Choosing A Modified Valuation Premium

Why the Valuation Premium Must Be Chosen Explicitly

A modified reserve is calculated using a specified valuation basis, which may use a premium different from the contract premium. The premium is not chosen by a universal rule: it depends on the modification being applied and on which benefits and expenses the basis recognizes. A sound calculation therefore begins by stating the cash flows and the rule used to select the valuation premium.

This tutorial uses a specific and transparent rule: choose a level annual valuation premium that equates the issue-date present value of valuation premiums with the present value of benefits and the expenses included in the modified basis. A reference basis in the examples values the same benefits but excludes expenses. These definitions make the comparison precise; other modifications require their own stated selection rule. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A modified valuation premium, denoted $P^m$, is the level premium used to value future premiums on a stated modified basis. In this tutorial, $P^m$ is selected by an issue-date equivalence that includes the specified benefits and valuation expenses. The reference valuation premium, denoted $P^r$, is selected on the same benefit and interest basis but excludes expenses. Neither premium is necessarily the amount charged by the insurer under the contract.

Issue-Date Equivalence on a Discrete Basis

Assume a term policy has a benefit $b_k$ payable at the end of policy year $k$ if death occurs during that year. Annual premiums are level and payable at the beginning of each year while the insured is in force, for at most $n$ years. Let $q_k$ be the probability of death during year $k$, conditional on being alive at its start, and let $p_k=1-q_k$. Use an effective annual interest rate $i$, with $v=1/(1+i)$. The examples use expenses payable at issue and at the start of renewal years while the policy is in force.

Write $e_0$ for the expense payable at issue and $e_k$ for the expense payable at the start of year $k+1$, for $1\leq k<n$. Thus $e_k$ is incurred only if the insured survives to that time. These expenses may be specified directly, or calculated from a stated contract premium and expense percentages.

B_0=\sum_{k=1}^{n}v^k\,{}_{k-1}p_x\,q_k\,b_k,\qquad \ddot a_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k\,{}_kp_x,\qquad E_0=e_0+\sum_{k=1}^{n-1}v^k\,{}_kp_x\,e_k.

Here $B_0$ is the expected present value of benefits, $\ddot a_{x:\overline{n}|}$ is the expected present value of the premium stream per dollar of level premium, and $E_0$ is the expected present value of the specified expenses. The reference and modified valuation premiums under the stated rule are therefore:

P^r=\frac{B_0}{\ddot a_{x:\overline{n}|}}, \qquad P^m=\frac{B_0+E_0}{\ddot a_{x:\overline{n}|}}, \qquad P^m-P^r=\frac{E_0}{\ddot a_{x:\overline{n}|}}.

This selection rule assumes the same mortality, benefit, and interest basis for both premiums. It also assumes that the modified premium is level and that the included expenses are exactly those specified in $E_0$. If benefits or expenses have different timing, or the modification imposes another constraint, the present values and the premium equation must be changed accordingly.

Derivation: At issue, the modified basis has expected outgo $B_0+E_0$. A level premium $P^m$ payable in advance while the policy is in force has expected present value $P^m\ddot a_{x:\overline{n}|}$. Equating these values gives $P^m\ddot a_{x:\overline{n}|}=B_0+E_0$, and division by the positive premium annuity gives the formula for $P^m$. Subtracting the reference equation $P^r\ddot a_{x:\overline{n}|}=B_0$ gives $P^m-P^r=E_0/\ddot a_{x:\overline{n}|}$. Thus, with nonnegative expenses, this particular expense-inclusive modified premium is at least as large as the reference premium.

Comparing Reserves at a Duration

State the reserve timing before comparing values. In the examples, the duration-$t$ reserve is immediately before the premium and any expense due at the start of year $t+1$, conditional on the insured being alive then. The modified reserve includes future expenses on the modified basis; the reference reserve excludes them. Both use the same remaining benefits and the same interest and mortality assumptions.

V_t^m-V_t^r = E_t-(P^m-P^r)\ddot a_{x+t:\overline{n-t}|},

Here $E_t$ is the conditional expected present value at duration $t$ of expenses still payable, including an expense due immediately at that duration. The annuity factor values the remaining premiums, including the premium due immediately. The equation follows by subtracting the two prospective reserve calculations: the benefit values cancel, while the modified calculation adds future expenses and uses a different premium. The sign is not determined by $P^m>P^r$ alone. Future expense outgo increases the modified reserve; the larger modified valuation premium decreases it.

