Sensitivity to the Expected Loss Ratio

Why Expected Loss Ratio Sensitivity Matters

A reserve estimate can change because the underlying claim experience changes, or because an assumption changes. Keeping those causes separate is essential when explaining a revision. In Bornhuetter-Ferguson (BF), the selected expected loss ratio directly determines the expected amount of loss not yet reported. In Cape Cod, the portfolio loss ratio is fitted from observed losses and exposure adjusted for maturity; it is not an independent prior selection that can be changed while all the data stay fixed.

This distinction gives two different sensitivity questions. For BF, ask how much the reserve changes when the selected expected loss ratio changes, with premiums and development assumptions held fixed. For Cape Cod, ask how a change in observed experience or in the exposure-and-maturity weights changes the fitted portfolio ratio and, in turn, the reserves. A different external BF selection alone does not change a Cape Cod estimate calculated from the same data.

Definition: For accident period $i$, let $EP_i$ be earned premium, $\lambda$ the selected expected loss ratio, and $q_i$ the estimated proportion of ultimate losses reported by the valuation date. Assume $0\leq q_i\leq 1$, and that the premium, reported losses, and $q_i$ refer to consistent bases. The BF expected unpaid loss is $EP_i\lambda(1-q_i)$. Under Cape Cod, the fitted portfolio loss ratio is $\hat{\lambda}_{CC}=\frac{\sum_i R_i}{\sum_i EP_iq_i}$, where $R_i$ is reported loss. The corresponding expected unpaid loss is $EP_i\hat{\lambda}_{CC}(1-q_i)$.

Here, “reserve” means the estimate of total unpaid loss under the stated method. In BF, if $R_i$ is reported loss, estimated ultimate loss is $R_i+EP_i\lambda(1-q_i)$. Consequently, changing $\lambda$ changes the BF reserve directly but does not change reported loss. These sensitivity results hold other inputs fixed; if premium, reported proportion, or the valuation basis also changes, that is a separate source of movement.

Bornhuetter-Ferguson: Direct Sensitivity

For a fixed accident period, the BF reserve is proportional to the selected expected loss ratio. If the selection changes from $\lambda_{\text{old}}$ to $\lambda_{\text{new}}$, the reserve change is the new reserve minus the old reserve:

\Delta \text{Reserve}_{BF,i} =EP_i(1-q_i)(\lambda_{\text{new}}-\lambda_{\text{old}}).

The factor $EP_i(1-q_i)$ is the premium-weighted unreported proportion. It converts a change in loss ratio into a change in expected unpaid loss. A five-percentage-point change is $0.05$, not $5$. The same calculation applies to multiple accident periods by summing their individual changes.

Worked Example: A Revised Selection for One BF Period

A hypothetical program has earned premium of $800,000. At the valuation date, 40% of ultimate losses are estimated to have been reported, and reported losses are $210,000. The selected expected loss ratio falls from 65% to 60%. Calculate the old and new BF reserves and ultimates, and verify the reserve change.

Method. The reported proportion is $q=0.40$, so the unreported proportion is $1-q=0.60$. The change in expected loss ratio is $0.60-0.65=-0.05$.

At the old selection, expected unpaid loss is $800{,}000(0.65)(0.60)=\$312{,}000$. Thus the old ultimate is $\$210{,}000+\$312{,}000=\$522{,}000$.

At the new selection, expected unpaid loss is $800{,}000(0.60)(0.60)=\$288{,}000$. The new ultimate is $\$210{,}000+\$288{,}000=\$498{,}000$.

The sensitivity formula gives $800{,}000(0.60)(-0.05)=-\$24{,}000$. Direct subtraction gives $\$288{,}000-\$312{,}000=-\$24{,}000$, so the reserve decreases by $24,000. The reported loss is unchanged. This method avoids applying the loss-ratio change to the entire premium when only the expected unpaid portion is being revised.

The reason for the formula is visible from the BF reserve itself. Holding premium and reported proportion fixed, the old and new reserves are $EP_i(1-q_i)\lambda_{\text{old}}$ and $EP_i(1-q_i)\lambda_{\text{new}}$. Subtracting the old from the new and factoring out the common terms gives the sensitivity expression. For multiple periods, addition is valid because total reserve is the sum of period reserves.

\begin{aligned} \Delta \text{Reserve}_{BF} &=\sum_i EP_i(1-q_i)\lambda_{\text{new}} -\sum_i EP_i(1-q_i)\lambda_{\text{old}}\\ &=\sum_i EP_i(1-q_i)(\lambda_{\text{new}}-\lambda_{\text{old}}). \end{aligned}

Worked Example: Aggregating BF Sensitivity Across Different Maturities

Two hypothetical accident periods have earned premiums and reported proportions of $1,000,000 and 30%, and $2,000,000 and 90%, respectively. The selected expected loss ratio decreases from 60% to 56%. Calculate the change in total BF reserve and verify it from the old and new reserves.

