Consistency Checks For Modified Reserves

Why Modified Reserves Need Independent Checks

A modified reserve is calculated using a specified valuation basis, which may differ from the basis used for another reserve or for the contract’s actual cash flows. Once the basis and cash-flow timing are fixed, however, its reserve values must be internally consistent. A prospective calculation, an annual recursion, and the boundary conditions are different ways to express the same valuation.

These checks are especially useful when premiums or benefits change by policy year, a basis changes during the contract, or a benefit is payable at the end of the term. A result can look plausible but still be wrong because a premium was placed at the wrong time or a terminal benefit was counted on the wrong side of the reserve date.

Definition: In an annual discrete model, let $V_k$ be the modified reserve at time $k$, immediately before any premium due at that time. Let $P_k$ be that premium, $i$ the effective annual interest rate, $q_k$ the probability of death during year $k$ conditional on being alive at time $k$, and $p_k=1-q_k$. If the death benefit $B_{k+1}$ is paid at the end of the year, and a survivor has reserve $V_{k+1}$ immediately before the next premium, then, with no other cash flows, $$ (V_k+P_k)(1+i)=q_kB_{k+1}+p_kV_{k+1}. $$

This convention treats a premium at time $k$ as an immediate inflow: it is added to the reserve before the year’s interest is earned. A reserve may instead be defined just after a premium or just before a benefit. Either convention can be used, but the prospective formula, recursion, and boundary values must all use the same one.

Three Core Consistency Checks

The first check is prospective: calculate the expected present value of future benefits and expenses, less the expected present value of future premiums, conditional on the policy being in force at the reserve date. The second is recursive: verify that one year of investment and mortality transitions produces the next reserve. The third is a boundary check: identify what cash flows remain at the end of the coverage period and whether they have already been paid at the stated reserve instant.

Write the timing convention.
State whether each reserve is before or after the premium, benefit, or expense due at that time.

Calculate prospectively.
Value all future modified-basis benefits and expenses, then subtract the value of future premiums, conditional on being in the required state.

Reconcile by recursion.
Apply the one-year interest and transition calculation and check that it gives the same reserve.

Check the boundary.
At maturity or termination, distinguish the value immediately before a final payment from the value immediately after it.

Why the Annual Recursion Holds

The recursion is a one-year decomposition of the prospective value. At time $k$, the reserve together with the premium due at that time forms the funds available for the year. Those funds accumulate at the valuation interest rate. At the end of the year, death occurs with probability $q_k$, producing the death benefit; survival occurs with probability $p_k$, leaving the next reserve. Taking the conditional expected value of those end-of-year obligations gives the stated equation.

(V_k+P_k)(1+i) =q_kB_{k+1}+p_kV_{k+1}.

Solving for $V_k$ gives the useful backward calculation:

V_k=\frac{q_kB_{k+1}+p_kV_{k+1}}{1+i}-P_k.

This equation assumes annual premiums at the start of the year, death benefits at year-end, a reserve carried for survivors at the next anniversary, and no other cash flows. If expenses or other benefits are present, they must be included at their actual payment times. Applying this equation with a different cash-flow timing is not a valid consistency check.

Worked Example: Matching Prospective and Recursive Values

A hypothetical two-year term contract pays $1,000 at the end of the year of death and nothing on survival to time 2. Modified-basis premiums of $60 are due at times 0 and 1 while the insured is alive. The effective annual interest rate is $i=25\%$. For each year, conditional on being alive at its start, the death indicator is Bernoulli with $q_k=0.20$, so $p_k=0.80$. There are no expenses. Find the reserves at times 0 and 1, immediately before any premium due then, and verify them both ways.

Prospective reserve at time 1. The year-2 death benefit has conditional expected amount $0.20(1{,}000)=200$, payable at time 2. The premium of $60 is due immediately at time 1. Thus:

V_1=\frac{200}{1.25}-60=160-60=100.

Prospective reserve at time 0. The probability of death in year 1 is $0.20$, and the probability of death in year 2 is $0.80(0.20)=0.16$. The present value of benefits is:

0.20(1{,}000)(0.8)+0.16(1{,}000)(0.8^2) =160+102.4=262.4.

