Temporary Annuity Values from Term Insurance

Why These Values Fit Together

A finite-term life annuity-due and a term insurance contract describe different payments, but they use the same survival probabilities and discount factors. That makes it possible to recover the annuity value from the term-insurance value and the value of a benefit paid at the end of the term if the life survives. This relationship is useful when a calculation supplies insurance factors but asks for an annuity value, or when you want an independent check on a direct annuity calculation.

The key is to match the term and payment timing. In this tutorial, payments are annual and discrete, interest is an effective annual rate $i$, a term-insurance benefit is paid at the end of the year of death, and an annuity-due payment is made at the start of a year if the life is then alive. The terminal survival benefit is paid at time $n$.

Definition: For a life aged $x$ and an $n$-year term, a unit temporary life annuity-due pays 1 at times $0,1,\ldots,n-1$, provided the life is alive at each payment time. Its expected present value is $\ddot a_{x:\overline{n}|}$. A unit discrete $n$-year term insurance pays 1 at the end of the year of death if death occurs within the $n$-year term; its expected present value is $A^1_{x:\overline{n}|}$. A unit $n$-year pure endowment pays 1 at time $n$ if the life survives the full term; its expected present value is ${}_nE_x=v^n{}_np_x$, where $v=1/(1+i)$.

Write ${}_kp_x$ for the probability that a life aged $x$ survives $k$ years, and ${}_kq_x=1-{}_kp_x$ for the probability of death within $k$ years. In particular, the probability of being alive at the start of year $k+1$ and dying during that year is ${}_kp_xq_{x+k}$, where $q_{x+k}$ is the one-year probability of death at age $x+k$. The annuity and insurance factors below are expected present values under the same mortality basis and interest rate.

The Finite-Term Identity

The finite-term identity connects the three values directly. It applies to a unit benefit and a unit annuity, with the timing just defined. The annual effective discount rate is $d=i/(1+i)=1-v$; do not confuse this discount rate with a deductible or another use of the letter $d$ in a different context.

Theorem: For an $n$-year discrete term insurance with benefit paid at the end of the year of death, a unit pure endowment payable at time $n$, and a unit temporary life annuity-due, all valued at the same interest rate and mortality basis, $$ d\,\ddot a_{x:\overline{n}|}+A^1_{x:\overline{n}|}+{}_nE_x=1, \qquad d=1-v=\frac{i}{1+i}. $$ Equivalently, $$ \ddot a_{x:\overline{n}|} =\frac{1-A^1_{x:\overline{n}|}-{}_nE_x}{d}, \qquad i>0. $$

Here is why the identity holds. At time 0, the life is alive and the annuity-due makes a payment of 1. At each later payment time $k$, for $1\leq k\leq n-1$, a payment is made if the life has survived $k$ years. The reduction in discounted annuity value from one payment time to the next is $d$ times the discounted survival probability. These reductions can be matched to the discounted payments on death during the term and to the discounted terminal payment if the life survives to time $n$.

\begin{aligned} d\,\ddot a_{x:\overline{n}|} &=\sum_{k=0}^{n-1}(v^k-v^{k+1})\,{}_kp_x\\ &=1-\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k} -v^n{}_np_x\\ &=1-A^1_{x:\overline{n}|}-{}_nE_x. \end{aligned}

For the second equality, use ${}_{k+1}p_x={}_kp_x(1-q_{x+k})$. The difference of the two finite sums leaves the initial value 1, subtracts the death probabilities discounted to their year-end payment times, and subtracts the probability of survival to time $n$ discounted to time $n$. Those subtracted terms are exactly the term-insurance and pure-endowment expected present values. This derivation also shows why the identity requires matching durations and payment times.

If $i=0$, then $d=0$, so the rearranged formula that divides by $d$ cannot be used. In that case the identity still holds in its first form, but it does not recover the annuity by division. A direct survival-probability sum is appropriate instead.

Worked Calculations

When one-year mortality probabilities are given, first build the survival probabilities. When insurance and pure-endowment factors are already supplied, use them directly in the identity. In either case, a direct annuity sum provides a useful check when enough survival information is available.

Worked Example: Recovering an Annuity From One-Year Mortality

A hypothetical three-year contract covers a life aged $x$. The effective annual interest rate is $i=5\%$. The one-year death probabilities at ages $x$, $x+1$, and $x+2$ are $0.10$, $0.20$, and $0.25$, respectively. Find the three-year temporary annuity-due value using the term-insurance and pure-endowment factors, then check it directly.

