Term Reserve Patterns Near Expiry

Why Term Reserves Change Near Expiry

A level-premium term policy has a fixed death benefit, but the period during which that benefit can be paid gets shorter each year. Its reserve therefore reflects both the remaining benefit exposure and the premiums still due. Near expiry, the terminal boundary becomes especially important: once the term ends, there are no future term benefits or premiums, so the term reserve is zero.

A reserve sequence does not have to rise steadily toward expiry. The last reserve before expiry depends on the expected cost of the final year’s coverage compared with the level premium due at that time. If mortality is higher in that final year than the premium was designed to fund, the reserve may be positive; if it is lower, the reserve may be negative. In either case, the reserve reaches zero at expiry.

Definition: Consider an $n$-year discrete term insurance policy on a life aged $x$, with death benefit $B$ payable at the end of the year of death and annual level net premiums payable at the beginning of each policy year while the insured is alive. Let $P$ be the annual premium. Define ${}_kV_x$ as the net prospective reserve at time $k$, immediately before the premium due at that time, for $0\leq k<n$. At expiry, define ${}_nV_x=0$. The reserve excludes expenses and any benefit payable on survival to expiry.

This timing convention matters. At time $k<n$, the premium due then is still a future payment from the reserve’s viewpoint, so it is subtracted in the prospective value. At time $n$, there is no premium due and no term coverage remaining. For annual mortality probabilities, write $q_{x+k}$ for the probability of death during the year from age $x+k$, and $p_{x+k}=1-q_{x+k}$ for the probability of surviving that year. Assume an effective annual interest rate $i$, with $v=1/(1+i)$.

The Recursion and the Expiry Boundary

At time $k$, conditional on being alive, the policy has one year of coverage ahead. At the end of that year, either the death benefit is paid, with probability $q_{x+k}$, or the life survives and the reserve at time $k+1$ applies. Discounting those end-of-year values to time $k$, then subtracting the premium due at time $k$, gives the recursion below.

{}_kV_x = v\left(q_{x+k}B+p_{x+k}{}_{k+1}V_x\right)-P, \qquad k=0,1,\ldots,n-1, \qquad {}_nV_x=0.

The terminal condition is not an extra assumption about an amount paid at expiry. It expresses the fact that the term policy has no remaining term-insurance cash flows at time $n$. In particular, setting ${}_nV_x=B$ would incorrectly treat a term policy as if it paid a maturity benefit.

Theorem: For the stated annual term-insurance model, the reserve in the final coverage year is $$ {}_{n-1}V_x=vq_{x+n-1}B-P. $$ It is the present value of the expected final-year death benefit less the premium due at the start of that year. The reserve at expiry itself is ${}_nV_x=0$.

The formula follows directly from the recursion at $k=n-1$: the survival branch leads to ${}_nV_x=0$, so its contribution vanishes. Thus only the discounted expected death benefit in the final year and the premium due then remain. This also explains why a positive reserve just before expiry does not carry beyond expiry: it is a value at time $n-1$ for that year’s remaining coverage, not a benefit paid at time $n$.

Start at expiry.
Set ${}_nV_x=0$, because no term-insurance benefits or premiums remain after the term ends.

Move backward one year at a time.
For each year, combine the death-benefit branch and survival-reserve branch, discount them, and subtract the premium due at that time.

Interpret the pattern.
Compare the reserve with the adjacent-year reserves and check the final-year value against $vq_{x+n-1}B-P$. Do not assume the sequence is monotone.

For a level net premium, the issue premium is set so that the expected present value of premiums equals the expected present value of benefits. With premiums due at times $0,1,\ldots,n-1$ while alive, the issue equation is

P\sum_{j=0}^{n-1}v^j\,{}_jp_x = B\sum_{j=0}^{n-1}v^{j+1}\,{}_jp_x\,q_{x+j}.

Here ${}_0p_x=1$, and ${}_jp_x$ is the probability of surviving from age $x$ to time $j$. In a fully specified net-premium model this equation gives ${}_0V_x=0$. It does not imply that every later reserve is zero: later reserves compare the remaining benefits and premiums conditional on survival to that duration.

