Projecting Triangle Cells and Future Diagonals

Introduction

A chain-ladder projection does more than estimate an ultimate amount for each accident period. It can also fill in the unobserved cumulative cells one development age at a time. Arranging those projected cells by calendar period shows which accident periods contribute to each future diagonal and how much new loss emerges in that period.

The essential discipline is to apply the factor for the development age being left. When a cell is projected again, its development age has advanced, so the next transition may require a different factor. This tutorial uses selected link ratios as given and focuses on applying them consistently.

Triangle Indices and Projection Rule

Let $C_{i,j}$ denote the cumulative loss for accident period $i$, observed at development age $j$. For example, $C_{2,3}$ is the cumulative amount for accident period 2 at development age 3. The age-to-age factor $f_j$ applies to the transition from development age $j$ to age $j+1$. Thus $f_2$ projects an age-2 cell to age 3; it does not mean “the factor for accident period 2.”

Definition: Given a cumulative cell $C_{i,j}$ and a selected factor $f_j$, the chain-ladder projection for the next development age is $ \widehat{C}_{i,j+1}=C_{i,j}f_j $. The hat indicates a projected value. Each subsequent transition uses the factor indexed by the new current development age.

For equally spaced annual accident periods and development ages, the calendar-period index associated with cell $(i,j)$ is $i+j-1$. Cells on the same calendar diagonal therefore have the same value of $i+j-1$. In an ordinary triangular data set, the latest observed cells lie on one such diagonal, while the first projected cells lie on the next.

These projections assume that the selected age-to-age factors are appropriate for the cumulative data being projected and are applied consistently. The calculations fill only transitions for which a factor has been supplied. They do not imply that a factor should be invented for an unavailable transition.

Why the Projection and Emergence Formulas Hold

The link ratio for a particular development age is a multiplicative relationship between cumulative amounts. If a selected factor $f_j$ represents the age-$j$-to-age-$j+1$ relationship, applying that relationship to the current cumulative amount gives the next cumulative amount. The incremental amount between those ages is the difference between the new cumulative amount and the old one:

\widehat{C}_{i,j+1}=C_{i,j}f_j,\qquad \widehat{C}_{i,j+1}-C_{i,j}=C_{i,j}(f_j-1).

The second equality follows by subtracting $C_{i,j}$ from both sides of the projection and factoring it out. It also explains why a factor greater than 1 produces positive emergence from a positive cumulative amount. A factor of 1 produces no increment. A factor below 1 produces a negative increment; the arithmetic still follows the selected factor, though whether that factor is suitable is a separate selection question.

For a future calendar diagonal, each included accident period advances by one development age. Its contribution to emergence is therefore its new cumulative cell minus its cumulative cell on the preceding diagonal. Summing those differences gives total emergence for the diagonal. Summing the cumulative cell amounts themselves does not give emergence.

Worked Examples

Worked Example: Projecting Two Diagonals

A hypothetical triangle has its latest observed diagonal at calendar-period index 3, with cumulative amounts of $100 thousand for accident period 1 at age 3, $160 thousand for accident period 2 at age 2, and $200 thousand for accident period 3 at age 1. Selected factors are $f_1=1.25$, $f_2=1.20$, $f_3=1.10$, and $f_4=1.05$. Project the next two diagonals and calculate the emergence on each.

Method. Project each cell using the factor indexed by its current age. Then subtract its preceding-diagonal cumulative amount to obtain that cell’s incremental emergence. All amounts below are in thousands of dollars.

First projected diagonal. Accident period 1 moves from age 3 to age 4, so use $f_3$: $100(1.10)=110$. Accident period 2 moves from age 2 to age 3, so use $f_2$: $160(1.20)=192$. Accident period 3 moves from age 1 to age 2, so use $f_1$: $200(1.25)=250$.

The first projected diagonal is therefore $110,\ 192,\ 250$. The cell increments are $110-100=10$, $192-160=32$, and $250-200=50$. Total emergence is $10+32+50=92$ thousand dollars.

