Choosing a Valuation Method
A life-annuity problem can often be solved in more than one way. The best method depends on two features of the information given: when payments are made and how survival is described. Payment dates determine the discount factors and survival probabilities to use. Mortality information may be supplied as conditional survival probabilities, life-table columns, or a survival model suitable for integration.
A reliable first step is to list the possible payment times and ask whether a payment is made at each time only if the life is then alive. Once those dates and conditions are explicit, the valuation method usually follows directly. This approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Let \(i\) be the effective annual interest rate and \(v=1/(1+i)\). For a life aged \(x\), \({}_tp_x\) is the probability of surviving \(t\) years. The whole-life annual annuity-due value, when the payments are one unit, can be written as \(\ddot{a}_x=\sum_{k=0}^{\infty}v^k\,{}_kp_x\). For an annual annuity-immediate, the corresponding expression is \(a_x=\sum_{k=1}^{\infty}v^k\,{}_kp_x\). These expressions are useful summaries; in a finite problem, writing out the payment dates is often clearest.
Begin with Payment Dates and Survival Events
For each possible payment, identify its time, amount, and survival condition. A payment at time \(k\) is discounted by \(v^k\), and, if it is conditional on being alive then, its expected value is multiplied by \({}_kp_x\). In a due annuity, the payment at time zero is certain under the usual assumption that the annuitant is alive at issue. In an immediate annuity, the first payment is at time one and depends on survival for one year.
Worked Example: Direct Summation for an Annual Annuity-Due
A hypothetical three-year annuity pays $1,000 at times 0, 1, and 2, provided the annuitant is alive at each payment time. The annual effective interest rate is \(i=5\%\). Conditional on survival to the start of each year, the probabilities of surviving the next year are 0.90, 0.80, and 0.75. Find the expected present value.
Method. Directly sum the expected present value of each possible payment. Only the first two conditional survival probabilities are needed because the final possible payment is at time 2.
The survival probabilities to payment times 0, 1, and 2 are \(1\), \(0.90\), and \(0.90(0.80)=0.72\). The discount factor is \(v=1/1.05=20/21\). Therefore,
The value exceeds $2,000 because the time-zero payment is certain and expected payments at times 1 and 2 are discounted, but remain substantial. The common mistake this method avoids is treating the conditional probability of surviving year 2, 0.80, as the probability of surviving from issue to time 2. The latter is the product \(0.90(0.80)=0.72\). The calculation is checked by confirming that the time-2 term uses both years of survival and two years of discounting.
When Life-Table Columns Are Given
A life table gives the number of lives \(\ell_y\) surviving to age \(y\) in a hypothetical cohort. Conditional on being alive at age \(x\), the probability of surviving \(k\) years is \({}_kp_x=\ell_{x+k}/\ell_x\), provided \(\ell_x>0\). Thus, a life-table problem can be converted into the same payment-by-payment sum as a direct-summation problem.
Worked Example: Using Life-Table Ratios for an Annuity-Immediate
A hypothetical three-year annuity-immediate pays $500 at the end of each year if the annuitant is alive then. The effective annual interest rate is \(i=4\%\). A life table gives \(\ell_{60}=10{,}000\), \(\ell_{61}=9{,}800\), \(\ell_{62}=9{,}500\), and \(\ell_{63}=9{,}100\). Find the expected present value for a life aged 60.
Method. Convert each life-table column to a survival probability from age 60, then discount the payment at the matching time. The probabilities are \({}_1p_{60}=9{,}800/10{,}000=0.98\), \({}_2p_{60}=9{,}500/10{,}000=0.95\), and \({}_3p_{60}=9{,}100/10{,}000=0.91\). Here \(v=1/1.04=25/26\).
The intermediate sum is approximately \(2.6296227823\), so multiplying by $500 gives approximately $1,314.811391 before rounding to cents. The common mistake avoided is dividing each later \(\ell\)-value by the previous year’s \(\ell\)-value and then using that one-year probability as though it were survival from age 60. The required probabilities are ratios to \(\ell_{60}\). Each term also uses the same time index for survival and discounting.
Decomposing a Payment Stream
A payment stream can sometimes be valued more efficiently by expressing it as a combination of simpler streams. For example, payments at times 2, 3, and 4 can be obtained by taking a five-payment annuity-due at times 0 through 4 and subtracting a two-payment annuity-due at times 0 and 1. This subtraction works because the overlapping early payments cancel exactly. Alternatively, the deferred stream can be valued conditionally at time 2 and then discounted and weighted by the probability of reaching time 2.
Worked Example: Decomposing a Deferred Three-Payment Annuity
A hypothetical annuity pays $1,000 at times 2, 3, and 4 if the annuitant is alive at the respective payment time. The annual effective interest rate is \(i=5\%\). The conditional probabilities of surviving successive years are 0.90, 0.80, 0.75, and 0.70. Find the expected present value, using a decomposition.
Method. The survival probabilities to times 2, 3, and 4 are \(0.90(0.80)=0.72\), \(0.72(0.75)=0.54\), and \(0.54(0.70)=0.378\). With \(v=20/21\), the direct payment-date expression is \(1000(0.72v^2+0.54v^3+0.378v^4)\).
