Timing Conventions for Annual Profits

Why the Timing Convention Matters

A profit test may calculate an annual profit for each policy year without specifying one exact date on which that profit emerges. To discount the profits, however, each amount needs a time index. A timing convention supplies that index: it places each annual profit at the beginning, middle, or end of its policy year. This is an attribution convention for the annual profit, not a recalculation of the underlying premiums, claims, expenses, or reserves.

The distinction matters because the same profit signature can have different present values under different timing conventions. It also prevents a common indexing error: policy year 1 is not automatically time 1. Its beginning is time 0, its midpoint is time 0.5, and its end is time 1.

Definition: Let $P_k$ be the profit assigned to policy year $k$, for $k=1,\ldots,n$. Under a timing convention with within-year position $\alpha$, where $\alpha=0$ denotes the beginning, $\alpha=0.5$ the midpoint, and $\alpha=1$ the end, the elapsed-time index for $P_k$ is $t_k=k-1+\alpha$. Thus, the beginning-, middle-, and end-of-year indices are respectively $k-1$, $k-0.5$, and $k$.

Assigning the Duration Index

The duration index describes when the annual profit is treated as emerging, measured from issue or the start of the profit test. It is not the policy-year number itself. For example, the profit for policy year 3 has index 2 under a beginning-of-year convention, 2.5 under a middle-of-year convention, and 3 under an end-of-year convention.

Policy year	Beginning of year	Middle of year	End of year
1	0	0.5	1
2	1	1.5	2
3	2	2.5	3
$k$	$k-1$	$k-0.5$	$k$

A profit signature may contain expected cohort profits, or another consistently defined annual profit amount. Whatever its construction, the timing convention is applied to the signature entries at the stated durations. If a signature already reflects in-force probabilities, changing the assigned profit date does not, by itself, change those probabilities or the profit amounts.

Worked Example: Building a Two-Year Signature and Assigning Its Dates

A hypothetical cohort has 100 policies at issue. The annual death probabilities, conditional on being in force at the start of each year, are $q_1=0.10$ and $q_2=0.20$; there are no other decrements. The annual profit is deterministic conditional on being in force at the start of the year: $40 per policy in year 1 and $60 per policy in year 2. The effective annual interest rate is $i=0.21$. Find the cohort profit signature and the elapsed-time indices under each convention.

Method. First weight each per-policy profit by the expected number of policies in force at the start of its year. Then attach the appropriate time index to each signature entry.

Year 1. There are 100 policies in force at the start, so the cohort profit is $100(40)=\$4{,}000$. The year-1 indices are 0, 0.5, and 1 for beginning, middle, and end of year.

Year 2. The expected number in force at the start is $100(1-0.10)=90$. Thus the year-2 cohort profit is $90(60)=\$5{,}400$. The year-2 indices are 1, 1.5, and 2, respectively. The second-year death probability does not alter the stated start-of-year profit amount; it would be relevant to subsequent in-force counts.

The signature is therefore $(\$4{,}000,\$5{,}400)$. Under the end-of-year convention the indexed entries are $(1,\$4{,}000)$ and $(2,\$5{,}400)$; under the middle-of-year convention they are $(0.5,\$4{,}000)$ and $(1.5,\$5{,}400)$; under the beginning-of-year convention they are $(0,\$4{,}000)$ and $(1,\$5{,}400)$.

Check. The expected start-of-year in-force counts are 100 and 90, so the signature amounts are consistent with the stated mortality and per-policy profits. The indices place each profit within its own policy year.

How Timing Changes Present Value

Let $i$ be the effective annual interest rate, with $i>-1$, and let $v=1/(1+i)$. A profit $P_k$ assigned to elapsed time $t_k$ has present value $P_kv^{t_k}$. For a whole signature, the present value under a common timing convention is the sum of these discounted entries. The convention changes the exponent, not the profit amount.