Worked Reserve Patterns

Worked Example: A Positive Final-Year Reserve

A hypothetical two-year term policy pays $B=\$1{,}000$ at the end of the year of death. The annual death probabilities are $q_x=0.10$ and $q_{x+1}=0.20$, so the corresponding survival probabilities are $0.90$ and $0.80$. The effective annual interest rate is $i=25\%$. Find the level annual net premium and the reserves at times 1 and 2.

Method. First apply the equivalence equation at issue. Then use the final-year formula and the expiry boundary. The premium is due at the start of each insured year, including at time 1 if the insured is alive.

The discount factor is $v=1/1.25=0.8$. The expected present value of benefits per dollar of benefit is

$$ 0.8(0.10)+0.8^2(0.90)(0.20) =0.08+0.1152 =0.1952. $$

The premium annuity factor is

$$ 1+0.8(0.90)=1.72. $$

Therefore $P=\$1{,}000(0.1952)/1.72=\$113.488372\ldots$, or $\$113.49$ to the nearest cent. In the final coverage year,

{}_1V_x=0.8(0.20)(\$1{,}000)-\$113.488372\ldots =\$160-\$113.488372\ldots =\$46.511628\ldots.

Thus ${}_1V_x=\$46.51$ to the nearest cent, while ${}_2V_x=0$ at expiry. Check the issue recursion: $0.8[0.10(\$1{,}000)+0.90(\$46.511628\ldots)]-\$113.488372\ldots=\$0$, as required by the net-premium equivalence equation.

Common mistake avoided: The reserve at time 1 is not just the discounted expected claim of $\$160$. The $\$113.49$ premium due at time 1 must also be subtracted.

Worked Example: A Reserve That Rises Before It Reaches Zero

A hypothetical three-year term policy pays $B=\$3{,}212$ at the end of the year of death. The annual death probabilities are $0.10$, $0.20$, and $0.30$, and the effective annual interest rate is $10\%$. Find the level annual net premium and the complete reserve sequence.

Method. Calculate the issue premium from the full three-year benefit and premium values, then use the reserve recursion backward from expiry. The survival probabilities are $0.90$, $0.80$, and $0.70$, and $v=10/11$.

The expected present value of benefits per dollar of benefit is

0.10v+(0.90)(0.20)v^2+(0.90)(0.80)(0.30)v^3 = \frac{121+198+216}{1331} = \frac{535}{1331}.

The premium annuity factor is $1+0.90v+(0.90)(0.80)v^2=1+9/11+72/121=292/121=3212/1331$. Hence

P=\$3{,}212\left(\frac{535/1331}{3212/1331}\right)=\$535.

Start with ${}_3V_x=0$. In the last insured year,

{}_2V_x = \frac{10}{11}(0.30)(\$3{,}212)-\$535 = \$876-\$535 = \$341.

One year earlier, the death branch has expected benefit $0.20(\$3{,}212)=\$642.40$, and the survival branch has value $0.80(\$341)=\$272.80$. Therefore

{}_1V_x = \frac{10}{11}(\$642.40+\$272.80)-\$535 = \$832-\$535 = \$297.

At issue, the recursion gives ${}_0V_x=(10/11)[0.10(\$3{,}212)+0.90(\$297)]-\$535=(10/11)(\$588.50)-\$535=\$0$. The sequence is ${}_0V_x=\$0$, ${}_1V_x=\$297$, ${}_2V_x=\$341$, and ${}_3V_x=\$0$. It rises over the two interior durations, then drops to zero at expiry. The terminal drop does not contradict the positive ${}_2V_x$: that reserve funds only the remaining year of coverage.

Common mistake avoided: Extending the upward pattern to ${}_3V_x$ ignores the absence of coverage after the third year. The boundary value is zero, not an extrapolation of earlier reserves.

Worked Example: A Negative Reserve Near Expiry

A hypothetical two-year term policy pays $B=\$9{,}750$ at the end of the year of death. The annual death probabilities are $q_x=0.30$ and $q_{x+1}=0.10$; the effective annual interest rate is $25\%$. Find the level annual net premium and the reserve at time 1.

Method. Use the two-year equivalence equation, then apply the last-year reserve formula. This example tests whether the final-year reserve is automatically positive; it is not.

