Sensitivity to the Selected Development Factors

Why Development-Factor Sensitivity Matters

A selected age-to-ultimate factor determines how much of an accident period’s ultimate losses are treated as reported at the valuation date. Changing that factor therefore changes the estimated unreported share. Even when observed losses and exposure remain exactly the same, a different factor can change a reserve estimate.

The effect depends on the reserving method. In Bornhuetter-Ferguson (BF), if the expected ultimate loss assumption is held fixed, changing a factor changes the expected unreported portion directly. In Cape Cod, the factor also affects the experience-based loss ratio: reported losses are compared with the amount of exposure expected to have emerged by the valuation date. A changed factor can therefore affect more than one period’s estimate.

Definition: Let $f_i$ be the selected age-to-ultimate factor for accident period $i$, on the same paid or incurred basis as its observed losses. The corresponding reported proportion is $q_i=1/f_i$, and the unpaid proportion is $1-q_i=1-1/f_i$. This interpretation assumes $f_i$ is a positive cumulative factor from the valuation age to ultimate and that the selected development pattern applies to the loss basis being analyzed.

For example, if $f_i=1.25$, then $q_i=0.80$: the factor implies that 80% of ultimate losses are reported and 20% remain unreported. A larger factor implies a smaller reported proportion and a larger unpaid proportion. These are proportions of ultimate losses, not percentages of the observed amount.

Bornhuetter-Ferguson: The Direct Effect

Let $C_i$ be the observed losses for period $i$, and let $E_i$ be its expected ultimate losses. In BF, the expected unreported losses are $E_i(1-q_i)$. Thus the estimated reserve is that expected unreported amount, while the estimated ultimate is observed losses plus the reserve. When the expected ultimate is fixed, the development factor acts through $q_i$ alone.

\widehat{R}^{BF}_i=E_i\left(1-\frac{1}{f_i}\right), \qquad \widehat{U}^{BF}_i=C_i+E_i\left(1-\frac{1}{f_i}\right).

The observed amount $C_i$ does not change when only the selected factor changes. It follows that the change in the BF ultimate equals the change in the BF reserve. For old and new factors $f_{\text{old}}$ and $f_{\text{new}}$, respectively,

\Delta \widehat{R}^{BF}_i =E_i\left(\frac{1}{f_{\text{old}}}-\frac{1}{f_{\text{new}}}\right).

The formula also gives a useful direction check. If the factor increases, its reciprocal decreases, so the estimated unpaid proportion and reserve increase. If the factor decreases, the reserve decreases, all else equal.

Worked Example: BF Reserve Under Two Selected Factors

A hypothetical property portfolio has $600,000 of incurred losses reported for an accident period. Its expected ultimate losses are $1,000,000. Compare the BF reserve and ultimate using an age-to-ultimate factor of 1.25 with those using a factor of 1.20. Hold the observed losses and expected ultimate fixed. There is no discounting ($i=0.000$); mortality and a probability distribution are not modeled.

Method. Convert each factor to a reported proportion, then apply the BF unpaid share to the fixed expected ultimate.

With $f=1.25$, $q=1/1.25=0.80$, so the unpaid proportion is $1-0.80=0.20$. The reserve is $1{,}000{,}000(0.20)=\$200{,}000$, and the ultimate is $\$600{,}000+\$200{,}000=\$800{,}000$.

With $f=1.20$, $q=1/1.20=0.833333\ldots$, so the unpaid proportion is $1-0.833333\ldots=0.166666\ldots$. The reserve is $1{,}000{,}000(0.166666\ldots)=\$166{,}666.67$, and the ultimate is $\$600{,}000+\$166{,}666.67=\$766{,}666.67$.

The factor decreases by 0.05, and the reserve decreases by $\$200{,}000-\$166{,}666.67=\$33{,}333.33$. The corresponding change in ultimate is also a decrease of $33,333.33 because the observed losses are unchanged. Amounts are rounded to the nearest cent only at the end.

Check and common mistake. The smaller factor produces the larger reported proportion (83.33% rather than 80.00%) and therefore the smaller unpaid proportion. Using the old 20% unpaid share with the new factor would incorrectly leave the reserve at $200,000.

Why the BF Formula Uses the Unpaid Share

The BF reserve formula follows from the interpretation of the selected factor. If the expected ultimate is $E_i$, and the reported proportion is $q_i$, the expected reported portion of that ultimate is $E_iq_i$. The complementary portion is $E_i-E_iq_i=E_i(1-q_i)$. BF uses that expected complementary portion as the reserve, then adds the already observed losses to obtain the estimated ultimate. Substituting $q_i=1/f_i$ gives the displayed formula.

