Term Insurance Retrospective Reserves

Why Term Insurance Reserves Need Careful Timing

A term insurance contract pays a benefit if death occurs during a specified coverage period and pays nothing if the insured survives to the end of that period. At an intermediate duration, its retrospective reserve measures the accumulated value of past premiums and past death benefits for the group of policies that has survived to that duration. The calculation depends on both the timing of cash flows and the fact that only survivors remain in the group being valued.

This tutorial considers a discrete annual model. Premiums are paid at the beginning of each coverage year while the insured is alive, and a death benefit is paid at the end of the year of death. The reserve at duration $k$ is taken immediately before any premium due at time $k$. In particular, premiums paid before expiry are included, but no premium is due at the expiry time.

Definition: For an $n$-year term insurance issued at age $x$, the retrospective reserve ${}_kV_x$ at duration $k$ is the accumulated net cash flow from the original policy cohort to time $k$, divided by the probability that an insured survives to time $k$. It is conditional on survival to the valuation duration. For this definition, premiums are paid at times $0,1,\ldots,n-1$ while the insured is alive, and a death benefit $B$ is paid at the end of the year of death.

Model and General Formula

Assume a fixed annual effective interest rate $i$, with $v=1/(1+i)$. Let ${}_jp_x$ be the probability of surviving $j$ years from age $x$, and let $q_{x+j}$ be the probability of dying during the year from age $x+j$, given survival to that age. Thus, the probability that a policy from the original cohort pays a benefit at the end of year $j+1$ is ${}_jp_xq_{x+j}$.

Suppose the contract has a level annual net premium $P$. Because each premium is due at the beginning of a year, the expected present value of the premiums per policy at issue is $P\sum_{j=0}^{n-1}v^j{}_jp_x$. The expected present value of the death benefits is $B\sum_{j=0}^{n-1}v^{j+1}{}_jp_xq_{x+j}$. Equating these values gives the net premium.

P = B\, \frac{\displaystyle\sum_{j=0}^{n-1}v^{j+1}{}_jp_xq_{x+j}} {\displaystyle\sum_{j=0}^{n-1}v^j{}_jp_x}

To value the past cash flows at time $k$, accumulate each premium and benefit to that date. A premium paid at time $j$ accumulates for $k-j$ years. A benefit paid at the end of year $j+1$ accumulates for $k-j-1$ years. Then divide the net accumulated cash flow for the original cohort by the probability ${}_kp_x$ that a policy survives to time $k$.

Theorem: For $0\leq k\leq n$, with the timing and assumptions above, $$ {}_kV_x = \frac{ P\displaystyle\sum_{j=0}^{k-1}{}_jp_x(1+i)^{k-j} - B\displaystyle\sum_{j=0}^{k-1}{}_jp_xq_{x+j}(1+i)^{k-j-1} } {{}_kp_x}. $$ At $k=0$, both sums are empty and ${}_0V_x=0$. The formula assumes premiums are paid at times $0,\ldots,n-1$ and the benefit is paid at the end of the year of death.

Why the Formula Conditions on Survival

Consider an original cohort represented by one policy at issue. At time $j$, the expected number of policies still in force per original policy is ${}_jp_x$, so expected premium income at that time is $P{}_jp_x$. At the end of year $j+1$, the expected number of deaths per original policy is ${}_jp_xq_{x+j}$, so expected benefits paid then are $B{}_jp_xq_{x+j}$. These are cohort expectations, not amounts for a policy known to have survived.

For a valuation at time $k$, accumulate each of those expected cash flows to time $k$, and subtract accumulated benefits from accumulated premiums. The cohort contains only a proportion ${}_kp_x$ of survivors at time $k$. Dividing by that proportion converts the cohort’s accumulated amount per original policy into an amount per surviving policy. Without this division, the result is not a reserve conditional on survival.

List past premiums.
Include premiums at times $j=0,\ldots,k-1$, weighted by the probability ${}_jp_x$ that they are paid.

Accumulate past cash flows.
Use $(1+i)^{k-j}$ for a premium at time $j$, and $(1+i)^{k-j-1}$ for a benefit paid at the end of year $j+1$.

Condition on survival.
Subtract accumulated benefits from accumulated premiums, then divide by ${}_kp_x$.