Here $v=0.8$, and the survival probability for the first year is $0.70$. The expected present value of benefits per dollar of benefit is

$$ 0.8(0.30)+0.8^2(0.70)(0.10) = 0.24+0.0448 = 0.2848. $$

The premium annuity factor is $1+0.8(0.70)=1.56$. Since $\$9{,}750(0.2848)=\$2{,}776.80$, the level premium is $P=\$2{,}776.80/1.56=\$1{,}780$. Thus

{}_1V_x = 0.8(0.10)(\$9{,}750)-\$1{,}780 = \$780-\$1{,}780 = -\$1{,}000.

The issue value checks: $0.8[0.30(\$9{,}750)+0.70(-\$1{,}000)]-\$1{,}780=0.8(\$2{,}225)-\$1{,}780=\$0$. The negative reserve means that, conditional on reaching the final year, the premium due exceeds the discounted expected cost of that year’s death benefit under these assumptions. It is a net prospective value, not a cash amount owed to the policyholder.

Common mistake avoided: A negative final-year reserve is not automatically an arithmetic error. Here the first-year death probability is much higher than the second-year probability, and the level premium reflects both years.

Interpreting the Pattern Without Assuming Monotonicity

The reserve sequence is driven by a balance of several effects, not by elapsed time alone. As expiry approaches, fewer years of benefits and premiums remain. At the same time, the conditional mortality rate can change with attained age or with the policy’s assumed mortality pattern. The final-year expression makes the local comparison explicit: ${}_{n-1}V_x$ is positive when $vq_{x+n-1}B>P$, zero when they are equal, and negative when $vq_{x+n-1}B<P$.

Earlier reserves also depend on the next reserve through the survival branch. A positive reserve in the final coverage year can be preceded by a smaller or larger reserve, as the three-year example showed. A negative final-year reserve can likewise occur even when the reserve at issue is zero. For this reason, inspect the recursion and the mortality assumptions instead of inferring the sign of a reserve from the policy being term insurance.

The key timing checks are straightforward: premiums are subtracted at the beginning of an insured year under the stated convention; death benefits are valued at the end of the year of death; and no term benefit or premium remains at expiry. Mixing an end-of-year reserve with a before-premium reserve changes the recursion. So does inserting a maturity payment that the term contract does not provide.

Key takeaway: For annual level-premium term insurance, work backward from ${}_nV_x=0$. In particular, the final-year reserve is $vq_{x+n-1}B-P$. Its sign and the overall reserve pattern depend on the remaining mortality exposure and premiums; neither steady growth nor a positive reserve is guaranteed.

Check Your Understanding

Use the same convention as in the examples: reserves are measured immediately before the premium due at that time, and the terminal reserve is zero. Round money to the nearest cent where needed.

A one-year term policy has $B=\$50{,}000$, $q_x=0.02$, and effective annual interest $i=5\%$. Find the net premium and the reserve at expiry. (Answer: $P=\$952.38$; ${}_1V_x=\$0$.)
A two-year term policy has $B=\$1{,}000$, $q_x=0.05$, $q_{x+1}=0.10$, and $i=0\%$. Find the level net premium and the final-year reserve. (Answer: $P=\$74.36$; ${}_1V_x=\$25.64$.)
A two-year term policy has $B=\$2{,}090$, annual death probabilities $0.20$ and $0.10$, and effective annual interest $i=10\%$. Find the net premium and the reserve at time 1. (Answer: $P=\$300$; ${}_1V_x=-\$110$.)
A three-year term policy has $B=\$2{,}710$, annual death probabilities $0.10$, $0.10$, and $0.20$, and $i=0\%$. Find the level net premium and reserves at times 1, 2, and 3. (Answer: $P=\$352$; ${}_1V_x=\$90$, ${}_2V_x=\$190$, ${}_3V_x=\$0$.)
A two-year term policy has $B=\$5{,}000$, $i=25\%$, and final-year death probability $q_{x+1}=0.08$. The level premium is $\$400$. Find the time-1 reserve and explain its sign. (Answer: ${}_1V_x=0.8(0.08)(\$5{,}000)-\$400=-\$80$; it is negative because the final-year premium exceeds the discounted expected final-year benefit.)

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