Annuity-Immediate Cash Flows at Integer Ages

What an Annuity-Immediate Cash Flow Means

An annuity-immediate makes payments at the ends of successive payment periods. In a life-contingent version, a scheduled payment is made only if the annuitant is alive at its payment date. The first possible payment is therefore at the end of year 1, not at the valuation date. To value the stream, each payment must be adjusted for both the chance that it is payable and the time value of money.

This tutorial considers annual payments at integer policy-year ends. Assume a fixed effective annual interest rate $i$, with $v=1/(1+i)$. Let the annuitant’s age at the valuation date be $x$, and let ${}_kp_x$ denote the probability that a life aged $x$ survives for at least $k$ years. A scheduled amount $C_k$ is paid at the end of year $k$ if the annuitant is alive then. We take “alive then” to mean alive immediately before that payment is made.

Definition: For a payment schedule with possible payments $C_1,C_2,\ldots,C_K$ at the ends of years $1,2,\ldots,K$, the expected present value is the expected value, at the valuation date, of the payments that are actually made. Under a fixed annual effective interest rate and the survival-contingent payment rule, it is $$ \operatorname{EPV}=\sum_{k=1}^{K} C_k v^k\,{}_kp_x. $$

The survival probability in the $k$th term is the probability of being alive at the date of that payment. It is not the probability of dying during year $k$. The discount factor is also tied to that same date: a payment at the end of year $k$ is discounted by $v^k$, not by $v^{k-1}$.

Why the Formula Works

For each payment date $k$, define $I_k$ to be 1 if the annuitant is alive at the end of year $k$, and 0 otherwise. The present value of the payment due at that date is $C_kv^kI_k$. The total present value is the sum of these individual present values. Taking expectations gives the formula directly.

\begin{aligned} \operatorname{EPV} &=E\!\left[\sum_{k=1}^{K}C_kv^kI_k\right]\\ &=\sum_{k=1}^{K}C_kv^kE[I_k]\\ &=\sum_{k=1}^{K}C_kv^kP(\text{alive at time }k)\\ &=\sum_{k=1}^{K}C_kv^k\,{}_kp_x. \end{aligned}

The second line uses linearity of expectation. It does not require the survival indicators at different dates to be independent; in fact, they are linked because survival to a later date implies survival to every earlier date. The result assumes the interest rate is deterministic, the payment amounts are specified, and there are no additional conditions such as expenses or a guarantee after death. If a contract uses a different payment rule, the indicator and therefore the expected value must be modified to reflect that rule.

Set out the payment dates.
For each scheduled payment, record its year $k$ and amount $C_k$. The first possible payment is at year 1.

Find survival to each date.
Use ${}_kp_x$, or multiply the successive conditional one-year survival probabilities to obtain it.

Discount and weight each payment.
Multiply $C_k$ by $v^k$ and by ${}_kp_x$, then sum the terms.

Worked Calculations

Worked Example: Equal Payments with Year-End Survival Probabilities

A hypothetical contract pays $1,000 at the end of each of the next three years if the annuitant is alive at the payment date. The effective annual interest rate is 25%. The survival probabilities are ${}_1p_x=0.90$, ${}_2p_x=0.80$, and ${}_3p_x=0.70$. Find the expected present value.

Method. Apply the survival-weighted discount factor separately at each payment date. The survival probabilities are cumulative from the valuation date, so use them directly; do not multiply them together again.

First calculate the discount factor and its powers:

v=\frac{1}{1.25}=0.8,\qquad v^2=0.64,\qquad v^3=0.512.

The three expected present-value contributions are:

\begin{aligned} \text{Year 1: }&\$1{,}000(0.8)(0.90)=\$720.00,\\ \text{Year 2: }&\$1{,}000(0.64)(0.80)=\$512.00,\\ \text{Year 3: }&\$1{,}000(0.512)(0.70)=\$358.40. \end{aligned}

Adding the contributions gives $\$720.00+\$512.00+\$358.40=\$1,590.40$. The expected present value is $1,590.40. As a check, the year-3 contribution is less than the year-2 contribution because both the survival probability and the discount factor are lower at year 3.

Worked Example: Unequal Payments and Conditional Survival

A hypothetical benefit pays $800 at the end of year 1, $1,200 at the end of year 2, and $1,600 at the end of year 3, provided the annuitant is alive at each payment date. The effective annual interest rate is 25%. The conditional probabilities of surviving each successive year, given survival to the beginning of that year, are 0.90, 0.80, and 0.875. Calculate the expected present value.

Method. First convert the conditional one-year survival probabilities into survival probabilities from the valuation date. Then pair each cumulative probability with the corresponding payment and discount factor.

The year-end survival probabilities are:

{}_1p_x=0.90,\qquad {}_2p_x=(0.90)(0.80)=0.72,\qquad {}_3p_x=(0.90)(0.80)(0.875)=0.63.