Method and assumptions. Mortality is represented by those annual death probabilities, interest is constant at 5% effective annually, and benefits are paid at the discrete times specified above. The corresponding survival probabilities for one, two, and three years are $$ {}_1p_x=0.90,\qquad {}_2p_x=0.90(0.80)=0.72,\qquad {}_3p_x=0.72(0.75)=0.54. $$ The discount quantities are $v=1/1.05=0.95238095$, $v^2=0.90702948$, $v^3=0.86383760$, and $d=0.05/1.05=0.04761905$.

Step 1: Calculate the term-insurance factor. Death in the first, second, or third year has probability $0.10$, $0.90(0.20)=0.18$, or $0.72(0.25)=0.18$, respectively. The benefit is discounted to each year-end: $$ A^1_{x:\overline{3}|} =(0.10)v+(0.18)v^2+(0.18)v^3 =0.09523810+0.16326531+0.15549077 =0.41399417. $$

Step 2: Calculate the pure-endowment factor. Survival to time 3 has probability $0.54$, so $$ {}_3E_x=(0.54)v^3=(0.54)(0.86383760)=0.46647230. $$

Step 3: Recover the annuity-due value. Substituting the two factors into the finite-term identity gives $$ \ddot a_{x:\overline{3}|} =\frac{1-0.41399417-0.46647230}{0.04761905} =\frac{0.11953353}{0.04761905} =2.5102. $$ The value is rounded to 4 decimal places.

Check and trap avoided. Directly, the annuity pays at times 0, 1, and 2 with survival probabilities $1$, $0.90$, and $0.72$: $$ \ddot a_{x:\overline{3}|} =1+(0.90)v+(0.72)v^2 =1+0.85714286+0.65306122 =2.5102. $$ The direct sum agrees. A common mistake is to discount the term-insurance benefit at the start, rather than the end, of the year of death. That would incorrectly replace each $v^{k+1}$ in the insurance value by $v^k$, breaking the identity.

Worked Example: Using a Short Mortality Table

A hypothetical two-year contract is valued at an effective annual interest rate of $5\%$. A life aged $x$ has one-year survival probability $0.90$, followed by one-year survival probability $0.80$ at age $x+1$. Find the temporary annuity-due value from the insurance and pure-endowment values, and verify it using the survival probabilities.

Method and assumptions. Death benefits are paid at year-end; annuity payments are made at times 0 and 1 if the life is alive. The survival probabilities are ${}_1p_x=0.90$ and ${}_2p_x=0.90(0.80)=0.72$. At $i=0.05$, $v=0.95238095$, $v^2=0.90702948$, and $d=0.04761905$.

Term-insurance value. The first-year death probability is $0.10$. The second-year death probability is $0.90(0.20)=0.18$. Thus $$ A^1_{x:\overline{2}|} =(0.10)v+(0.18)v^2 =0.09523810+0.16326531 =0.25850340. $$

Pure-endowment value and recovery. The probability of surviving two years is $0.72$, so $$ {}_2E_x=(0.72)v^2=(0.72)(0.90702948)=0.65306122. $$ Then $$ \ddot a_{x:\overline{2}|} =\frac{1-0.25850340-0.65306122}{0.04761905} =\frac{0.08843538}{0.04761905} =1.8571. $$ This result is rounded to 4 decimal places.

Check and trap avoided. The direct sum is $1+(0.90)v=1+0.85714286=1.8571$, which agrees. Notice that the pure endowment uses survival for the full two years and discounting by $v^2$. Using one-year survival or discounting to time 1 would mismatch the term and give the wrong recovery.

Worked Example: Recovering an Annuity From Supplied Factors

For a hypothetical four-year valuation at $i=4\%$, the discrete term-insurance factor is $A^1_{x:\overline{4}|}=0.28$, and the matching pure-endowment factor is ${}_4E_x=0.52$. Find $\ddot a_{x:\overline{4}|}$. The factors are assumed to come from the same annual mortality basis, with the benefit timings used in the identity.

Method and assumptions. Because the two expected present values are supplied, the one-year mortality probabilities are not needed. The interest rate is effective annually, so $$ v=\frac{1}{1.04}=0.96153846,\qquad d=1-v=\frac{0.04}{1.04}=0.03846154. $$

Apply the identity. The amount left after subtracting the two supplied values from 1 is $$ 1-0.28-0.52=0.20. $$ Therefore $$ \ddot a_{x:\overline{4}|} =\frac{0.20}{0.03846154}=5.2. $$ The recovered value is $5.2000$ to 4 decimal places.

