Comparing Annual and Continuous Payment Values

Why Payment Timing Changes an Annuity Value

A life annuity can provide the same annual benefit in different ways: one payment at the end of each year, one at the beginning of each year, or a continuous stream paid at a constant rate. The mortality and interest assumptions may be identical in all three cases, yet their expected present values differ because the money is paid at different times.

A fair comparison must keep the benefit rate, duration, mortality model, and interest model fixed. In particular, an annual payment of one unit is not the same payment pattern as a continuous rate of one unit per year: the annual payment arrives all at once, while the continuous benefit is spread through the year.

Definition: For a life aged $x$, an annual life annuity-immediate pays one unit at each integer time $k=1,2,\ldots$, provided the life is then alive. An annual life annuity-due pays one unit at times $k=0,1,2,\ldots$, provided the life is then alive. A continuous life annuity at rate one per year pays continuously while the life is alive. A temporary annuity with term $n$ stops at time $n$; its continuous payments occur over $0\le t<n$.

The corresponding whole-life present values are denoted $a_x$ for the annual annuity-immediate, $\ddot{a}_x$ for the annual annuity-due, and $\bar{a}_x$ for the continuous annuity. The bar distinguishes the continuous value. Values are conditional on the life being alive at age $x$ at the valuation date. There is no guarantee of payments after death in these definitions.

Common Model for a Direct Comparison

For the formulas and examples below, assume a constant force of interest $\delta$ and a constant force of mortality $\mu$ from age $x$ onward. Thus the discount factor to time $t$ is $e^{-\delta t}$, and the survival probability to time $t$ is ${}_tp_x=e^{-\mu t}$. These assumptions imply a constant force for the entire period being valued; the formulas do not automatically apply when interest or mortality varies with time.

Model formulas: Under constant $\delta$ and $\mu$, let $c=\delta+\mu$, where $c>0$. For a term of $n$ years, the annual annuity-immediate, annual annuity-due, and continuous annuity values are, respectively, $$ a_{x:\overline{n}|}=\sum_{k=1}^{n}e^{-ck} =\frac{1-e^{-cn}}{e^c-1},\qquad \ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}e^{-ck} =\frac{1-e^{-cn}}{1-e^{-c}},\qquad \bar{a}_{x:\overline{n}|}=\int_0^n e^{-ct}\,dt =\frac{1-e^{-cn}}{c}. $$ For whole life, the corresponding values are $a_x=1/(e^c-1)$, $\ddot{a}_x=1/(1-e^{-c})$, and $\bar{a}_x=1/c$.

Each annual payment at time $k$ contributes its amount times the probability of being alive then and the discount factor to that time. The product is $e^{-\mu k}e^{-\delta k}=e^{-ck}$. For continuous payments, a payment of rate one over a short interval of length $dt$ contributes approximately $e^{-\mu t}e^{-\delta t}dt$, so the expected present value is the integral of $e^{-ct}$. In both cases mortality and interest affect each payment through the same combined exponent $c$; the payment dates or payment interval determine how those contributions are aggregated.

Worked Example: Whole-Life Values Under One Mortality and Interest Model

A hypothetical whole-life annuity is valued for a life aged $x$. The force of interest is $\delta=0.04$ and the force of mortality is $\mu=0.06$, both constant. Find the values for an annual annuity-immediate, an annual annuity-due, and a continuous annuity, each at a rate of one unit per year. Also compare their ordering.

Method and model check. The effective annual interest rate corresponding to $\delta=0.04$ is $i=e^{0.04}-1=0.04081$, or about $4.081\%$. The one-year survival probability is $e^{-0.06}=0.9418$, so the one-year death probability is $1-e^{-0.06}=0.0582$. Both forces are constant as required. Set $c=\delta+\mu=0.04+0.06=0.10$.

