Sensitivity Analysis for Reserve Reporting

Why Test a Selected Liability?

A selected reserve is a point estimate based on data, methods, and assumptions. Reporting only that point can conceal how much the estimate depends on a particular assumption. Sensitivity analysis makes that dependence visible: it recalculates the same selected liability under a plausible alternative assumption while holding the other inputs fixed. This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

The purpose is not to replace the selection with a second point or to imply that every value between two results is equally likely. It is to quantify the effect of a specified change, explain the comparison, and identify its limits. A useful result states the baseline, the changed assumption, the recalculated liability, and the impact in both dollars and relative terms.

Definition: Let $L_0$ be a selected liability calculated using baseline assumptions. A sensitivity result is a recalculated liability $L_1$ obtained by changing a stated assumption, or a stated set of assumptions, while keeping the remaining inputs and calculation method fixed. The absolute impact is $L_1-L_0$; the percentage impact is $(L_1-L_0)/L_0$, provided $L_0\ne 0$.

A positive impact means the alternative assumption produces a higher liability than the baseline; a negative impact means it produces a lower one. Always use the same baseline in the numerator and denominator of the percentage comparison. Report the units and rounding convention so a reader can distinguish, for example, dollars from thousands of dollars.

How the Assumption Enters the Calculation

A sensitivity is meaningful only if the changed input is connected clearly to the selected liability. In general, write the calculation as $L=g(\theta;D)$, where $D$ denotes the data and other fixed inputs and $\theta$ is the assumption being tested. The baseline is $L_0=g(\theta_0;D)$; the alternative is $L_1=g(\theta_1;D)$. The formula $g$ may be linear or nonlinear, so the change in the assumption need not produce a proportional change in liability.

For example, if a selected liability is the present value of fixed future payments, changing the effective annual interest rate changes each discounted payment. Assume payments $C_t$ are deterministic, occur at the end of year $t$, and are discounted at a constant effective annual rate $i>-1$. Then:

R(i)=\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}

The difference between two rates follows directly by subtracting the two present-value calculations:

R(i_1)-R(i_0) =\sum_{t=1}^{n} C_t\left[\frac{1}{(1+i_1)^t}-\frac{1}{(1+i_0)^t}\right]

If all $C_t$ are positive and $i_1>i_0>-1$, then $(1+i_1)^{-t}<(1+i_0)^{-t}$ for every $t\ge 1$. Each term in the difference is negative, so $R(i_1)<R(i_0)$. This establishes the direction of the impact under these assumptions; it does not establish how plausible either rate is.

Differentiating the same expression also shows the local direction and size of the response:

\frac{dR}{di} =-\sum_{t=1}^{n}\frac{tC_t}{(1+i)^{t+1}}

With positive payments, the derivative is negative. Later payments have a larger factor $t$, so they contribute more to the local sensitivity to the discount rate. This derivative is a local measure; for a finite change in rate, calculate both present values rather than treating the derivative as the exact change.

Worked Applications

Worked Example: Discount-Rate Sensitivity

A hypothetical insurer reports a selected liability for three deterministic year-end payments: $1,000 thousand at the end of year 1, $2,000 thousand at the end of year 2, and $3,000 thousand at the end of year 3. The baseline effective annual interest rate is 4%; test 3% and 5%. Assume no mortality model or probability distribution is used, and no other cash-flow or timing assumption changes. Calculate the baseline and impacts. Report amounts to the nearest $1 thousand and percentages to three decimal places.

Method. Discount each payment at its stated year and sum. First calculate the baseline:

\begin{aligned} R(0.04) &=\frac{1{,}000}{1.04}+\frac{2{,}000}{1.04^2}+\frac{3{,}000}{1.04^3}\\ &=961.538462+1{,}849.112426+2{,}666.989174\\ &=5{,}477.640062\text{ thousand} \end{aligned}

The 3% alternative is:

\begin{aligned} R(0.03) &=\frac{1{,}000}{1.03}+\frac{2{,}000}{1.03^2}+\frac{3{,}000}{1.03^3}\\ &=970.873786+1{,}885.191818+2{,}745.424977\\ &=5{,}601.490581\text{ thousand} \end{aligned}

Its absolute impact is $5{,}601.490581-5{,}477.640062=123.850519$ thousand. Its percentage impact is $123.850519/5{,}477.640062=0.022612$, or 2.261% to three decimal places.

