The Continuous Whole-Life Insurance-Annuity Identity

Why the Continuous Identity Holds

A whole-life insurance benefit paid immediately at death and a continuous life annuity have complementary payment patterns. At each moment before death, the annuity is still paying; at death, the insurance pays. Discounting both streams at the same constant force of interest makes their relationship precise. This identity is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $T_x$ be the future lifetime of a life aged $x$, and let the force of interest be the constant $\delta$. For a unit benefit payable immediately at death, the continuous whole-life insurance present value is $e^{-\delta T_x}$, and its expected present value is $\bar{A}_x=E[e^{-\delta T_x}]$. For a continuous annuity paying at rate 1 per year while the life is alive, the present value is $\int_0^{T_x}e^{-\delta t}\,dt$, and its expected present value is $\bar{a}_x=E[\int_0^{T_x}e^{-\delta t}\,dt]$.

The annuity payment rate is one unit of currency per year, not one payment at the start or end of each year. The insurance benefit is also one unit, but it is paid at the instant of death. Both present values use the same interest basis and the same future lifetime. Those details are essential: changing the benefit timing or the annuity payment pattern changes the present-value expression.

Theorem: If the force of interest $\delta$ is constant and the stated present values exist, then $$ \bar{A}_x+\delta\bar{a}_x=1. $$

A Pathwise Derivation

The identity follows before taking any expectation. Fix one possible lifetime $T_x=t$. The present value of the unit insurance payment is $e^{-\delta t}$. The present value of the annuity payments made before death is the integral from time 0 to $t$. For $\delta\ne 0$, evaluate that integral:

\int_0^t e^{-\delta s}\,ds =\frac{1-e^{-\delta t}}{\delta}, \qquad e^{-\delta t}+\delta\int_0^t e^{-\delta s}\,ds=1.

This is a pathwise statement: it holds for every finite death time $t$, not just on average. Taking expectations of both sides gives

E[e^{-\delta T_x}] +\delta E\left[\int_0^{T_x} e^{-\delta s}\,ds\right] =1, \qquad \bar{A}_x+\delta\bar{a}_x=1.

The proof explains why the same mortality assumption must be used for both factors: each factor averages a present value over the same random lifetime. For $\delta=0$, the integral is $t$, so the identity as written reduces to $\bar{A}_x=1$. This holds when death occurs in finite time with probability 1, since $e^0=1$. In applications with a positive force of interest, the displayed complement form is directly usable whenever the expected present values are finite.

An equivalent rearrangement is $\bar{A}_x=1-\delta\bar{a}_x$, or $\bar{a}_x=(1-\bar{A}_x)/\delta$ when $\delta\ne0$. These are not additional assumptions; they are algebraic forms of the same relationship. In particular, $\delta$ is a force of interest, not an effective annual rate to be substituted without conversion.

Using the Identity Under Constant-Force Mortality

The identity does not require a particular mortality model. A useful model for calculation is constant-force mortality: assume $T_x$ has an exponential distribution with constant force of mortality $\mu>0$, so its density is $f_{T_x}(t)=\mu e^{-\mu t}$ for $t\ge0$. Assume also a constant interest force $\delta\ge0$. Then the insurance value can be evaluated by integrating its discounted benefit against the death-time density:

\bar{A}_x =\int_0^\infty e^{-\delta t}\mu e^{-\mu t}\,dt =\frac{\mu}{\mu+\delta}.

The integral converges because $\mu+\delta>0$. The identity then gives $\bar{a}_x=1/(\mu+\delta)$ when $\delta>0$. Equivalently, direct integration of the annuity present value gives that same value. These constant-force formulas apply only when the specified mortality and interest assumptions hold; they are not general formulas for every survival model.

Worked Example: Finding Both Values from Constant Forces

A hypothetical life aged $x$ has constant force of mortality $\mu=0.05$. The effective annual interest rate is $i=0.03$. Assuming interest is represented by its equivalent constant force, find the unit continuous whole-life insurance value and the unit continuous life annuity value. Verify the identity.

Method. Convert the effective annual rate to a force, use the constant-force mortality formulas, and then check the complement identity. This avoids treating an effective annual rate as though it were already a force.

