Comparing Pricing Decisions Across Economic Scenarios

Pricing Across Economic Scenarios

A premium indication depends on assumptions about both the claims being funded and the money available to fund them. Higher claim frequency or severity inflation increases projected claims; a lower investment return reduces the value accumulated from premium income. When assumptions change, the indication and the risk attached to a proposed rate should be recalculated rather than carried forward unchanged.

This tutorial uses a one-year hypothetical pricing model. It compares indications under specified scenarios and then tests a proposed premium against expected-surplus and deficit-probability criteria. The scenarios are decision assumptions, not forecasts.

Model: Let $P$ be premium collected at the start of the year, $\alpha$ the fraction of premium consumed by variable expenses, $E$ fixed expenses paid at the start of the year, $i$ the effective annual investment return, and $L$ aggregate claims paid at year-end. Terminal surplus is $$ U_1=(1+i)\bigl((1-\alpha)P-E\bigr)-L. $$ This model assumes premium and fixed expenses are invested, net of variable expenses, for the full year; claims are paid at year-end; and there are no other cash flows. It does not represent within-year claim timing or multiple-year cash flows.

If baseline expected claims are $C_0$, frequency increases by $g_f$, and severity increases by $g_s$, the projected expected claims are $C=C_0(1+g_f)(1+g_s)$, provided the trends apply multiplicatively to the same baseline period. Investment return does not reduce the claim estimate; it changes the terminal value of the funds available to pay claims.

Indication for a Target Expected Surplus

Taking expectations in the terminal-surplus equation gives $E[U_1]=(1+i)((1-\alpha)P-E)-E[L]$. If the required expected terminal surplus is $T$, set $E[U_1]=T$ and solve for $P$:

P=\frac{E+\dfrac{E[L]+T}{1+i}}{1-\alpha}.

The order of operations follows the cash-flow timing. The claims and target surplus are year-end amounts, so they are discounted by $1+i$ to the start of the year. Fixed expenses are already paid at the start, so they are added without discounting. Finally, the amount needed before variable expenses is divided by $1-\alpha$.

Derivation: Starting from $E[U_1]=T$, substitute the model and rearrange: $$ (1+i)\bigl((1-\alpha)P-E\bigr)-E[L]=T, $$ $$ (1+i)(1-\alpha)P=(1+i)E+E[L]+T, $$ $$ P=\frac{E+\dfrac{E[L]+T}{1+i}}{1-\alpha}. $$ Thus, the formula is the premium that makes the model's expected terminal surplus equal to the stated target. It is not, by itself, a guarantee about the probability of a deficit.

Worked Example: Re-indicating After Frequency and Severity Trends

A hypothetical portfolio has baseline expected annual claims of $400,000. For the coming year, assume a 1% frequency trend and 2% severity inflation. The effective annual investment return is 2%, variable expenses are 10% of premium, fixed expenses are $15,000, and the target expected terminal surplus is $20,000. Calculate the indicated premium.

Method. Project expected claims first, then apply the one-year target-surplus formula.

The claim trends compound: $$ E[L]=400{,}000(1.01)(1.02) =400{,}000(1.0302) =412{,}080. $$

Substitute $E=15{,}000$, $T=20{,}000$, $i=0.02$, and $\alpha=0.10$: $$ P=\frac{15{,}000+\dfrac{412{,}080+20{,}000}{1.02}}{0.90}. $$

The year-end amount to fund is $412{,}080+20{,}000=432{,}080$. Its start-of-year value is $432{,}080/1.02=423{,}607.8431$. Adding fixed expenses gives $438{,}607.8431$, and dividing by $0.90$ gives $487{,}342.048$. Rounded to the nearest cent, the indicated premium is $487,342.05.

Check. Variable-expense-adjusted premium less fixed expenses is $0.90(487{,}342.048)-15{,}000=423{,}607.843$. Accumulating this at 2% gives $432,080, exactly the projected claims plus target surplus. The common mistake avoided here is reporting $479,210.32: it does not follow from the stated trend factors and target-surplus formula.

Comparing Scenario Risk Measures

An indication addresses an expected value. A decision-maker may also need to know how variable aggregate claims make a deficit possible. For a compound Poisson model, suppose claim count $N$ is Poisson with mean $\lambda$, and individual claim amounts $X_j$ are independent and identically distributed, independent of $N$, with finite second moment. Then $L=\sum_{j=1}^N X_j$ has

E[L]=\lambda E[X], \qquad \operatorname{Var}(L)=\lambda E[X^2].

