Modified Reserve Recursions

Introduction

An annual reserve is a value at a specified time, calculated using a specified valuation basis. The recursion for that reserve depends not only on interest and transition probabilities, but also on which premiums, expenses, and benefits the valuation recognizes. Those valuation amounts need not equal the cash flows specified in the contract.

This distinction matters when interpreting a reserve movement. A larger valuation premium tends to reduce a reserve, while a larger valuation expense or benefit tends to increase it. The recursion makes those signs precise, provided the timing convention is stated and followed consistently. The methods here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Timing and Valuation Basis

First fix the annual timing convention. At time $t$, consider a policy in a specified state, immediately before the scheduled premium and expense for that time are paid. The premium is received and the expense is paid at time $t$. Transitions occur during the year, and a transition benefit is paid at the end of the year. The reserve at time $t+1$ is measured after that year’s transition and any end-of-year benefit payment.

Use a superscript $v$ to label quantities on the valuation basis. For example, $P_t^v$, $e_t^v$, and $B_t^v$ are the valuation premium, expense, and benefit. They are not automatically the contract premium, expense, and benefit. Interest and transition probabilities used in the recursion must also be those of the valuation basis.

Definition: Under the timing convention above, $V_t^v(s)$ is the reserve at time $t$, immediately before the time-$t$ premium and expense, for a policy currently in state $s$. A modified reserve is calculated using a specified modified valuation basis, which may use cash-flow amounts different from those in the contract.

For a single active state, let $i^v$ be the annual effective valuation interest rate, $q_t^v$ the probability of death during year $t$, and $p_t^v=1-q_t^v$ the probability of survival. Let $B_t^v$ be payable at the end of the year of death. If the policy survives, its reserve at time $t+1$ is $V_{t+1}^v$. The terminal reserve is specified by the contract and valuation problem; for a term contract ending at time $n$ with no maturity benefit, $V_n^v=0$.

Theorem: With the timing and single-state assumptions just stated, the annual reserve recursion is $$ V_t^v=e_t^v-P_t^v+\frac{q_t^v B_t^v+p_t^vV_{t+1}^v}{1+i^v}. $$

The expense increases the reserve needed at time $t$; the premium received reduces it. The expected end-of-year amount is the death probability times the death benefit, plus the survival probability times the next reserve. Discounting that expected amount to time $t$ gives the final term.

Why the Recursion Holds

The recursion follows by separating the reserve’s prospective value into time-$t$ cash flows and the value at the end of the year. This is the same prospective-value logic used in modified reserve equivalence, now written one year at a time.

Record time-$t$ cash flows.
The valuation expense is an outgo of $e_t^v$, and the valuation premium is an income of $P_t^v$. Their net contribution to the reserve is $e_t^v-P_t^v$.

Enumerate the end-of-year outcomes.
Death occurs with probability $q_t^v$ and generates $B_t^v$. Survival occurs with probability $p_t^v$ and leaves reserve $V_{t+1}^v$. Thus the expected end-of-year amount is $q_t^v B_t^v+p_t^vV_{t+1}^v$.

Discount and combine.
One year’s discount factor is $1/(1+i^v)$. Adding the discounted expected amount to the net time-$t$ cash flow gives the stated recursion.

The derivation requires the stated payment times. If a premium is paid at the end of the year, or a benefit is paid immediately on transition, that cash flow must be discounted according to its own timing. The displayed recursion does not apply unchanged to such a payment.

Reserve Movement for a Single Active State

The recursion can be rearranged to show how the reserve changes for a policy that survives the year. This is useful for interpreting a calculation, but it does not mean that every policy experiences the expected transition: the formula conditions on survival.

(1+i^v)(V_t^v+P_t^v-e_t^v) =q_t^v B_t^v+p_t^vV_{t+1}^v

Solving first for the next reserve and then subtracting the current reserve gives

p_t^v(V_{t+1}^v-V_t^v) =i^vV_t^v+(1+i^v)(P_t^v-e_t^v)-q_t^vB_t^v, \qquad V_{t+1}^v-V_t^v =\frac{i^vV_t^v+(1+i^v)(P_t^v-e_t^v)-q_t^vB_t^v}{p_t^v}.

