Deferred Annuity-Due Valuations

Why Deferment Changes an Annuity Value

A life annuity-due pays at the beginning of each payment period, provided the annuitant is alive on the payment date. If payments are deferred, the first payment does not occur at issue: the contract must first reach its specified starting date, and the annuitant must be alive then. Later payments also depend on survival to their own dates. A correct present value therefore reflects both when each payment is made and the probability the payment is made.

This tutorial considers annual payments at integer times and a constant effective annual interest rate $i$. Assume the life is alive at time 0. A payment due at time $k$ is made only if the life is alive at that time. An annuity-due deferred $m$ years has its first payment at time $m$, not at time $m+1$. For a finite annuity with $n$ payments, the payment dates are $m,m+1,\ldots,m+n-1$.

Definition: The expected present value of a deferred life annuity-due with annual payment amount 1 and payments at times $m,m+1,\ldots$, while the annuitant is alive, is $$ {}_{m|}\ddot{a}_x=\sum_{k=m}^{\infty}v^k\,{}_kp_x,\qquad v=\frac{1}{1+i}. $$ Here $m$ is a nonnegative integer, $x$ is the age at issue, and ${}_kp_x$ is the probability that a life aged $x$ survives $k$ years. For a finite annuity with $n$ payments, the corresponding value is $$ {}_{m|}\ddot{a}_{x:\overline{n}|}=\sum_{k=m}^{m+n-1}v^k\,{}_kp_x. $$ Multiply either factor by the stated annual payment amount to obtain the expected present value of the payments.

Valuing Each Payment at Its Own Date

The sum makes the timing explicit. The payment at time $k$ contributes its amount multiplied by $v^k$, to discount it to time 0, and by ${}_kp_x$, the probability it is payable. The payment at time 0, if any, is certain for a life alive at issue because ${}_0p_x=1$; when $m>0$, there is no payment at time 0. Mortality probabilities should be calculated to each payment date rather than applied just once to the entire stream.

For example, if a life table gives the number of survivors $\ell_x$ at age $x$, then, when the table represents the same mortality basis, ${}_kp_x=\ell_{x+k}/\ell_x$. Alternatively, probabilities can be built from one-year survival probabilities: ${}_kp_x=p_xp_{x+1}\cdots p_{x+k-1}$, where $p_y={}_1p_y$. These are probabilities conditional on being alive at age $x$.

Worked Example: Three Payments After a Two-Year Deferment

A hypothetical contract pays $1,000 annually at the beginning of each payment year, with three payments beginning two years from issue. Assume a constant annual survival probability of $0.9$ at every age and an effective annual interest rate $i=1/9$. Find the expected present value.

Method. List the payment dates, then multiply each payment by the probability of survival to that date and its discount factor. The discount factor is $v=1/(1+1/9)=0.9$. The payment dates are $2,3,4$, and the survival probabilities are ${}_2p_x=0.9^2=0.81$, ${}_3p_x=0.9^3=0.729$, and ${}_4p_x=0.9^4=0.6561$.

The discounted survival-weighted factors at the three dates are $v^2{}_2p_x=0.9^2(0.81)=0.6561$, $v^3{}_3p_x=0.9^3(0.729)=0.531441$, and $v^4{}_4p_x=0.9^4(0.6561)=0.43046721$.

Thus the expected present value is $1{,}000(0.6561+0.531441+0.43046721)=\$1{,}618.00821$, or $1,618.01 to the nearest cent.

As a check, survival to time 2 is $0.81$. Conditional on being alive then, the three-payment annuity-due factor is $1+0.9(0.9)+0.9^2(0.9^2)=2.4661$. Multiplying by $v^2{}_2p_x=0.6561$ gives $0.6561(2.4661)=1.61800821$, the same factor as the direct sum. The common mistake avoided here is to treat the first payment as occurring at time 3: an annuity-due beginning after a two-year deferment pays at time 2.

