Separation Method

Separation Method models each incremental cell of a development triangle as a development-age effect times a calendar-year (diagonal) effect, separating claims inflation from the pattern of delays. Venter (1998) presents it as the way to build diagonal effects into the emergence model itself, citing Taylor's "Separation of Inflation and Other Effects from the Distribution of Non-Life Insurance Claim Delays" (1977):

E[q(w,d)] = f(d)\,g(w+d)

• The parameters. q(w,d) is the incremental loss of accident year w at age d, and f(d) is the age term. g(w+d) is the diagonal term, and every diagonal has one. Its usual reading is the cumulative claims inflation that applies to diagonal w + d since the first accident year. A constant rate j leaves a single inflation parameter:

g(w+d) = (1+j)^{w+d}

• Adding accident years. Accident-year levels can be added, E[q(w,d)] = f(d)h(w)g(w+d). That is too many parameters unless many are set equal, or written as trends: f(d) = (1+i)^d and h(w) = h(1+k)^w. With all three trends constant, j cannot be separated, because the model collapses to h(1+i+j+ij)^d(1+k+j+jk)^w. Venter suggests keeping the calendar-year trend and dropping the accident-year one, especially once the triangle has been normalized for accident-year changes.
• Fitting. Start from reasonable values, fix all but one set of parameters and fit that set by least squares, then rotate until the estimates converge. A non-linear search does the same job.
• Forecasting. Future cells lie on diagonals the data has not reached, so g must be projected. The latest diagonal effects can be carried forward as the estimate of future inflation, informed by what drives them in company operations. Mack (1994) notes that the chain ladder already extrapolates a constant inflation rate left in the data. The separation model makes that rate explicit, and changeable.

Separating Inflation, and Projecting It Incremental paid losses ($000), already adjusted for differences in exposure between accident years:

AY	Age 0	Age 1	Age 2
2022	1{,}000.0	630.0	220.5
2023	1{,}050.0	661.5
2024	1{,}102.5

(a) Fit q(w,d) = f(d)g(w+d), taking g = 1 on the 2022 diagonal. (b) Estimate the unpaid claims if inflation continues at the rate the diagonals show, and compare the chain ladder. (c) Re-estimate them if inflation runs at 10\% from 2025.

Answer (a) The three age-0 cells lie on the three diagonals, so they give f(0) and the diagonal terms directly, and the 2022 row then gives the other age terms:

\begin{align*}
f(0) &= 1{,}000 \\
g(2023) &= 1{,}050/1{,}000 \\
&= 1.05 \\
g(2024) &= 1{,}102.5/1{,}000 \\
&= 1.1025 \\
f(1) &= 630/1.05 \\
&= 600 \\
f(2) &= 220.5/1.1025 \\
&= 200
\end{align*}

Check the one remaining cell: 600 \times 1.1025 = 661.5, as observed. The triangle fits the model exactly, with inflation of 5\% a year.

(b) At 5\%, g(2025) = 1.157625 and g(2026) = 1.215506:

\begin{align*}
\text{AY 2023, age 2} &= 200 \times 1.157625 \\
&= 231.5 \\
\text{AY 2024, age 1} &= 600 \times 1.157625 \\
&= 694.6 \\
\text{AY 2024, age 2} &= 200 \times 1.215506 \\
&= 243.1 \\
\text{Unpaid} &= 231.5 + 694.6 + 243.1 \\
&= 1{,}169.2
\end{align*}

The chain ladder factors on the cumulative triangle are 3{,}341.5/2{,}050 = 1.63000 and 1{,}850.5/1{,}630 = 1.13528. They give 1{,}711.5 \times 0.13528 = 231.5 for 2023 and 1{,}102.5 \times (1.63000 \times 1.13528 - 1) = 937.7 for 2024, the same 1{,}169.2. A steady 5\% inflation in the data is simply carried forward by the chain ladder.

(c) At 10\% from 2025, g(2025) = 1.1025 \times 1.10 = 1.21275 and g(2026) = 1.334025:

\begin{align*}
\text{Unpaid} &= 200(1.21275) + 600(1.21275) \\
&\quad + 200(1.334025) \\
&= 242.6 + 727.7 + 266.8 \\
&= 1{,}237.0
\end{align*}

That is 67.8 more than the chain ladder, which cannot respond to a change in the inflation rate because it has no separate inflation parameter to change.