Worked Calculations

Worked Example: A Three-Year Term Reserve at Duration Two

A hypothetical insurer issues a three-year term policy with benefit $B=\$10{,}000$. The annual effective interest rate is $i=5\%$. The annual death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.30$. Premiums are level, paid at the beginning of each coverage year while the insured is alive. Calculate the net premium and the retrospective reserve at duration $k=2$.

Method and assumptions. This is annual discrete term insurance, with the death benefit paid at the end of the year of death. The survival probabilities for the first two years are ${}_1p_x=0.90$ and ${}_2p_x=0.90(0.80)=0.72$. Also, $v=1/1.05$.

First calculate the present value of benefits per dollar of benefit:

0.10v+0.90(0.20)v^2+0.90(0.80)(0.30)v^3 = \frac{2}{21}+\frac{8}{49}+\frac{64}{343} = \frac{3206}{7203}.

The present value of the premium annuity per dollar of annual premium is

1+0.90v+0.72v^2 = 1+\frac{6}{7}+\frac{32}{49} = \frac{123}{49}.

Therefore,

P = \$10{,}000\left(\frac{3206}{7203}\right)\left(\frac{49}{123}\right) = \$10{,}000\left(\frac{3206}{18081}\right) = \$1{,}773.13.

Keep the unrounded premium in the reserve calculation. By time 2, the time-0 premium has accumulated for two years, and the time-1 premium is paid only if the insured survives the first year. Accumulated premiums per original policy are therefore

P\left(1.05^2+0.90(1.05)\right) = P(1.1025+0.945) = 2.0475P.

The first-year death benefit accumulates for one year, and the second-year death benefit is paid at time 2. Accumulated benefits per original policy are

\$10{,}000\left(0.10(1.05)+0.90(0.20)\right) = \$10{,}000(0.105+0.180) = \$2{,}850.

There are $0.72$ survivors per original policy at time 2. Thus,

{}_2V_x = \frac{2.0475P-\$2{,}850}{0.72} = \frac{2.0475(\$1{,}773.13198\ldots)-\$2{,}850}{0.72} = \$1{,}084.01.

The reserve is positive: for the surviving policies, accumulated premiums exceed the share of accumulated claims assigned to them. The calculation avoids the mistake of treating all original policies as if they survived to duration 2.

Worked Example: The Reserve at the End of a Two-Year Term

A hypothetical two-year term policy has benefit $B=\$10{,}000$, annual effective interest $i=5\%$, and death probabilities $q_x=0.10$ and $q_{x+1}=0.20$. It has level annual net premiums due at times 0 and 1 if the insured is alive. Find the net premium, the reserve at duration 1, and the reserve at expiry.

Method and assumption. Use the net premium equivalence principle for the two-year coverage, then accumulate the cohort cash flows. The premium due at time 1 is included in past premiums at expiry, but no premium is due at time 2.

The benefit present value per dollar of benefit is

0.10v+0.90(0.20)v^2 = \frac{2}{21}+\frac{8}{49} = \frac{38}{147}.

The premium annuity present value per dollar of premium is $1+0.90v=13/7$. Hence the net premium is

P = \$10{,}000\left(\frac{38}{147}\right)\left(\frac{7}{13}\right) = \$10{,}000\left(\frac{266}{1911}\right) = \$1{,}391.94.

At duration 1, the original cohort has accumulated the time-0 premium for one year and paid first-year benefits to the $0.10$ who die. There are $0.90$ survivors. Thus,

{}_1V_x = \frac{1.05P-\$10{,}000(0.10)}{0.90} = \frac{1.05(\$1{,}391.9414\ldots)-\$1{,}000}{0.90} = \$512.82.

At expiry, the accumulated premiums per original policy are

P\left(1.05^2+0.90(1.05)\right)=2.0475P.

The accumulated benefits per original policy are

\$10{,}000\left(0.10(1.05)+0.90(0.20)\right)=\$2{,}850.

Using the unrounded net premium, $2.0475P=\$2{,}850$. Therefore the cohort net accumulated amount at expiry is zero, and dividing by the survival probability ${}_2p_x=0.72$ still gives

{}_2V_x=\frac{\$2{,}850-\$2{,}850}{0.72}=\$0.00.

The zero expiry reserve follows from net premium equivalence for this contract. It is not a reason to include a premium at expiry: the last premium was due at time 1, at the beginning of the final coverage year.