The second formula uses the second raw moment, not just the severity variance. It follows from conditioning on $N$: $E[L\mid N]=NE[X]$ and $\operatorname{Var}(L\mid N)=N\operatorname{Var}(X)$. The law of total variance gives $$ \operatorname{Var}(L)=E[N]\operatorname{Var}(X)+\operatorname{Var}(N)(E[X])^2 =\lambda\bigl(\operatorname{Var}(X)+(E[X])^2\bigr) =\lambda E[X^2]. $$

For an approximate decision measure, one can approximate $L$ by a normal random variable with the same mean and variance. Then $U_1$ is also approximated as normal, with mean $E[U_1]$ and standard deviation $\sqrt{\operatorname{Var}(L)}$. This is an approximation, not an exact distributional result; it should be identified as such, especially when claim counts are small or the aggregate distribution is strongly skewed.

Worked Example: Testing One Premium Across Three Scenarios

A hypothetical portfolio has baseline claim count $N\sim\operatorname{Poisson}(1000)$ and exponential claim severity with mean $400. In each scenario, frequency and severity trends multiply the baseline count mean and severity mean, respectively; the severity distribution remains exponential after scaling. A proposed premium is $500,000, variable expenses are 10%, and fixed expenses are $15,000. The scenarios are: A, $i=2\%$, frequency trend 1%, severity inflation 2%; B, $i=0\%$, frequency trend 2%, severity inflation 4%; and C, $i=4\%$, frequency trend 0%, severity inflation 1%. For each scenario, calculate expected terminal surplus and the normal-approximation probability of a deficit. Treat the decision as acceptable only if expected surplus is at least $20,000 and the deficit probability is at most 5%.

Method. For an exponential severity with mean $m$, $E[X^2]=2m^2$. Under trend multipliers $1+g_f$ and $1+g_s$, the scenario parameters are $\lambda=1000(1+g_f)$ and $m=400(1+g_s)$. Thus, $E[L]=\lambda m$ and $\operatorname{Var}(L)=2\lambda m^2$. For the proposed premium, the amount accumulated before claims is $1.02(0.90(500{,}000)-15{,}000)=443{,}700$ in A; it is $435,000 in B and $1.04(435{,}000)=452{,}400$ in C.

Scenario A. Here $\lambda=1010$, $m=408$, and $E[L]=1010(408)=412{,}080$. The aggregate variance is $2(1010)(408^2)=336{,}257{,}280$, so the standard deviation is approximately $18,337.32. Expected surplus is $443{,}700-412{,}080=$31,620$. Under the normal approximation, the standardized surplus is $31{,}620/18{,}337.32=1.724$, giving deficit probability $\Phi(-1.724)\approx0.0423$.

Scenario B. Here $\lambda=1020$, $m=416$, and $E[L]=1020(416)=424{,}320$. The aggregate variance is $2(1020)(416^2)=353{,}034{,}240$, so the standard deviation is approximately $18,789.21. Expected surplus is $435{,}000-424{,}320=$10,680$. The standardized surplus is $10{,}680/18{,}789.21=0.5684$, giving deficit probability $\Phi(-0.5684)\approx0.2850$.

Scenario C. Here $\lambda=1000$, $m=404$, and $E[L]=1000(404)=404{,}000$. The aggregate variance is $2(1000)(404^2)=326{,}432{,}000$, so the standard deviation is approximately $18,067.43. Expected surplus is $452{,}400-404{,}000=$48,400$. The standardized surplus is $48{,}400/18{,}067.43=2.678$, giving deficit probability $\Phi(-2.678)\approx0.0037$.

Decision and check. A meets both criteria: $31,620 exceeds $20,000 and 0.0423 is below 0.05. B meets neither: $10,680 is below the expected-surplus target and 0.2850 exceeds the probability limit. C meets both. Therefore, the $500,000 decision is not acceptable across all three scenarios. The common mistake avoided is using the investment return from A or C when evaluating B; each scenario must use its own return in the accumulation calculation.

Finding a Premium That Works in Every Scenario

When the same premium must meet requirements under several specified scenarios, calculate the required premium separately in each scenario and use the largest requirement. For an expected-surplus target $T$, the scenario-specific indication is $$ P_{\mathrm{expected}}= \frac{E+\dfrac{E[L]+T}{1+i}}{1-\alpha}. $$ If the additional requirement is a normal-approximation deficit probability no greater than $\varepsilon$, then $E[U_1]/\sqrt{\operatorname{Var}(L)}$ must be at least $z_{1-\varepsilon}$, where $z_{1-\varepsilon}$ is the corresponding standard normal quantile. Equivalently, the mean terminal surplus must be at least $z_{1-\varepsilon}\sqrt{\operatorname{Var}(L)}$. This yields $$ P_{\mathrm{risk}}= \frac{E+\dfrac{E[L]+z_{1-\varepsilon}\sqrt{\operatorname{Var}(L)}}{1+i}}{1-\alpha}. $$

Both conditions apply, so the scenario-specific premium must be at least the larger of these two indications. To work across scenarios, take the largest such premium across all scenarios. This conclusion holds only for the stated model and criteria; it does not remove model risk or make the normal approximation exact.

