Matching Deferred Insurance and Annuity Values

Why Matching Deferred Values Are Related

A deferred whole-life insurance and a deferred life annuity can have different payment triggers, but if they share the same deferral period, both values contain the same probability and discount factor for reaching the start of coverage. That common factor lets the whole-life insurance-annuity identity at the attained age relate their present values directly.

The key is to compare benefits beginning at the same time. A deferred insurance that begins after \(m\) years should be paired with a deferred annuity that also begins after \(m\) years. The identity is applied at age \(x+m\), conditional on survival to that age, and then both values are brought back to age \(x\).

Definition: In the annual discrete model, a unit deferred whole-life insurance with deferral period \(m\) pays 1 at the end of the year of death if the life aged \(x\) survives \(m\) years. Its present value is denoted \({}_{m|}A_x\). A unit deferred life annuity-due with deferral period \(m\), denoted \({}_{m|}\ddot{a}_x\), pays 1 at times \(m,m+1,\ldots\), provided the life is alive at each payment time. Define the pure endowment factor \({}_mE_x=v^m{}_mp_x\), where \(v=1/(1+i)\) and \({}_mp_x\) is the probability of surviving \(m\) years.

These definitions use annual payments and annual effective interest. For the insurance, death during the \(k\)th year after deferral produces a payment at time \(m+k\). For the annuity, the first payment is at time \(m\), conditional on survival to that time. Neither contract pays if the life dies before the deferral period ends.

The Matching Deferred Identity

At any attained age \(y\), the standard discrete whole-life identity is \(A_y+d\ddot{a}_y=1\), where \(d=i/(1+i)\) is the effective annual discount rate. Apply it at \(y=x+m\). The deferred insurance and annuity values are each the attained-age value multiplied by the same factor \(v^m{}_mp_x\).

{}_{m|}A_x = v^m{}_mp_x A_{x+m} = {}_mE_x A_{x+m}, \qquad {}_{m|}\ddot{a}_x = v^m{}_mp_x\ddot{a}_{x+m} = {}_mE_x\ddot{a}_{x+m}.

Substituting the attained-age identity gives the matching deferred relation:

Theorem: For a unit deferred whole-life insurance and a unit deferred life annuity-due with the same deferral period \(m\), under the annual discrete model,

{}_{m|}A_x+d\,{}_{m|}\ddot{a}_x={}_mE_x.

Equivalently, if the pure endowment factor is known, either deferred value can be recovered from the other:

{}_{m|}A_x={}_mE_x-d\,{}_{m|}\ddot{a}_x, \qquad {}_{m|}\ddot{a}_x=\frac{{}_mE_x-{}_{m|}A_x}{d}.

This identity requires a common deferral period and consistent benefit timing. It does not equate an insurance that starts at \(m\) with an annuity that starts at a different time. It also uses annual discrete whole-life values; continuous benefits require a different identity.

Why the Formula Holds

The attained-age whole-life identity has a cash-flow explanation. Suppose death occurs in year \(k\) after age \(y\), so the insurance payment has present value \(v^k\). The life annuity-due has payments at times \(0,1,\ldots,k-1\). Multiplying that annuity's present value by \(d\) gives:

d\sum_{j=0}^{k-1}v^j =d\frac{1-v^k}{1-v} =1-v^k,

because \(1-v=d\). Thus the insurance present value plus \(d\) times the annuity present value is \(v^k+(1-v^k)=1\) for that death year. Taking expectations gives \(A_y+d\ddot{a}_y=1\). If the life survives the deferral period, the remaining contract is valued at age \(x+m\); discounting that value to age \(x\) and multiplying by the probability of survival gives the factor \(v^m{}_mp_x\). Applying this same factor to both sides proves the deferred identity.

Find the attained age.
For a deferral of \(m\) years, the remaining whole-life contracts begin at age \(x+m\).