Method. The unreported proportions are $1-0.30=0.70$ and $1-0.90=0.10$. For the first period, the reserve change is $1{,}000{,}000(0.70)(-0.04)=-\$28{,}000$. For the second, it is $2{,}000{,}000(0.10)(-0.04)=-\$8{,}000$. The total change is $-\$28{,}000-\$8{,}000=-\$36{,}000$.

For a direct check, the old total reserve is $1{,}000{,}000(0.70)(0.60)+2{,}000{,}000(0.10)(0.60)=\$420{,}000+\$120{,}000=\$540{,}000$. The new total is $1{,}000{,}000(0.70)(0.56)+2{,}000{,}000(0.10)(0.56)=\$392{,}000+\$112{,}000=\$504{,}000$. The difference is $\$504{,}000-\$540{,}000=-\$36{,}000$, as calculated.

The less mature period contributes more to sensitivity because its unreported proportion is larger. This calculation avoids treating both periods as equally sensitive merely because they share the same selected loss-ratio change.

Cape Cod: Sensitivity Through the Fitted Ratio

Cape Cod estimates a portfolio loss ratio from the observed reported losses and the amount of premium expected to have generated reported losses by the valuation date. In the notation above, the denominator is $\sum_i EP_iq_i$. It is the maturity-adjusted premium base; it is not total premium and it is not the premium-weighted unreported amount. Once the fitted ratio is obtained, it is applied to each period’s premium to estimate ultimate losses, and the unreported proportion gives the reserve.

For fixed premiums and reported proportions, write $D=\sum_i EP_iq_i$ and $U=\sum_i EP_i(1-q_i)$. The total Cape Cod reserve is $U\hat{\lambda}_{CC}$. Thus, if the fitted portfolio ratio changes by $\Delta\hat{\lambda}_{CC}$, the total reserve changes by $U\Delta\hat{\lambda}_{CC}$. If reported losses change by $\Delta R$ while the denominator stays fixed, then $\Delta\hat{\lambda}_{CC}=\Delta R/D$. This identifies the route from experience movement to reserve movement.

\begin{aligned} \text{Reserve}_{CC} &=\left(\sum_i EP_i(1-q_i)\right) \frac{\sum_i R_i}{\sum_i EP_iq_i},\\ \Delta\text{Reserve}_{CC} &=U\Delta\hat{\lambda}_{CC} =\frac{U}{D}\Delta R \quad\text{when } EP_i \text{ and } q_i \text{ are fixed.} \end{aligned}

Worked Example: Updating Cape Cod After a Change in Observed Losses

A hypothetical portfolio has the following data, with amounts in thousands of dollars. Calculate the Cape Cod fitted loss ratio and total reserve. Then suppose the total reported losses increase by $47,000, with premiums and reported proportions unchanged. Recalculate the fitted ratio and reserve.

Period	Earned premium	Reported proportion	Reported loss
A	$1,000	0.50	$250
B	$2,000	0.80	$900
C	$1,000	0.25	$166

Method. The total reported losses are $250+900+166=1{,}316$. The maturity-adjusted premium denominator is $1{,}000(0.50)+2{,}000(0.80)+1{,}000(0.25)=500+1{,}600+250=2{,}350$. Therefore, $\hat{\lambda}_{CC}=1{,}316/2{,}350=0.56$, or 56%.

The reserve by period is $1{,}000(0.56)(1-0.50)=280$ for A, $2{,}000(0.56)(1-0.80)=224$ for B, and $1{,}000(0.56)(1-0.25)=420$ for C. The total is $280+224+420=924$, or $924,000.

After the reported-loss increase, total reported losses are $1{,}316+47=1{,}363$. The denominator remains 2,350, so the new fitted ratio is $1{,}363/2{,}350=0.58$, or 58%. The fitted ratio increases by $0.02$. The total premium-weighted unreported amount is $1{,}000(0.50)+2{,}000(0.20)+1{,}000(0.75)=500+400+750=1{,}650$. The reserve change is $1{,}650(0.02)=33$, or $33,000.