The expected present value of premiums is $60+0.80(60)(0.8)=60+38.4=98.4$. Therefore $V_0=262.4-98.4=164$.

Recursive check. At time 1, $(100+60)(1.25)=200$, which equals the expected year-2 death benefit. At time 0, the expected year-end amount is $0.20(1{,}000)+0.80(100)=200+80=280$. The available funds accumulate to $(164+60)(1.25)=224(1.25)=280$. Both values reconcile. The method avoids the common mistake of discounting the time-1 premium as if it were payable at the end of year 1.

Boundary Conditions and Payment Timing

A boundary condition is a reserve value at a contract endpoint or other specified transition time. It depends on the reserve’s exact timing. For a term contract with no payment on survival to the end, the reserve immediately after the final year’s death benefit or coverage termination is zero. For an endowment contract, a maturity benefit is still due to a survivor immediately before the payment. The reserve after that payment is zero if no further obligations remain.

Thus, a statement such as “the reserve at maturity is $500” is incomplete unless it specifies whether the reserve is before or after the maturity payment. In a recursive calculation, use the value that matches the endpoint of the year being projected.

Worked Example: Checking an Endowment Boundary

A hypothetical two-year endowment pays $1,000 at the end of each year of death and $500 at time 2 if the insured survives. Premiums of $80 are due at times 0 and 1 while alive. The effective annual interest rate is $i=25\%$. In each year, conditional on being alive at the start, death is Bernoulli with $q_k=0.20$ and $p_k=0.80$. There are no expenses. Calculate the time-1 and time-0 reserves before premiums, and state the time-2 boundary values.

Boundary. Immediately before the maturity payment, the survivor is owed $500, so $V_2^-=500$. Immediately after payment, with no further obligations, $V_2^+=0$. Use $V_2^-=500$ in the year-2 recursion.

Reserve at time 1. The expected year-end amount is $0.20(1{,}000)+0.80(500)=200+400=600$. Discounting one year and subtracting the premium due immediately gives:

V_1=\frac{600}{1.25}-80=480-80=400.

Reserve at time 0. The year-1 expected year-end amount is $0.20(1{,}000)+0.80(400)=200+320=520$. Hence:

V_0=\frac{520}{1.25}-80=416-80=336.

Prospective check at time 0. The present value of the year-1 death benefit is $0.20(1{,}000)(0.8)=160$. The year-2 death benefit has probability $0.80(0.20)=0.16$, giving present value $0.16(1{,}000)(0.8^2)=102.4$. The maturity benefit has probability $0.80^2=0.64$, giving present value $0.64(500)(0.8^2)=204.8$. Total benefit value is $160+102.4+204.8=467.2$.

The premium value is $80+0.80(80)(0.8)=80+51.2=131.2$. The prospective reserve is $467.2-131.2=336$, matching the recursion. Setting $V_2=0$ in the recursion would omit the $500 maturity benefit and fail this check.

Checking a Basis Change Mid-Contract

When a modified basis changes during a contract, the reserve at the change date is the continuation value under the basis that applies from that date onward. The earlier reserve must then value that continuation amount using the mortality, interest, premium, and benefit assumptions applicable before the change. It is not consistent to use one basis for the prospective calculation and another for the recursion over the same period.

A useful procedure is to calculate the reserve at the change date first, using only the future basis. Then work backward through the earlier period using the earlier basis and the reserve at the change date as the survivor value. This makes the transition explicit and provides a direct check against a prospective valuation over the combined periods.

Worked Example: Reconciling Reserves Across a Basis Change

A hypothetical two-year term contract pays $1,000 at the end of the year of death, with no benefit on survival to time 2. A modified basis changes at time 1. In year 1, $i=25\%$, $q_0=0.20$, $p_0=0.80$, and a $60 premium is due at time 0. In year 2, the new basis has $i=25\%$, $q_1=0.25$, $p_1=0.75$, and a $100 premium due at time 1. Each yearly death indicator is Bernoulli with its stated conditional probability. There are no expenses. Calculate the reserves before premiums and verify the reserve at time 0 prospectively.