Conditioning on Survival to the Start Date

A deferred value can also be expressed in terms of an annuity beginning at the age reached at the end of the deferment. This is useful when a value at that later age is known or easier to calculate. The expression separates the chance of reaching the payment start date from the value of payments thereafter.

Theorem: For integer $m\geq 0$, under a constant effective annual interest rate and a consistent mortality model, $$ {}_{m|}\ddot{a}_x=v^m\,{}_mp_x\,\ddot{a}_{x+m}. $$ For $n$ payments, replace $\ddot{a}_{x+m}$ with the $n$-payment temporary annuity-due factor at age $x+m$. The value $\ddot{a}_{x+m}$ is conditional on being alive at age $x+m$.

To justify the result, start with the payment-by-payment sum and write each payment date as $k=m+j$, where $j\geq0$. Survival to time $m+j$ requires survival to $m$, followed by survival for another $j$ years conditional on reaching age $x+m$. Thus ${}_{m+j}p_x={}_mp_x\,{}_jp_{x+m}$. Substitution gives

\begin{aligned} {}_{m|}\ddot{a}_x &=\sum_{j=0}^{\infty}v^{m+j}\,{}_{m+j}p_x\\ &=v^m\,{}_mp_x\sum_{j=0}^{\infty}v^j\,{}_jp_{x+m}\\ &=v^m\,{}_mp_x\,\ddot{a}_{x+m}. \end{aligned}

The same argument for a finite stream ends at $j=n-1$, producing the $n$-payment temporary annuity-due factor. The decomposition does not assume that survival probabilities are constant by year; it requires only that the conditional survival probabilities are calculated consistently.

Worked Example: A Deferred Whole-Life Annuity-Due

A hypothetical whole-life annuity pays $600 annually beginning three years from issue and continues while the annuitant is alive. Assume a constant annual survival probability of $0.8$ and an effective annual interest rate of $25\%$. Find the expected present value.

Method. The discount factor is $v=1/1.25=0.8$. For a payment at time $k$, the combined discount and survival factor is $v^k{}_kp_x=(0.8)^k(0.8)^k=(0.64)^k$. The first payment is at time 3, so the unit-benefit value is the geometric series $0.64^3+0.64^4+0.64^5+\cdots$.

The first term is $0.64^3=0.262144$, and the common ratio is $0.64$. Since its magnitude is less than 1, the series sums to $0.262144/(1-0.64)=0.262144/0.36=0.728177777\ldots$.

Multiplying by the annual benefit gives $600(0.728177777\ldots)=\$436.906666\ldots$, or $436.91 to the nearest cent.

As a check by conditioning, $v^3{}_3p_x=0.8^3(0.8^3)=0.262144$. The whole-life annuity-due factor at the attained age is $1+0.64+0.64^2+\cdots=1/(1-0.64)=2.777777\ldots$. Their product is $0.728177777\ldots$, as above. The geometric-series method relies on the stated constant survival probability; it would not apply with a changing annual survival probability in this form.

Changing Mortality and a Finite Payment Period

When mortality varies by age, calculate survival to each payment date by multiplying the intervening one-year survival probabilities. Do not extend a constant-mortality shortcut beyond the assumptions that support it. For a finite annuity, also count the payment dates carefully: a deferred $n$-payment annuity-due has $n$ payments, starting at time $m$ and ending at time $m+n-1$.

Worked Example: Age-Dependent Survival with Three Deferred Payments

A hypothetical contract pays $1,500 at times 2, 3, and 4, provided the annuitant is alive at each payment date. The one-year survival probabilities from issue age onward are $p_x=0.9$, $p_{x+1}=0.8$, $p_{x+2}=0.7$, and $p_{x+3}=0.6$. The effective annual interest rate is $25\%$. Calculate the expected present value.