This reasoning relies on the specific assumption that $E_i$ is fixed while the factor changes. If the expected ultimate is recalculated as part of a separate assumption change, the resulting difference is not solely the factor effect considered here.

Cape Cod: Direct and Portfolio Effects

Cape Cod estimates an experience-based expected loss ratio using observed losses divided by exposure adjusted for the proportion expected to be reported. Let $P_i$ denote the exposure measure for period $i$, expressed on a consistent basis, and let $L_i$ denote its observed losses. The Cape Cod loss ratio is $\widehat{\lambda}_{CC}=\sum_i L_i/\sum_i P_iq_i$. The quantity $P_iq_i$ is the reported portion of exposure, so the denominator reflects the exposure associated with losses expected to have emerged by the valuation date.

\widehat{\lambda}_{CC} =\frac{\sum_i L_i}{\sum_i P_iq_i}, \qquad \widehat{R}^{CC}_i =P_i\widehat{\lambda}_{CC}(1-q_i), \qquad \widehat{U}^{CC}_i=L_i+\widehat{R}^{CC}_i.

When one reported proportion increases and all observed losses and exposures remain fixed, the denominator of the Cape Cod loss ratio increases. The estimated loss ratio consequently decreases. The reserve for that period also has a smaller unpaid proportion. Moreover, the changed loss ratio affects reserves for every period included in the same Cape Cod calculation, even if their own factors did not change.

Key distinction: In BF with fixed expected ultimates, changing a period’s factor changes that period’s unpaid share. In Cape Cod, changing a factor changes both that period’s unpaid share and the portfolio loss ratio used to estimate expected losses. Recalculate the ratio before calculating any Cape Cod reserve.

Worked Example: One Factor Change in a Cape Cod Estimate

A hypothetical portfolio has two accident periods. Exposures and observed losses are in thousands of dollars: $P_1=100$, $L_1=50$, $P_2=200$, and $L_2=120$. Initially, the reported proportions are $q_1=0.50$ and $q_2=0.80$. Recalculate the Cape Cod estimates when $q_1$ changes to 0.60 and all other inputs remain fixed. There is no discounting ($i=0.000$); mortality and a probability distribution are not modeled.

Method. For each set of reported proportions, recalculate the common Cape Cod loss ratio from total observed losses divided by exposure adjusted for reporting. Then calculate each period’s reserve and ultimate.

Initially, the adjusted exposure denominator is $100(0.50)+200(0.80)=50+160=210$. Total observed losses are $50+120=170$, so the loss ratio is $170/210=0.8095238$, or approximately 80.95%.

The initial period 1 reserve is $100(0.8095238)(1-0.50)=40.4762$ thousand dollars. Its ultimate is $50+40.4762=90.4762$ thousand dollars. The initial period 2 reserve is $200(0.8095238)(1-0.80)=32.3810$ thousand dollars. Its ultimate is $120+32.3810=152.3810$ thousand dollars. The total reserve is $40.4762+32.3810=72.8572$ thousand dollars.

After the change, the adjusted exposure denominator is $100(0.60)+200(0.80)=60+160=220$. The observed loss total remains 170, so the revised loss ratio is $170/220=0.7727273$, or approximately 77.27%.

The revised period 1 reserve is $100(0.7727273)(1-0.60)=30.9091$ thousand dollars. Its ultimate is $50+30.9091=80.9091$ thousand dollars. The revised period 2 reserve is $200(0.7727273)(1-0.80)=30.9091$ thousand dollars. Its ultimate is $120+30.9091=150.9091$ thousand dollars. The revised total reserve is $30.9091+30.9091=61.8182$ thousand dollars.

The total reserve decreases by $72.8572-61.8182=11.0390$ thousand dollars. Both periods’ reserves fall: period 1 has a smaller unpaid share, and the lower recalculated loss ratio reduces the estimated losses for both periods. The check is that total ultimate equals observed losses plus total reserve: initially $170+72.8572=242.8572$ thousand dollars, and after the change $170+61.8182=231.8182$ thousand dollars. Amounts are rounded to four decimal places in thousands only at the end.

Common mistake avoided. Keeping the original 80.95% loss ratio after changing $q_1$ would ignore the change to the adjusted exposure denominator. Cape Cod requires recalculating its loss ratio whenever a factor used in that denominator changes.

A Sensitivity Check for Cape Cod

The direction of the portfolio effect can be checked algebraically. Write $N=\sum_i L_i$ for total observed losses and $D=\sum_i P_iq_i$ for adjusted exposure. Then $\widehat{\lambda}_{CC}=N/D$. If $q_k$ increases while all other inputs are fixed, $D$ increases by $P_k$ times the increase in $q_k$, while $N$ stays fixed. For positive exposures and losses, the recalculated loss ratio decreases.