The first equality follows by multiplying the recursion by $1+i^v$. For the second, subtract $p_t^vV_t^v$ from both sides and use $1-p_t^v=q_t^v$. This rearrangement requires $p_t^v>0$; if survival has probability zero, a survivor reserve is not defined by this equation.

The numerator separates the main effects: $i_t^v V_t^v$ is interest on the opening reserve; $(1+i^v)(P_t^v-e_t^v)$ reflects the net premium and expense, accumulated to year-end; and $q_t^vB_t^v$ is the probability-weighted death benefit. Dividing by $p_t^v$ conditions the movement on survival. This is a reserve identity, not a measure of actual profit: actual experience may differ from the valuation assumptions.

When Valuation Cash Flows Differ from Contract Cash Flows

Suppose, for comparison, that the interest rate and transition probabilities are the same for the valuation and contract quantities. Write $P_t^c$, $e_t^c$, and $B_t^c$ for the contract amounts, and define the valuation differences by $\Delta P_t=P_t^v-P_t^c$, $\Delta e_t=e_t^v-e_t^c$, and $\Delta B_t=B_t^v-B_t^c$. Substitution into the recursion gives

V_t^v=e_t^c-P_t^c+ \frac{q_t^v(B_t^c+\Delta B_t)+p_t^vV_{t+1}^v}{1+i^v} +\Delta e_t-\Delta P_t.

Relative to the contract amounts, the direct time-$t$ effect is $\Delta e_t-\Delta P_t$, and the death-benefit difference contributes $q_t^v\Delta B_t/(1+i^v)$. Therefore, holding all else fixed, increasing the valuation expense increases the current reserve, increasing the valuation premium decreases it, and increasing the valuation death benefit increases it. If the valuation interest or transition probabilities also differ, these cash-flow comparisons alone do not capture the whole change in reserve.

This comparison is an algebraic way to identify the terms in the recursion, not a claim that the contract must pay the valuation amounts. A reserve is based on its stated valuation basis. When the valuation basis specifies a premium, expense, or benefit amount different from the contract amount, use the valuation amount in the reserve calculation.

Worked Example: Two-Year Term Reserve with Modified Amounts

A hypothetical two-year term contract is valued at an annual effective interest rate of $5\%$. The valuation probability of death is $0.10$ in each year, independently conditional on being alive at the start of that year. The valuation basis uses a premium of $\$600$ and an expense of $\$20$ at the start of each year, and a death benefit of $\$10{,}000$ at the end of the year of death. There is no maturity benefit, so $V_2^v=0$. The contract instead specifies a premium of $\$700$, expense of $\$30$, and death benefit of $\$12{,}000$. Calculate the modified reserves at times 1 and 0.

Method. Apply the one-year recursion backward using the valuation amounts. The contract amounts are not substituted into this valuation recursion.

At time 1, $p_1^v=0.90$, and the terminal reserve is zero:

V_1^v=20-600+ \frac{0.10(10{,}000)+0.90(0)}{1.05} =-580+\frac{1{,}000}{1.05} =372.380952\ldots

At time 0, the expected end-of-year amount is the death benefit weighted by the death probability plus the time-1 reserve weighted by the survival probability:

V_0^v=20-600+ \frac{0.10(10{,}000)+0.90(372.380952\ldots)}{1.05}

The weighted amount is $1{,}000+335.142857\ldots=1{,}335.142857\ldots$. Its present value is $1{,}271.564626\ldots$, so $V_0^v=691.564626\ldots$. Rounded to the nearest cent, $V_1^v=\$372.38$ and $V_0^v=\$691.56$.

Check. Substituting the unrounded $V_1^v$ into the time-0 recursion reproduces $V_0^v$. A common mistake would be to use the contract’s $\$12{,}000$ death benefit in the calculation. That would value a different basis, not the modified basis specified here.