Annual annuity-immediate. Payments occur at the ends of years, beginning at time 1. Its whole-life value is $$ a_x=\frac{1}{e^{0.10}-1} =\frac{1}{1.105170-1} =\frac{1}{0.105170} =9.5083. $$

Annual annuity-due. The payment at time 0 is certain, and subsequent payments occur at the beginnings of later years. Its value is $$ \ddot{a}_x=\frac{1}{1-e^{-0.10}} =\frac{1}{1-0.904837} =\frac{1}{0.095163} =10.5083. $$

Continuous annuity. Payments flow at rate one per year while the life is alive: $$ \bar{a}_x=\int_0^\infty e^{-0.10t}\,dt =\frac{1}{0.10} =10.0000. $$

The ordering is $a_x<\bar{a}_x<\ddot{a}_x$: $9.5083<10.0000<10.5083$. The annual immediate payments are delayed until year-end, while the due annuity includes a certain payment at time 0. The continuous stream falls between those two schedules. The method compares all three on the same unit-rate, whole-life basis; it avoids treating an annual end-of-year payment as if it were paid at the start of the year.

Why the Values Have This Ordering

The ordering is not limited to the whole-life example. For a finite term, each value contains the same positive factor $1-e^{-cn}$; the remaining denominator determines the comparison. For whole life, the same comparison applies with the numerator equal to 1.

Compare annual immediate with continuous.
For $c>0$, $e^c>1+c$. Therefore $e^c-1>c$, so $1/(e^c-1)<1/c$. Multiplying by the common positive term factor $1-e^{-cn}$ gives $a_{x:\overline{n}|}<\bar{a}_{x:\overline{n}|}$.

Compare continuous with annual due.
For $c>0$, $e^{-c}>1-c$, so $1-e^{-c}<c$. Thus $1/c<1/(1-e^{-c})$. Multiplying by the same positive term factor gives $\bar{a}_{x:\overline{n}|}<\ddot{a}_{x:\overline{n}|}$.

Interpret the result.
The later annual-immediate dates tend to reduce present value relative to a stream paid throughout each year. The annuity-due starts with a certain payment at time 0, so it has the greatest value under these matched assumptions.

The inequalities rely on $c>0$, level benefits, and the stated timing conventions. If $c=0$, the finite-term values all equal $n$; for whole life, the values are not finite. A payment guarantee, a different benefit rate, or a different payment schedule changes the comparison and must be reflected in the cash flows.

Worked Example: Comparing Five-Year Temporary Annuities

Use the same hypothetical mortality and interest assumptions as in the preceding example: $\delta=0.04$ and $\mu=0.06$, each constant. Compare the present values of a five-year annual annuity-immediate, a five-year annual annuity-due, and a five-year continuous annuity, all with a benefit rate of one unit per year.

Method. First combine the forces: $c=0.04+0.06=0.10$. Since the annuities end after five years, use $n=5$ and calculate the common numerator: $$ 1-e^{-cn}=1-e^{-0.10(5)} =1-e^{-0.5} =1-0.606531 =0.393469. $$

Annual annuity-immediate. Payments are at times 1, 2, 3, 4, and 5 if the life is then alive: $$ a_{x:\overline{5}|} =\frac{0.393469}{e^{0.10}-1} =\frac{0.393469}{0.105170} =3.7413. $$

Annual annuity-due. Payments are at times 0, 1, 2, 3, and 4 if the life is then alive: $$ \ddot{a}_{x:\overline{5}|} =\frac{0.393469}{1-e^{-0.10}} =\frac{0.393469}{0.095163} =4.1347. $$

Continuous annuity. The payment stream runs from time 0 up to time 5: $$ \bar{a}_{x:\overline{5}|} =\frac{0.393469}{0.10} =3.9347. $$

The values are $3.7413<3.9347<4.1347$, consistent with the timing result. The calculation uses the same five-year endpoint for each pattern: the final annual-immediate payment is at year 5, whereas the last annual-due payment is at year 4. Omitting either endpoint distinction would value a different contract.

Translating Values into Payment Amounts

An actuarial annuity value is a present value per unit of the stated payment amount or payment rate. If the benefit is a constant $B$ dollars per year, multiply the corresponding unit value by $B$. For a continuous annuity, $B$ is a payment rate per year, not a lump sum paid once per year.

Worked Example: Dollar Values for a Two-Year Benefit

A hypothetical contract pays a life annuity for at most two years at a rate of \$1,000 per year. Use constant forces $\delta=0.04$ and $\mu=0.06$. Find the expected present value for each of three schedules: \$1,000 at the end of each year while alive, \$1,000 at the beginning of each year while alive, and a continuous rate of \$1,000 per year while alive.