At 5%, the calculation is:

\begin{aligned} R(0.05) &=\frac{1{,}000}{1.05}+\frac{2{,}000}{1.05^2}+\frac{3{,}000}{1.05^3}\\ &=952.380952+1{,}814.058957+2{,}591.512794\\ &=5{,}357.952703\text{ thousand} \end{aligned}

The impact is $5{,}357.952703-5{,}477.640062=-119.687359$ thousand, or $-119.687359/5{,}477.640062=-0.021854$, a decrease of 2.185% to three decimal places. Rounded as requested, the baseline is $5,478 thousand; the 3% result is $5,601 thousand, up $124 thousand or 2.261%; and the 5% result is $5,358 thousand, down $120 thousand or 2.185%. The exact calculations confirm that the lower rate increases the present value and the higher rate decreases it.

Common mistake avoided. Using $3{,}000/1.04^3=2{,}667.011$ thousand gives an incorrect baseline. The correct denominator is $1.04^3=1.124864$, so the discounted amount is $3{,}000/1.124864=2{,}666.989174$ thousand. The corrected baseline is 5,477.640 thousand; comparisons must use that same baseline.

Worked Example: Sensitivity to Future Severity Inflation

A hypothetical liability consists of expected payments stated in current cost levels: $2.000 million at the end of year 1 and $3.000 million at the end of year 2. The baseline annual severity inflation assumption is 4%; test 6%. Apply the annual trend once to the year-1 payment and twice to the year-2 payment. Do not discount the payments. No mortality model or probability distribution is used. Calculate the liability and its impact, reporting amounts to the nearest $1,000 and the percentage to three decimal places.

Method. With no discounting, apply the assumed inflation factor to each payment according to its projection year. At 4%:

\begin{aligned} L_0 &=2.000(1.04)+3.000(1.04)^2\\ &=2.080+3.2448\\ &=5.3248\text{ million} \end{aligned}

At 6%:

\begin{aligned} L_1 &=2.000(1.06)+3.000(1.06)^2\\ &=2.120+3.3708\\ &=5.4908\text{ million} \end{aligned}

The liability increases by $5.4908-5.3248=0.1660$ million, or $166,000. The relative increase is $0.1660/5.3248=0.031174$, or 3.117%. Rounded as specified, the baseline is $5,325,000 and the alternative is $5,491,000, with an increase of $166,000, or 3.117%. The calculation checks because both future payments rise under the higher inflation assumption, and the year-2 payment receives two years of trend.

Common mistake avoided. Applying only one year of inflation to both payments would use $3.000(1.06)$ for the year-2 amount. That contradicts the stated timing convention: the year-2 payment is projected two years and must use $(1.06)^2$.

Worked Example: Sensitivity to Open-Claim Severity

A hypothetical book has 80 open claims. The selected unpaid indemnity is based on an average expected indemnity of $30,000 per claim, and allocated claim expense is $5,000 per claim. Test a 10% increase in expected indemnity per claim while holding the number of claims and expense per claim fixed. There is no discounting, mortality assumption, or probability distribution in this calculation. Report dollar amounts to the nearest dollar and the percentage to three decimal places.

Method. The baseline selected liability is the claim count multiplied by expected indemnity plus expense per claim:

\begin{aligned} L_0 &=80(30{,}000+5{,}000)\\ &=80(35{,}000)\\ &=\$2{,}800{,}000 \end{aligned}

A 10% increase changes expected indemnity per claim to $30{,}000(1.10)=\$33{,}000$. Expense remains $5,000 per claim. Therefore:

\begin{aligned} L_1 &=80(33{,}000+5{,}000)\\ &=80(38{,}000)\\ &=\$3{,}040{,}000 \end{aligned}

The dollar impact is $\$3{,}040{,}000-\$2{,}800{,}000=\$240{,}000$. The percentage impact is $\$240{,}000/\$2{,}800{,}000=0.085714$, or 8.571%. The result checks another way: the indemnity increase is $3,000 per claim, so $80(\$3,000)=\$240,000$; expenses do not change.

