Prove Agreement of Prospective and Retrospective Reserves

Why Two Reserve Methods Should Agree

A reserve at duration \(k\) can be viewed from either side of the contract. The prospective reserve values the benefits and premiums still to come, conditional on the insured being alive at time \(k\). The retrospective reserve accumulates the premiums and benefits already paid or expected from the original group, then divides by the proportion of that group still in force. These approaches look different, but under the equivalence principle they measure the same net amount.

The equality depends on consistent cash-flow timing and on the contract satisfying equivalence at issue. It is not a claim that every policyholder’s individual past premiums, or every realized group’s cash flows, equal a conditional expected future value. It is an identity between expected values under the stated interest and mortality assumptions.

Definition: Consider an \(n\)-year discrete life contract issued at age \(x\). Assume a deterministic effective annual interest rate \(i\), with \(v=1/(1+i)\), and annual mortality described by \({}_jp_x\) and \(q_{x+j}\). A premium \(P_j\) is paid at time \(j\) if the insured is alive then. A death benefit \(b_{j+1}\) is paid at the end of year \(j+1\) if death occurs in that year; a maturity benefit \(m\) is paid at time \(n\) if the insured survives to time \(n\). A net reserve at time \(k\) is the conditional expected present value of future benefits less future premiums, given survival to time \(k\), or the corresponding accumulated past net cash flow per survivor. Expenses, lapses, and other cash flows are excluded.

For this convention, the probabilities at issue are \({}_jp_x\) for survival to time \(j\) and \({}_jp_xq_{x+j}\) for death during year \(j+1\). The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits at issue. All reserves below are net reserves; adding expenses or other cash flows requires including them consistently in both sides.

The Two Reserve Expressions

At duration \(k<n\), the prospective reserve is conditional on being alive at that time. Future death benefits are weighted by the conditional probability of death in each future year. Future premiums are weighted by the conditional probability of being alive when each premium is due. The maturity benefit is included only if the insured survives to time \(n\).

{}_kV_x^{\mathrm{pros}} = \sum_{j=k}^{n-1} b_{j+1}v^{j+1-k}{}_{j-k}p_{x+k}q_{x+j} +mv^{n-k}{}_{n-k}p_{x+k} -\sum_{j=k}^{n-1}P_jv^{j-k}{}_{j-k}p_{x+k}.

The retrospective reserve uses the original group of policies. Accumulate each premium paid before time \(k\) to time \(k\), and subtract each death benefit paid before time \(k\), also accumulated to time \(k\). Divide the expected net accumulation for the original group by the probability \({}_kp_x\) that a policy is still in force at time \(k\).

{}_kV_x^{\mathrm{retro}} = \frac{ \displaystyle \sum_{j=0}^{k-1}P_j{}_jp_x(1+i)^{k-j} - \sum_{j=0}^{k-1}b_{j+1}{}_jp_xq_{x+j}(1+i)^{k-j-1} }{ {}_kp_x }.

The different exponents reflect payment dates. A premium due at time \(j\) earns interest for \(k-j\) years by time \(k\). A death benefit paid at time \(j+1\) earns interest for only \(k-j-1\) years. A premium at time \(k\) belongs in the prospective calculation, not in the retrospective sum of premiums strictly before \(k\).

Deriving Agreement from Equivalence

Write expected values at issue, then separate the cash flows before time \(k\) from those at or after time \(k\). Under equivalence, the issue-date expected present value of premiums minus benefits is zero. Therefore, the expected present value of future benefits minus premiums must equal the past premiums minus benefits, both measured at issue. Discounting the future flows to time \(k\) and conditioning on survival will match the retrospective accumulation per survivor.

Apply equivalence at issue.
The expected present value of all premiums equals that of all benefits. In the stated model, the equation is \(\sum_{j=0}^{n-1}P_jv^j{}_jp_x=\sum_{j=0}^{n-1}b_{j+1}v^{j+1}{}_jp_xq_{x+j}+mv^n{}_np_x\).

Separate past and future.
Move the expected present values of premiums and benefits before time \(k\) to one side. Equivalence implies that future benefits minus premiums, in issue-date expected present value, equal past premiums minus benefits.

Express the future side at time \(k\).
Every future cash flow has issue-date expected present value equal to \({}_kp_xv^k\) times its value in the prospective reserve. This factor accounts for reaching time \(k\) and discounting from \(k\) to issue.

Accumulate the past side.
Multiplying past issue-date present values by \(v^{-k}=(1+i)^k\) accumulates them to time \(k\). Dividing by \({}_kp_x\) gives the retrospective reserve per policy still in force.