First, calculate the force of interest:

\delta=\ln(1+i)=\ln(1.03)=0.0295588\ldots

Next, add the mortality force and interest force:

\mu+\delta=0.05+0.0295588\ldots=0.0795588\ldots

Thus the unit insurance value is

\bar{A}_x=\frac{0.05}{0.0795588\ldots}=0.6285\ldots

The unit continuous annuity value is

\bar{a}_x=\frac{1}{0.0795588\ldots}=12.5696\ldots

Using unrounded values, $\delta\bar{a}_x=0.0295588\ldots/0.0795588\ldots=0.3715\ldots$. Therefore $\bar{A}_x+\delta\bar{a}_x=0.6285\ldots+0.3715\ldots=1$, as required. Rounded only at the final step, the factors are $\bar{A}_x=0.6285$ and $\bar{a}_x=12.57$. The common mistake this method avoids is substituting $i=0.03$ for $\delta$; the equivalent force is $\ln(1.03)$, not 0.03.

Recovering One Factor from the Other

Often a problem supplies an insurance or annuity value rather than a complete mortality model. The identity is then a direct way to recover the missing value, provided both values refer to a unit benefit or payment rate, the same life, the same interest force, and the specified continuous payment timing. A calculated value can also be used as a consistency check on a pair of supplied factors.

Worked Example: Recovering Insurance from an Annuity Value

On a common valuation basis, a hypothetical life has continuous whole-life annuity value $\bar{a}_x=15$. The constant force of interest is $\delta=0.04$. Find the continuous whole-life insurance value and check that the result is in the possible range for a unit insurance value.

Method. Rearrange the identity to $\bar{A}_x=1-\delta\bar{a}_x$. The values are supplied on one common basis, so no mortality model needs to be imposed.

Multiply the annuity value by the force:

\delta\bar{a}_x=0.04(15)=0.60.

Subtract this complement from 1:

\bar{A}_x=1-0.60=0.40.

The check is $\bar{A}_x+\delta\bar{a}_x=0.40+0.60=1.00$. Also, $0\le \bar{A}_x\le1$, as expected for the expected present value of a unit benefit discounted at a nonnegative force. The common mistake this method avoids is using the annuity value itself as the complement without multiplying by $\delta$.

Worked Example: Checking Values and Inferring Mortality

A hypothetical life is valued using constant-force mortality and constant interest. The continuous whole-life insurance value is $\bar{A}_x=0.70$, and the force of interest is $\delta=0.03$. Find the continuous annuity value and the force of mortality $\mu$. Verify both factors under the stated model.

Method. First use the identity to recover the annuity value. Then use the constant-force insurance formula to solve for mortality. This separates the general identity from the additional model assumption needed to infer $\mu$.

From $\bar{A}_x+\delta\bar{a}_x=1$,

\bar{a}_x=\frac{1-\bar{A}_x}{\delta} =\frac{1-0.70}{0.03} =\frac{0.30}{0.03} =10.

Under constant-force mortality, $\bar{A}_x=\mu/(\mu+\delta)$. Substitute the given values and solve:

0.70=\frac{\mu}{\mu+0.03}, \qquad 0.70\mu+0.021=\mu, \qquad 0.30\mu=0.021, \qquad \mu=0.07.

Verify the insurance value using $\mu=0.07$:

\frac{0.07}{0.07+0.03}=0.70.

Verify the annuity value using the constant-force annuity formula:

\frac{1}{0.07+0.03}=10.

Finally, $0.70+0.03(10)=1$. All three checks agree. The common mistake this method avoids is using the identity to infer a mortality force without first stating a mortality model; the identity alone determines the annuity factor, but not $\mu$.

Scaling the Paired Cash Flows

The theorem concerns a unit death benefit and a unit annual annuity payment rate. The same pathwise argument also clarifies how to interpret contracts with other amounts. If the death benefit is $B$ and the continuous annuity payment rate is $c$, their combined expected present value is $B\bar{A}_x+c\bar{a}_x$. This equals $B$ when $c=B\delta$, because

B\bar{A}_x+B\delta\bar{a}_x =B(\bar{A}_x+\delta\bar{a}_x)=B.