\operatorname{PV}_{\text{begin}}=\sum_{k=1}^{n}P_kv^{k-1},\qquad \operatorname{PV}_{\text{mid}}=\sum_{k=1}^{n}P_kv^{k-0.5},\qquad \operatorname{PV}_{\text{end}}=\sum_{k=1}^{n}P_kv^k.

The relationship between the conventions follows directly by factoring out the same multiplier from every term. Since $v^{-0.5}=\sqrt{1+i}$ and $v^{-1}=1+i$, moving all entries from year-end to midyear multiplies the end-year present value by $\sqrt{1+i}$; moving them to the beginning of each year multiplies it by $1+i$.

\operatorname{PV}_{\text{mid}}=\sqrt{1+i}\operatorname{PV}_{\text{end}}, \qquad \operatorname{PV}_{\text{begin}}=(1+i)\operatorname{PV}_{\text{end}}.

This is an algebraic relationship for a common shift of every entry in the signature. It remains true when profits have mixed signs: the total is multiplied by the stated factor. It does not mean that an earlier date always gives a larger present value. If $i>0$, the factor from end to midyear is greater than 1; if $i=0$, it is 1; and if $-1<i<0$, it is less than 1. For a negative profit, multiplying by a factor greater than 1 makes its present value more negative, while multiplying by a factor below 1 makes it less negative.

Identify the annual profit amounts.
Use the profit signature as defined for the problem. Do not alter its mortality weighting merely because a timing convention changes.

Translate each duration into elapsed time.
For policy year $k$, use $k-1$, $k-0.5$, or $k$ for beginning, middle, or end of year.

Apply the discount factor at that time.
Discount $P_k$ by $v^{t_k}$, retaining the sign of the profit.

Check the direction using the interest rate.
For positive interest, moving a positive amount earlier increases its present value; for negative interest, that comparison reverses. A zero rate makes the timing irrelevant to present value.

Worked Example: Comparing All Three Conventions

Use the two-year cohort signature $(\$4{,}000,\$5{,}400)$ from the preceding example, with effective annual interest $i=0.21$, so $v=1/1.21$. Calculate the present value under each convention. The annual death probabilities remain $q_1=0.10$ and $q_2=0.20$, and the per-policy profits remain deterministic conditional on start-of-year in-force status as specified above.

End of year. The indices are 1 and 2. Therefore,

\operatorname{PV}_{\text{end}} =\frac{4{,}000}{1.21}+\frac{5{,}400}{1.21^2} =3{,}305.7851+3{,}688.2727 =\$6{,}994.0578.

Middle of year. The indices are 0.5 and 1.5. Since $\sqrt{1.21}=1.1$, the first term is $4{,}000/1.1=\$3{,}636.3636$, and the second is $5{,}400/1.331=\$4{,}057.0999$. Thus,

\operatorname{PV}_{\text{mid}} =3{,}636.3636+4{,}057.0999 =\$7{,}693.4636.

Beginning of year. The indices are 0 and 1. The first profit is not discounted, and the second is discounted one year:

\operatorname{PV}_{\text{begin}} =4{,}000+\frac{5{,}400}{1.21} =4{,}000+4{,}462.8099 =\$8{,}462.8099.

The values are rounded to four decimal places in dollars to make the intermediate arithmetic visible. As a check, $1.1(\$6{,}994.0578)$ is approximately $\$7{,}693.4636$, and $1.21(\$6{,}994.0578)$ is approximately $\$8{,}462.8099$. Both agree with the direct calculations, subject to rounding.

Signed Profits and Negative Interest

The timing multiplier applies to positive and negative entries alike, but the interpretation of a comparison depends on both the interest rate and the sign. A negative annual profit is a loss in the signature; it must remain negative when discounted. Taking its absolute value before applying the timing factor would change the signature.

