Deriving Annuity Reserves from Insurance-Annuity Identities

Why Insurance-Annuity Identities Help with Reserves

A life annuity reserve is the expected present value of the payments still due, conditional on the annuitant being alive at the valuation date. One way to calculate it is to value those payments directly. Another is to use an identity connecting an annuity value with a life insurance value. The second method can be efficient when the insurance value is already available, but only if the benefit timing and annuity payment timing match.

The key identity in this tutorial is for a unit whole-life insurance benefit paid at the end of the year of death and a unit whole-life annuity-due payment made at the start of each year while alive. Its factor is the annual discount rate \(d\), not 1. Temporary and deferred annuities require further attention to the maturity benefit and to survival during the deferral period.

Definitions and timing convention: \(A_y\) is the expected present value at age \(y\) of a unit whole-life insurance benefit paid at the end of the year of death. \(\ddot{a}_y\) is the expected present value of unit annuity-due payments at times \(0,1,2,\ldots\), payable while the life is alive at each payment time. \(a_y\) is the corresponding annuity-immediate value, with payments at times \(1,2,3,\ldots\) while alive. The effective annual interest rate is \(i\), \(v=1/(1+i)\), and \(d=1-v\).

Deriving the Whole-Life Identity

Suppose death occurs in policy year \(m\), so the end-of-year insurance benefit is paid at time \(m\). The annuity-due payments have then been made at times \(0,1,\ldots,m-1\). Their present value is \(1+v+\cdots+v^{m-1}\), while the insurance payment has present value \(v^m\).

Because \(d=1-v\), multiplying the annuity-due present value by \(d\) makes the cash flows add to 1 on that path:

d(1+v+\cdots+v^{m-1})+v^m =d\frac{1-v^m}{1-v}+v^m =1-v^m+v^m =1.

Taking expected present values gives the insurance-annuity identity. Equivalently, it follows by expanding the values in terms of survival probabilities. Here \({}_jp_y\) is the probability that a life aged \(y\) survives \(j\) years.

A_y =v-d\sum_{j=1}^{\infty}v^j\,{}_jp_y, \qquad d\ddot{a}_y =d+d\sum_{j=1}^{\infty}v^j\,{}_jp_y, \qquad A_y+d\ddot{a}_y=1.

The expansion for \(A_y\) can be understood by starting from a benefit of 1 at time 1 and reducing its expected present value by \(d v^j\) for each year \(j\) that the life survives beyond that time. The annuity expansion includes the certain payment at time 0 and the later payments conditional on survival. The terms involving survival cancel when the two expressions are added.

Theorem: For a unit whole-life insurance payable at the end of the year of death and a unit whole-life annuity-due payment payable at the start of each year while alive, \(A_y+d\ddot{a}_y=1\). The identity uses the same interest and mortality assumptions for both values.

Since the annuity-due value includes the payment at time 0 and the annuity-immediate value does not, their relationship is \(\ddot{a}_y=1+a_y\). Therefore:

\ddot{a}_y=\frac{1-A_y}{d}, \qquad a_y=\frac{1-A_y}{d}-1 =\frac{v-A_y}{d}.

The final expression for \(a_y\) follows because \(1-d=v\). It is not \((i-A_y)/d\). That distinction matters: \(i\) is the effective interest rate, whereas \(d=1-v\) is the effective discount rate.

Worked Example: Whole-Life Annuity Values from Insurance

A hypothetical life is age \(y\). The effective annual interest rate is \(i=25\%\). The probability of death in the first year is \(0.25\); anyone surviving that year dies by the end of the second year. Find \(A_y\), \(\ddot{a}_y\), and \(a_y\). Use the method of matching the whole-life insurance and annuity-due cash flows.

Step 1: Set the discount quantities. The discount factor is \(v=1/1.25=0.8\), and the discount rate is \(d=1-v=0.2\).

Step 2: Calculate the insurance value. The probability of death in year 1 is \(0.25\). The probability of death in year 2 is the probability of surviving year 1, \(0.75\), because all those survivors die by the end of year 2. Thus:

$$ A_y=(0.25)(0.8)+(0.75)(0.8^2) =0.20+0.48 =0.68. $$

Step 3: Calculate the annuity-due value from the identity.

\ddot{a}_y=\frac{1-A_y}{d} =\frac{1-0.68}{0.2} =\frac{0.32}{0.2} =1.60.

Step 4: Convert to an annuity-immediate value.

a_y=\ddot{a}_y-1=1.60-1=0.60.

