Reserve Differences From Expense Basis Changes

Why Separate Expense Changes?

A modified reserve basis may change the expenses allowed for in valuation while leaving the premium, benefit, interest, and mortality assumptions unchanged. Isolating that effect makes the reserve difference easier to calculate and interpret: an increase in future expenses increases the prospective liability, while a decrease reduces it. The timing of an expense matters just as much as its amount.

This tutorial considers a discrete annual model. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The focus is the reserve difference caused solely by changed valuation expenses, not a simultaneous change in premiums or benefits.

Definition: Let $V_k^{(A)}$ be the prospective reserve at duration $k$ on a reference basis and $V_k^{(B)}$ the reserve on a modified basis. Define the reserve difference by $D_k=V_k^{(B)}-V_k^{(A)}$. An expense difference is $\Delta e=e^{(B)}-e^{(A)}$. Thus a positive $\Delta e$ means the modified basis assumes a larger expense.

Assume the two bases use the same interest rate, mortality assumptions, future premiums, and benefits. Expenses are paid at the end of a policy year only if the insured is alive then. An acquisition expense, if present, is paid immediately at issue. Under these assumptions, future premiums and benefits cancel when the two prospective reserves are subtracted; only the expense cash flows remain. If the bases also differ in benefits, premiums, or interest, this expense-only calculation does not give the total reserve difference.

Prospective Formula for an Expense Difference

Suppose the policy is issued at age $x$, has a finite term of $n$ years, and the modified basis changes the expense paid at the end of year $j$ by $\Delta e_j$. The expense is paid only if the policy is in force at that time. Write $v=1/(1+i)$, where $i$ is the effective annual interest rate, and let ${}_jp_x$ be the probability of surviving $j$ years from age $x$.

D_0=\Delta e_0+\sum_{j=1}^{n}v^j\,{}_jp_x\,\Delta e_j

Here $\Delta e_0$ is the difference in acquisition expense paid at issue. It is not multiplied by a discount or survival factor because it is paid immediately. At duration $k$, conditional on the policy being in force at that duration, the future expense difference is

D_k=\sum_{j=1}^{n-k}v^j\,{}_jp_{x+k}\,\Delta e_{k+j},\qquad k\geq 1.

This formula includes only expenses after duration $k$. An acquisition expense already paid at issue is not a future cash flow at a later valuation date. The formulas also rely on the stated end-of-year payment condition: if an expense is paid at the start of a year, on death, or only when a claim occurs, its timing and probability weight must be changed accordingly.

Why the Formula Holds

A prospective reserve is the expected present value of future outgo less the expected present value of future income. Consider the difference between the two bases at issue. The future benefit and premium cash flows are identical, so their differences are zero. At the end of year $j$, the expense difference $\Delta e_j$ is payable only if the insured survives to that time. Its expected present value is therefore the amount $\Delta e_j$, multiplied by the probability ${}_jp_x$ of being alive then and the discount factor $v^j$. Summing those expected present values and adding the immediate acquisition expense difference gives the issue formula.

\text{Expected present value of the year-}j\text{ expense difference} =v^j\,{}_jp_x\,\Delta e_j

At duration $k$, the valuation is conditional on the policy having reached that duration. The relevant probability is consequently ${}_jp_{x+k}$, the probability of surviving a further $j$ years from age $x+k$, not ${}_{k+j}p_x$. This conditional viewpoint is why the later-duration formula starts at the current duration and omits all expenses already incurred.

A One-Year Check

The prospective sum can be checked using a one-year recursion. Let $p_{x+k}$ and $q_{x+k}$ be the one-year survival and death probabilities at duration $k$, with $p_{x+k}+q_{x+k}=1$. Under the end-of-year-survival expense convention, there is no expense difference if death occurs during the year, and the end-of-year expense difference is paid if the policy survives.

D_k=v\,p_{x+k}\bigl(\Delta e_{k+1}+D_{k+1}\bigr)

The recursion discounts one year and weights the future amount by the probability of surviving that year. It is a useful check on the prospective calculation. It would need an additional death-state term if the two bases also differed in an expense payable on death.

