Recovering Annuity-Due Values from Whole-Life Insurance

Why a Whole-Life Insurance Value Determines an Annuity-Due Value

A whole-life insurance value and a whole-life annuity-due value describe different payment patterns, but their payments fit together neatly when they use the same interest and mortality basis. That connection is useful when a problem provides one value and asks for the other. The key is to match the timing: the insurance benefit is paid at the end of the year of death, while the annuity-due payment is made at the start of each year the life is alive.

This tutorial develops the identity for a level annual annuity-due and a discrete whole-life insurance benefit. It also shows how to use a fixed benefit amount other than one unit, and why the benefit must be normalized before applying the identity. The results are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $i>0$ be the effective annual interest rate, $v=1/(1+i)$, and $d=1-v=i/(1+i)$ the effective annual discount rate. $A_x$ is the expected present value of a unit whole-life insurance benefit payable at the end of the year of death for a life aged $x$. $\ddot{a}_x$ is the expected present value of an annuity-due paying one unit at the start of each year, including time 0, provided the life is then alive.

The identity assumes the insurance and annuity use the same age, mortality basis, and effective annual interest rate. It also assumes the insurance benefit is level and paid at the end of the year of death, and the annuity payments are level and made annually in advance while the life is alive. There are no expenses or other cash flows in these values. No particular mortality law is required: the identity follows from the payment timing for each possible year of death.

The Whole-Life Identity

With these conventions, the unit values satisfy

\ddot{a}_x = \frac{1-A_x}{d}.

Equivalently, $A_x=1-d\ddot{a}_x$. The first form recovers an annuity-due value from an insurance value; the second expresses the insurance value in terms of the annuity-due value. Here $d$, not $i$, is the rate in the denominator. Since $d=i/(1+i)$, using the interest rate itself in its place generally gives a different answer.

Why the Identity Holds

Let death occur in the $(k+1)$st year after age $x$, so that the life is alive at times $0,1,\ldots,k$. On this outcome, the annuity-due pays at those $k+1$ times, and the insurance pays at time $k+1$. The present value of the annuity payments is a finite geometric sum:

1+v+v^2+\cdots+v^k =\frac{1-v^{k+1}}{1-v} =\frac{1-v^{k+1}}{d}.

The insurance present value on this same outcome is $v^{k+1}$. Therefore,

d(1+v+\cdots+v^k)+v^{k+1}=1.

This holds for every possible year of death. Taking expectations over the mortality outcomes gives $d\ddot{a}_x+A_x=1$, and rearranging gives the identity. In probability notation, the same values can be represented as

\ddot{a}_x=\sum_{k=0}^{\infty}v^k\,{}_kp_x, \qquad A_x=\sum_{k=0}^{\infty}v^{k+1}\,{}_kp_x\,{}_1q_{x+k}.

The annuity sum counts a payment at time $k$ when the life survives to that time. The insurance sum counts a payment at time $k+1$ when the life survives $k$ years and then dies in the following year. The pathwise argument explains why the two expected present values are connected without needing to calculate these sums separately.

Worked Example: Recovering a Unit Annuity-Due Value

For a hypothetical life aged $x$, a unit whole-life insurance benefit is payable at the end of the year of death. Its value is $A_x=0.70$ at an effective annual interest rate of $5\%$. Find $\ddot{a}_x$. Give the annuity factor to two decimal places, and check the result using the identity.

Method. Apply the whole-life identity using the effective annual discount rate $d$, which must first be calculated from $i$.

Step 1: Calculate the discount rate.

i=0.05,\qquad v=\frac{1}{1.05}=\frac{20}{21},\qquad d=1-v=\frac{1}{21}.

Step 2: Substitute the unit insurance value.

\ddot{a}_x =\frac{1-A_x}{d} =\frac{1-0.70}{1/21} =\frac{0.30}{1/21} =0.30(21) =6.30.

Check. $d\ddot{a}_x=(1/21)(6.30)=0.30$, so $1-d\ddot{a}_x=1-0.30=0.70=A_x$. The recovered annuity-due factor is $6.30$. The calculation uses $d$, not $i$; substituting $0.05$ for $d$ would incorrectly give $0.30/0.05=6.00$.

Using a Fixed Nonunit Insurance Benefit

The symbol $A_x$ is the value for a unit insurance benefit. If the benefit is a fixed amount $B$, the expected present value of the insurance benefit is $B A_x$, because every possible death payment is multiplied by $B$. Thus, if the problem gives the dollar value of the insurance benefit rather than the unit value, divide by $B$ first. Then use the identity to recover the annuity factor.

\text{Insurance value}=B A_x, \qquad A_x=\frac{\text{Insurance value}}{B}, \qquad \ddot{a}_x =\frac{1-\text{Insurance value}/B}{d}.

The annuity factor is still a value per one unit of annual annuity payment. If the annuity pays $R$ dollars at the beginning of each year while the life is alive, its expected present value is $R\ddot{a}_x$. Keep the insurance benefit amount, the insurance value, and the annuity payment amount distinct until the unit factor has been found.

Worked Example: Converting a Dollar Insurance Value to a Unit Value

A hypothetical policy pays a fixed death benefit of $ \$60{,}000$ at the end of the year of death. On a specified mortality basis and at an effective annual interest rate of $4\%$, the expected present value of this insurance benefit is $ \$39{,}000$. Find the corresponding unit whole-life annuity-due factor. State the factor to two decimal places.

Method. Divide the insurance value by the fixed benefit to obtain $A_x$. Do not insert the dollar-valued insurance value directly into a unit-value identity.

Step 1: Normalize the insurance value.

