Modified Reserves For Deferred Benefits

Why Deferred Benefits Need Careful Reserve Timing

A deferred benefit creates two distinct periods for valuation. During the deferment period, the policy may collect premiums and incur expenses, but the specified benefit is not yet payable. Once the deferral ends, a different set of benefit, premium, and expense cash flows may apply. A modified valuation basis can also use different assumptions in those two periods.

The central task is to value each period on its own terms and connect them at the transition date. A premium due at the start of the final deferment year is not the same cash flow as a premium due when the benefit period begins. Likewise, the probability of reaching the benefit period must be valued using the deferment-period assumptions.

Definition: A deferred benefit is a benefit that is payable only after a specified deferment period, subject to the contract’s conditions. A modified reserve for a deferred benefit is the prospective value, on the stated modified valuation basis, of future benefits and expenses less future premiums, conditional on the policy being in force at the valuation time. The basis may specify different interest, mortality, premium, and expense assumptions in the deferment and benefit periods.

For the calculations below, use an annual discrete model. At each integer time, the reserve is measured immediately before any premium or expense due at that time. During deferment, premiums and expenses are payable at the start of a year while the insured is alive. Death during deferment terminates premiums, and no deferred benefit is payable for such a death. In the benefit period, a death benefit is paid at the end of the year of death; any stated premium or expense for that period is payable at the start of each year while the insured is alive.

These timing conventions are assumptions for the formulas that follow, not universal contract features. If a policy instead pays a benefit on death during deferment, collects premiums at year-end, or uses a different benefit-period payment pattern, the cash-flow valuation must be changed accordingly.

A Two-Phase Valuation Formula

Let the deferment last $n$ years. In the deferment period, let $p_D$ be the one-year probability of survival on the valuation basis, $i_D$ the effective annual interest rate, $P_D$ the premium due at each year-start, and $e_D$ the expense due at each year-start. Write $v_D=1/(1+i_D)$. In the benefit period, use $p_B$, $i_B$, $v_B=1/(1+i_B)$, and one-year death probability $q_B=1-p_B$. These are valuation assumptions; they need not equal the assumptions used to determine the contract’s actual premiums.

Suppose that, after deferment, a whole-life death benefit $B$ is payable at the end of the year of death. Let $P_B$ and $e_B$ be, respectively, the premium and expense payable at the start of each benefit-period year while the insured is alive. The reserve at the transition time $n$, just before those time-$n$ cash flows, is

V_n=\frac{v_Bq_BB+e_B-P_B}{1-v_Bp_B}.

This expression values cash flows from the start of the benefit period onward. It can be negative if the specified future premiums exceed the value of future benefits and expenses. That result is a property of the stated basis and timing, not by itself a calculation error.

To value the deferment period, put $a_D=v_Dp_D$. The probability-weighted discount factor for surviving one deferment year is $a_D$. Thus, at time 0, just before the initial premium and expense, the reserve is

V_0=\sum_{j=0}^{n-1}a_D^j(e_D-P_D)+a_D^nV_n.

The sum covers the net cash flows at times $0,1,\ldots,n-1$. The final term is the transition-period reserve, weighted by the probability and discount factor for surviving all $n$ deferment years. If premiums or expenses vary by year, replace the constant $e_D-P_D$ with the appropriate net cash flow for each year.

Why the Formulas Hold

The benefit-period formula follows by conditioning on what happens during the next year. At the start of a benefit-period year, the current expense is paid and the current premium is received. At year-end, a death claim is paid with probability $q_B$; with probability $p_B$, the policy survives and the reserve continues. Discounting year-end values at rate $i_B$ gives

V_n=e_B-P_B+v_B\bigl(q_BB+p_BV_n\bigr).

Collecting the terms in $V_n$ gives $(1-v_Bp_B)V_n=e_B-P_B+v_Bq_BB$, which yields the stated formula. The denominator is positive because $v_Bp_B<1$ for a positive interest rate and a survival probability no greater than one.

For the deferment formula, each net cash flow at time $j$ is reached only if the insured survives the first $j$ years. Its expected present-value factor is therefore $(v_Dp_D)^j$. The transition reserve is reached only after survival through all $n$ years, giving factor $(v_Dp_D)^n$. Adding the values of these mutually timed cash flows gives the formula. This argument also shows why applying benefit-period mortality to the deferment years, or discounting the transition reserve without allowing for survival, is incorrect.