For the decomposition, first value the five-payment annuity-due at times 0 through 4 and subtract the two-payment annuity-due at times 0 and 1. The five-payment value, in units of $1,000, is
The two-payment value in the same units is \(1+0.90v\). Subtracting cancels the payments at times 0 and 1, leaving
As a check, the same three remaining terms come from the direct payment-date expression, and their sum is approximately \(1.4305150671\). The common mistake avoided is subtracting only the nominal amounts of the early payments without subtracting their expected present values. The decomposition must use the same survival and discount assumptions in both component streams.
A second useful form is to condition on survival to time 2. Given survival to time 2, the conditional survival probabilities for the next two years are 0.75 and \(0.75(0.70)=0.525\). The time-2 value of the three payments per $1,000 is \(1+0.75v+0.525v^2\). Discounting back and allowing for survival to time 2 gives \(1000(0.72)v^2(1+0.75v+0.525v^2)\), which is the same expression.
When Payments Are Continuous
For a continuous life annuity paying at a constant rate \(b\) per year while the life is alive, each small payment over an interval of length \(dt\) has amount approximately \(b\,dt\). If the survival function from age \(x\) is \({}_tp_x\), and the force of interest is constant at \(\delta\), the expected present value over a term of \(n\) years is
The reason is that a payment at time \(t\) is made only if the life survives to \(t\). Its expected amount is therefore \(b\,dt\,{}_tp_x\), and its discount factor is \(e^{-\delta t}\). Integrating these expected discounted amounts over all possible payment times gives the value. This expression requires a suitable survival function and an integrable payment stream; it is not a substitute for an annual sum when the contract pays only at discrete dates.
Worked Example: Integration Under Constant Forces
A hypothetical continuous annuity pays at a rate of $1,000 per year while the annuitant is alive, for at most five years. Mortality follows a constant force \(\mu=0.04\), so \({}_tp_x=e^{-0.04t}\). Interest follows a constant force \(\delta=0.06\), equivalent to an effective annual rate \(i=e^{0.06}-1\). Find the expected present value.
Method. The joint discount-and-survival factor is \(e^{-0.06t}e^{-0.04t}=e^{-0.10t}\). Therefore,
The check is that the integral is positive and less than the five-year certain-payment value \(1000\int_0^5e^{-0.06t}\,dt\), because survival reduces expected payments. The common mistake avoided is adding \(\mu\) to the effective annual rate \(i\). The calculation uses the force of interest \(\delta\) and the force of mortality \(\mu\), so the combined exponent is \(\delta+\mu=0.10\).
Checks That Keep the Method on Track
The following checks help identify whether the selected method matches the contract and the data. They are particularly useful when a problem combines a deferment, a finite term, or an unusual payment frequency.
Write time 0 for a due payment and time 1 for the first immediate payment. For continuous payments, specify the integration interval.
Use \({}_kp_x\) for a payment at time \(k\). From a life table, calculate this as \(\ell_{x+k}/\ell_x\), not as a one-year ratio unless \(k=1\).
A payment at time \(k\) receives \(v^k\); a continuous payment at time \(t\) receives \(e^{-\delta t}\) under constant force of interest.
Sum individual terms when the dates are few, convert life-table columns to survival probabilities, decompose matching segments, or integrate a continuous stream under a specified model.
A compact way to justify the annual formula is to define an indicator \(I_k\) that equals 1 if the annuitant is alive at time \(k\), and 0 otherwise. A payment \(b\) at time \(k\) has present value \(bv^kI_k\). Since \(E[I_k]=P(I_k=1)={}_kp_x\), its expected present value is \(bv^k{}_kp_x\). Adding the expected values across the payment dates gives the direct-summation formula. This also explains why the value is based on marginal survival probabilities at each payment date, rather than on multiplying all payment indicators together.
Check Your Understanding
For each problem, identify the appropriate valuation structure and keep the survival time and discount time aligned.
- A hypothetical three-payment annuity-due pays $200 at times 0, 1, and 2 if the annuitant is alive. The effective annual interest rate is 4%, and conditional one-year survival probabilities are 0.95 and 0.80. Find the expected present value. (Answer: $523.22)
- A two-year annuity-immediate pays $1,000 at times 1 and 2 if the annuitant is alive. At age 50, a life table has \(\ell_{50}=5{,}000\), \(\ell_{51}=4{,}800\), and \(\ell_{52}=4{,}500\). At \(i=5\%\), find the value. (Answer: $1,730.61)
- A deferred annuity pays $400 at times 2, 3, and 4 if the annuitant is alive. Conditional annual survival probabilities for years 1 through 4 are 0.90, 0.80, 0.75, and 0.80. At \(i=4\%\), use a deferred-segment calculation to find the expected present value. (Answer: $606.01)
- A continuous annuity pays $600 per year while alive for four years. The force of mortality is constant at \(\mu=0.03\), and the force of interest is constant at \(\delta=0.05\). Find its expected present value to the nearest cent. (Answer: $2,053.88)