In equations, let “past” include premiums at times \(0,\ldots,k-1\) and death benefits paid at times \(1,\ldots,k\). Equivalence gives the identity between future benefits minus premiums and past premiums minus benefits. The future cash flows have issue-date present value \({}_kp_xv^k{}_kV_x^{\mathrm{pros}}\). The past premiums-minus-benefits have issue-date present value equal to \(v^k{}_kp_x{}_kV_x^{\mathrm{retro}}\), by the retrospective formula. Thus:

{}_kp_xv^k{}_kV_x^{\mathrm{pros}} = \sum_{j=0}^{k-1}v^jP_j{}_jp_x -\sum_{j=0}^{k-1}v^{j+1}b_{j+1}{}_jp_xq_{x+j} = {}_kp_xv^k{}_kV_x^{\mathrm{retro}}.

For \({}_kp_x>0\), division by \({}_kp_xv^k\) proves agreement. The result does not require level premiums: premiums may vary by duration, provided the same schedule and timing are used in the equivalence equation and both reserve calculations. If the contract does not satisfy equivalence, the past and future sides need not match.

Theorem: For the stated discrete contract, if issue-date expected present values of premiums and benefits are equal, then at each duration \(k\) with \({}_kp_x>0\), the prospective and retrospective net reserves agree. This conclusion uses the same mortality assumptions, interest rate, benefit schedule, and premium timing throughout.

Worked Calculations

Worked Example: A Two-Year Term Contract

A hypothetical two-year term policy pays \(1{,}000\) at the end of the year of death. The effective annual interest rate is \(i=0\), \(q_x=0.10\), and \(q_{x+1}=0.20\). A level net premium \(P\) is due at times 0 and 1 if the insured is alive. Find \(P\), then calculate the time-1 reserve by both methods.

Method. First apply equivalence at issue. Then value only the remaining year prospectively and compare it with the expected past net cash flow per survivor.

The probability of death in year 1 is \(0.10\). The probability of death in year 2 is \(0.90(0.20)=0.18\). Thus the expected benefit present value is \(1{,}000(0.10+0.18)=280\). The expected number of premiums is \(1+0.90=1.90\), since the time-1 premium is paid only if the insured survives year 1. Therefore:

P(1.90)=280,\qquad P=\frac{280}{1.90}=\frac{2800}{19}\approx147.37.

Given survival to time 1, the year-2 death probability is \(0.20\). The prospective reserve is the expected benefit less the premium due at time 1:

{}_1V_x^{\mathrm{pros}}=1{,}000(0.20)-\frac{2800}{19} =200-\frac{2800}{19} =\frac{1000}{19}\approx52.63.

For the retrospective value, the original group has expected time-0 premiums of \(P\), and expected year-1 benefits of \(1{,}000(0.10)=100\). There is no interest accumulation because \(i=0\). The survival proportion at time 1 is \(0.90\), so:

{}_1V_x^{\mathrm{retro}} =\frac{P-100}{0.90} =\frac{\frac{2800}{19}-100}{0.90} =\frac{\frac{900}{19}}{0.90} =\frac{1000}{19}\approx52.63.

Both methods give \(52.63\) dollars to the nearest cent. The check is exact before rounding: both values equal \(1000/19\). A common mistake is to count the time-1 premium as certain for every policy at issue; its expected contribution is \(0.90P\), not \(P\).

Worked Example: Including Interest in Both Methods

A hypothetical two-year term policy pays \(1{,}000\) at the end of the year of death. Let \(i=0.10\), \(q_x=0.10\), and \(q_{x+1}=0.20\). Level premiums are due at times 0 and 1 while the insured is alive. Find the net premium and the reserve at time 1 by both methods.

Method. Use \(v=1/1.10=10/11\) in the issue-date equivalence equation. For the retrospective reserve, accumulate the time-0 premium one year; the year-1 death benefit is already paid at time 1 and is not accumulated further.

The expected present value of benefits is:

1{,}000(0.10)v+1{,}000(0.90)(0.20)v^2 =1{,}000(0.10)\frac{10}{11}+180\left(\frac{10}{11}\right)^2 =\frac{11{,}000+18{,}000}{121} =\frac{29{,}000}{121}.

The expected present value of premiums is \(P(1+0.90v)\), where the factor 1 is for the time-0 premium and \(0.90v\) is for the time-1 premium. Hence:

P\left(1+0.90\frac{10}{11}\right) =P\frac{20}{11} =\frac{29{,}000}{121}, \qquad P=\frac{1{,}450}{11}\approx131.82.

At time 1, the prospective value of the remaining death benefit is \(1{,}000(0.20)v=2{,}000/11\). Subtract the premium due at time 1:

{}_1V_x^{\mathrm{pros}} =\frac{2{,}000}{11}-\frac{1{,}450}{11} =\frac{550}{11}=50.00.

Retrospectively, the time-0 premium accumulates to \(1.10P=145\). Expected year-1 benefits are \(100\), paid at time 1. Divide the net accumulated amount by the \(0.90\) survival proportion:

{}_1V_x^{\mathrm{retro}} =\frac{1.10P-100}{0.90} =\frac{145-100}{0.90} =50.00.

The methods agree at \(50.00\) dollars. The timing check is important: accumulating the year-1 benefit for another year, or omitting interest on the time-0 premium, would use a different valuation date.

