Why Survival and Interest Both Matter
A life annuity provides payments according to a schedule, but a payment may be made only if the annuitant is alive at its scheduled time. Its expected present value therefore depends on two separate questions: how much is the payment worth at the valuation date after discounting for interest, and how likely is it to be paid? A payment far in the future is generally discounted more, and it may also have a smaller probability of being received.
The basic method is to consider each scheduled payment separately. Discount its amount to the valuation date, multiply by the probability that the payment condition is met, and add the resulting expected present values. This approach works for equal or unequal payments and for schedules with gaps. It is the foundation for more specialized life annuity valuations.
For a finite schedule, the random present value \(Z\) is the sum of the individual payment present values. Its expected value is
This expression assumes the payment amount and schedule are fixed, interest is deterministic, and each listed payment is made if and only if the annuitant is alive at its scheduled time. A payment guaranteed regardless of survival needs no survival weighting. If benefits depend on other events, those payment conditions must also be reflected in the expected value.
Why the Formula Works
For one scheduled payment, the present value is a fixed discounted amount multiplied by an indicator. The indicator is 1 with probability \({}_{t_j}p_x\) and 0 otherwise. Thus its expected value is the discounted amount times that probability. Adding across payment dates gives the expected value of the total present value.
At time \(t_j\), a payment of \(b_j\) has present value \(b_jv^{t_j}I_j\).
Since \(I_j=1\) exactly when the annuitant survives to the payment time, \(E[I_j]={}_{t_j}p_x\). Therefore \(E[b_jv^{t_j}I_j]=b_jv^{t_j}{}_{t_j}p_x\).
Expectation is linear, so the expected value of a finite sum is the sum of the expected values. Independence between survival indicators is not required.
The survival probability in the formula is cumulative from the valuation age to the payment time. If the one-year conditional survival probabilities are given separately, multiply them along the path. For example, if the probabilities of surviving successive years are \(p_0,p_1,p_2\), then survival through year 3 is \(p_0p_1p_2\). A conditional one-year probability should not be mistaken for survival from the valuation date.
Applying Survival-Weighted Discounting
The order of calculation is flexible because the amount, discount factor, and survival probability are multiplied. A reliable workflow is to list each payment date, determine cumulative survival to that date, calculate the discount factor, and then calculate the expected present value for that row. Payments at time zero, if any, have discount factor 1 and survival probability 1 when the annuitant is alive at valuation.
Worked Example: Equal Payments with Constant Annual Survival
A hypothetical contract pays $1,000 at the end of each of the next three years, provided the annuitant is alive on the payment date. The effective annual interest rate is 25%. The probability of surviving each year, conditional on being alive at its beginning, is 0.80. Find the expected present value. Use the stated annual probabilities and round the final answer to the nearest cent.
Method. First find cumulative survival probabilities. With the same conditional survival probability each year, survival to years 1, 2, and 3 is \(0.80\), \(0.80^2=0.64\), and \(0.80^3=0.512\). The discount factor is \(v=1/1.25=0.80\), so the discount factors for years 1, 2, and 3 are \(0.80\), \(0.64\), and \(0.512\).
Year 1. The expected present value is \(1{,}000(0.80)(0.80)=\$640.00\).
Year 2. The expected present value is \(1{,}000(0.64)(0.64)=\$409.60\).
Year 3. The expected present value is \(1{,}000(0.512)(0.512)=\$262.144\).
Adding the three amounts gives \(\$640.00+\$409.60+\$262.144=\$1{,}311.744\), or $1,311.74 to the nearest cent. As a check, each successive expected payment value is \(0.80\times0.80=0.64\) times the preceding one, because both survival and discount factors are multiplied by 0.80 each year.
Worked Example: Unequal Survival Probabilities and Payment Amounts
A hypothetical annuity makes payments of $700, $900, and $1,100 at the ends of years 1, 2, and 3, respectively, if the annuitant is alive on each payment date. The successive conditional probabilities of surviving years 1, 2, and 3 are 0.95, 0.90, and 0.80. The effective annual interest rate is 5%. Calculate the expected present value, rounding to the nearest cent.
Method. The cumulative probabilities of being alive at the three payment dates are \(0.95\), \(0.95(0.90)=0.855\), and \(0.95(0.90)(0.80)=0.684\). The discount factors are \(v=1/1.05\), \(v^2=1/1.05^2=1/1.1025\), and \(v^3=1/1.05^3=1/1.157625\).
Year 1 payment. Its expected present value is \(700(0.95)/1.05=665/1.05=\$633.3333\ldots\).
Year 2 payment. Its expected present value is \(900(0.855)/1.1025=769.5/1.1025=\$697.9592\ldots\).
Year 3 payment. Its expected present value is \(1{,}100(0.684)/1.157625=752.4/1.157625=\$649.9514\ldots\).
The total is \(\$633.3333\ldots+\$697.9592\ldots+\$649.9514\ldots=\$1{,}981.2439\ldots\), or $1,981.24 to the nearest cent. The method avoids treating each conditional one-year survival probability as though it were the probability of surviving from the valuation date.
