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Exam FAM · Prospective Reserves · Tutorial 406 of 988

Term Insurance Reserves During the Coverage Period

Learn to value the remaining term benefits and premiums at a policy duration, with careful attention to premium timing.

Intermediate 12 min read

What You'll Learn

  • Define the reserve timing convention for annual-premium term insurance
  • Calculate remaining-term death-benefit and premium values
  • Apply the prospective reserve formula at an interior duration
  • Distinguish reserves just before and just after an anniversary premium
  • Check a term reserve using a hypothetical mortality model
  • Identify common errors in the remaining term and premium count

Why Term Reserves Depend on Duration

A term insurance policy covers death only during a specified period. At an interior duration, its prospective reserve therefore depends on the remaining coverage period, not the original term. It also depends on premiums that are still due, including whether a premium is payable immediately at the valuation date.

The calculations below assume a hypothetical policy with a level death benefit, annual premiums, and no expenses. The benefit is paid at the end of the policy year of death. Premiums are paid at the beginning of each policy year if the insured is alive and the coverage remains in force. These timing assumptions are essential: a different benefit or premium timing changes the present values.

Definition: Consider an \(n\)-year term policy issued at age \(x\), with benefit \(B\) paid at the end of the year of death. Let \(P\) be its annual level premium. For this tutorial, \(V_k^-\) denotes the prospective reserve at duration \(k\), immediately before the premium due at that date. It is conditional on the insured being alive at duration \(k\). Thus, if \(k<n\), the premium at duration \(k\) is included among future premiums. The duration \(k\) is interior when \(0<k<n\).

We use \(i\) for the effective annual interest rate and \(v=1/(1+i)\). At age \(y\), \(q_y\) is the probability of death during the next year and \(p_y=1-q_y\) is the probability of surviving that year. The symbols \({}_jp_y\) and \({}_jq_y\) denote the probabilities of surviving or dying, respectively, within \(j\) years from age \(y\), with \({}_0p_y=1\).

Values for the Remaining Term

For a term of \(m\) years beginning at age \(y\), define the present value of a unit end-of-year-of-death benefit and the present value of an annual premium annuity-due as follows:

$$ A^1_{y:\overline{m}|} =\sum_{j=0}^{m-1}v^{j+1}{}_jp_yq_{y+j}, \qquad \ddot{a}_{y:\overline{m}|} =\sum_{j=0}^{m-1}v^j{}_jp_y. $$

In the insurance sum, the term for year \(j+1\) includes the probability of surviving the first \(j\) years and then dying in the next year. Its benefit is discounted \(j+1\) years because it is paid at that year's end. In the annuity sum, the premium at the start of year \(j+1\) is paid only if the insured has survived to that premium date, and it is discounted \(j\) years.

Prospective term reserve: Under the stated timing assumptions, at duration \(k\) immediately before the premium due, $$ V_k^- =B A^1_{x+k:\overline{n-k}|} -P\ddot{a}_{x+k:\overline{n-k}|}, \qquad 0\leq k<n. $$ There are \(n-k\) years of coverage remaining, and the premium due at duration \(k\) is included. This is a conditional value: all probabilities and present values are evaluated from age \(x+k\).

At issue, before the initial premium, the same formula gives \(V_0^-=0\) if \(P\) is the net level premium determined by the equivalence principle:

$$ P=\frac{B A^1_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}}. $$

If the reserve is instead needed immediately after the premium at duration \(k\) has been paid, that premium is no longer an outstanding future payment. With the benefit and other cash-flow assumptions unchanged, the reserve is

$$ V_k^+=V_k^-+P. $$

The plus sign matters. Before payment, the prospective value subtracts the premium due now; after payment, it does not. Removing that future premium from the subtraction increases the reserve by \(P\). At duration \(n\), coverage has ended, so this term-insurance reserve is zero, assuming no other benefits or payments.

Worked Example: A Three-Year Policy at Duration One

Worked Example: A Three-Year Policy at Duration One

A hypothetical policy issued at age 40 pays \(B=\$10{,}000\) at the end of the year of death during its three-year term. Annual premiums are paid in advance. Assume \(i=25\%\), with \(q_{40}=0.10\), \(q_{41}=0.20\), and \(q_{42}=0.25\). Find the net annual premium and the reserve at duration 1 immediately before and after the premium due then.

Method. First value the full three-year term to determine \(P\). Then revalue only the two years remaining from age 41. The main timing trap is to omit the duration-1 premium from the reserve just before it is paid.