Worked Calculations

Worked Example: Deferred Whole-Life Benefit With Deferment Premiums

A hypothetical contract pays a death benefit of $10,000 only for deaths after a two-year deferment. Premiums of $1,000 are due at times 0 and 1 while the insured is alive; there are no premiums or expenses after the deferment. On the modified valuation basis, each deferment year has $q_D=0.10$ and $i_D=0.05$. In the benefit period, $q_B=0.20$ and $i_B=0.05$. Find the reserve at time 0 before the initial premium.

Method. First calculate the reserve at the transition into the benefit period, then discount and survival-weight it through deferment, and finally subtract the value of the deferment premiums. This separates the two mortality assumptions.

In the benefit period, $p_B=0.80$ and $v_B=1/1.05=20/21$. With no ongoing premium or expense,

V_2=\frac{(20/21)(0.20)(10{,}000)}{1-(20/21)(0.80)} =\frac{4{,}000/21}{5/21} =8{,}000.

For deferment, $p_D=0.90$, so $a_D=v_Dp_D=(20/21)(0.90)=6/7$. The time-0 value of the transition reserve is

\left(\frac67\right)^2(8{,}000)=\frac{288{,}000}{49}=\$5{,}877.55.

The value of premiums at times 0 and 1 is

1{,}000\left[1+\frac67\right] =1{,}000\left(\frac{13}{7}\right) =\$1{,}857.14.

Therefore $V_0=5{,}877.55-1{,}857.14=\boxed{\$4{,}020.41}$, rounded to the nearest cent. As a check, the exact difference is $197{,}000/49$, or approximately $4,020.4082. A common mistake is to use $q_B=0.20$ during deferment; the transition value is instead weighted by the deferment survival and interest factor $(6/7)^2$.

Worked Example: Expenses and a Premium at the Transition

A hypothetical contract defers a $10,000 whole-life death benefit for two years. At time 1, immediately before the second deferment-year cash flows, the insured is alive. A $700 premium is due then, and a $100 expense is incurred then. There is one remaining deferment year with $q_D=0.10$ and $i_D=0.05$. After deferment, the valuation basis is $q_B=0.20$, $i_B=0.05$, with an annual expense of $50 payable at each benefit-period year-start. No premiums are payable after deferment. Find the reserve at time 1 before its premium and expense.

Method. Include the cash flows due immediately at time 1 separately, then value the transition reserve one year ahead. The benefit-period expense is a year-start payment while alive, so it must be included in the benefit-period reserve.

As in the previous calculation, $v_B=20/21$, $p_B=0.80$, and the death-benefit component at the transition is $8,000. The present-value factor for a benefit-period annuity of year-start expenses is

\frac{1}{1-v_Bp_B} =\frac{1}{1-(20/21)(0.80)} =\frac{21}{5}=4.2.

Thus the value of the $50 annual expenses is $50(4.2)=\$210$, and the transition reserve is $V_2=8{,}000+210=\$8{,}210$. For the remaining deferment year, $v_Dp_D=(20/21)(0.90)=6/7$. At time 1 the current expense is $100 and the current premium is $700, so

V_1=100-700+\frac67(8{,}210) =-600+7{,}037.142857\ldots =\boxed{\$6{,}437.14}.

The result is rounded to the nearest cent. The calculation includes the transition reserve only after one year of deferment survival. A common mistake is to omit the $100 expense because it is not a benefit; it is still a future outflow from the insurer and therefore increases the reserve.

Worked Example: Premiums Continue After the Deferral

A hypothetical policy has a two-year deferment followed by a $10,000 whole-life death benefit. A $500 premium is due at every year-start while the policy is in force, including at the start of the benefit period. There are no expenses. During deferment, $q_D=0.10$ and $i_D=0.05$; during the benefit period, $q_B=0.20$ and $i_B=0.05$. Find the reserve at time 0, before the initial premium.

Method. Include the premium due at the transition in the benefit-period reserve, then value the two deferment premiums separately. This avoids treating the time-2 premium as though it were a deferment-period payment.

