The formula accumulates cohort totals first and then expresses the result per policy in force at the valuation time. It is not a calculation for an average policy that ignores the number of policies contributing each cash flow.

Locate the valuation time.
Write $t=k+s$, with $k$ the completed policy years and $s$ the elapsed fraction of the current year.

List the cash flows already made.
Include premiums paid at durations $j\leq k$, and benefits paid at the ends of years $j<k$, under the stated timing convention.

Accumulate each cohort total to $t$.
A cash flow at time $c$ is multiplied by $(1+i)^{t-c}$. Keep each cash flow’s own time rather than assigning one accumulation period to all of them.

Condition on policies in force.
Subtract accumulated claims from accumulated premiums, then divide by $n_k$, the number of policies in force at $t$ in this model.

Why the Formula Works

The justification follows directly from the cohort definition of a retrospective reserve. Consider all policies issued as one cohort. At duration $j$, $n_j$ policies are in force and each pays $P_j$, so the cohort receives $n_jP_j$. If $m_j$ policies die during year $j$, the cohort pays $m_jB_j$ at duration $j+1$. Accumulating each receipt and payment separately to $t$ gives the cohort’s net accumulated balance.

\text{cohort balance at }t = \sum_{j=0}^{k} n_jP_j(1+i)^{t-j} - \sum_{j=0}^{k-1}m_jB_j(1+i)^{t-(j+1)}.

By definition, the retrospective reserve per policy in force is the cohort’s accumulated balance divided by the number of policies in force at the valuation date. In this annual discrete model, no deaths occur between anniversaries, so that number at $t=k+s$ is $n_k$. Dividing the balance by $n_k$ gives the stated formula. This also explains why using $n_0$, the original issue count, in the denominator would produce a cohort balance per original policy rather than a reserve per policy currently in force.

Worked Calculations

Worked Example: A Level-Premium Contract at Duration 1.5

A hypothetical annual discrete contract has an effective annual interest rate of $i=21\%$. It charges a level premium of $P=\$100$ at durations 0 and 1 and pays a level death benefit of $B=\$250$ at the end of the year of death. The mortality experience assumed for the cohort is $n_0=1{,}000$, $n_1=900$, and $n_2=810$. Thus the first two annual death counts are $m_0=100$ and $m_1=90$, corresponding to annual death probabilities $q_0=0.10$ and $q_1=0.10$. Find the reserve at $t=1.5$.

Method. Accumulate the two premium receipts and the first-year claims to duration 1.5, then divide by the 900 policies in force during the second policy year. The benefit for deaths during year 1 is not yet paid at time 1.5.

The accumulation factors are

(1.21)^{1.5}=1.331,\qquad (1.21)^{0.5}=1.1.

At duration 0, 1,000 premiums of \$100 are received. Their accumulated value is

1{,}000(100)(1.331)=\$133{,}100.

At duration 1, 900 premiums are received. Their accumulated value is

900(100)(1.1)=\$99{,}000.

The first-year death count is $m_0=1{,}000-900=100$. Claims of $100(250)=\$25{,}000$ are paid at duration 1 and accumulate for half a year:

100(250)(1.1)=\$27{,}500.

Therefore, the accumulated cohort balance is $\$133{,}100+\$99{,}000-\$27{,}500=\$204{,}600$. The reserve per policy in force is

{}_{{1.5}}V=\frac{\$204{,}600}{900}=\$227.3333\ldots

To the nearest cent, the reserve is \$227.33. This calculation avoids the mistake of including year-1 claims before their scheduled payment at duration 2.

Worked Example: Unequal Annual Death Counts at Duration 2.5

A hypothetical contract has $i=21\%$, a level annual premium of $P=\$80$, and a death benefit of $B=\$500$, paid at the end of the year of death. The cohort counts are $n_0=500$, $n_1=450$, and $n_2=360$. Thus $m_0=50$ and $m_1=90$; the annual death probabilities for years 0 and 1 are $q_0=0.10$ and $q_1=0.20$. Calculate the reserve at $t=2.5$.

Method. Accumulate premiums paid at durations 0, 1, and 2, subtract claims paid at durations 1 and 2, and divide by $n_2=360$. No year-2 claims have been paid by time 2.5.

The required factors are

(1.21)^{2.5}=1.61051,\qquad (1.21)^{1.5}=1.331,\qquad (1.21)^{0.5}=1.1.

The accumulated premiums are, by payment duration,

\begin{aligned} j=0:&\quad 500(80)(1.61051)=\$64{,}420.40,\\ j=1:&\quad 450(80)(1.331)=\$47{,}916.00,\\ j=2:&\quad 360(80)(1.1)=\$31{,}680.00. \end{aligned}

Their total is $\$64{,}420.40+\$47{,}916.00+\$31{,}680.00=\$144{,}016.40$. The accumulated claims are

\begin{aligned} \text{claims paid at }1:&\quad 50(500)(1.331)=\$33{,}275.00,\\ \text{claims paid at }2:&\quad 90(500)(1.1)=\$49{,}500.00. \end{aligned}

The claims total $\$82{,}775.00$, so the cohort balance is $\$144{,}016.40-\$82{,}775.00=\$61{,}241.40$. Dividing by 360 gives

{}_{{2.5}}V=\frac{\$61{,}241.40}{360}=\$170.115.