The annual discount factor is \(v=1/1.25=0.8\). The probabilities of death in policy years 1, 2, and 3, conditional on issue, are

$$ 0.10,\qquad 0.90(0.20)=0.18,\qquad 0.90(0.80)(0.25)=0.18. $$

The unit-benefit value is therefore

$$ A^1_{40:\overline{3}|} =0.10(0.8)+0.18(0.8^2)+0.18(0.8^3) =0.08000+0.11520+0.09216 =0.28736. $$

Premiums at times 0, 1, and 2 are due with probabilities \(1\), \(0.90\), and \(0.90(0.80)=0.72\). Their present-value annuity factor is

$$ \ddot{a}_{40:\overline{3}|} =1+0.90(0.8)+0.72(0.8^2) =1+0.72+0.4608 =2.1808. $$

Thus the net premium is

$$ P=\frac{\$10{,}000(0.28736)}{2.1808} =\frac{\$2{,}873.60}{2.1808} =\$1{,}317.6816\ldots $$

At duration 1, two years remain. Conditional on survival to age 41, the death probabilities in those years are \(0.20\) and \(0.80(0.25)=0.20\). Hence

$$ A^1_{41:\overline{2}|} =0.20(0.8)+0.20(0.8^2) =0.1600+0.1280 =0.2880. $$

The premium annuity factor at duration 1 includes the premium then due and the next premium, payable if the insured survives one more year:

$$ \ddot{a}_{41:\overline{2}|} =1+0.80(0.8)=1.64. $$

The reserve immediately before the duration-1 premium is

$$ V_1^-=\$10{,}000(0.2880)-\$1{,}317.6816\ldots(1.64) =\$2{,}880.00-\$2{,}160.9978\ldots =\$719.00. $$

Immediately after that premium is paid, \(V_1^+=V_1^-+P=\$719.0022\ldots+\$1{,}317.6816\ldots=\$2{,}036.68\). Check: after payment, only the premium at duration 2 remains outstanding; its conditional present value is \(0.80(0.8)P=0.64P\). Thus the post-premium reserve is \(\$2{,}880.00-0.64P=\$2{,}036.68\), as calculated. Amounts are rounded to cents only at the final step. The common error avoided here is using the after-premium premium count for a before-premium reserve.

Why the Prospective Formula Holds

The formula follows by taking conditional expected present values from the valuation date. The insured is assumed alive at duration \(k\), so the policy is viewed as a fresh \(n-k\)-year contract beginning at age \(x+k\). For a death in the \(j+1\)st remaining year, the conditional probability is \({}_jp_{x+k}q_{x+k+j}\), and its benefit value at duration \(k\) is discounted by \(v^{j+1}\). Summing these mutually exclusive death-year outcomes gives \(B A^1_{x+k:\overline{n-k}|}\).

A premium at the start of the \(j+1\)st remaining year is paid only if the insured survives the first \(j\) years. Its expected present value is therefore \(P v^j{}_jp_{x+k}\). Summing over the \(n-k\) remaining premium dates gives \(P\ddot{a}_{x+k:\overline{n-k}|}\). The prospective reserve is the expected present value of future benefits minus that of future premiums, which yields the stated formula.

1
Fix the valuation instant.
Decide whether the reserve is just before or just after the anniversary premium. The formula for \(V_k^-\) includes the premium due at duration \(k\).
2
Count remaining coverage years.
At duration \(k\) in an \(n\)-year term, use \(n-k\) years, beginning at attained age \(x+k\).
3
Value benefits and premiums conditionally.
Use the remaining-term insurance value for benefits and the remaining temporary annuity-due value for premiums.
4
Subtract and check timing.
Calculate benefits minus premiums. If the premium at duration \(k\) has already been paid, add \(P\) to the before-premium reserve.

Worked Example: A Four-Year Term at Duration Two

Worked Example: A Four-Year Term at Duration Two

A hypothetical four-year term policy issued at age 50 pays \(\$50{,}000\) at the end of the year of death. It charges a net annual premium in advance. Let \(i=0\), with \(q_{50}=0.10\), \(q_{51}=0.20\), \(q_{52}=0.30\), and \(q_{53}=0.40\). Find the net premium and the reserve at duration 2 immediately before the premium due.

Method. At zero interest, discount factors equal 1, but survival probabilities still determine whether premiums and benefits occur. The trap is to count all four original years rather than only the two remaining years when finding the reserve.

The unconditional probabilities of death in the four years are \(0.10\), \(0.90(0.20)=0.18\), \(0.90(0.80)(0.30)=0.216\), and \(0.90(0.80)(0.70)(0.40)=0.2016\). Their sum is \(0.6976\), so the expected benefit at issue is \(\$50{,}000(0.6976)=\$34{,}880\).

The expected number of premiums is \(1+0.90+0.90(0.80)+0.90(0.80)(0.70)=1+0.90+0.72+0.504=3.124\). Therefore, \(P=\$34{,}880/3.124=\$11{,}165.172855\ldots\).

At duration 2, the insured is age 52 and two years of coverage remain. The conditional probability of death in the first remaining year is \(0.30\); in the second it is \(0.70(0.40)=0.28\). The expected benefit is \(\$50{,}000(0.30+0.28)=\$29{,}000\). The expected number of premiums still due, including the duration-2 premium, is \(1+0.70=1.70\). Thus

$$ V_2^-=\$29{,}000-\$11{,}165.172855\ldots(1.70) =\$29{,}000-\$18{,}980.793854\ldots =\$10{,}019.21. $$

Check: the remaining benefit probability is \(0.30+0.28=0.58\), and the premium count is between 1 (if death occurs in the first remaining year) and 2 (if the insured survives that year). The calculated reserve uses exactly those two remaining benefit years and those two possible premium dates. The final amount is rounded to cents.