Worked Example: A Rising Maintenance Expense Schedule

A hypothetical four-year contract has effective annual interest $i=10\%$ and a constant annual mortality probability $q=0.10$, so $p=0.90$. The modified basis has an acquisition expense at issue that is $10 higher than the reference basis. Its end-of-year maintenance expense differences for years 1 through 4 are $2, $3, $4, and $5, respectively. Benefits and premiums are unchanged. Find the reserve differences at issue and at durations 1 and 2.

Method. Discount each future expense difference and weight it by the probability of surviving to that payment time. First calculate the one-year discounted survival factor:

r=vp=\frac{0.90}{1.10}=0.8181818\ldots

At issue, the four future expense contributions are:

\begin{aligned} \text{Year 1: }&2r=2(0.8181818\ldots)=1.6363636\ldots,\\ \text{Year 2: }&3r^2=3(0.6694215\ldots)=2.0082645\ldots,\\ \text{Year 3: }&4r^3=4(0.5477085\ldots)=2.1908340\ldots,\\ \text{Year 4: }&5r^4=5(0.4481251\ldots)=2.2406256\ldots. \end{aligned}

The maintenance contribution is $1.6363636+2.0082645+2.1908340+2.2406256=8.0760877$, using unrounded values in the sum. Add the $10 acquisition expense difference:

D_0=10+8.0760877\ldots=18.0760877\ldots

At duration 1, the acquisition expense and the year-1 maintenance expense have already occurred. The remaining expense differences are $3, $4, and $5:

\begin{aligned} D_1 &=3r+4r^2+5r^3\\ &=2.4545455\ldots+2.6776860\ldots+2.7385425\ldots\\ &=7.8707739\ldots. \end{aligned}

At duration 2, only the $4 and $5 differences remain:

D_2=4r+5r^2=3.2727273\ldots+3.3471074\ldots=6.6198347\ldots.

Check. The recursion gives $D_0=10+r(2+D_1)=10+0.8181818\ldots(2+7.8707739\ldots)=18.0760877\ldots$. It also gives $D_1=r(3+D_2)=0.8181818\ldots(3+6.6198347\ldots)=7.8707739\ldots$. Both checks agree. Rounded to the nearest cent, the three reserve differences are $D_0=\$18.08$, $D_1=\$7.87$, and $D_2=\$6.62$.

Common mistake avoided: At duration 1, the initial expense and the year-1 maintenance expense must not be included again. Also, the year-4 contribution at issue uses $r^4=0.4481251\ldots$, not $0.4480855$; retaining sufficient precision prevents a small arithmetic error from entering the total.

Worked Example: A Lower Acquisition Expense but Higher Renewal Expenses

For a hypothetical three-year contract, let $i=25\%$ and let the annual survival probability be $p=0.80$ in each year. The modified basis has an acquisition expense $8 lower than the reference basis, but its end-of-year maintenance expense is $10 higher in each of years 1, 2, and 3. Find the reserve difference at issue and at duration 1.

Method. Use signed expense differences: $\Delta e_0=-8$, and $\Delta e_j=10$ for $j=1,2,3$. The discounted survival factor is

r=vp=\frac{0.80}{1.25}=0.64.

At issue, the three maintenance contributions are $10(0.64)=6.40$, $10(0.64^2)=10(0.4096)=4.096$, and $10(0.64^3)=10(0.262144)=2.62144$. Their total is $13.11744$. Including the lower acquisition expense gives

$$ D_0=-8+13.11744=5.11744. $$

At duration 1, only two maintenance expenses remain. Thus $D_1=10(0.64+0.4096)=10(1.0496)=10.496$. Rounded to the nearest cent, $D_0=\$5.12$ and $D_1=\$10.50$.

Check. The recursion gives $D_0=-8+0.64(10+D_1)=-8+0.64(20.496)=5.11744$. The negative acquisition difference is included only at issue; it is not carried forward as a future cash flow. The positive issue result shows that the higher expected value of future maintenance expenses more than offsets the acquisition expense reduction.

Common mistake avoided: Taking the absolute values of the expense changes would wrongly add $8 at issue. Expense differences must retain their signs.