To the nearest cent, the reserve is \$170.12. The main timing trap is to use the same accumulation period for cash flows made at different durations; here the three premiums have accumulated for 2.5, 1.5, and 0.5 years.

Worked Example: Varying Premiums and Benefits

A hypothetical annual discrete contract earns $i=21\%$. The cohort counts are $n_0=200$, $n_1=180$, and $n_2=150$, so $m_0=20$ and $m_1=30$. The annual death probabilities are $q_0=20/200=0.10$ and $q_1=30/180=1/6$. Premiums at durations 0, 1, and 2 are respectively $\$40$, $\$60$, and $\$80$. Benefits for deaths in years 0 and 1 are respectively $\$300$ and $\$400$, payable at the end of the applicable year. Find the reserve at $t=2.5$.

Method. Use the actual premium and benefit amount for each year, rather than treating the contract as level. The same accumulation factors as in the preceding example apply.

The three accumulated premium amounts are

\begin{aligned} 200(40)(1.61051)&=\$12{,}884.08,\\ 180(60)(1.331)&=\$14{,}374.80,\\ 150(80)(1.1)&=\$13{,}200.00. \end{aligned}

Their sum is $\$40{,}458.88$. The accumulated claims are

\begin{aligned} 20(300)(1.331)&=\$7{,}986.00,\\ 30(400)(1.1)&=\$13{,}200.00. \end{aligned}

Their sum is $\$21{,}186.00$. The accumulated cohort balance is therefore $\$40{,}458.88-\$21{,}186.00=\$19{,}272.88$. With $n_2=150$ policies in force,

{}_{{2.5}}V=\frac{\$19{,}272.88}{150}=\$128.485866\ldots

To the nearest cent, the reserve is \$128.49. This method avoids the mistake of using a level benefit or premium when the cash flows actually change by year.

Common Timing Errors and Model Checks

A fractional valuation date changes the accumulation period even when it does not change which scheduled cash flows have occurred. For example, at duration 1.5 in the first example, a common incorrect calculation is to accumulate every cash flow only to duration 1, giving

\frac{100{,}000(1.21)+90{,}000-25{,}000}{900} = \frac{\$186{,}000}{900} = \$206.67.

That is a value at duration 1, not at duration 1.5. It fails to accumulate the duration-0 premium and the duration-1 premium and benefit to the actual valuation date. The correct duration-1.5 result is $\$227.33$.

Before accepting a result, check the timeline and the following details:

Premium at the current anniversary: If $t=k+s$ with $s>0$, the premium at duration $k$ has been paid under the stated start-of-year convention. Its accumulation period is $s$, not zero.
Claims in the current year: Under the year-end payment convention, claims for deaths during year $k$ are not yet paid at $k+s$. Do not subtract them as if they were paid immediately.
Survivor denominator: Divide by the in-force count at the valuation time, $n_k$, not by the original issue count $n_0$.
Fractional interest period: A cash flow at time $c$ accumulates for $t-c$ years. The exponent must reflect the difference between its actual payment time and the exact valuation time.
Consistency of assumptions: If the contract instead pays claims at death or allows fractional-year mortality, this annual discrete formula needs to be replaced by a calculation using those actual timings and survival assumptions.

These steps make a fractional-duration retrospective calculation auditable: each term can be tied to a payment date, an in-force count, and an accumulation period. The approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

Use the annual discrete model stated above. Round each final dollar reserve to the nearest cent.

A cohort has $n_0=100$, a premium of $\$20$ at duration 0, and no claim payment before duration 1. With $i=21\%$, find the reserve at $t=0.5$. (Answer: $\$22.00$)
A contract has $i=21\%$, $n_0=1{,}000$, $n_1=900$, $n_2=810$, a level premium of $\$50$, and a level benefit of $\$200$ paid at each year-end. Find the reserve at $t=1.5$. (Answer: $\$104.50$)
At $i=21\%$, a cohort has counts $n_0=100$, $n_1=80$, and $n_2=60$. Premiums at durations 0, 1, and 2 are $\$10$, $\$20$, and $\$30$; benefits for deaths in years 0 and 1 are $\$50$ and $\$100$. Find the reserve at $t=2.5$. (Answer: $\$36.49$)
A contract earns $i=21\%$, with $n_0=400$, $n_1=320$, and a level premium of $\$25$. The benefit is $\$100$, paid at the end of the year of death. Find the reserve at $t=1.5$. (Answer: $\$41.59$)
A cohort has $i=21\%$, counts $n_0=200$, $n_1=160$, and $n_2=120$, and a level premium of $\$15$. Benefits are $\$40$ for deaths in year 0 and $\$60$ for deaths in year 1, paid at each year-end. Find the reserve at $t=2.5$. (Answer: $\$43.64$)

What You'll Learn