Premium Timing: Advance and Arrears

Why Premium Timing Matters

Two contracts can provide the same insurance benefit but require different level premiums because their premiums are due at different times. A premium paid at issue is discounted less than a premium due a year later, and it is payable before any future mortality can occur. A premium due at the end of a year, by contrast, is paid only if the insured is alive on that due date under the convention used here.

The key is to model the payment dates and survival conditions explicitly. This tutorial considers annual premiums for discrete life insurance. Benefits are paid at the end of the year of death. Premiums in advance are paid at the start of a year while the policy is in force; premiums in arrears are paid at the end of a year only if the insured survives to that date. No expenses are included, so the premiums are net premiums.

Definition: An advance premium is due at the beginning of a coverage period. For a policy issued at age $x$, the first annual premium is due at time 0, with later premiums due at times $1,2,\ldots$ if the insured is then alive and the policy remains in force. An arrears premium is due at the end of a coverage period, and is paid only if the insured survives to that due date. Thus the first such premium is due at time 1, conditional on survival to time 1.

The timing convention must match the contract. In particular, an arrears premium is not paid at the end of the year of death in the model used below. A contract with a different rule for a premium due on a death date must be valued using that specific rule instead.

Survival-Contingent Premium Factors

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write ${}_kp_x$ for the probability that a life aged $x$ survives for $k$ years, with ${}_0p_x=1$. The expected present value of a level annual premium stream is the premium amount multiplied by its survival-contingent annuity factor.

\ddot{a}_x=\sum_{k=0}^{\infty}v^k\,{}_kp_x, \qquad a_x=\sum_{k=1}^{\infty}v^k\,{}_kp_x.

The first factor, $\ddot{a}_x$, values whole-life premiums in advance: the time-0 premium is certain, and each later premium is conditional on survival to its due date. The second, $a_x$, values whole-life premiums in arrears: its first term is discounted one year and requires survival for that year.

For premiums payable for at most $n$ years, the corresponding factors are

\ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k\,{}_kp_x, \qquad a_{x:\overline{n}|}=\sum_{k=1}^{n}v^k\,{}_kp_x.

These sums specify exactly which premium dates are included. The advance stream has a possible payment at time 0 and at most $n-1$ further payments. The arrears stream has possible payments at times 1 through $n$, each conditional on survival to that payment date.

How the Two Timing Conventions Relate

The advance factor contains the time-0 payment, which has present value 1. Every later term in the advance factor is also present in the arrears factor, but the arrears factor stops at time $n$ for a temporary stream. Consequently, for an $n$-year premium period,

\ddot{a}_{x:\overline{n}|} =1+\sum_{k=1}^{n-1}v^k\,{}_kp_x, \qquad a_{x:\overline{n}|} =\sum_{k=1}^{n}v^k\,{}_kp_x.

The difference between these finite-period factors is not generally just 1: the arrears factor includes a possible payment at time $n$, while the advance factor does not. For whole-life factors, the sums have no final term, so the relationship simplifies.

\ddot{a}_x=1+a_x.

This identity follows directly by separating the $k=0$ term, $v^0{}_0p_x=1$, from the whole-life advance sum. The remaining terms are exactly those in $a_x$. Since $a_x=\ddot{a}_x-1<\ddot{a}_x$, a given whole-life benefit present value requires a larger level premium in arrears than in advance.

Level Net Premium Equations

Under the equivalence principle, the expected present value of premiums equals the expected present value of benefits. If $B$ is the death benefit and $A_x$ is the present-value factor for a whole-life insurance benefit paid at the end of the year of death, then the whole-life net premium $P$ satisfies one of the following equations, according to the premium timing:

P_{\text{advance}}\ddot{a}_x=BA_x, \qquad P_{\text{arrears}}a_x=BA_x.

Thus $P_{\text{advance}}=BA_x/\ddot{a}_x$ and $P_{\text{arrears}}=BA_x/a_x$, provided the corresponding factor is positive and finite. For premiums payable only during a limited period, use the matching finite-period factor in the denominator, while keeping the benefit present value appropriate to the coverage.