Common mistake avoided. Increasing the full liability by 10% would produce $3,080,000, but the stated assumption changes indemnity only. Because baseline indemnity is $2,400,000, its 10% increase is $240,000; the fixed $400,000 expense is not inflated in this test.

Interpreting the Result and Its Limits

A sensitivity result answers a conditional question: if the stated alternative assumption were used, with the specified inputs and method otherwise unchanged, what liability would result? It does not establish that the alternative is more likely, that the baseline is wrong, or that the actual unpaid amount must lie between the two estimates.

If several plausible assumptions are tested separately, the lowest and highest calculated results can be reported as an assumption-based range. That range summarizes the alternatives actually tested; it is not automatically a statistical confidence interval, a probability distribution, or a guaranteed bound. It does not account for untested assumptions, data limitations, or changes in the method. Nor does it imply that all values within the range are equally plausible.

Sensitivity also depends on what is held fixed. In the open-claim example, the claim count and expense per claim were fixed deliberately to isolate indemnity severity. If those inputs also changed, the result would answer a different question. State the fixed items and the changed item so readers can interpret the impact correctly.

Present both absolute and relative impacts when useful. A $100,000 change may be material for a small liability but modest for a large one; a percentage alone can hide the dollar amount relevant to financial reporting. Conversely, a large dollar impact may be a small share of a large portfolio liability. Always identify whether values are gross, ceded, or net, and compare like with like.

State the baseline.
Name the selected liability, its units, the method, and the assumptions used for the comparison.

Specify the perturbation.
Identify the assumption changed and its alternative value; say explicitly what is held fixed.

Recalculate and compare.
Show the alternative liability, the signed dollar difference, and the percentage difference using the baseline as denominator.

Qualify the conclusion.
Explain what the result demonstrates and what it does not establish about likelihood, adequacy, or possible outcomes.

Key takeaway: Sensitivity analysis measures how a selected liability responds to explicitly stated assumption changes. The result is conditional on the chosen data, method, and fixed inputs; it is evidence about dependence on assumptions, not by itself a probability statement or a bound on the eventual liability.

Check Your Understanding

For each problem, identify the stated change and compare the alternative with its baseline.

A hypothetical reserve comprises payments of $800 thousand at year 1 and $1,200 thousand at year 2. Discount at an effective annual rate of 5%, then recalculate at 6%. Assume deterministic payments and no mortality model or probability distribution. What is the alternative less the baseline, and what is the percentage impact? Round amounts to the nearest $1,000 and the percentage to three decimals. (Answer: Baseline $1,850,000; alternative $1,823,000; decrease $28,000, or 1.493%.)
A hypothetical portfolio has 50 open claims with expected indemnity of $24,000 and expense of $3,000 per claim. Indemnity severity increases by 6%, while claim count and expense remain fixed. With no discounting, mortality model, or probability distribution, calculate the alternative liability and percentage increase. (Answer: Baseline $1,350,000; alternative $1,422,000; increase $72,000, or 5.333%.)
Two deterministic future payments, stated at current cost levels, are $1.500 million at year 1 and $2.500 million at year 2. With no discounting, compare annual inflation assumptions of 2% and 4%, applying trend for each payment's full number of years. No mortality model or probability distribution is used. What is the dollar and percentage impact? (Answer: Baseline $4.131 million; alternative $4.264 million; increase $133,000, or 3.220%.)
A selected liability is $3.200 million, made up of $2.400 million of indemnity and $800,000 of fixed expense. Test a 7% increase in indemnity only, with all else unchanged and no discounting, mortality model, or probability distribution. What is the alternative liability and percentage impact? (Answer: Alternative $3.368 million; increase $168,000, or 5.250%.)

What You'll Learn