Check and trap avoided. Multiplying the recovered value by $d$ gives $5.2(0.03846154)=0.20$. Adding the two supplied factors gives $0.28+0.52=0.80$, and $0.80+0.20=1$, as required by the identity. Do not divide by $i=0.04$ in place of $d$: the identity uses $d=i/(1+i)$, not the effective interest rate itself. Dividing by $0.04$ would give $5.0000$, which fails this check.

How to Apply the Relationship Reliably

A disciplined calculation makes it easier to detect a wrong duration, mismatched benefit timing, or incorrect rate conversion. The process below applies whether you calculate the insurance factors from a mortality table or receive them as supplied values.

Match the contract terms.
Confirm that the annuity, term insurance, and pure endowment all use the same life, term $n$, interest rate, and mortality basis. The insurance benefit must be payable at the end of the year of death, and the pure endowment at time $n$.

Establish the discount rate.
For an effective annual interest rate $i$, calculate $v=1/(1+i)$ and $d=1-v=i/(1+i)$. Keep these separate: the rearranged identity divides by $d$.

Obtain the two benefit values.
Calculate or identify $A^1_{x:\overline{n}|}$ and ${}_nE_x$. If using mortality probabilities, discount each death benefit at its year-end and the survival benefit at time $n$.

Recover and check the annuity.
Use $\ddot a_{x:\overline{n}|}=(1-A^1_{x:\overline{n}|}-{}_nE_x)/d$. When survival probabilities are available, check against $\sum_{k=0}^{n-1}v^k{}_kp_x$.

The direct annuity sum is particularly helpful as an audit, but it is not a substitute for matching contract definitions. For example, an annuity-immediate pays at the end of each year rather than the start, so its payment times differ. Similarly, an insurance benefit paid at the moment of death is not the discrete year-end benefit used here. Those products require their own timing relationships; applying this identity without adjusting the benefit timing is not valid.

Common Errors and Practical Checks

Omitting the pure endowment. The finite-term relation includes both death during the term and survival to its end. Using only $1-A^1_{x:\overline{n}|}$ does not recover the annuity.
Mixing terms. An $n$-year term-insurance factor must be paired with the $n$-year pure endowment and annuity. A value for a different duration cannot be substituted just because it uses the same age.
Using the wrong discount rate. The divisor is $d=i/(1+i)$, not $i$. The difference matters even when the interest rate is familiar.
Misplacing payments. The annuity-due pays at times $0$ through $n-1$, the discrete death benefit at the end of the year of death, and the pure endowment at time $n$. The survival and discount factors must reflect those times.
Rounding too early. Keep sufficient precision in $v$, the insurance factor, and the pure-endowment factor. Round the final annuity value only after using the identity, and verify with a direct sum when possible.

Key takeaway: For matched discrete contracts, the temporary annuity-due value is $(1-A^1_{x:\overline{n}|}-{}_nE_x)/d$. The finite-term identity works because the annuity’s discounted payment increments correspond to death within the term and survival to the term’s end. Matching duration, basis, and payment timing is essential.

Check Your Understanding

For each problem, use the discrete payment timings and the same mortality and interest basis for all factors. Answers are rounded as indicated.

At $i=5\%$, a two-year valuation has survival probabilities ${}_1p_x=0.80$ and ${}_2p_x=0.60$. Find the temporary annuity-due value directly from survival probabilities. (Answer: 1.7619)
At $i=4\%$, a three-year life has survival probabilities ${}_1p_x=0.90$ and ${}_2p_x=0.70$. Find $\ddot a_{x:\overline{3}|}$ using the direct sum and give 4 decimal places. (Answer: 2.5126)
At $i=6\%$, matching four-year factors are $A^1_{x:\overline{4}|}=0.28$ and ${}_4E_x=0.42$. Recover the temporary annuity-due value to 4 decimal places. (Answer: 5.3000)
At $i=5\%$, a three-year term-insurance factor is $0.25$, and the matching pure-endowment factor is $0.65$. Recover $\ddot a_{x:\overline{3}|}$ to 4 decimal places. (Answer: 2.1000)
At $i=5\%$, a matching valuation has $\ddot a_{x:\overline{3}|}=2.5$ and $A^1_{x:\overline{3}|}=0.38$. Find ${}_3E_x$ to 4 decimal places. (Answer: 0.5010)

What You'll Learn