Worked Example: A Schedule with a Gap Between Payments
A hypothetical annuity pays $400 at the end of year 1, $600 at the end of year 2, and $900 at the end of year 4, provided the annuitant is alive at each payment time. The conditional probabilities of surviving years 1 through 4 are 0.90, 0.80, 0.75, and 0.50. The effective annual interest rate is 4%. Find the expected present value, rounded to the nearest cent.
Method. The cumulative survival probabilities at years 1, 2, and 4 are \(0.90\), \(0.90(0.80)=0.72\), and \(0.90(0.80)(0.75)(0.50)=0.27\). There is no payment in year 3, but survival through year 3 is still part of the probability of reaching the year 4 payment date. The discount factors are \(1/1.04\), \(1/1.04^2=1/1.0816\), and \(1/1.04^4=1/1.16985856\).
Year 1 payment. Its expected present value is \(400(0.90)/1.04=360/1.04=\$346.1538\ldots\).
Year 2 payment. Its expected present value is \(600(0.72)/1.0816=432/1.0816=\$399.4083\ldots\).
Year 4 payment. Its expected present value is \(900(0.27)/1.16985856=243/1.16985856=\$207.7174\ldots\).
The total is \(\$346.1538\ldots+\$399.4083\ldots+\$207.7174\ldots=\$953.2795\ldots\), or $953.28 to the nearest cent. The gap in payments does not shorten the survival period: the year 4 payment is still contingent on survival through all four years.
Timing, Assumptions, and Common Errors
A survival-weighted valuation is only as accurate as the payment timing and survival event used. The time \(t_j\) in the discount factor must be measured from the valuation date in years when \(i\) is an effective annual rate. The survival probability must refer to being alive at that same payment time. For fractional-year payment times, \(v^{t_j}\) uses the fractional power under the effective annual interest convention; the appropriate fractional-time survival probability must also be supplied by the survival model or contract assumptions.
If mortality is described by a force \(\mu\) that is constant over the interval, the probability of surviving \(t\) years is \(e^{-\mu t}\). If the force of interest is a constant \(\delta\), the discount factor is \(e^{-\delta t}\). In that particular model, the expected present value of a payment \(b\) at time \(t\), contingent on survival, is \(be^{-(\mu+\delta)t}\). This compact expression requires both constant-force assumptions; it should not be substituted for given annual survival probabilities or an effective annual interest rate without justification.
A frequent exam trap is discounting a payment correctly but using only the last year’s conditional survival probability. Suppose the successive annual survival probabilities are 0.95, 0.90, and 0.80. A payment at the end of year 3 has probability \(0.95(0.90)(0.80)=0.684\) of being paid, not 0.80. The incorrect calculation weights the payment by \(0.80\) and overstates its expected present value. The correction is to multiply all conditional survival probabilities from the valuation date through the payment date before applying the discount factor.
Another error is applying a survival factor to every cash flow automatically. A payment guaranteed at a specified time even if the annuitant has died is not multiplied by the probability of survival. Conversely, a payment conditional on being alive must be survival-weighted even if the payment amount is certain. The contract’s payment condition, rather than the label on the cash flow, determines the indicator.
| Cash flow condition | Survival factor | Expected present value of payment \(b\) at time \(t\) |
|---|---|---|
| Paid only if alive at time \(t\) | \({}_tp_x\) | \(bv^t{}_tp_x\) |
| Guaranteed at time \(t\), regardless of survival | None | \(bv^t\) |
| Paid only if alive at an earlier date \(s\), with \(s\le t\) | \({}_sp_x\), if that is the full condition | \(bv^t{}_sp_x\) |
For a long or infinite schedule, the same reasoning applies term by term, but the sum must converge for the expected present value to be finite. In practical calculations, the payment schedule, discount basis, and survival probabilities should all be stated explicitly. This makes it possible to audit each term and to detect whether a payment is being counted at the wrong date.
Check Your Understanding
For each problem, identify the survival probability appropriate to each payment date before combining it with discounting.
- A hypothetical annuity pays $100 at the ends of years 1 and 2 if the annuitant is alive. The annual effective interest rate is 10%, and the conditional probability of surviving each year is 0.90. Find its expected present value to the nearest cent. (Answer: $148.76)
- A contract pays $500 at the end of year 1 and $1,000 at the end of year 2, each only if the annuitant is alive. The effective annual interest rate is 25%; the conditional survival probabilities in years 1 and 2 are 0.80 and 0.75. Find the expected present value. (Answer: $704.00)
- A hypothetical annuity pays $200 at the end of year 1 and $600 at the end of year 3, if the annuitant is alive on each payment date. The effective annual interest rate is 0%; the conditional survival probability in each year is 0.90. Find the expected present value. (Answer: $617.40)
- A life annuity makes payments of $1,000 at times 0.5 and 1.5 years, each contingent on survival to its payment time. The force of interest is constant at 0.04 and the force of mortality is constant at 0.03. Calculate the expected present value to the nearest cent. (Answer: $1,865.93)
- A hypothetical contract pays $500 at the end of year 1 regardless of whether the annuitant is alive, and $800 at the end of year 2 only if the annuitant is alive at that time. The effective annual interest rate is 5%, and the conditional survival probabilities in years 1 and 2 are 0.90 and 0.80. Find the expected present value to the nearest cent. (Answer: $998.64)