At 25% interest, $v=0.8$, $v^2=0.64$, and $v^3=0.512$. Thus the payment contributions are:

\begin{aligned} \text{Year 1: }&\$800(0.90)(0.8)=\$576.00,\\ \text{Year 2: }&\$1{,}200(0.72)(0.64)=\$552.96,\\ \text{Year 3: }&\$1{,}600(0.63)(0.512)=\$516.096. \end{aligned}

Using unrounded contributions, the total is $\$576.00+\$552.96+\$516.096=\$1,645.056$, which rounds to $1,645.06 to the nearest cent. The survival probabilities decrease from 0.90 to 0.72 to 0.63, as required for survival to progressively later dates.

Worked Example: Recovering a Conditional Survival Probability

A hypothetical annuity pays $500, $1,000, and $1,500 at the ends of years 1, 2, and 3, respectively, if the annuitant is alive at the relevant payment date. The effective annual interest rate is 25%. The conditional survival probabilities for years 1 and 2 are 0.90 and 0.80. The expected present value is $1,235.52. Find the conditional probability of surviving year 3, given survival to its beginning.

Method. Write the cumulative survival probabilities using the unknown conditional probability, $r$, and solve the expected-present-value equation. This avoids treating a conditional probability as though it were already cumulative.

The survival probabilities to the three payment dates are $0.90$, $(0.90)(0.80)=0.72$, and $0.72r$. With $v=0.8$, the first two expected present-value contributions are:

\$500(0.90)(0.8)=\$360.00,\qquad \$1{,}000(0.72)(0.64)=\$460.80.

The year-3 contribution is:

\$1{,}500(0.72r)(0.512)=\$552.96r.

Equate the sum to the given expected present value and solve:

\begin{aligned} 360+460.80+552.96r&=1{,}235.52,\\ 552.96r&=414.72,\\ r&=\frac{414.72}{552.96}=0.75. \end{aligned}

The conditional probability of surviving year 3 is 0.75. Checking, the cumulative survival probability to year 3 is $0.72(0.75)=0.54$, and the third contribution is $\$1,500(0.54)(0.512)=\$414.72$. Then $\$360.00+\$460.80+\$414.72=\$1,235.52$, as required.

Common Exam Trap: Using Death Probability or the Wrong Year

A frequent error is to multiply a scheduled payment by the probability of dying during the year rather than the probability of surviving to its payment date. Another is to discount a year-2 payment by $v$ instead of $v^2$. Both errors mismatch the payment condition or timing.

For example, in the first worked example, the year-2 payment is due only if the annuitant survives two years. The correct contribution is $\$1,000(0.64)(0.80)=\$512$. Using the one-year death probability $1-0.90=0.10$ and discounting only one year gives the incorrect amount $\$1,000(0.10)(0.80)=\$80$. That calculation weights the payment by a death event and places it at the wrong time. The correction is to use the year-2 survival probability, $0.80$, and the two-year discount factor, $v^2=0.64$.

Always check that each term uses the same year $k$ for all three items: payment amount $C_k$, discount factor $v^k$, and survival probability ${}_kp_x$. If conditional one-year survival probabilities are supplied, multiply them through time to obtain ${}_kp_x$ before valuing the payments.

Key takeaway: A payment made at the end of year $k$ while the annuitant is alive contributes $C_kv^k\,{}_kp_x$ to expected present value. Sum one such term for every scheduled payment date.

Check Your Understanding

For each problem, use the payment date to select both the discount factor and the survival probability.

A hypothetical annuity pays $200 at the ends of years 1 and 2 if the annuitant is alive. The effective annual interest rate is 10%, with ${}_1p_x=0.80$ and ${}_2p_x=0.60$. Find the expected present value. (Answer: $244.63)
A benefit pays $400 at year 1 and $600 at year 2. The effective annual interest rate is 25%; the conditional survival probabilities for years 1 and 2 are 0.90 and 0.80. Find the expected present value. (Answer: $564.48)
At 0% interest, a hypothetical annuity pays $1,000 at each of years 1 through 4 if the annuitant is alive. The conditional survival probabilities in those years are 0.90, 0.80, 0.75, and 0.50. Find the expected present value. (Answer: $2,430.00)
At 25% effective annual interest, a contract pays $1,000 at the ends of years 1, 2, and 3 if the annuitant is alive. The cumulative survival probabilities are 0.80, 0.70, and 0.50. Find the expected present value. (Answer: $1,344.00)
A hypothetical annuity pays $1,000 at the ends of years 1 and 2, and an unknown amount $X$ at the end of year 3, provided the annuitant is alive at each date. The effective annual interest rate is 25%, the cumulative survival probabilities are 0.90, 0.80, and 0.50, and the expected present value is $1,300. Find $X$. (Answer: $265.63)

What You'll Learn