Worked Example: A Mixed Signature at a Negative Interest Rate

A hypothetical cohort has 100 policies at issue. The first-year death probability is $q_1=0.20$, and there are no other decrements before the second-year profit is assigned. The deterministic profit conditional on being in force at the start of year 1 is $1 per policy. In year 2, the deterministic profit conditional on being in force at the start of that year is a loss of $0.50 per policy. The effective annual interest rate is $i=-0.10$, so $v=1/0.9$. Find the signature and its end-, middle-, and beginning-of-year present values.

Form the signature. The year-1 cohort profit is $100(\$1)=\$100$. Expected in-force count at the start of year 2 is $100(1-0.20)=80$, so its cohort profit is $80(-\$0.50)=-\$40$. The signature is therefore $(\$100,-\$40)$.

End of year. The indices are 1 and 2:

\operatorname{PV}_{\text{end}} =\frac{100}{0.9}-\frac{40}{0.9^2} =111.1111-49.3827 =\$61.7284.

Middle of year. Both entries move one-half year earlier, so the end-year present value is multiplied by $\sqrt{0.9}$:

\operatorname{PV}_{\text{mid}} =\sqrt{0.9}(\$61.7284) \approx \$58.5607.

Beginning of year. Both entries move one year earlier, so the end-year present value is multiplied by $0.9$:

\operatorname{PV}_{\text{begin}} =0.9(\$61.7284) \approx \$55.5556.

The factors are below 1 because the interest rate is negative. The earlier-timing values are smaller than the end-year value here. The check is that $0.9(61.728395\ldots)=55.555556\ldots$, which also equals $100-40/0.9$ when the profits are assigned at times 0 and 1.

Common Mistakes and Practical Limits

Using the policy-year number as the elapsed time. Year 1 begins at time 0, not time 1. Under the midyear convention, its index is 0.5.
Changing the amount instead of its date. A timing convention relocates the annual profit used in the signature. It does not automatically recalculate the profit, the in-force probability, or the mortality assumption.
Dropping a negative sign. A loss must be discounted as a negative amount. The sign affects the interpretation of the resulting present value.
Claiming earlier timing always increases present value. The comparison depends on the effective interest rate. For $i>0$, the end-to-midyear multiplier $\sqrt{1+i}$ exceeds 1; for $i=0$, it equals 1; and for $-1<i<0$, it is below 1.
Applying one multiplier to only part of a signature. The simple whole-signature relationships require every entry to be shifted by the same amount. If different entries use different timing conventions, discount each at its own stated time.

These conventions are useful when an annual profit test needs a consistent emergence date but does not model every cash flow at its exact date. They are approximations: if a problem specifies irregular cash-flow dates or distinct dates for individual profit components, use those dates rather than imposing a single annual convention. Correct date assignment is a foundational step for profit-signature calculations aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the relevant duration index and apply the stated convention carefully.

A profit is assigned to policy year 4 under the middle-of-year convention. What elapsed-time index is used? (Answer: 3.5)
A hypothetical cohort has 70 policies at issue and a first-year death probability of 0.20. There are no other decrements before the second-year profit is assigned. The deterministic second-year profit is $5 per policy in force at the start of year 2. What is the year-2 cohort profit, and what is its end-of-year index? (Answer: $280 at time 2)
A two-year signature is $(\$100,-\$50)$, and every entry is moved from year-end to midyear. If $i=0.05$, what factor multiplies its end-year present value? (Answer: $\sqrt{1.05}\approx1.0247$)
A signature has end-year present value of $200. All entries are moved to midyear, and the effective annual rate is $i=-0.04$. What is the midyear present value, rounded to the nearest cent? (Answer: $\sqrt{0.96}(200)\approx\$195.96$)
A positive profit in policy year 3 is assigned to the beginning of that year. At $i=0.10$, what is its time index, and by what factor does its present value compare with assigning the same profit at the end of year 3? (Answer: time 2; beginning-date present value is $1.10$ times the end-date present value)

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