Value the benefits.
Apply the specified death probabilities, survival probabilities, benefit amounts, and discount factors.

Value the premiums and expenses.
Include each cash flow only when it is due, and weight conditional cash flows by the probability of being in force then.

Select the premium.
Use the stated issue-date equivalence, or another explicitly specified modification rule.

Compare prospective reserves.
Use the same duration and cash-flow timing for both bases; account for both remaining expenses and the change in valuation premiums.

Worked Example: Renewal Expenses Lower the Modified Reserve

A hypothetical two-year term policy pays $1,000 at the end of the year of death. The conditional death probabilities are $q_1=0.10$ and $q_2=0.20$, and $i=0$. The contract premium is $200 per year. For this modified basis, the issue expense is 30% of the contract premium and the renewal expense is 5% of it. Find $P^m$, then compare the reserves at duration 1, immediately before the second premium and renewal expense.

Method. Calculate the issue-date equivalence, then use prospective values at duration 1. This avoids treating a valuation premium as the contract premium or omitting the renewal expense from the modified reserve.

The survival probabilities are ${}_0p_x=1$ and ${}_1p_x=0.90$. Since $v=1$, the expected present value of benefits is

B_0=1{,}000(0.10)+1{,}000(0.90)(0.20)=100+180=280.

The premium annuity factor and expenses are

\ddot a_{x:\overline{2}|}=1+0.90=1.90,\qquad e_0=0.30(200)=60,\qquad e_1=0.05(200)=10,

E_0=60+0.90(10)=69,\qquad P^r=\frac{280}{1.90}=147.3684,\qquad P^m=\frac{280+69}{1.90}=183.6842.

At duration 1, conditional on survival, the expected benefit value is $1{,}000(0.20)=200$. The reference reserve subtracts the reference premium due immediately:

$$ V_1^r=200-147.3684=52.6316. $$

The modified reserve also includes the renewal expense due immediately:

$$ V_1^m=200+10-183.6842=26.3158. $$

Rounded to the nearest cent, $P^m=\$183.68$, $V_1^r=\$52.63$, and $V_1^m=\$26.32$. The modified reserve is lower by $\$26.32$: the additional premium deduction, $183.6842-147.3684=36.3158$, exceeds the $10 renewal expense. As a check, at issue $P^m(1.90)=349=B_0+E_0$, so the modified prospective reserve is zero under this equivalence.

Worked Example: Large Renewal Expenses Raise the Modified Reserve

Use the same hypothetical two-year policy, benefit, mortality, and zero interest rate. Now the modified basis has no issue expense and a $100 renewal expense payable at the start of year 2. Calculate the modified premium and compare duration-1 reserves.

Method. Revalue the specified expense stream, then compare conditional prospective values immediately before the renewal premium and expense. The expense is incurred only if the insured survives year 1.

The benefit value and premium annuity factor remain $B_0=280$ and $\ddot a_{x:\overline{2}|}=1.90$. The issue-date expense value is

$$ E_0=0+0.90(100)=90. $$

Therefore,

P^m=\frac{280+90}{1.90}=194.7368,\qquad P^r=\frac{280}{1.90}=147.3684.

At duration 1, the reference reserve is still $200-147.3684=52.6316$. The modified reserve is

$$ V_1^m=200+100-194.7368=105.2632. $$

Rounded to the nearest cent, $P^m=\$194.74$, $V_1^r=\$52.63$, and $V_1^m=\$105.26$. The modified reserve is higher by $\$52.63$. At this duration the $100 expense exceeds the $47.3684 increase in the premium deducted. This example shows why a higher modified valuation premium does not, by itself, imply a lower modified reserve.

Worked Example: Interest and Expense Timing Both Matter

A hypothetical two-year policy pays $1,000 at the end of the year of death, with $q_1=0.20$, $q_2=0.25$, and effective annual interest $i=0.10$. The contract premium is $250 annually. The valuation basis includes an issue expense of 20% of that contract premium and a renewal expense of 4% of it. Calculate the modified valuation premium and the duration-1 modified reserve, immediately before the second premium and expense.