For a small change in $q_k$, the rate of change in the loss ratio is $\partial\widehat{\lambda}_{CC}/\partial q_k=-NP_k/D^2$, which is negative when $N$, $P_k$, and $D$ are positive. For period $i$, the reserve is $P_i\widehat{\lambda}_{CC}(1-q_i)$. A change in another period’s $q_k$ therefore changes period $i$’s reserve through $\widehat{\lambda}_{CC}$. For $i=k$, there is an additional direct effect because that period’s unpaid proportion also changes. This explains why a portfolio recalculation cannot be reduced to changing only the reserve for the period with the revised factor.

Worked Example: Separating the Cape Cod Effects

Use the same hypothetical two-period portfolio, with exposures of 100 and 200 thousand dollars and observed losses of 50 and 120 thousand dollars. Keep $q_2=0.80$, and change $q_1$ from 0.50 to 0.60. Quantify the effect on period 2’s reserve, even though period 2’s own reported proportion does not change. There is no discounting ($i=0.000$); mortality and a probability distribution are not modeled.

Method. Isolate period 2’s indirect change by comparing its reserve under the old and new Cape Cod loss ratios. Its exposure and unpaid proportion remain fixed.

The old adjusted exposure is $100(0.50)+200(0.80)=210$ thousand dollars, and the old loss ratio is $170/210=0.8095238$. Period 2’s old reserve is $200(0.8095238)(0.20)=32.3810$ thousand dollars.

The new adjusted exposure is $100(0.60)+200(0.80)=220$ thousand dollars, and the new loss ratio is $170/220=0.7727273$. Period 2’s new reserve is $200(0.7727273)(0.20)=30.9091$ thousand dollars. Its reserve falls by $32.3810-30.9091=1.4719$ thousand dollars, or approximately $1,471.90.

The period 2 change is entirely indirect: $q_2$ stays at 0.80, so its unpaid proportion remains 0.20, but the common loss ratio falls. The result agrees with the portfolio recalculation in the preceding example. Amounts are rounded to four decimal places in thousands only at the end.

Common mistake avoided. Holding period 2’s reserve unchanged because its own factor is unchanged misses the portfolio-wide loss-ratio effect.

Common Exam Traps

Reversing the factor direction. A larger age-to-ultimate factor means a smaller reported proportion, not a larger one. Since the unpaid share grows, a fixed-expected-ultimate BF reserve increases.
Applying the factor to observed losses as if they were the reserve. The BF reserve is $E_i(1-1/f_i)$, not $C_i(1-1/f_i)$. The expected ultimate and observed losses have different roles in the formula.
Updating only the affected Cape Cod period. A changed reported proportion changes the adjusted exposure denominator and therefore the common loss ratio. Recalculate all period reserves based on that ratio.
Mixing loss bases. The observed losses and development factors must be on the same basis. A paid factor should not be applied to incurred observations without an appropriate, consistent method.
Confusing the factor with a percentage. A factor of 1.25 corresponds to a reported proportion of 80%, not 125%.

Key takeaway: Convert each selected age-to-ultimate factor to its reported proportion before assessing sensitivity. Under BF, with expected ultimate fixed, the factor changes the unpaid share directly. Under Cape Cod, it also changes the experience-based loss ratio, so the entire portfolio calculation must be updated.

Check Your Understanding

For each problem, assume all losses and exposures are on consistent bases, there is no discounting, and only the stated development-factor input changes.

A hypothetical period has expected ultimate losses of $900,000 and a factor of 1.50. Calculate the BF reserve and ultimate if observed losses are $500,000. (Answer: Reserve $300,000; ultimate $800,000.)
Expected ultimate losses are $1,200,000. A BF factor changes from 2.00 to 1.60 while expected ultimate and observed losses remain fixed. What is the change in reserve? (Answer: Reserve decreases from $600,000 to $450,000, a decrease of $150,000.)
A BF estimate uses expected ultimate losses of $750,000. Its factor changes from 1.25 to 1.20. Find the old and new reported proportions and the reserve change. (Answer: Reported proportions increase from 0.8000 to 0.8333; reserve decreases from $150,000 to $125,000, a decrease of $25,000.)
A two-period Cape Cod estimate has exposures of 80 and 120 thousand dollars, observed losses of 32 and 72 thousand dollars, and reported proportions of 0.40 and 0.75. Calculate the Cape Cod loss ratio and the total reserve. (Answer: Loss ratio $104/122=0.8525$; total reserve approximately $66.49 thousand.)
For the portfolio in question 4, change only the first reported proportion from 0.40 to 0.50. Recalculate the loss ratio and total reserve, and state the reserve change. (Answer: Loss ratio $104/130=0.8000$; total reserve $56.00 thousand, a decrease of approximately $10.49 thousand.)

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