Value survival to the start.
Calculate \({}_mE_x=v^m{}_mp_x\), including both discounting and survival.

Apply the attained-age identity.
Use \(A_{x+m}+d\ddot{a}_{x+m}=1\), then multiply by \({}_mE_x\).

Check the relationship.
Verify that \({}_{m|}A_x+d\,{}_{m|}\ddot{a}_x={}_mE_x\).

Worked Applications

Worked Example: Finding Deferred Insurance from an Attained-Age Annuity

A hypothetical contract has an annual effective interest rate of \(25\%\). A life aged \(x\) has probability \(0.75\) of surviving two years. At age \(x+2\), the unit life annuity-due value is \(4\). Find the two-year deferred whole-life insurance value and the matching deferred annuity value.

Method. Calculate the common survival-discount factor, use the attained-age identity, and then check the deferred relation. The mortality assumption is \({}_2p_x=0.75\); the interest assumption is \(i=0.25\).

First, \(v=1/1.25=0.8\), so \(v^2=0.64\). The effective discount rate is \(d=i/(1+i)=0.25/1.25=0.20\). Therefore:

{}_2E_x=v^2{}_2p_x=0.64(0.75)=0.48.

At age \(x+2\), the attained-age identity gives \(A_{x+2}=1-d\ddot{a}_{x+2}=1-0.20(4)=0.20\). Thus the deferred insurance value is:

{}_{2|}A_x={}_2E_x A_{x+2}=0.48(0.20)=0.096.

The deferred annuity value is \({}_{2|}\ddot{a}_x={}_2E_x\ddot{a}_{x+2}=0.48(4)=1.92\). Check: \(0.096+0.20(1.92)=0.096+0.384=0.480={}_2E_x\). The values agree with the matching deferred identity. This method avoids using an insurance value at age \(x\) instead of age \(x+2\).

Worked Example: Recovering a Deferred Annuity from Insurance

A hypothetical insurer values a one-year deferred whole-life insurance at \(0.16\). The annual effective interest rate is \(25\%\), and the probability of surviving one year from age \(x\) is \(0.80\). Find the value of the matching one-year deferred life annuity-due.

Method. Rearrange the matching deferred identity. The insurance payment is at the end of the year of death after the deferral period, and the annuity's first payment is at time 1 if the life is then alive.

Here \(v=1/1.25=0.8\), \(d=0.25/1.25=0.20\), and the pure endowment factor is:

{}_1E_x=v\,{}_1p_x=0.8(0.80)=0.64.

Using \({}_{1|}A_x+d\,{}_{1|}\ddot{a}_x={}_1E_x\):

{}_{1|}\ddot{a}_x =\frac{{}_1E_x-{}_{1|}A_x}{d} =\frac{0.64-0.16}{0.20} =\frac{0.48}{0.20} =2.4.

Check: \(0.16+0.20(2.4)=0.16+0.48=0.64\), equal to the pure endowment factor. The answer is the deferred annuity value at age \(x\), not the annuity value conditional on reaching age \(x+1\); the latter is \(2.4/0.64=3.75\).

Worked Example: Recovering Survival from Matching Deferred Values

For a hypothetical life aged \(x\), the annual effective interest rate is \(25\%\). A two-year deferred whole-life insurance has value \(0.128\), and the matching two-year deferred annuity-due has value \(2.24\). Find the probability of surviving two years.

Method. First recover the pure endowment factor from the identity, then remove the discount factor to isolate the survival probability. Assume annual discrete benefits and a common two-year deferral for both contracts.

The effective discount rate is \(d=0.25/1.25=0.20\). Therefore:

{}_2E_x={}_{2|}A_x+d\,{}_{2|}\ddot{a}_x =0.128+0.20(2.24) =0.128+0.448 =0.576.

The discount factor is \(v=0.8\), so \(v^2=0.64\). Since \({}_2E_x=v^2{}_2p_x\):

{}_2p_x=\frac{{}_2E_x}{v^2} =\frac{0.576}{0.64} =0.90.

