Recovering Whole-Life Insurance from Annuity-Due Values

Why an Annuity-Due Value Determines an Insurance Value

A discrete whole-life insurance pays a benefit at the end of the year of death. A whole-life annuity-due pays at the start of each year, provided the life is then alive. These contracts have different payment patterns, but their values are linked: knowing the annuity-due value is enough to calculate the insurance value at the same age and interest rate.

This tutorial focuses on using that relationship in the direction from an annuity-due value to an insurance value. The key is to use the effective discount rate, not the effective interest rate. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $i$ be the effective annual interest rate, $v=1/(1+i)$ the annual discount factor, and $d=1-v=i/(1+i)$ the effective annual discount rate. For a life aged $x$, $A_x$ is the actuarial present value of a unit discrete whole-life insurance benefit paid at the end of the year of death. The whole-life annuity-due value is $\ddot{a}_x$, with payments of one at times $0,1,2,\ldots$ while the life is alive at each payment time.

For clarity, the discrete insurance benefit is paid at time $k+1$ if death occurs during the $k$th year of future life, while an annuity-due payment at time $k$ is made if the life survives to time $k$. Thus the first annuity payment is certain at time 0, while the insurance benefit is not paid until the end of the year of death.

Using standard survival notation, ${}_kp_x$ is the probability of surviving $k$ years from age $x$, and ${}_kq_x=1-{}_kp_x$ is the probability of dying within $k$ years. The annuity-due value and discrete whole-life insurance value can be written as

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

Here $q_{x+k}$ is the probability of dying during the year after reaching age $x+k$. Both values use the same mortality assumptions and the same effective annual interest rate. The formula connecting them is

Theorem: For a unit discrete whole-life insurance and a whole-life annuity-due valued at the same age $x$ and effective annual interest rate $i$, $$ A_x=1-d\ddot{a}_x, \qquad d=\frac{i}{1+i}=1-v. $$

Why the Identity Holds

The relationship follows by comparing the discounted cash flows for one possible lifetime. Suppose death occurs at the end of year $K+1$, where $K$ is a nonnegative integer. The insurance pays $v^{K+1}$ in present value. The annuity-due pays at times $0$ through $K$, so its present value on that same lifetime is $1+v+\cdots+v^K$.

d(1+v+\cdots+v^K) =d\frac{1-v^{K+1}}{1-v} =1-v^{K+1},

because $d=1-v$. Rearranging gives $v^{K+1}=1-d(1+v+\cdots+v^K)$. Taking expectations over the lifetime turns the left side into $A_x$ and the annuity present value into $\ddot{a}_x$, proving the identity.

The argument depends on the specified timing: the annuity is due at the beginning of each year, and the insurance benefit is paid at the end of the year of death. It does not directly give the corresponding formula for an annuity-immediate, a continuous insurance, or a benefit paid at the beginning of the year of death. Those payment patterns have different discounted cash flows.

A Reliable Calculation Method

When the annuity-due value is given or has already been calculated, use the same sequence each time. First identify the effective annual interest rate and calculate $d=i/(1+i)$. Then multiply $d$ by the annuity-due value and subtract that amount from 1. Keep full precision until the final reported value.

Match the contract definitions.
Confirm that the insurance is discrete whole life with a unit benefit at the end of the year of death, and that the annuity is a whole-life annuity-due.

Find the effective discount rate.
Calculate $d=1/(1+i)$ subtracted from 1, equivalently $d=i/(1+i)$. Do not substitute $i$ for $d$.

Apply the complement.
Calculate $A_x=1-d\ddot{a}_x$, using an annuity value based on the same age, interest, and mortality assumptions.

Check the result.
Confirm that the answer is between 0 and 1 for a unit benefit and, when practical, verify it by summing the discounted death benefits under the mortality model.

Worked Example: Constant Annual Mortality

A hypothetical life has a constant annual probability of death $q=0.20$, independently applied in each future year while the life is alive. Thus the annual survival probability is $p=0.80$. The effective annual interest rate is $i=0.25$. Find $A_x$ for a unit benefit paid at the end of the year of death. Report the final value to four decimal places.

Method. Calculate the annuity-due value from the constant survival model, then use the discount-rate identity. Here $v=1/1.25=0.80$, and the probability of surviving $k$ years is ${}_kp_x=0.80^k$. Therefore each term of the annuity-due series has ratio $vp=0.80(0.80)=0.64$:

\ddot{a}_x=\sum_{k=0}^{\infty}(0.64)^k =\frac{1}{1-0.64} =\frac{25}{9} =2.777777\ldots

The effective discount rate is

d=\frac{0.25}{1.25}=0.20.

Thus

A_x=1-(0.20)\left(\frac{25}{9}\right) =1-\frac{5}{9} =\frac{4}{9} =0.4444.

Check. The probability of death in year $k+1$ is $0.20(0.80)^k$, and the benefit is discounted by $v^{k+1}=(0.80)^{k+1}$. The direct insurance series is therefore

A_x=\sum_{k=0}^{\infty}0.20(0.80)^k(0.80)^{k+1} =\frac{0.16}{1-0.64} =\frac{4}{9} =0.4444.

The direct calculation agrees. This check also confirms that the benefit is discounted one full year in the first year of coverage.