Worked Example: Duration-Varying Valuation Amounts

A hypothetical two-year contract has annual effective valuation interest $i^v=4\%$. The death probabilities are $q_0^v=0.05$ and $q_1^v=0.10$. The valuation amounts are $P_0^v=\$500$, $e_0^v=\$40$, $B_0^v=\$6{,}000$, and $P_1^v=\$300$, $e_1^v=\$25$, $B_1^v=\$9{,}000$. There is no maturity benefit and $V_2^v=0$. Find $V_1^v$ and $V_0^v$.

Method. Work backward, using the probability and valuation amounts for each duration separately. In year 1, $p_1^v=0.90$, so

V_1^v=25-300+\frac{0.10(9{,}000)+0.90(0)}{1.04} =-275+\frac{900}{1.04} =590.384615\ldots

In year 0, $p_0^v=0.95$:

V_0^v=40-500+ \frac{0.05(6{,}000)+0.95(590.384615\ldots)}{1.04}.

The two weighted amounts are $300$ and $560.865384\ldots$, for a total of $860.865384\ldots$. Discounting gives $827.755177\ldots$, and adding the time-0 net cash flow, $40-500=-460$, gives $V_0^v=367.755177\ldots$. To the nearest cent, $V_1^v=\$590.38$ and $V_0^v=\$367.76$.

Check. The time-0 calculation uses $q_0^v=0.05$, not the year-1 death probability $0.10$. Using one mortality probability for both durations would ignore the stated duration dependence.

Worked Example: Rearranging for the Survivor Reserve

At time $t$, a hypothetical policy has $V_t^v=\$1{,}200$. The annual effective valuation interest rate is $4\%$, the valuation death probability is $0.08$, the valuation premium is $\$500$, the valuation expense is $\$40$, and the end-of-year valuation death benefit is $\$10{,}000$. The contract amounts are instead a premium of $\$540$, expense of $\$25$, and death benefit of $\$11{,}000$. Find the reserve at $t+1$ conditional on survival.

Method. Rearrange the valuation recursion. Since $p_t^v=1-0.08=0.92$,

V_{t+1}^v =\frac{(1+i^v)(V_t^v+P_t^v-e_t^v)-q_t^vB_t^v}{p_t^v} =\frac{1.04(1{,}200+500-40)-0.08(10{,}000)}{0.92}.

The amount inside the first product is $1{,}200+500-40=1{,}660$, and $1.04(1{,}660)=1{,}726.40$. The probability-weighted benefit is $0.08(10{,}000)=800$. Thus

V_{t+1}^v=\frac{1{,}726.40-800}{0.92} =\frac{926.40}{0.92} =1{,}006.956521\ldots

Rounded to the nearest cent, the survivor reserve is $\$1{,}006.96$. Substitution checks the result: $25$ is not used here; using the specified valuation amounts gives $40-500+(800+0.92(1{,}006.956521\ldots))/1.04=1{,}200$. The common mistake is to substitute the contract amounts into a recursion explicitly using valuation amounts.

Multiple-State Recursion

The same reasoning applies when a policy can move among several states. Let $p_{sr,t}^v$ be the valuation probability that a policy in state $s$ at time $t$ is in state $r$ at time $t+1$. Let $b_{sr,t}^v$ be a benefit paid at the end of the year for that transition, and $V_{t+1}^v(r)$ the reserve in the destination state. Under the same timing convention, the recursion is

V_t^v(s)=e_t^v(s)-P_t^v(s)+ \frac{\displaystyle\sum_r p_{sr,t}^v \left[b_{sr,t}^v+V_{t+1}^v(r)\right]}{1+i^v}.

A transition benefit belongs inside the sum because it is paid only when that transition occurs. A death state with no continuing liability has reserve zero after the death benefit, but that does not mean the death benefit itself is zero. It appears as $b_{sr,t}^v$ on the transition into that state.