Worked Example: Whole-Life Premiums Under Constant Mortality

A hypothetical whole-life policy pays $5,000 at the end of the year of death. Assume annual independent one-year survival probabilities are constant at $p=0.8$, so $q=1-p=0.2$, and the effective annual interest rate is $i=25\%$. Calculate the level net premium if payable in advance and if payable in arrears.

Method and assumptions. The benefit is paid at the end of the year of death. Premiums in advance are paid at time 0 and at later anniversaries if alive; arrears premiums are paid at year-end only if alive then. The model has $v=1/1.25=0.8$, and the annual survival probability remains $0.8$ at every duration.

The probability of death in year $k+1$ is $p^kq$. Therefore the benefit present-value factor is

A_x=\sum_{k=0}^{\infty}p^kqv^{k+1} =\frac{qv}{1-pv} =\frac{0.2(0.8)}{1-0.8(0.8)} =\frac{0.16}{0.36} =\frac{4}{9}.

The expected present value of the benefit is $5{,}000(4/9)=\$2{,}222.22$, rounded to the nearest cent. The premium factors are

\ddot{a}_x=\sum_{k=0}^{\infty}(pv)^k =\frac{1}{1-0.64} =\frac{25}{9}, \qquad a_x=\sum_{k=1}^{\infty}(pv)^k =\frac{0.64}{0.36} =\frac{16}{9}.

The advance net premium is $(2{,}222.22)/(25/9)$, or, using unrounded values, $5{,}000(4/9)/(25/9)=\$800.00$. The arrears net premium is $5{,}000(4/9)/(16/9)=\$1{,}250.00$. To check, $800(25/9)=\$2{,}222.22$ and $1{,}250(16/9)=\$2{,}222.22$, to the nearest cent. The larger arrears premium reflects its smaller present-value factor.

Common mistake avoided: The numerator in the benefit factor is $qv=0.2(0.8)=0.16$, not $0.2$ or $0.2(0.8)(5{,}000)$ before the factor is simplified. Keeping the probability, discount factor, and benefit amount distinct prevents an arithmetic error.

Worked Example: A Different Survival Rate Changes Both Premiums

A hypothetical whole-life policy pays $6,000 at the end of the year of death. Assume constant annual survival probability $p=0.9$, death probability $q=0.1$, and effective annual interest $i=25\%$. Find the advance and arrears net premiums.

Method and assumptions. Use the constant-mortality geometric series, with annual premium payments as defined above. Here $v=1/1.25=0.8$, so $pv=0.9(0.8)=0.72$ and $qv=0.1(0.8)=0.08$.

The benefit factor is $qv/(1-pv)=0.08/(1-0.72)=0.08/0.28=2/7$. Thus the benefit expected present value is $6{,}000(2/7)=\$1{,}714.29$, rounded to cents. The premium factors are

\ddot{a}_x=\frac{1}{1-0.72}=\frac{25}{7}, \qquad a_x=\frac{0.72}{1-0.72}=\frac{18}{7}.

The advance premium is $6{,}000(2/7)/(25/7)=\$480.00$. The arrears premium is $6{,}000(2/7)/(18/7)=\$666.67$, rounded to cents. Checking with unrounded values, $480(25/7)=\$1{,}714.29$, and $(2{,}000/3)(18/7)=\$1{,}714.29$. Both premium streams therefore satisfy the equivalence equation.

Common mistake avoided: The benefit present value is not itself the level premium. It must be divided by the premium factor for the selected due-date convention. Using the advance factor for an arrears contract would understate its premium.

Worked Example: Two-Year Premium Period With Age-Dependent Mortality

A hypothetical two-year term policy pays $6,000 at the end of the year of death if death occurs during the term. The first-year death probability is $q_x=0.1$; conditional on surviving the first year, the second-year death probability is $q_{x+1}=0.2$. The effective annual interest rate is $10\%$. Find the level net premium for two annual premiums in advance and for two annual premiums in arrears.

Method and assumptions. The first premium in advance is certain at time 0, and a second is due at time 1 only if the insured survives. The two arrears premiums are due at times 1 and 2, each only if the insured is alive at its due date. Here $v=1/1.1=10/11$, and survival through year 1 is $p_x=0.9$; survival through year 2 is $0.9(0.8)=0.72$.