At time 1 under the new basis. The expected year-2 death benefit is $0.25(1{,}000)=250$. The reserve before the time-1 premium is:

V_1=\frac{250}{1.25}-100=200-100=100.

At time 0 under the earlier basis. The expected year-end amount is the year-1 death benefit in the death state plus the time-1 reserve in the survival state: $0.20(1{,}000)+0.80(100)=200+80=280$. Thus:

V_0=\frac{280}{1.25}-60=224-60=164.

Hybrid prospective check. The year-1 death benefit has present value $0.20(1{,}000)(0.8)=160$. If the insured survives, probability $0.80$, the value at time 1 of the remaining contract is $V_1=100$; its time-0 value is $0.80(100)(0.8)=64$. Subtract the time-0 premium of $60: $160+64-60=164$. This agrees with the recursion. The common mistake is to apply the new year-2 mortality assumption to year 1 as well; the time-0 calculation must use the earlier year-1 basis.

Common Inconsistencies to Look For

A numerical check is most useful when it tests assumptions as well as arithmetic. Before accepting a modified reserve, inspect the following points:

Premium timing: A premium due at the reserve date is included in the prospective value if the reserve is before that premium. It is not discounted as a future end-of-year payment.
Benefit timing: A year-end benefit is discounted for the full year. A benefit paid immediately at the reserve date is not discounted for a year.
Survival value: The recursion uses the reserve at the next date conditional on survival, with its stated before-premium or after-premium convention.
Endpoint value: A maturity benefit belongs in a reserve immediately before payment, but not in the reserve immediately after payment if the contract then ends.
Basis by duration: If assumptions change mid-contract, use each basis only for the period to which it applies.

A quick diagnostic is to substitute the calculated reserves into each one-year recursion. If the two sides do not agree, check the cash-flow dates and the selected endpoint value before redoing the arithmetic. A separate prospective calculation then checks that the full remaining contract has been valued consistently.

Key takeaway: A modified reserve should pass three linked checks: it equals the prospective value on the modified basis, it satisfies the annual recursion under the stated timing convention, and it uses the correct reserve boundary before or after any terminal cash flow.

Check Your Understanding

For each problem, use the stated reserve convention and verify the cash-flow timing as well as the arithmetic.

A one-year term contract pays $500 at year-end on death, with $q=0.20$, $p=0.80$, $i=0$, and a $50 premium due immediately at time 0. There are no expenses. Find the reserve at time 0 before the premium. (Answer: $50.)
A two-year term contract pays $800 at year-end on death and nothing on survival to time 2. The annual death probability is $q=0.25$ in each year, $p=0.75$, and $i=25\%$. A $100 premium is due at times 0 and 1 while alive. Find $V_1$ and $V_0$, both before premium. (Answer: $V_1=\$60$, $V_0=\$96$.)
A two-year endowment pays $1,000 at year-end on death and $500 at time 2 on survival. In each year $q=0.20$, $p=0.80$, and $i=25\%$. Premiums of $80 are due at times 0 and 1. Find the reserve at time 1 before premium and distinguish the time-2 values immediately before and after maturity payment. (Answer: $V_1=\$400$; $V_2^-=\$500$ and $V_2^+=\$0$.)
A two-year term contract pays $1,000 at year-end on death. In year 1, $i=25\%$, $q_0=0.20$, and the time-0 premium is $60. At time 1 the basis changes: year-2 $i=25\%$, $q_1=0.25$, and the time-1 premium is $100. Find the reserves before premiums at times 1 and 0. (Answer: $V_1=\$100$, $V_0=\$164$.)
A one-year endowment pays $400 at year-end on death or survival. Let $q=0.25$, $p=0.75$, $i=0$, and let a premium $P_0$ be due immediately at time 0. If the reserve before premium is $50 and there are no expenses, find $P_0$ using the recursion. (Answer: $P_0=\$350$.)

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