Method. Here $v=1/1.25=0.8$. The survival probabilities to the payment dates are ${}_2p_x=0.9(0.8)=0.72$, ${}_3p_x=0.9(0.8)(0.7)=0.504$, and ${}_4p_x=0.9(0.8)(0.7)(0.6)=0.3024$.

At time 2, the survival-weighted discount factor is $v^2{}_2p_x=0.64(0.72)=0.4608$, giving an expected present value contribution of $1{,}500(0.4608)=\$691.20$. At time 3, the factor is $v^3{}_3p_x=0.512(0.504)=0.258048$, giving $1{,}500(0.258048)=\$387.072$. At time 4, the factor is $v^4{}_4p_x=0.4096(0.3024)=0.12386304$, giving $1{,}500(0.12386304)=\$185.79456$.

Adding the contributions gives $\$691.20+\$387.072+\$185.79456=\$1{,}264.06656$, or $1,264.07 to the nearest cent.

A conditional check gives $v^2{}_2p_x=0.4608$. For a life alive at age $x+2$, the three-payment due factor is $1+0.8(0.7)+0.8^2(0.7)(0.6)=1+0.56+0.2688=1.8288$. Then $0.4608(1.8288)=0.84271104$, and $1{,}500(0.84271104)=\$1{,}264.06656$. This agrees with the direct calculation.

Common Timing and Survival Errors

For the changing-mortality example, a tempting but incorrect calculation is to apply survival to the deferment date to every payment: $1{,}500(0.64)(0.72)(1+0.8+0.8^2)=\$1{,}686.528$. This treats being alive at time 2 as if it guaranteed being alive at times 3 and 4. The correction is to use ${}_2p_x$, ${}_3p_x$, and ${}_4p_x$ for the respective payment dates, giving $1,264.07.

Other checks help prevent errors:

Confirm the first payment date. A deferment of $m$ years for an annuity-due means a first payment at time $m$.
Count the payments. If there are $n$ payments, the final payment is at time $m+n-1$, not $m+n$.
Use the right survival probability. The payment at time $k$ requires survival for $k$ years, so its factor uses ${}_kp_x$.
Keep the interest and mortality roles distinct. Discounting uses $v^k$; survival weighting uses ${}_kp_x$. Both refer to the same payment date.

Key takeaway: For a deferred annuity-due, begin with the actual payment dates. For each date $k$, multiply the payment by $v^k{}_kp_x$, then add the contributions. Equivalently, discount and weight for survival to the deferment date, then use the annuity-due value conditional on survival to that date.

Check Your Understanding

For each problem, use annual payment dates and state the expected present value or annual payment as requested.

A hypothetical two-payment annuity-due pays $500 at times 1 and 2, provided the life is alive. The annual survival probability is $0.9$ at every age and the effective annual interest rate is $25\%$. Find its expected present value. (Answer: $619.20)
A hypothetical two-payment annuity-due pays $1,000 at times 2 and 3. The effective annual interest rate is $10\%$, and the one-year survival probabilities are $p_x=0.9$, $p_{x+1}=0.8$, and $p_{x+2}=0.7$. Find the expected present value to the nearest cent. (Answer: $973.70)
A hypothetical whole-life annuity-due pays $900 annually beginning at time 2. Assume constant annual survival probability $0.8$ and effective annual interest rate $i=1/3$. Find its expected present value. (Answer: $810.00)
A hypothetical contract pays $2,000 at times 2 and 3 if the annuitant is alive. The effective annual interest rate is $25\%$, and the one-year survival probabilities are $p_x=0.95$, $p_{x+1}=0.8$, and $p_{x+2}=0.5$. Find its expected present value. (Answer: $1,361.92)
A hypothetical annuity-due pays the same amount at times 1, 2, 3, and 4, provided the annuitant is alive. The annual survival probability is $0.9$ at every age and the effective annual interest rate is $25\%$. If the expected present value is $2,000, find the annual payment to the nearest cent. (Answer: $1,063.61)

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