Death in year 1 has probability $0.1$; death in year 2 has probability $0.9(0.2)=0.18$. The benefit present-value factor and benefit expected present value are

0.1v+0.18v^2 =\frac{1}{11}+\frac{18}{121} =\frac{29}{121}, \qquad 6{,}000\left(\frac{29}{121}\right) =\$1{,}438.02.

The two-year advance factor is $1+0.9v=1+9/11=20/11$. The arrears factor is $0.9v+0.72v^2=99/121+72/121=171/121$. Therefore,

P_{\text{advance}} =\frac{6{,}000(29/121)}{20/11} =\frac{174{,}000}{220} =\$790.91, \qquad P_{\text{arrears}} =\frac{6{,}000(29/121)}{171/121} =\frac{174{,}000}{171} =\$1{,}017.54.

To check, using unrounded premiums gives $P_{\text{advance}}(20/11)=\$1{,}438.02$ and $P_{\text{arrears}}(171/121)=\$1{,}438.02$, to cents. The two-year factors differ by more than the whole-life identity $\ddot{a}_x=1+a_x$ might suggest: the temporary advance stream has a time-0 payment, while the temporary arrears stream includes a possible payment at time 2.

Common mistake avoided: The probability of surviving to the second premium date is $0.72$, not $0.8$. The second-year survival probability is conditional on first-year survival, so it must be multiplied by $0.9$.

Exam Traps and Practical Checks

The most reliable way to avoid timing errors is to list the premium dates before writing a factor. For each date, ask whether the premium is due then and what survival event is required. Then discount that payment to time 0. This also makes clear why changing only the annuity factor, rather than the benefit valuation, is appropriate when the insurance benefit is unchanged.

Counting the first arrears premium at time 0. It is due at time 1 and is conditional on survival to time 1.
Using the wrong survival condition. An advance premium at time $k$ requires survival to time $k$; an arrears premium at time $k$ also requires survival to that due date, but it is discounted by $v^k$.
Assuming finite-period factors differ by 1. The whole-life identity $\ddot{a}_x=1+a_x$ does not carry over unchanged to an $n$-year period, because the two finite streams have different final dates.
Mixing the benefit and premium timing conventions. A death benefit paid at year-end and an arrears premium due at year-end are not both automatically payable on a death during that year. Follow the stated contract rule.

A final check is to multiply the calculated level premium by the factor for its own timing. The product should equal the benefit expected present value. If it does not, revisit the survival probabilities, discount exponents, or payment dates before rounding.

Check Your Understanding

For each problem, use the stated convention that an arrears premium is payable only if the insured survives to its due date. Round money to the nearest cent.

A hypothetical whole-life policy pays $4,000 at the end of the year of death. Annual survival probability is constantly $0.75$, and the effective annual interest rate is $25\%$. Find the net premiums in advance and in arrears. (Answer: advance $800.00; arrears $1,333.33.)
A two-year term policy pays $5,000 at the end of the year of death. Assume constant annual survival probability $0.8$, death probability $0.2$, and effective annual interest rate $1/9$. Find the two-year net premiums in advance and in arrears. (Answer: advance $900.00; arrears $1,250.00.)
A two-year term policy pays $8,000 at the end of the year of death. The first-year death probability is $0.2$, and the second-year death probability conditional on survival to year 1 is $0.1$. At an effective annual interest rate of $10\%$, find the level premiums for two payments in advance and two payments in arrears. (Answer: advance $1,148.33; arrears $1,500.00.)
A three-year term policy has constant annual death probability $0.15$, so annual survival probability is $0.85$. The interest rate is $0\%$, and the death benefit is $12,000 at the end of the year of death. Calculate the net premiums for three annual payments in advance and three in arrears. (Answer: advance $1,800.00; arrears $2,117.65.)
A whole-life policy pays $3,000 at the end of the year of death. Assume constant annual survival probability $0.9$ and effective annual interest $25\%$. Calculate both net premiums, then verify that each premium multiplied by its matching whole-life factor equals the benefit expected present value. (Answer: advance $240.00; arrears $333.33; benefit expected present value $857.14.)

What You'll Learn