Check by reconstructing the deferred values. The attained-age annuity value implied by the data is \(2.24/0.576=3.8889\) to four decimal places, and the attained-age insurance value is \(0.128/0.576=0.2222\) to four decimal places. Their identity check is \(0.2222+0.20(3.8889)\approx1.0000\), with the small displayed discrepancy due only to rounding. The method avoids treating the pure endowment factor itself as a survival probability.

Common Trap: Omitting Survival to the Deferral Date

The common factor is \(v^m{}_mp_x\), not \(v^m\) alone. Discounting accounts for the time value of money, while \({}_mp_x\) accounts for the possibility that the life does not survive to the point when the deferred benefits begin. Both are needed.

For the first example, omitting survival would incorrectly give a deferred annuity value of \(v^2\ddot{a}_{x+2}=0.64(4)=2.56\). Substituting that incorrect value into the matching identity would give an insurance value of \(0.48-0.20(2.56)=-0.032\). A negative value is an immediate warning: the proposed annuity value exceeds what is consistent with the available pure endowment factor. Including survival gives the correct annuity value \(0.48(4)=1.92\) and insurance value \(0.096\).

Another indexing error is to use \(A_x+d\ddot{a}_x=1\) directly for the deferred pair. The deferred contracts are conditional on reaching age \(x+m\), so the attained-age values are \(A_{x+m}\) and \(\ddot{a}_{x+m}\). The deferred values are those attained-age quantities multiplied by the same pure endowment factor.

Quantity	Meaning in the matching identity	Check
\({}_{m\|}A_x\)	Deferred insurance value at age \(x\)	Includes survival and discounting to time \(m\)
\({}_{m\|}\ddot{a}_x\)	Deferred annuity-due value at age \(x\)	Uses the same deferral period as the insurance
\({}_mE_x\)	Value of 1 payable at time \(m\) if alive	\({}_mE_x=v^m{}_mp_x\)
\(d\)	Effective annual discount rate	\(d=i/(1+i)\)

This relationship is useful when one deferred value is easier to calculate than the other, or when a problem supplies both deferred values and asks for a survival probability. It also gives a consistency check: for positive \(d\), both deferred values are nonnegative, and their weighted sum must equal \({}_mE_x\). The relation is for unit benefits; if benefits are not unit amounts, first account for their stated amounts rather than applying the unit identity without adjustment.

Key takeaway: For matching annual discrete deferred whole-life insurance and life annuity-due values, apply the whole-life identity at age \(x+m\), then multiply by \({}_mE_x=v^m{}_mp_x\). This gives \({}_{m|}A_x+d\,{}_{m|}\ddot{a}_x={}_mE_x\).

Check Your Understanding

Use annual discrete values unless stated otherwise. Each deferred insurance and annuity pair has the same deferral period.

At \(i=5\%\), \({}_1p_x=0.90\), and \(\ddot{a}_{x+1}=10\). Find \({}_{1|}A_x\). (Answer: \(0.4490\))
At \(i=25\%\), \({}_2p_x=0.75\), and \(A_{x+2}=0.40\). Find \({}_{2|}\ddot{a}_x\). (Answer: \(1.4400\))
At \(i=25\%\), a two-year deferred insurance is valued at \(0.128\) and its matching deferred annuity-due at \(2.24\). Find \({}_2p_x\). (Answer: \(0.9000\))
At \(i=10\%\), \({}_2p_x=0.80\), and \(\ddot{a}_{x+2}=6\). Find \({}_{2|}A_x\), rounding to four decimals. (Answer: \(0.3005\))
At \(i=5\%\), \({}_2p_x=0.90\), and \(\ddot{a}_{x+2}=14\). Find \({}_{2|}A_x\), rounding to four decimals. (Answer: \(0.2721\))

What You'll Learn