As a check, the revised period reserves are $1{,}000(0.58)(0.50)=290$, $2{,}000(0.58)(0.20)=232$, and $1{,}000(0.58)(0.75)=435$. Their sum is $290+232+435=957$, or $957,000; $957-924=33$. All amounts are in thousands until the final dollar interpretation. This method avoids treating an observed-loss change as a direct reserve change without first refitting the portfolio ratio.

Separating Assumption Changes from Experience Changes

A selected expected loss ratio and a Cape Cod fitted ratio may both be expressed as percentages, but they have different roles. The BF selection is an assumption supplied to the method. A change in that selection changes the expected unpaid component even if reported losses are unchanged. The Cape Cod ratio is calculated from the data and maturity weights. A change to an external BF assumption, with $R_i$, $EP_i$, and $q_i$ unchanged, leaves the Cape Cod ratio and reserve unchanged.

Conversely, a change to reported losses affects Cape Cod’s numerator. A change to the reported proportions or premiums affects its denominator and may also change the unreported-premium factor $U$. When several inputs change at once, recompute the fitted ratio and the reserve from the revised inputs rather than applying only one of the fixed-input sensitivity formulas.

Identify the method and the changed input.
For BF, determine whether the selected loss ratio changed. For Cape Cod, determine whether reported losses, premiums, or reported proportions changed.

Hold other inputs fixed when measuring one effect.
State explicitly which premiums, reported proportions, reported losses, and selections are unchanged.

Apply the matching sensitivity calculation.
Use $EP_i(1-q_i)\Delta\lambda$ for BF. For Cape Cod, first find the change in the fitted ratio, then multiply by the portfolio’s premium-weighted unreported proportion.

Verify by recalculating the estimates.
Compute old and new reserves directly where practical, and confirm that their difference agrees with the sensitivity result.

Common Exam Traps

Using the full premium for BF reserve sensitivity. The reserve responds to $EP_i(1-q_i)$, not $EP_i$. The remaining reported portion is not part of the BF expected unpaid loss.
Mixing percentage points and percentages. A decrease from 60% to 56% is a change of $-0.04$, not $-4$ or $-0.4$.
Applying an external selection change to Cape Cod without refitting. With all Cape Cod inputs fixed, changing a separate BF selection has no effect on the Cape Cod result.
Using total premium as Cape Cod’s fitted-ratio denominator. The denominator is $\sum_i EP_iq_i$, which accounts for how much loss is expected to be reported by the valuation date.
Confusing reserve movement with ultimate-loss movement. A change in BF expected ultimate loss is $EP_i\Delta\lambda$; the reserve movement is only $EP_i(1-q_i)\Delta\lambda$. The difference corresponds to the reported-proportion component.

Key takeaway: BF reserve sensitivity to a selected expected loss ratio is direct and linear: multiply the loss-ratio change by premium and the unreported proportion. Cape Cod sensitivity is data-driven: changes in its fitted portfolio ratio flow through the same unreported-premium factor, while an external expected loss ratio change alone does not alter the fitted result.

Check Your Understanding

For each problem, keep the stated basis fixed except for the input being changed.

A hypothetical BF period has earned premium of $750,000 and a reported proportion of 40%. The selected expected loss ratio decreases by 3 percentage points. What is the reserve change? (Answer: $13,500 decrease)
Two BF periods have earned premiums and reported proportions of $1,200,000 and 25%, and $800,000 and 75%. The expected loss ratio increases from 50% to 55%. Find the total reserve change. (Answer: $55,000 increase)
A Cape Cod portfolio has periods with earned premiums, reported proportions, and reported losses of ($600,000, 50%, $180,000) and ($1,400,000, 75%, $630,000). Calculate the fitted ratio and total reserve. (Answer: 60%; $390,000)
In a Cape Cod portfolio, fixed premiums and reported proportions give $D=\sum_i EP_iq_i=\$900{,}000$ and $U=\sum_i EP_i(1-q_i)=\$300{,}000$. Reported losses increase by $27,000. Find the fitted-ratio change and reserve change. (Answer: 3 percentage points; $9,000 increase)
A BF period has earned premium of $2,000,000 and a reported proportion of 20%. The selected expected loss ratio falls by 5 percentage points. A Cape Cod estimate for the same exposure has fitted ratio 53%, with all its data and maturity weights unchanged. Find the BF reserve change and state the change in the Cape Cod reserve. (Answer: BF decreases by $80,000; Cape Cod change is $0)

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