Annuity-Due Cash Flows at Integer Ages

What Makes an Integer-Age Cash Flow Annuity-Due?

An annuity-due cash flow makes payments at the beginnings of policy years, provided the annuitant is alive when each payment is due. At integer policy time $k$, the payment is $k$ years after issue. Thus, a payment at time 0 is made immediately, while a payment at time 1 is made one year after issue. This timing matters: the first payment is not discounted, and later payments are discounted according to how many full years after issue they occur.

The survival condition matters just as much. A payment scheduled for time $k$ is included only if the annuitant survives to that payment time. The expected present value therefore combines two weights for each payment: the probability of being alive when it is due and the interest discount factor.

Definition: Suppose an annuitant is age $x$ at issue, and $c_k$ is the payment scheduled at integer time $k$, for $k=0,1,\ldots,n$. If the payment is made only when the annuitant is alive at time $k$, its expected present value is $$ \operatorname{EPV}=\sum_{k=0}^{n} c_k v^k\,{}_kp_x, $$ where $v=1/(1+i)$, $i$ is the effective annual interest rate, and ${}_kp_x$ is the probability that a life aged $x$ survives for $k$ years. In particular, ${}_0p_x=1$.

The payment $c_k$ may vary by year, and some scheduled payments may be zero. This form makes the timing and survival condition explicit for each cash flow. It applies when interest is represented by the stated annual effective rate and the payment amounts and times are known. For an indefinite sequence of payments, the corresponding infinite sum can be used only when it converges.

Getting Survival Weights from Annual Mortality

Let $q_{x+j}$ be the probability that a life aged $x+j$ dies within the following year, conditional on being alive at the start of that year. Then the one-year survival probability is $1-q_{x+j}$. To survive $k$ years, the life must survive each of the $k$ successive years, so the survival probability is the product of those one-year probabilities:

{}_kp_x=\prod_{j=0}^{k-1}(1-q_{x+j}),\qquad k\geq 1.

This product uses a different age-specific mortality probability for each year when the probabilities vary by age. If a life table gives the number of survivors $\ell_x$ at age $x$, the equivalent survival weight is ${}_kp_x=\ell_{x+k}/\ell_x$, provided both columns refer to the same life table and basis. Either method gives the probability needed for the payment at time $k$.

Why the Survival-Weighted Sum Gives the EPV

For each scheduled payment, consider whether it is actually paid. Let $I_k$ equal 1 if the annuitant is alive at time $k$, and 0 otherwise. The present value of the entire finite payment stream is the sum of the discounted payments that occur. The expectation of that sum gives the EPV.

Represent each contingent payment.
The payment at time $k$, if made, has present value $c_kv^k$. Including the survival indicator, its random present value is $c_kv^kI_k$.

Take the expectation of one payment.
Since $I_k$ is 1 with probability ${}_kp_x$ and 0 otherwise, $E[I_k]={}_kp_x$. Therefore $E[c_kv^kI_k]=c_kv^k{}_kp_x$.

Add the expected values.
For a finite schedule, linearity of expectation gives $E[\sum_{k=0}^{n}c_kv^kI_k]=\sum_{k=0}^{n}c_kv^k{}_kp_x$. This step does not require the survival indicators to be independent.

The result is a survival-weighted discount sum: discount each scheduled amount to issue, multiply by the probability it is payable, and then add the contributions. A payment at time 0 has $v^0{}_0p_x=1$, so its full amount is included at issue.

Worked Example: Two Payments, Including the Payment at Issue

A hypothetical annuity pays $600 at time 0 and $600 at time 1, provided the annuitant is alive when each payment is due. The one-year death probability from age $x$ is $q_x=0.10$, and the effective annual interest rate is $i=25\%$. Find the expected present value. Method: calculate each payment’s survival-weighted discounted value separately. This avoids omitting the payment at issue or applying survival to it.

First, $v=1/(1+i)=1/1.25=0.80$. The time-0 payment is due immediately, so its survival probability and discount factor are both 1:

$$ 600(1)(1)=600. $$

For time 1, the probability of surviving the year is ${}_1p_x=1-q_x=1-0.10=0.90$. The discount factor for one year is $v^1=0.80$, so this payment contributes

$$ 600(0.90)(0.80)=432. $$

Adding the two contributions gives $600+432=\$1{,}032$. Check: the time-1 payment contributes less than $600 because it is both contingent on survival and discounted. The payment at time 0 is not reduced for either reason. The expected present value is $1,032.00, rounded to the nearest cent.

Payments at Selected Integer Times

A payment stream need not include a payment every year. A zero payment at a time contributes nothing, while a later payment still requires survival through every year from issue to its due date. For example, a payment at time 4 is weighted by ${}_4p_x$, not just by the probability of surviving the year immediately before time 4.

Worked Example: Gaps Between Scheduled Payments

A hypothetical contract pays $250 at time 1, $500 at time 2, and $1,000 at time 4, provided the annuitant is alive at each payment time. Assume $q_x=0.20$, $q_{x+1}=0.10$, $q_{x+2}=0.20$, and $q_{x+3}=0.25$. The effective annual interest rate is $i=25\%$. Find the expected present value, rounded to the nearest cent. Method: calculate the full survival probability and discount factor for each payment time. This avoids treating a skipped payment year as a skipped survival year.