Check. The due annuity pays 1 at time 0 and has probability \(0.75\) of paying 1 at time 1, giving \(1+(0.75)(0.8)=1.60\). The immediate annuity pays at time 1 with probability \(0.75\), giving \((0.75)(0.8)=0.60\). Both direct values agree with the identities. The method avoids confusing the interest rate \(i\) with the discount rate \(d\).

Temporary Annuities and Endowment Insurance

For a temporary annuity-due with \(n\) possible payments, payments occur at times \(0,1,\ldots,n-1\) if the life is alive. Match it with an \(n\)-year endowment insurance: a unit benefit is paid at the end of the year of death if death occurs within the term, or at time \(n\) if the life survives the term. Denote the expected present value of this endowment benefit by \(A_{y:\overline{n}|}\), and the temporary annuity-due value by \(\ddot{a}_{y:\overline{n}|}\).

If death occurs in year \(m\leq n\), the discounted insurance benefit is \(v^m\) and the annuity-due payments have present value \(1+v+\cdots+v^{m-1}\). If the life survives to time \(n\), the annuity payments have present value \(1+v+\cdots+v^{n-1}\) and the endowment benefit has present value \(v^n\). In either case, multiplying the annuity-due value by \(d\) and adding the insurance value gives 1. Thus:

A_{y:\overline{n}|}+d\ddot{a}_{y:\overline{n}|}=1, \qquad \ddot{a}_{y:\overline{n}|} =\frac{1-A_{y:\overline{n}|}}{d}.

For an annuity-immediate limited to payments at times \(1,\ldots,n\), the first \(n-1\) payment times align with the later payments of the temporary annuity-due, but the final payment at time \(n\) is made only if the life survives to \(n\). If \({}_np_y\) is the probability of surviving \(n\) years, then:

a_{y:\overline{n}|} =\ddot{a}_{y:\overline{n}|}-1+v^n{}_np_y.

The subtraction removes the certain payment at time 0. The final term adds the time-\(n\) payment, discounted and conditional on survival, which is not part of the annuity-due. This adjustment is why the annuity-immediate value is not always just the annuity-due value minus 1.

Worked Example: A Two-Year Temporary Annuity

For a hypothetical life aged \(y\), let \(i=25\%\). The probability of surviving one year is \(0.8\), and the probability of surviving two years is \(0.4\). Find the temporary annuity-due value for payments at times 0 and 1, and the annuity-immediate value for payments at times 1 and 2. Use the matching two-year endowment insurance identity, then adjust for payment timing.

Step 1: Calculate the endowment insurance value. The probability of death in year 1 is \(1-0.8=0.2\). The probability of death in year 2 is \(0.8-0.4=0.4\), and the maturity benefit is paid with probability \(0.4\). With \(v=0.8\):

A_{y:\overline{2}|} =(0.2)(0.8)+(0.4)(0.8^2)+(0.4)(0.8^2) =0.16+0.256+0.256 =0.672.

Step 2: Apply the identity. Here \(d=0.2\), so:

\ddot{a}_{y:\overline{2}|} =\frac{1-0.672}{0.2} =\frac{0.328}{0.2} =1.64.

Step 3: Adjust for the immediate payment pattern.

a_{y:\overline{2}|} =1.64-1+(0.8^2)(0.4) =0.64+0.256 =0.896.

Check. Directly, the immediate annuity pays at time 1 with probability \(0.8\), and at time 2 with probability \(0.4\). Its value is \((0.8)(0.8)+(0.4)(0.64)=0.64+0.256=0.896\). The final survival adjustment is essential; subtracting 1 alone would give an incorrect value of \(0.64\).

Deferred Annuity Reserves at a Policy Duration

A deferred annuity has no payments during its deferral period. Suppose payments begin at time \(n\), with the first payment made at time \(n\) if the life is alive then. Its value at time 0 is the probability of surviving to time \(n\), multiplied by the discount factor to time \(n\), multiplied by the annuity-due value at the commencement age:

{}_{n|}\ddot{a}_x =v^n{}_np_x\,\ddot{a}_{x+n}.

For a reserve at duration \(k<n\), condition on the life being alive at duration \(k\). The remaining deferral is \(n-k\) years, survival is measured from age \(x+k\), and the commencement age remains \(x+n\). Therefore:

{}_kV_x =v^{n-k}{}_{n-k}p_{x+k}\,\ddot{a}_{x+n}, \qquad k<n.

This is a prospective value: it discounts the future payments from the valuation date, conditional on survival to that date. If valuation is immediately after a payment at duration \(k\), that payment is already past and is not included. Once payments have commenced, the future value depends on the exact valuation timing: immediately before a due payment, include it; immediately after it, do not.