Worked Example: A One-Year Reserve and a Timing Check

Use the three-year policy in the first example, with $B=\$10{,}000$, $i=5\%$, annual death probabilities $0.10, 0.20, 0.30$, and net premium $P=\$1{,}773.13198\ldots$. Find the reserve at duration 1, just before the premium due at time 1.

Method. Only the time-0 premium and first-year death benefit have occurred by the valuation time. The time-1 premium is not a past cash flow because the reserve is measured immediately before it is due.

The time-0 premium accumulates for one year to $1.05P$. First-year benefits per original policy equal $\$10{,}000(0.10)=\$1{,}000$, and ${}_1p_x=0.90$. Therefore,

{}_1V_x = \frac{1.05P-\$1{,}000}{0.90} = \frac{1.05(\$1{,}773.13198\ldots)-\$1{,}000}{0.90} = \$957.54.

As a check, the prospective value at time 1 is the present value of the remaining two years of benefits less the present value of the premiums due at times 1 and 2, conditional on survival to time 1. The benefit value is

\$10{,}000\left(0.20v+0.80(0.30)v^2\right)=\$4{,}081.63.

The premium value is $P(1+0.80v)=\$3{,}124.09$, so the difference is $\$957.54$, agreeing with the retrospective calculation. The check catches the common timing error of adding the premium due at time 1 to the retrospective accumulation.

Expiry, Premium Cessation, and Exam Traps

For an $n$-year term contract, the last premium is due at time $n-1$, provided the insured is alive then. At time $n$, coverage has ended and no further premium is due. Under net premium pricing, the total expected accumulated premiums and benefits for the original cohort balance at expiry. Consequently, the retrospective reserve for surviving policies at expiry is zero. The result depends on using the net premium for that same term, benefit, mortality model, and interest assumption.

The reserve formula also makes the premium timing explicit. A valuation at time $k$ immediately before the premium due at $k$ includes premiums only through time $k-1$. If a problem instead defines the value immediately after the premium at time $k$, that premium must be included in the accumulated past cash flows. State the valuation instant before substituting values.

Forgetting survival conditioning. The cohort accumulation is per original policy. Divide by ${}_kp_x$ to obtain the amount per policy that has survived to time $k$.
Accumulating a benefit for the wrong number of years. A benefit paid at the end of year $j+1$ accumulates for $k-j-1$ years to time $k$, not $k-j$.
Adding a premium due at the valuation time. Under the before-premium convention used here, a time-$k$ premium is not yet part of past cash flows.
Assuming every expiry reserve is zero. Zero follows here from net premium equivalence. A premium different from the net premium need not produce a zero accumulated balance for survivors at expiry.

Key takeaway: For term insurance, accumulate premiums and death benefits to the valuation date using their actual payment times, then divide the cohort balance by the probability of survival to that date. At expiry, net premium equivalence gives a zero retrospective reserve when the final premium was paid at the start of the final coverage year and no premium is due after coverage ends.

Check Your Understanding

For each problem, use the annual discrete timing convention stated in the problem. Round money to the nearest cent.

A hypothetical one-year term policy pays $\$8{,}000$ at the end of the year of death. The annual effective interest rate is $4\%$, and $q_x=0.08$. A level net premium is paid at issue. Find the premium and the retrospective reserve at expiry for surviving policies. (Answer: premium $\$615.38$; expiry reserve $\$0.00$)
A two-year term policy has benefit $\$10{,}000$, annual effective interest $5\%$, and death probabilities $q_x=0.10$, $q_{x+1}=0.20$. Premiums are level and due at the beginning of each coverage year while alive. Find the net premium and the reserve at duration 1, immediately before the second premium. (Answer: net premium $\$1{,}391.94$; reserve $\$512.82$)
A three-year term policy has benefit $\$10{,}000$, annual effective interest $5\%$, and death probabilities $0.10, 0.20, 0.30$ in successive years. Find its level net premium and retrospective reserve at duration 2. (Answer: net premium $\$1{,}773.13$; reserve $\$1{,}084.01$)
A two-year term contract has benefit $\$10{,}000$, annual effective interest $5\%$, and death probabilities $0.10$ and $0.20$. The annual premium is $\$1{,}500$, payable at the start of each coverage year while alive. Find the retrospective reserve at duration 1 immediately before the second premium. (Answer: $\$638.89$)

What You'll Learn