Worked Example: A Different First-Year Survival Probability

For a hypothetical life, the first-year survival probability is $0.50$. Conditional on surviving that year, the probability of surviving each subsequent year is $0.80$. The effective annual interest rate is $i=0.25$. Find the unit discrete whole-life insurance value and report it to four decimal places.

Method. Use the survival probabilities to calculate $\ddot{a}_x$, being careful not to apply the later-year survival probability to the first year. With $v=0.80$, the first three annuity terms are $1$, $v(0.50)=0.40$, and $v^2(0.50)(0.80)=0.256$. After the second term, successive terms have ratio $v(0.80)=0.64$. The tail starting at $k=2$ is

0.256+0.256(0.64)+0.256(0.64)^2+\cdots =\frac{0.256}{1-0.64} =\frac{32}{45}.

Consequently,

\ddot{a}_x=1+0.40+\frac{32}{45} =\frac{19}{9} =2.111111\ldots

As in the first example, $d=0.25/1.25=0.20$. Then

A_x=1-(0.20)\left(\frac{19}{9}\right) =1-\frac{19}{45} =\frac{26}{45} =0.5778.

Check. The probability of death in the first year is $0.50$, giving a discounted benefit of $0.50(0.80)=0.40$. The probability of death in year 2 is $0.50(0.20)=0.10$, and in year 3 it is $0.50(0.80)(0.20)=0.08$. Discounting these later benefits and continuing the geometric tail gives

0.40+\frac{0.064}{1-0.64} =0.40+\frac{8}{45} =\frac{26}{45}.

This matches the annuity-based calculation. The first-year mortality assumption changes the initial terms, but the identity itself is unchanged.

Worked Example: Mortality Ends After Three Years

A hypothetical life has probability $0.50$ of surviving the first year and, conditional on that, probability $0.50$ of surviving the second year. No one survives to the start of year 4. The effective annual interest rate is $i=0.25$. Find the unit discrete whole-life insurance value, reporting the final value to four decimal places.

Method. The survival probabilities at times 0, 1, and 2 are $1$, $0.50$, and $0.25$; the probability of survival to time 3 is zero. With $v=0.80$, the annuity-due value has just three nonzero terms:

\ddot{a}_x=1+(0.80)(0.50)+(0.80)^2(0.25) =1+0.40+0.16 =1.56.

The discount rate remains $d=0.20$, so

$$ A_x=1-(0.20)(1.56)=1-0.312=0.6880. $$

Check. Death occurs in year 1 with probability $0.50$, in year 2 with probability $0.50-0.25=0.25$, and in year 3 with probability $0.25$. The discounted insurance benefits are

$$ (0.50)(0.80)+(0.25)(0.80)^2+(0.25)(0.80)^3 =0.400+0.160+0.128 =0.688. $$

The direct value agrees. The phrase “no one survives to the start of year 4” means the third-year death benefit is still possible; it must not be omitted.

Common Mistake: Using the Interest Rate Instead

A frequent error is to replace $d$ by $i$ in the identity. These rates are related but not equal: $d=i/(1+i)$. For example, at $i=0.25$, the correct discount rate is $d=0.20$, not $0.25$. With $\ddot{a}_x=25/9$, the incorrect calculation would be

1-i\ddot{a}_x =1-(0.25)\left(\frac{25}{9}\right) =\frac{11}{36} =0.3056.

The correct value is $1-d\ddot{a}_x=4/9=0.4444$, as verified directly from the death-benefit cash flows. The incorrect result fails because the cash-flow identity uses $1-v=d$, not $i$. A useful check is that $d<i$ for positive $i$; substituting $i$ therefore subtracts too much and understates the insurance value.

Also check the contract timing before applying the formula. A benefit payable at the beginning of the year of death, or an annuity payable at the end of each year, is not the payment pattern used in this identity. Confirming the timing and using the same mortality basis for both values avoids many otherwise plausible errors.

Key takeaway: For a discrete whole-life insurance with a unit benefit at the end of the year of death, use $A_x=1-d\ddot{a}_x$, where $d=i/(1+i)$. The age, mortality assumptions, and interest rate underlying $\ddot{a}_x$ must match those for $A_x$.

Check Your Understanding

For each problem, identify the discount rate before applying the identity. Answers are rounded to four decimal places where needed.

A whole-life annuity-due value is $2.5$, and the effective annual interest rate is $20\%$. Find the value of the matching unit discrete whole-life insurance. (Answer: $7/12=0.5833$)
A hypothetical life has the same annual survival probability, $0.60$, in every year, and the effective annual interest rate is $25\%$. Calculate the annuity-due value from the survival model and then find the unit insurance value. (Answer: $A_x=8/13=0.6154$)
A hypothetical life has ${}_1p_x=0.80$, ${}_2p_x=0.40$, and ${}_3p_x=0$. At an effective annual interest rate of $10\%$, calculate $\ddot{a}_x$ and the matching unit whole-life insurance value. (Answer: $\ddot{a}_x=249/121$, $A_x=1082/1331=0.8129$)
A hypothetical life has first-year survival probability $0.70$, followed by conditional survival probability $0.75$ in each later year. At $i=25\%$, find the annuity-due value and the matching unit discrete whole-life insurance value. (Answer: $\ddot{a}_x=2.4$, $A_x=0.5200$)
A matching annuity-due value is $12$, and the effective annual interest rate is $5\%$. Find the unit discrete whole-life insurance value, using the effective discount rate rather than the interest rate. (Answer: $A_x=3/7=0.4286$)

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