This scaling is useful for checking the cash-flow pattern, but it should not obscure the units: $B$ is a benefit amount, whereas $c$ is an annual payment rate. If the rate is not $B\delta$, the unit identity still holds, but the scaled insurance and annuity values do not sum to $B$.

Worked Example: Checking a Paired Dollar Benefit and Annuity

A hypothetical contract pays a death benefit of $300,000 immediately at death and a continuous annuity at $12,000 per year while the insured is alive. On the common valuation basis, $\delta=0.04$ and $\bar{A}_x=0.50$. Find the annuity factor, calculate both expected present values, and test whether their total equals the death benefit.

Method. Recover $\bar{a}_x$ using the unit identity, then scale each factor by its actual payment amount. The test is whether the annuity rate equals the death benefit multiplied by $\delta$.

First find the annuity factor:

\bar{a}_x=\frac{1-0.50}{0.04}=12.5.

The insurance expected present value is

\$300{,}000(0.50)=\$150{,}000.

The annuity expected present value is

\$12{,}000(12.5)=\$150{,}000.

The combined expected present value is $\$150{,}000+\$150{,}000=\$300{,}000$. This matches the death benefit because $\delta B=0.04(\$300{,}000)=\$12{,}000$, exactly the stated annual annuity rate. The common mistake this method avoids is assuming that any insurance and annuity amounts must add to the death benefit; that conclusion follows here specifically because $c=B\delta$.

Common Traps and Scope

The identity is compact, but its conditions are easy to overlook. Before applying it, check that the contract pays the insurance benefit at death, the annuity is continuous while the life is alive, and both factors use the same constant force of interest and mortality basis.

Using the wrong interest quantity. The formula uses the force $\delta$, not the effective annual rate $i$. When $i$ is given, convert using $\delta=\ln(1+i)$.
Changing the payment timing. An annuity-due or annuity-immediate value is not $\bar{a}_x$. A benefit paid at the end of the year of death is not the immediate-at-death benefit represented by $\bar{A}_x$.
Mixing valuation bases. An insurance factor from one mortality assumption cannot be paired with an annuity factor from another and expected to satisfy the identity.
Forgetting units when scaling. The annuity factor multiplies a rate per year; the insurance factor multiplies a benefit amount.
Applying a model-specific formula universally. The relation $\bar{A}_x=\mu/(\mu+\delta)$ requires constant-force mortality. The complement identity does not.

Key takeaway: For a unit whole-life benefit payable immediately at death and a unit continuous annuity payable while alive, on the same constant-force interest basis, $\bar{A}_x+\delta\bar{a}_x=1$. The identity follows by adding the two discounted present values for each lifetime before taking expectations.

Check Your Understanding

For each problem, use the stated timing and assumptions carefully.

A hypothetical life has constant force of mortality $\mu=0.06$ and constant force of interest $\delta=0.04$. Find $\bar{A}_x$ and $\bar{a}_x$, and verify the identity. (Answer: $\bar{A}_x=0.6000,\ \bar{a}_x=10,\ 0.6000+0.04(10)=1$.)
On a common basis, $\delta=0.025$ and $\bar{a}_x=20$. Find $\bar{A}_x$. (Answer: $\bar{A}_x=0.5000$.)
Under constant-force mortality, $\mu=0.04$ and $\delta=0.01$. Find both factors and check them using the identity. (Answer: $\bar{A}_x=0.8000,\ \bar{a}_x=20,\ 0.8000+0.01(20)=1$.)
A common-basis valuation gives $\delta=0.05$ and $\bar{A}_x=0.60$. If mortality is constant-force, find $\bar{a}_x$ and $\mu$, then verify the insurance factor. (Answer: $\bar{a}_x=8,\ \mu=0.075,\ 0.075/(0.075+0.05)=0.60$.)
A hypothetical contract pays $240,000 immediately at death and a continuous annuity of $9,600 per year while the life is alive. On the valuation basis, $\delta=0.04$ and $\bar{A}_x=0.60$. Find $\bar{a}_x$, each expected present value, and their total. (Answer: $\bar{a}_x=10$; insurance expected present value is $144,000; annuity expected present value is $96,000; total is $240,000.)

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