A_x=\frac{\$39{,}000}{\$60{,}000}=0.65.

Step 2: Calculate the effective annual discount rate.

i=0.04,\qquad v=\frac{1}{1.04}=\frac{25}{26},\qquad d=1-v=\frac{1}{26}.

Step 3: Recover the annuity-due factor.

\ddot{a}_x =\frac{1-0.65}{1/26} =\frac{0.35}{1/26} =0.35(26) =9.10.

Check. $1-d\ddot{a}_x=1-(1/26)(9.10)=1-0.35=0.65$, matching the normalized unit insurance value. The result is a factor per one dollar of annual annuity payment; it is not a dollar amount of annuity income.

Worked Example: Applying the Factor to a Nonunit Annuity Payment

A hypothetical insurer considers a contract that pays $ \$90{,}000$ at the end of the year of death. At an effective annual interest rate of $6\%$, the expected present value of that insurance benefit is $ \$63{,}000$. The same life would receive an annuity-due payment of $ \$1{,}500$ at the start of each year alive. Find the expected present value of those annuity payments. Round the final dollar amount to the nearest cent.

Method. First recover the unit insurance value by dividing the insurance value by the death benefit. Then calculate the unit annuity-due factor and multiply by the annual annuity payment.

Step 1: Convert to a unit insurance value.

A_x=\frac{\$63{,}000}{\$90{,}000}=0.70.

Step 2: Find the discount rate.

i=0.06,\qquad v=\frac{1}{1.06}=\frac{50}{53},\qquad d=1-v=\frac{3}{53}.

Step 3: Calculate the unit annuity-due factor.

\ddot{a}_x =\frac{1-0.70}{3/53} =\frac{0.30(53)}{3} =\frac{15.9}{3} =5.3.

Step 4: Apply the annual payment amount.

\text{Expected present value} =\$1{,}500(5.3) =\$7{,}950.00.

Check. $1-(3/53)(5.3)=1-0.30=0.70$, which recovers the unit insurance value. Multiplying the annuity factor by $ \$1{,}500$ gives the expected present value of the stated payment stream. The calculation avoids confusing the $ \$90{,}000$ insurance benefit with the $ \$1{,}500$ annuity payment.

Assumptions, Limits, and Exam Traps

The identity is convenient because it avoids a separate mortality calculation when $A_x$ is already known. Its simplicity depends on matching the contract definitions. In particular, the life insurance must be discrete, level, and paid at the end of the year of death, while the annuity must be annual, level, and due at the start of each year alive. Changing a payment date changes its discount factor and therefore changes the relationship.

Using $i$ instead of $d$. The denominator is $d=i/(1+i)$, not $i$. At $i=5\%$, $d=0.05/1.05$, which is not $0.05$.
Forgetting to normalize a fixed benefit. If the benefit is $B$, use $A_x=(\text{insurance value})/B$ before finding $\ddot{a}_x$.
Mixing payment timing. An insurance payment at the moment of death is not the end-of-year payment assumed in $A_x$. Likewise, an annuity paid at each year-end is not the annuity-due defined here.
Assuming the formula handles every benefit pattern. If the death benefit changes by duration, the insurance value is not simply a fixed benefit times $A_x$. A single aggregate insurance value generally does not identify the unit annuity-due factor through the fixed-benefit conversion above.
Ignoring the zero-interest boundary. The displayed formula divides by $d$, so it requires $d>0$. At $i=0$, do not divide by zero; value the cash flows directly if that case is relevant.

A useful check is to substitute the recovered factor back into $A_x=1-d\ddot{a}_x$. If it does not reproduce the unit insurance value, check the interest-rate conversion, the benefit normalization, and whether the payment timing matches the identity.

Key takeaway: For a unit discrete whole-life insurance benefit payable at the end of the year of death and a unit annual life annuity-due on the same basis, $\ddot{a}_x=(1-A_x)/d$, where $d=i/(1+i)$. With a fixed nonunit death benefit, first divide its expected present value by the benefit amount to recover $A_x$.

Check Your Understanding

For each problem, match the payment timing to the whole-life identity and keep dollar amounts separate from unit factors.

A unit whole-life insurance value is $A_x=0.82$ at an effective annual interest rate of $5\%$. Find the annual annuity-due factor, to two decimal places. (Answer: $\ddot{a}_x=3.78$)
A hypothetical policy pays $ \$60{,}000$ at the end of the year of death and has an expected present value of $ \$39{,}000$. At an effective annual interest rate of $4\%$, find the unit annuity-due factor, to two decimal places. (Answer: $\ddot{a}_x=9.10$)
A unit whole-life insurance value is $0.74$ at an effective annual interest rate of $8\%$. Find the annuity-due factor to two decimal places, and verify by calculating $1-d\ddot{a}_x$. (Answer: $\ddot{a}_x=3.51$; $1-d\ddot{a}_x=0.74$)
A hypothetical fixed death benefit of $ \$150{,}000$ has an expected present value of $ \$102{,}000$. At an effective annual interest rate of $5\%$, what is the expected present value of an annuity-due paying $ \$2{,}000$ each year while the life is alive? Give the final dollar amount to the nearest cent. (Answer: $\ddot{a}_x=6.72$; expected present value $=\$13{,}440.00$)
At an effective annual interest rate of $4\%$, a unit annual annuity-due factor is $8.4$. A hypothetical policy pays $ \$75{,}000$ at the end of the year of death. Find the unit insurance value and the expected present value of the policy benefit; give the dollar value to the nearest cent. (Answer: $A_x=0.6769231$; expected present value $=\$50{,}769.23$)

What You'll Learn