Worked Example: Nonconstant Survival Probabilities

A hypothetical three-year contract has $i=25\%$. Its conditional survival probabilities are $p_x=0.80$, $p_{x+1}=0.75$, and $p_{x+2}=0.60$. There is no acquisition expense difference. The modified basis has end-of-year maintenance expenses higher by $4, $6, and $8 in years 1, 2, and 3. Calculate the reserve differences at issue and duration 1.

Method. Here $v=1/1.25=0.80$. At issue, the probability of surviving to the end of year 1 is $0.80$; to the end of year 2 it is $0.80(0.75)=0.60$; and to the end of year 3 it is $0.80(0.75)(0.60)=0.36$. The discounted contributions are:

\begin{aligned} \text{Year 1: }&4(0.80)(0.80)=2.56,\\ \text{Year 2: }&6(0.80^2)(0.60)=2.304,\\ \text{Year 3: }&8(0.80^3)(0.36)=1.47456. \end{aligned}

Therefore $D_0=2.56+2.304+1.47456=6.33856$. At duration 1, use conditional survival probabilities from age $x+1$: $0.75$ to the end of year 2 and $0.75(0.60)=0.45$ to the end of year 3. Hence

$$ D_1=6(0.80)(0.75)+8(0.80^2)(0.75)(0.60)=3.60+2.304=5.904. $$

Rounded to the nearest cent, $D_0=\$6.34$ and $D_1=\$5.90$. As a check, $D_0=vp_x(4+D_1)=0.80(0.80)(4+5.904)=6.33856$. The survival probability at duration 1 is conditional on reaching that duration; using issue-to-year-2 survival in the duration-1 calculation would apply an extra year of survival.

Common Exam Traps and Interpretation

Including a sunk expense in a later reserve. An acquisition expense paid at issue contributes to $D_0$, but not to $D_k$ for $k\geq1$.
Using the wrong survival probability. At duration $k$, use survival from the current duration to the future expense date. Do not apply an unconditional issue-to-date probability to a reserve already conditional on the policy being in force.
Discounting without weighting for survival. An expense payable only if the policy survives must be multiplied by the appropriate survival probability as well as discounted.
Ignoring signs. A lower expense on the modified basis gives a negative expense difference and reduces the reserve difference.
Changing the timing convention silently. An expense at the beginning of a year does not have the same discount and survival factors as one paid at the end of that year.
Calling the expense-only difference the total reserve change when other assumptions differ. If premiums, benefits, interest, or mortality also change, their effects must be included separately.

Key takeaway: With unchanged premiums, benefits, interest, and mortality, the reserve difference is the prospective value of the expense differences. Include an acquisition difference only when it is still a future payment, and weight each future expense by its correct timing and conditional probability of payment.

Check Your Understanding

For each problem, define the reserve difference as modified basis minus reference basis. Assume all stated future expenses are paid at the end of the year only if the insured survives to that time.

A two-year contract has $i=10\%$ and annual survival probability $p=0.90$. There is no acquisition expense difference, and the modified basis has maintenance expenses $7 higher at the end of each year. Find the issue reserve difference. (Answer: $10.41)
A contract has an acquisition expense $15 higher on the modified basis, with no difference in future expenses. Find the reserve differences at issue and at duration 1. (Answer: $D_0=\$15.00$, $D_1=\$0.00$)
A two-year contract has $i=25\%$, $p_x=0.80$, and $p_{x+1}=0.90$. The acquisition expense difference is $-\$2$; maintenance expense differences are $3 in year 1 and $5 in year 2. Find the issue reserve difference. (Answer: $2.22)
A three-year contract has $i=0\%$ and survival probability $0.80$ each year. The acquisition expense difference is $5, and the maintenance expense differences for years 1, 2, and 3 are $4, $6, and $8. Find reserve differences at issue and duration 1. (Answer: $D_0=\$16.14$, $D_1=\$9.92$)
A two-year contract has $i=25\%$, $p_x=0.90$, and $p_{x+1}=0.80$. The acquisition expense difference is $4, and the same positive maintenance expense difference $c$ applies at the end of both years. If $c=\$5$, find the issue reserve difference. (Answer: $9.90)

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