The benefit component at the transition is $8,000, as calculated from the stated benefit-period assumptions. The value of benefit-period premiums is

\frac{500}{1-v_Bp_B} =500\left(\frac{21}{5}\right) =\$2{,}100.

Consequently, $V_2=8{,}000-2{,}100=\$5{,}900$. In deferment, $a_D=6/7$. The two premiums due at times 0 and 1 have value

500\left(1+\frac67\right)=\frac{6{,}500}{7}=\$928.571428\ldots.

The transition reserve has time-0 value

\left(\frac67\right)^2(5{,}900) =\frac{212{,}400}{49} =\$4{,}334.693877\ldots.

Subtracting the deferment premiums gives $V_0=4{,}334.693877\ldots-928.571428\ldots=\boxed{\$3{,}406.12}$, rounded to the nearest cent. The reserve incorporates the time-2 premium through $V_2$, not through the two-premium deferment annuity. Omitting that transition premium would overstate the reserve by its benefit-period value.

Common Timing Traps

Deferred-benefit calculations are especially sensitive to whether cash flows occur at the beginning or end of a year. Before calculating, write a short timeline showing the deferral end date, the valuation instant, and each premium, expense, and benefit payment. Then apply the assumptions that govern the interval in question.

Using the wrong mortality period. The probability of reaching the benefit period comes from deferment-period survival assumptions. Benefit-period mortality is used to value claims after the transition.
Forgetting survival weighting. A transition reserve at time $n$ is not simply discounted by $v_D^n$; it is weighted by $(v_Dp_D)^n$ when reaching the transition requires survival.
Misplacing the transition premium. Decide whether a premium is due at time $n$. Under the convention used here, that cash flow belongs in the benefit-period reserve if the premium is due when the benefit period begins.
Omitting expenses or reversing their sign. Future expenses increase the reserve, while future premiums reduce it. Their timing and survival conditions must match the contract.
Applying a formula to a different payment pattern. A benefit payable during deferment, an end-of-year premium, or a benefit-period term instead of whole-life coverage requires a different cash-flow valuation.

A useful check is to calculate the transition-period reserve first and state explicitly which cash flows it includes. Then list the deferment-period cash flows by time. This makes it easier to identify double-counted or omitted premiums, especially at the transition date. These methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated annual timing conventions: premiums and expenses are due at the beginning of a year while alive, and the death benefit is paid at year-end. Round dollar answers to the nearest cent.

A hypothetical contract pays $4,000 for death after a one-year deferment. There are no premiums or expenses. During deferment, $q_D=0.20$ and $i_D=0$; in the benefit period, $q_B=0.25$ and $i_B=0$. Find the reserve at time 0. (Answer: $3,200.00)
A hypothetical contract defers a $6,000 whole-life death benefit for two years. Premiums of $300 are due at times 0 and 1 only. There are no expenses. During deferment, $q_D=0.10$ and $i_D=0$; in the benefit period, $q_B=0.10$ and $i_B=0$. Find the reserve at time 0. (Answer: $4,290.00)
A hypothetical contract pays a $9,000 whole-life death benefit after a one-year deferment. A premium of $800 is due at time 0 only. There are no expenses. In deferment, $q_D=0.20$ and $i_D=0.25$; in the benefit period, $q_B=0.20$ and $i_B=0.25$. Find the reserve at time 0. (Answer: $1,760.00)
A hypothetical contract defers a $5,000 whole-life benefit for one year. A $600 premium and a $40 expense are due at time 0, with no premiums after deferment. Deferment assumptions are $q_D=0.10$, $i_D=0.05$. Benefit-period assumptions are $q_B=0.20$, $i_B=0.05$, with a $25 expense due at each benefit-period year-start. Find the reserve at time 0. (Answer: $2,958.57)
A hypothetical contract defers an $8,000 whole-life benefit for two years. Premiums of $300 are due at times 0 and 1, and premiums of $400 continue at every benefit-period year-start. There are no expenses. During deferment, $q_D=0.05$, $i_D=0$; in the benefit period, $q_B=0.25$, $i_B=0$. Find the reserve at time 0. (Answer: $5,191.00)

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