Set up the model. The forces combine to give $c=0.04+0.06=0.10$. The annual survival probabilities used at years 1 and 2 are ${}_1p_x=e^{-0.06}=0.941765$ and ${}_2p_x=e^{-0.12}=0.886920$. The annual discount factors are $e^{-0.04}=0.960789$ and $e^{-0.08}=0.923116$. The expected discounted unit payments at those times are therefore $e^{-0.10}=0.904837$ and $e^{-0.20}=0.818731$.

Annual annuity-immediate. There are two possible payments, at years 1 and 2: $$ 1{,}000(e^{-0.10}+e^{-0.20}) =1{,}000(0.904837+0.818731) =1{,}723.57. $$

Annual annuity-due. The payment at time 0 is certain, and the payment at time 1 is made only if the life survives: $$ 1{,}000(1+e^{-0.10}) =1{,}000(1+0.904837) =1{,}904.84. $$

Continuous annuity. The expected present value is the rate multiplied by the two-year continuous unit value: $$ 1{,}000\int_0^2 e^{-0.10t}\,dt =1{,}000\left(\frac{1-e^{-0.20}}{0.10}\right) =1{,}000(1.812692) =\$1{,}812.69. $$

Rounded to the nearest cent, the values are \$1,723.57, \$1,812.69, and \$1,904.84 for the immediate, continuous, and due schedules, respectively. They have the expected ordering. As a check, the immediate value uses two payments, the due value includes a certain payment at time 0, and the continuous value uses a payment rate throughout the two-year interval.

Common Mistakes and Practical Checks

The most common error is to compare formulas that do not describe the same timing or term. A useful check is to write down every payment date, or the interval over which a continuous rate is paid, before calculating. Then confirm that survival and discounting are both measured from the valuation date to that payment time.

Using the due formula for an immediate annuity. The due value includes a payment at time 0. The immediate value does not.
Adding a time-0 payment to a continuous stream. Continuous payments have no separate lump-sum payment at time 0; the value of a single instant is zero in the integral.
Using the wrong term endpoint. A temporary annual-immediate annuity has payments at $1,\ldots,n$; a temporary annual-due annuity has payments at $0,\ldots,n-1$.
Mixing annual interest with a force. If the effective annual rate $i$ is given, the constant force is $\delta=\ln(1+i)$, not $i$ itself. Use a consistent interest model.
Confusing annual amount with continuous rate. A continuous benefit of $B$ dollars per year accumulates over time; it is not a payment of $B$ dollars at every instant.

These comparisons are useful when translating between contract designs or checking a valuation. They are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus: the key is to specify the payment pattern, condition each payment on survival, and discount it to the valuation date under the stated interest assumption.

Check Your Understanding

For each problem, state the timing convention before comparing or calculating values. Unless otherwise stated, assume level benefits and no payment guarantee.

A hypothetical whole-life annuity has constant force of interest $\delta=0.03$ and constant force of mortality $\mu=0.07$. Find the unit whole-life values for the annual-immediate, continuous, and annual-due schedules, and give their order. (Answer: $a_x=9.5083,\ \bar{a}_x=10.0000,\ \ddot{a}_x=10.5083$; $a_x<\bar{a}_x<\ddot{a}_x$.)
For a hypothetical three-year temporary annuity, let $\delta=0.04$, $\mu=0.06$, and the payment rate be one unit per year. Find the continuous annuity value. (Answer: $(1-e^{-0.30})/0.10=2.5918$.)
A hypothetical whole-life annuity has $\delta=0.02$ and $\mu=0.08$, both constant, and pays \$500 at each year-end while the life is alive. Find its expected present value. (Answer: $500/(e^{0.10}-1)=\$4{,}754.17$.)
A two-year hypothetical annuity has $\delta=0.04$, $\mu=0.06$, and a continuous payment rate of \$800 per year. Find its expected present value and identify the time interval of payment. (Answer: $800(1-e^{-0.20})/0.10=\$1{,}450.15$, with payments over $0\le t<2$.)
For a one-year hypothetical temporary life annuity, $\delta=0.10$ and $\mu=0.10$. Compare the unit annual-immediate, continuous, and annual-due values. Explain why the immediate value is not 1. (Answer: $a_{x:\overline{1}|}=e^{-0.20}=0.8187,\ \bar{a}_{x:\overline{1}|}=(1-e^{-0.20})/0.20=0.9063,\ \ddot{a}_{x:\overline{1}|}=1$; the immediate payment is discounted and contingent on survival to year 1.)

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