Worked Example: Testing an Unequal Premium Pattern

A hypothetical two-year term policy pays \(400\) at the end of year 1 on death and \(600\) at the end of year 2 on death. Let \(i=0\), \(q_x=0.20\), and \(q_{x+1}=0.25\). The premium at time 0 is \(150\); find the time-1 premium \(P_1\) that satisfies equivalence and calculate the reserve at time 1 by both methods.

Method. Equate expected premiums and benefits at issue, accounting for the chance that the second premium is paid. Then evaluate the remaining year prospectively and the past year retrospectively.

The expected benefit is \(400(0.20)+600(0.80)(0.25)=80+120=200\). Expected premiums are \(150+0.80P_1\), because the second premium is paid only if the insured survives year 1. Equivalence requires:

150+0.80P_1=200,\qquad P_1=\frac{50}{0.80}=62.50.

Given survival to time 1, the expected year-2 benefit is \(600(0.25)=150\). Subtract the premium due at time 1:

{}_1V_x^{\mathrm{pros}}=150-62.50=87.50.

For the retrospective value, the expected time-0 premium is \(150\), and the expected year-1 benefit is \(400(0.20)=80\). Divide the net amount by the \(0.80\) survival proportion:

{}_1V_x^{\mathrm{retro}} =\frac{150-80}{0.80} =\frac{70}{0.80} =87.50.

The reserves agree. If \(P_1\) were incorrectly set to \(50\), expected premiums would be \(150+0.80(50)=190\), not \(200\). That contract would not satisfy equivalence, so the agreement theorem would not apply; its time-1 prospective value would be \(150-50=100\), while its retrospective value would remain \(70/0.80=87.50\).

Common Traps and Checks

The retrospective formula is not simply “premiums paid minus claims paid.” It is an expected amount for the original group, accumulated to the valuation time and divided by the probability of remaining in force. Likewise, the prospective reserve is conditional on survival to the valuation time, so its probabilities must be conditional probabilities from age \(x+k\), not the original issue-age probabilities.

Leaving out the survival denominator. The retrospective numerator is for the original group. Divide by \({}_kp_x\) to express the amount per policy still in force.
Using the wrong accumulation period. A cash flow paid at time \(j\) accumulates \(k-j\) years to time \(k\), not \(k-j+1\).
Misplacing a premium at the valuation date. A premium due at time \(k\) is a future premium in the prospective reserve, but it is not among premiums paid before time \(k\) in the retrospective reserve.
Assuming agreement without equivalence. If the expected present values at issue do not balance, the proof’s key identity is absent. Calculate both sides rather than asserting equality.

Key takeaway: Under issue-date equivalence, future benefits minus future premiums equal past premiums minus past benefits when both are expressed consistently. Accumulating the past amount to time \(k\) and conditioning on survival gives the same reserve as valuing future net obligations from time \(k\).

Check Your Understanding

For each problem, use the stated discrete timing and check equivalence before relying on reserve agreement.

A two-year term policy pays \(500\) at the end of the year of death. Let \(i=0\), \(q_x=0.10\), and \(q_{x+1}=0.30\). Level premiums are due at times 0 and 1 while alive. Find the premium and both reserves at time 1. (Answer: \(P=1850/19\approx97.37\); both reserves are \(1000/19\approx52.63\).)
A two-year endowment pays \(80\) at the end of year 1 on death, or \(160\) at time 2 if alive at time 1, whether death occurs in year 2 or the insured survives to maturity. Let \(i=0\), \(q_x=0.25\), and \(q_{x+1}=0.20\). Level premiums are due at times 0 and 1 while alive. Find the net premium and the time-1 reserve by both methods. (Answer: \(P=80\); both reserves are \(80\).)
A three-year term policy pays \(400\) at the end of the year of death. Let \(i=0\), \(q_x=0.10\), \(q_{x+1}=0.20\), and \(q_{x+2}=0.25\). Level premiums are due at times 0, 1, and 2 while alive. Find the premium and both reserves at time 2. (Answer: \(P=9200/131\approx70.23\); both time-2 reserves are \(3900/131\approx29.77\).)
A two-year term policy pays \(400\) at the end of year 1 on death and \(600\) at the end of year 2 on death. Let \(i=0\), \(q_x=0.20\), and \(q_{x+1}=0.25\). If the time-0 premium is \(150\), find the time-1 premium required by equivalence and the reserve at time 1 by both methods. (Answer: the time-1 premium is \(62.50\); both reserves are \(87.50\).)
A two-year term policy pays \(1{,}000\) at the end of the year of death. Let \(i=0.10\), \(q_x=0.10\), and \(q_{x+1}=0.20\). Level premiums are due at times 0 and 1 while alive. Find the net premium and the time-1 reserve by both methods. (Answer: \(P=1450/11\approx131.82\); both reserves are \(50.00\).)

What You'll Learn