Here $v=1/1.25=0.80$. For the time-1 payment, ${}_1p_x=1-0.20=0.80$, so the contribution is

$$ 250(0.80)(0.80)=160. $$

For time 2, the annuitant must survive both the year from age $x$ and the following year. Thus, ${}_2p_x=(0.80)(0.90)=0.72$, and $v^2=0.64$. The contribution is

$$ 500(0.72)(0.64)=230.40. $$

For time 4, survival is required in all four years, including the years with no scheduled payment:

{}_4p_x=(0.80)(0.90)(0.80)(0.75)=0.432.

Also $v^4=(0.80)^4=0.4096$. The time-4 contribution is

$$ 1{,}000(0.432)(0.4096)=176.9472. $$

The total is $160+230.40+176.9472=567.3472$, which rounds to $567.35. Check: the survival probability to time 4 is smaller than the probability to time 2, and the time-4 discount factor is also smaller. The later payment is therefore substantially reduced from its face amount.

Using Life-Table Values

When the life-table survivor counts are supplied, a separate list of one-year death probabilities is not necessary. For a life aged $x$, the fraction of the age-$x$ group expected to remain at age $x+k$ is $\ell_{x+k}/\ell_x$. This is exactly the survival weight for a payment at time $k$. It is important to use the ratio with the issue-age column in the denominator; the payment is conditional on a life being age $x$ at issue.

Worked Example: Valuing Payments from Life-Table Columns

A hypothetical annuity pays $500 at time 0, $700 at time 1, and $900 at time 2 if the annuitant is alive at the payment time. A life table gives $\ell_{50}=1{,}000$, $\ell_{51}=900$, and $\ell_{52}=720$. The effective annual interest rate is $i=25\%$. Find the expected present value to the nearest cent. Method: obtain each survival weight as a ratio of life-table columns, then discount. This avoids confusing the number of survivors with the probability of survival.

At time 0, the survival weight is $\ell_{50}/\ell_{50}=1$, and the discount factor is 1. The contribution is $500.

At time 1, ${}_1p_{50}=\ell_{51}/\ell_{50}=900/1{,}000=0.90$. Since $v=1/1.25=0.80$, the contribution is

$$ 700(0.90)(0.80)=504. $$

At time 2, ${}_2p_{50}=\ell_{52}/\ell_{50}=720/1{,}000=0.72$, while $v^2=0.64$. The contribution is

$$ 900(0.72)(0.64)=414.72. $$

Adding the contributions gives $500+504+414.72=\$1{,}418.72$. Check: the time-2 survival probability is also consistent with the two annual survival ratios, $(900/1{,}000)(720/900)=0.72$. The expected present value is $1,418.72.

Common Exam Trap: Mixing Up the Payment Time and Survival Period

A frequent error is to count only the mortality risk in the year immediately before a payment. That is not enough for a payment at time $k$: the annuitant must have survived all $k$ years from issue. Another error is to discount a time-0 payment or to apply the time-1 survival probability to it. The first payment of an annuity-due stream is made at issue under the stated contract condition.

Common mistake: For the two-payment example above, an incorrect calculation might use $600(0.90)+600(0.90)(0.80)=1{,}080$, applying the one-year survival probability to the time-0 payment as well as to the time-1 payment. The time-0 payment is due immediately, so it contributes $600$, not $600(0.90)$. The correct calculation is $600+600(0.90)(0.80)=\$1{,}032.00$.

A reliable calculation records four items for every payment: its scheduled time, its amount, its survival probability, and its discount factor. The survival probability is based on the time elapsed since issue; the discount factor is based on that same time, but for a different reason. Keeping these two weights separate makes it easier to detect an incorrect age, year, or exponent.

Check Your Understanding

For each hypothetical contract, assume payments are made only if the annuitant is alive when due. Use the stated annual effective interest rate and round money answers to the nearest cent.

A contract pays $400 at time 0 and $500 at time 1. If $q_x=0.10$ and $i=25\%$, find the expected present value. (Answer: $760.00)
A contract pays $1,000 at times 0, 1, and 2. If $q_x=0.10$, $q_{x+1}=0.20$, and $i=25\%$, find the expected present value. (Answer: $2,180.80)
A contract pays $200 at time 0, $400 at time 1, and $600 at time 2. The life table has $\ell_{60}=1{,}000$, $\ell_{61}=950$, and $\ell_{62}=855$, and $i=25\%$. Find the expected present value. (Answer: $832.32)
A contract pays $500 at time 0, $700 at time 1, and $900 at time 2. If $q_x=0.20$, $q_{x+1}=0.10$, and $i=25\%$, find the expected present value. (Answer: $1,362.72)
A contract pays $300 at time 0, $400 at time 1, and an unknown amount $K$ at time 3. Given $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.25$, and $i=25\%$, what value of $K$ makes the expected present value $700? (Answer: $405.09)

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