Method. Discount year-1 and year-2 cash flows at their actual times, and apply survival weighting to the renewal cash flows. The common error avoided here is using the contract premium directly as the valuation premium.

Here $v=1/1.10=10/11$, $p_1=0.80$, $e_0=0.20(250)=50$, and $e_1=0.04(250)=10$. The benefit value and premium annuity factor are

B_0=1{,}000(0.20)\frac{10}{11} +1{,}000(0.80)(0.25)\left(\frac{10}{11}\right)^2 =\frac{42{,}000}{121}=347.1074, $$ $$ \ddot a_{x:\overline{2}|}=1+0.80\frac{10}{11}=\frac{19}{11}=1.7273.

The issue expense is payable at time 0. The renewal expense is payable at time 1 only if the insured survives, so

E_0=50+0.80\frac{10}{11}(10)=\frac{630}{11}=57.2727.

Thus

P^m=\frac{347.1074+57.2727}{1.7273}=234.1148.

At duration 1, the remaining death benefit has conditional expected present value $1{,}000(0.25)(10/11)=227.2727$. The renewal expense of $10 is due immediately, as is the valuation premium. Therefore

$$ V_1^m=227.2727+10-234.1148=3.1579. $$

Rounded to the nearest cent, $P^m=\$234.11$ and $V_1^m=\$3.16$. As a check on the issue-date selection, the reference premium is $347.1074/1.7273=200.9569$, and the premium difference is $57.2727/1.7273=33.1579$. The reserve difference at duration 1 is $10-33.1579=-23.1579$, consistent with the modified reserve being below the reference reserve.

Common Mistakes and Interpretation

Using the contract premium without checking the rule. The contract premium determines actual policy cash flow. A valuation premium is the amount specified by the valuation basis or selection rule; the two need not agree.
Forgetting survival weighting. A renewal expense payable only while the policy is in force must be weighted by the probability of reaching its payment time. In the first example, $E_0=60+10=70$ incorrectly treats the renewal expense as certain; the correct value is $60+0.90(10)=69$.
Mixing premium timing conventions. The examples value reserves immediately before a premium and expense due at that duration. A reserve immediately after those cash flows would omit them. Either convention can be used if applied consistently.
Assuming reserve direction from premium direction alone. Although the expense-inclusive rule gives $P^m\geq P^r$ for nonnegative expenses, the modified reserve also includes future expenses. Compare both effects at the duration in question.

Key takeaway: A modified valuation premium follows from a stated selection rule, not from the contract premium alone. Calculate its benefit, premium, and expense values on consistent timing and survival assumptions; then compare reserves using both the future expenses and the premium difference.

Check Your Understanding

For each problem, use the stated equivalence rule and reserve timing. Round dollar amounts to the nearest cent.

A two-year policy pays $800 at the end of the year of death. Let $q_1=0.20$, $q_2=0.10$, and $i=0$. Expenses are $30 at issue and $20 at the start of year 2 if in force. Find the modified valuation premium. (Answer: $150.00)
A two-year policy pays $1,000 at the end of the year of death, with $q_1=0.10$, $q_2=0.20$, and $i=0$. There is no issue expense and the renewal expense is $60. Find the modified valuation premium and the amount by which the duration-1 modified reserve exceeds the reference reserve. (Answer: Modified premium $175.79; reserve excess $31.58)
A two-year policy has $q_1=0.10$, $q_2=0.20$, a $1,000 death benefit, and $i=0.05$. The contract premium is $200; the issue expense is 20% of it and the renewal expense is 5% of it. Find the modified valuation premium. (Answer: $165.35)
For a two-year policy with $q_1=0.10$, $q_2=0.20$, $i=0$, and a $1,000 death benefit, expenses are $20 at issue and $30 at the start of year 2 if in force. Find the duration-1 modified reserve immediately before the renewal premium and expense. (Answer: $57.89)
For a two-year policy with $q_1=0.10$, $q_2=0.20$, $i=0$, and a $1,000 death benefit, suppose the only expense is a renewal expense of $38 at the start of year 2 if in force. By how much does the duration-1 modified reserve exceed the reference reserve? (Answer: $20.00)

What You'll Learn