Timing Traps and Practical Checks

A term reserve is not calculated by simply carrying forward the issue-date value. At each duration, mortality is conditional on survival to that date, the attained age changes, and the number of remaining benefit years shrinks. The premium stream shrinks as well, but its count must match the stated valuation instant.

  • Wrong remaining term: At duration \(k\), using \(n\) years instead of \(n-k\) includes coverage that has already elapsed.
  • Wrong premium date: For \(V_k^-\), the premium due at time \(k\) is included. For \(V_k^+\), it is excluded, so \(V_k^+=V_k^-+P\).
  • Unconditional mortality: A duration-\(k\) reserve is conditional on being alive then. Use mortality from age \(x+k\), not the original unconditional probabilities from issue.
  • Wrong benefit timing: The formula here discounts an end-of-year death benefit by \(v^{j+1}\). A benefit paid immediately on death would require a different present-value model.

One useful check is to examine the cash flows listed in the sums: the benefit sum must cover each remaining year exactly once, and the premium sum must include precisely the dates on which a premium is still due under the chosen convention. These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Worked Example: A Two-Year Term and the Premium-Date Check

A hypothetical two-year term policy issued at age 60 has a \(\$20{,}000\) end-of-year death benefit. Annual premiums are paid in advance. Assume \(i=10\%\), \(q_{60}=0.05\), and \(q_{61}=0.25\). Find the net premium and the reserve at duration 1 just before the premium due.

Method. Find the two-year issue-date benefit and premium values, then value the one remaining year conditionally at age 61. This example highlights that the duration-1 premium is included in the before-premium reserve.

Here \(v=1/1.10=10/11\). The probability of death in year 1 is \(0.05\), and the probability of death in year 2 is \(0.95(0.25)=0.2375\). Therefore

$$ A^1_{60:\overline{2}|} =0.05\left(\frac{10}{11}\right) +0.2375\left(\frac{10}{11}\right)^2 =\frac{1}{22}+\frac{95}{484} =\frac{117}{484}. $$

The premium annuity factor is \(\ddot{a}_{60:\overline{2}|}=1+0.95(10/11)=41/22\). The net premium is consequently

$$ P=\$20{,}000\frac{117/484}{41/22} =\$20{,}000\frac{117}{902} =\$2{,}594.2350\ldots. $$

At duration 1, only the age-61 year remains. Its benefit value is \(\$20{,}000(0.25)(10/11)=\$4{,}545.4545\ldots\), and its premium annuity factor is 1 because the premium is due immediately. Thus \(V_1^-=\$4{,}545.4545\ldots-\$2{,}594.2350\ldots=\$1{,}951.22\). Check: after the premium is paid, no future premium remains, so \(V_1^+=\$4{,}545.45\). This equals \(V_1^-+P\), to the cent. Amounts are rounded to cents only at the final step.

Key takeaway: For an annual-premium term policy, the prospective reserve at duration \(k\) is the conditional value of the \(n-k\) years of remaining death benefits minus the conditional value of the premiums still due. State whether the valuation is before or after the premium at duration \(k\) before counting cash flows.

Check Your Understanding

For each problem, use the stated convention and round money to cents where needed.

  1. A two-year term policy has \(B=\$1{,}000\), \(i=0\), and \(q_x=0.10,\ q_{x+1}=0.20\). Annual net premiums are paid in advance. Find the net premium and the reserve at duration 1 immediately before the premium due. (Answer: \(P=\$147.37\); \(V_1^-=\$52.63\))
  2. At an interior duration, a one-year term has \(B=\$20{,}000\), \(i=4\%\), \(q=0.15\), and annual premium \(P=\$2{,}500\) due immediately. Find the reserve just before that premium. (Answer: \(\$384.62\))
  3. For a policy with annual premium \(P=\$600\), the reserve at duration 2 immediately before the premium is \(\$900\). What is the reserve immediately after that premium is paid? (Answer: \(\$1{,}500\))
  4. A three-year term has \(B=\$10{,}000\), \(i=0\), and \(q_x=0.10,\ q_{x+1}=0.20,\ q_{x+2}=0.30\). Find the net annual premium and the reserve at duration 1 immediately before the premium due. (Answer: \(P=\$1{,}893.13\); \(V_1^-=\$992.37\))
  5. At duration 1, a hypothetical two-year term has \(B=\$8{,}000\), \(i=25\%\), and mortality probabilities \(q=0.20\) for the first remaining year and \(0.25\) for the second. The annual premium is \(P=\$1{,}000\), payable immediately and again one year later if alive. Find \(V_1^-\). (Answer: \(\$664.00\))

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