Worked Example: Transition Benefits and Destination Reserves

A hypothetical policy starts the year in a healthy state. The annual effective valuation interest rate is $5\%$. By year-end, the policy remains healthy with probability $0.90$, enters a disability state with probability $0.08$, or enters a death state with probability $0.02$. The destination reserves are $\$1{,}000$ in the healthy state, $\$1{,}500$ in the disability state, and zero in the death state. A transition benefit of $\$200$ is paid on entry to disability and $\$10{,}000$ on death; there is no benefit on remaining healthy. At the start of the year, the valuation expense is $\$30$ and the valuation premium is $\$500$. Find the reserve in the healthy state.

Method. Sum the benefit plus destination reserve for each transition, weight by its probability, and discount. The weighted amounts are $0.90(0+1{,}000)=900$, $0.08(200+1{,}500)=136$, and $0.02(10{,}000+0)=200$. Their total is $1{,}236$. Therefore,

V_t^v(\text{healthy}) =30-500+\frac{1{,}236}{1.05} =-470+1{,}177.142857\ldots =707.142857\ldots

Rounded to the nearest cent, the reserve is $\$707.14$. The probabilities sum to $0.90+0.08+0.02=1.00$, and the transition benefits have been included only in the outcomes where they are payable.

Common Mistakes and Checks

Mixing timing conventions. The formulas here assume premiums and expenses at the start of the year and transition benefits at year-end. A different payment time requires a different discounting term.
Using contract amounts instead of valuation amounts. Use the amounts specified for the reserve basis, even when they differ from the contract cash flows.
Reversing signs. A premium is income and enters as $-P_t^v$; an expense is an outgo and enters as $+e_t^v$.
Forgetting transition probabilities. Each transition’s benefit and destination reserve must be multiplied by that transition’s probability.
Using an unconditional movement for a survivor. The rearranged single-state movement divides by $p_t^v$, because it describes the reserve conditional on survival.

A reliable calculation starts by writing the reserve time, payment times, and valuation basis. Then write the one-year outcome amounts, weight them by the correct probabilities, discount, and include the time-$t$ premium and expense with their correct signs. Finally, substitute a calculated reserve back into the original recursion.

Check Your Understanding

For each problem, use the stated valuation basis and timing convention. Round monetary answers to the nearest cent.

A one-year term policy has $i^v=5\%$, $q^v=0.10$, valuation premium $\$450$, expense $\$15$, and death benefit $\$8{,}000$. The terminal reserve is zero. Find the time-0 reserve. (Answer: $\$326.90$)
A policy has $V_t^v=\$900$, $i^v=3\%$, $q_t^v=0.04$, valuation premium $\$400$, expense $\$20$, and death benefit $\$5{,}000$. Find the survivor reserve $V_{t+1}^v$. (Answer: $\$1{,}165.00$)
A two-year term policy has $i^v=5\%$, $q_0^v=0.02$, $q_1^v=0.08$, and $V_2^v=0$. At time 0, $P_0^v=\$300$, $e_0^v=\$10$, and $B_0^v=\$5{,}000$; at time 1, $P_1^v=\$250$, $e_1^v=\$20$, and $B_1^v=\$7{,}000$. Find $V_1^v$ and $V_0^v$. (Answer: $V_1^v=\$303.33$, $V_0^v=\$88.35$)
At a given year, $i^v=4\%$, $q_t^v=0.10$, and $V_{t+1}^v=\$700$. Contract amounts are $P_t^c=\$400$, $e_t^c=\$20$, and $B_t^c=\$5{,}000$. Valuation amounts are $P_t^v=\$450$, $e_t^v=\$30$, and $B_t^v=\$6{,}000$. Calculate the reserve using valuation amounts, then compare it with the reserve using contract amounts and the same other assumptions. (Answer: valuation reserve $\$762.69$; contract-amount reserve $\$706.54$; difference $\$56.15$)
A policy starts in a healthy state. At year-end, it remains healthy with probability $0.90$, enters disability with probability $0.08$, or dies with probability $0.02$. The corresponding destination reserves are $\$1{,}000$, $\$1{,}500$, and $0$; transition benefits are $0$, $\$200$, and $\$10{,}000$. If $i^v=5\%$, $P_t^v=\$500$, and $e_t^v=\$30$, find the healthy-state reserve. (Answer: $\$707.14$)

What You'll Learn