At every duration, the insurance-annuity identity uses the mortality basis from that duration onward. For example, a whole-life annuity reserve at duration \(k\) is valued using age \(x+k\), not age \(x\). A temporary contract also has only its remaining term at duration \(k\). The identity does not permit carrying forward an initial-age annuity value unchanged.

Worked Example: Deferred Annuity Reserve Before Commencement

A hypothetical contract begins at age \(x\), with annuity-due payments scheduled to start at time 4. At duration 2, immediately after any payment scheduled at that duration (there is none), calculate the prospective reserve. The effective annual interest rate is \(25\%\). From age \(x+2\), the probabilities of surviving years 1 and 2 are \(0.9\) and \(0.72\), respectively. At age \(x+4\), the probability of surviving one year is \(0.75\), and anyone surviving that year dies by the end of the following year. Payments are 1 per year while alive.

Step 1: Value the annuity at its commencement age. At age \(x+4\), the whole-life insurance value is:

A_{x+4} =(0.25)(0.8)+(0.75)(0.8^2) =0.20+0.48 =0.68.

With \(d=1-0.8=0.2\), the whole-life annuity-due identity gives:

\ddot{a}_{x+4} =\frac{1-0.68}{0.2} =1.60.

Step 2: Use the remaining deferral and survival from duration 2. There are \(4-2=2\) years until the first payment. The probability of surviving those two years from age \(x+2\) is \(0.72\), and the discount factor is \(v^2=0.8^2=0.64\).

{}_2V_x =(0.64)(0.72)(1.60) =0.73728.

Check. The product of the three factors is \(0.64\times0.72=0.4608\), then \(0.4608\times1.60=0.73728\). The reserve is positive and less than the commencement-age annuity value because the payments are both deferred and contingent on surviving the deferral period. The method avoids using survival from age \(x\) or discounting for all four years when valuing at duration 2.

Common Timing and Basis Errors

The pathwise identity is a useful check on the factor \(d\). If death occurs in year \(m\), the annuity-due payments have present value \(1+v+\cdots+v^{m-1}\), not \(v+v^2+\cdots+v^m\). The insurance payment occurs at time \(m\). Consequently, the correct relation is:

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

Omitting \(d\) does not produce an identity. For example, with \(v=0.8\), if death occurs in year 1, the annuity-due value is 1 and the insurance value is \(0.8\); their sum is \(1.8\), not 1. Multiplying the annuity value by \(d=0.2\) gives \(0.2+0.8=1\).

Using the wrong insurance benefit. The temporary identity uses endowment insurance, including the maturity benefit on survival. A term insurance value alone does not match the temporary annuity-due cash flows.
Ignoring the last immediate payment. A temporary annuity-immediate can pay at time \(n\) if the life survives. Include \(v^n{}_np_y\) when converting from the annuity-due value.
Using the wrong duration basis. A reserve at duration \(k\) is conditional on survival to \(k\). Use age \(x+k\) and the remaining term, not the original age and original term.
Mixing \(i\) and \(d\). The identity uses \(d=1-v\). Substituting \(i\) for \(d\) changes the value.

Key takeaway: Match the insurance benefit and annuity payment times first. Then apply the identity on the mortality and interest basis applicable at the valuation duration, adding a survival adjustment when the payment pattern requires it.

Check Your Understanding

For each problem, identify the relevant payment timing before using an identity.

A hypothetical life has \(i=20\%\), probability \(0.4\) of dying in year 1, and all year-1 survivors die by the end of year 2. Find \(A_y\), \(\ddot{a}_y\), and \(a_y\). (Answer: \(A_y=0.7500,\ \ddot{a}_y=1.5000,\ a_y=0.5000\).)
For a three-year temporary annuity-due, \(i=25\%\), and survival probabilities \({}_1p_y=0.8,\ {}_2p_y=0.6,\ {}_3p_y=0.3\), find its value using the three-year endowment insurance identity. (Answer: \(2.0240\).)
A deferred annuity-due begins at time 2. At duration 0, \(i=10\%\), survival to time 2 is \(0.72\), and the annuity-due value at age \(x+2\) is \(19/11\). Find the reserve at time 0. (Answer: \(1.0278\), to four decimals.)
A two-year temporary annuity-immediate has \(i=25\%\), \({}_1p_y=0.9\), and \({}_2p_y=0.54\). Use the endowment identity to find its value, including the possible payment at time 2. (Answer: \(1.0656\).)
A whole-life annuity-due is valued at duration 5 for a policy issued at age \(x\). State the age and payment timing that should be used for its reserve if valuation is immediately before the payment due at duration 5. (Answer: Use age \(x+5\) and include the payment due at duration 5.)

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