Worked Example: Setting a Premium for the Adverse Scenario

Use Scenario B from the previous example. Require expected terminal surplus of at least $20,000 and a normal-approximation deficit probability no greater than 5%. The scenario has $i=0$, $\alpha=0.10$, $E=15{,}000$, $E[L]=424{,}320$, and $\sqrt{\operatorname{Var}(L)}\approx18{,}789.21$. Find the minimum premium satisfying both requirements.

Method. First find the expected-surplus requirement: $$ P_{\mathrm{expected}} =\frac{15{,}000+424{,}320+20{,}000}{0.90} =\frac{459{,}320}{0.90} =\$510{,}355.56. $$

For a 5% upper-tail deficit probability, use $z_{0.95}=1.645$. The required mean surplus for the approximate probability condition is $1.645(18{,}789.21)=\$30{,}908.24$. Consequently, $$ P_{\mathrm{risk}} =\frac{15{,}000+424{,}320+30{,}908.24}{0.90} =\$522{,}475.82. $$

The risk requirement is larger, so the minimum premium meeting both criteria is $522,475.82. At this premium, expected surplus is approximately $30,908.24, which exceeds $20,000, and the normal-approximation deficit probability is 5%. The calculation is checked by substituting the premium into $0.90P-15{,}000-424{,}320$, which gives approximately $30,908.24. This remains an approximate risk-based indication because the probability calculation uses a normal approximation.

Common Traps in Scenario Comparisons

A scenario comparison is only useful if each assumption is applied in the correct place. In particular, claim trends affect the projected claim distribution, investment return affects the accumulated funds, and expense assumptions affect the premium available for investment. Keeping these roles separate makes the calculations easier to audit.

Adding trend rates instead of compounding them. A 1% frequency trend and 2% severity trend multiply expected claims by $1.01(1.02)=1.0302$, not by $1.03$. The latter gives $412,000 on a $400,000 baseline, rather than the correct $412,080.
Applying investment return to claims as if it were a claim trend. In this model, claims are paid at year-end. The return accumulates premium net of expenses; it does not change $E[L]$.
Using the wrong expense timing. Fixed expenses are paid at the start of the year, while claims and target surplus are year-end amounts. Discounting fixed expenses in the indication formula would contradict the model.
Confusing expected surplus with risk acceptability. A premium may meet an expected-surplus target yet fail a deficit-probability limit, or the reverse. Check each stated criterion separately.
Treating an approximation as an exact probability. A normal approximation to aggregate claims should be labelled and judged in light of the model. Its result is not an exact compound Poisson tail probability.

Key takeaway: Recalculate claim expectations and risk measures under each scenario, use that scenario's investment return in the terminal-surplus model, and test every stated decision criterion. A premium is robust across the specified scenarios only if it meets the requirements in each one.

Check Your Understanding

Each problem is hypothetical. Give money answers to the nearest cent and probabilities to four decimal places.

A portfolio has baseline expected claims of $250,000, a 3% frequency trend, and 2% severity inflation. Calculate projected expected claims using multiplicative trends. (Answer: $262,650)
Projected expected claims are $262,650, fixed expenses are $12,000, variable expenses are 8% of premium, the effective annual investment return is 3%, and the target expected terminal surplus is $15,000. Find the indicated premium. (Answer: $306,046.86)
A proposed premium is $360,000, variable expenses are 10%, fixed expenses are $18,000, and $i=2\%$. If projected year-end claims are $300{,}000(1.02)(1.03)$, calculate expected terminal surplus. (Answer: $-$3,060$)
Suppose aggregate claims are approximately $N(180{,}000,24{,}000^2)$. A proposed premium of $250,000 has 8% variable expenses and $10,000 fixed expenses; $i=2\%$. Calculate expected terminal surplus and the approximate probability of a deficit. Does it meet a $20,000 expected-surplus minimum and a 5% deficit-probability maximum? (Answer: Expected surplus $44,400; deficit probability 0.0322; it meets both.)
Two scenarios share fixed expenses of $12,000, variable expenses of 10%, and a $18,000 target expected terminal surplus. Scenario A has expected claims of $400,000 and $i=3\%$; Scenario B has expected claims of $420,000 and $i=0\%$. Find the minimum common premium that meets the expected-surplus target in both scenarios. (Answer: $500,000)

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