Second projected diagonal. Each accident period has advanced one age. Accident period 1 now moves from age 4 to age 5, so use $f_4$: $110(1.05)=115.5$. Accident period 2 moves from age 3 to age 4, so use $f_3$: $192(1.10)=211.2$. Accident period 3 moves from age 2 to age 3, so use $f_2$: $250(1.20)=300$.

The second projected diagonal is $115.5,\ 211.2,\ 300$. Its increments are $115.5-110=5.5$, $211.2-192=19.2$, and $300-250=50$. Total emergence is $5.5+19.2+50=74.7$ thousand dollars.

Check. The latest observed cells all have calendar index 3: $1+3-1=3$, $2+2-1=3$, and $3+1-1=3$. The first and second projected diagonals have indices 4 and 5, respectively. The second projection correctly uses $f_4$, $f_3$, and $f_2$, not the factors used for those rows on the first projection.

Worked Example: Completing a Triangle Across Future Calendar Periods

A hypothetical cumulative triangle, in thousands of dollars, has these latest observed cells: accident period 1 at age 4 is 200; accident period 2 at age 3 is 190; accident period 3 at age 2 is 196; and accident period 4 at age 1 is 160. Selected factors are $f_1=1.20$, $f_2=1.10$, $f_3=1.05$, $f_4=1.02$, and $f_5=1.01$. Fill the next two calendar diagonals and calculate emergence for each.

First future diagonal. The four transitions are age 4 to 5, age 3 to 4, age 2 to 3, and age 1 to 2. The projected cumulative cells are:

\begin{aligned} \widehat{C}_{1,5}&=200(1.02)=204,\\ \widehat{C}_{2,4}&=190(1.05)=199.5,\\ \widehat{C}_{3,3}&=196(1.10)=215.6,\\ \widehat{C}_{4,2}&=160(1.20)=192. \end{aligned}

The first diagonal’s emergence is $(204-200)+(199.5-190)+(215.6-196)+(192-160)=4+9.5+19.6+32=65.1$ thousand dollars.

Second future diagonal. The current ages are now 5, 4, 3, and 2. The next transitions therefore use $f_5$, $f_4$, $f_3$, and $f_2$, respectively:

\begin{aligned} \widehat{C}_{1,6}&=204(1.01)=206.04,\\ \widehat{C}_{2,5}&=199.5(1.02)=203.49,\\ \widehat{C}_{3,4}&=215.6(1.05)=226.38,\\ \widehat{C}_{4,3}&=192(1.10)=211.2. \end{aligned}

The second diagonal’s emergence is $(206.04-204)+(203.49-199.5)+(226.38-215.6)+(211.2-192)=2.04+3.99+10.78+19.2=36.01$ thousand dollars.

Check. The starting cells share calendar index 4. The first projected cells have index 5 and the second projected cells have index 6. For example, accident period 2 moves from age 3 to age 4 using $f_3$, then from age 4 to age 5 using $f_4$. The calculations use each age transition exactly once.

Worked Example: Tracking Which Factor Applies After a Projection

In a hypothetical two-period triangle, the latest diagonal contains $240 thousand for accident period 1 at age 2 and $150 thousand for accident period 2 at age 1. The selected factors are $f_1=1.30$, $f_2=1.15$, and $f_3=1.08$. Project two future diagonals and find the emergence on the second.

Method. Start with each cell’s current age, apply that age’s factor, and increase the age by one before choosing a factor for the next projection.

On the first future diagonal, accident period 1 uses $f_2$, giving $240(1.15)=276$. Accident period 2 uses $f_1$, giving $150(1.30)=195$. The first diagonal cells are therefore 276 and 195.

On the second future diagonal, accident period 1 is at age 3 and uses $f_3$: $276(1.08)=298.08$. Accident period 2 is at age 2 and uses $f_2$: $195(1.15)=224.25$. The second diagonal cells are 298.08 and 224.25.

The second-period emergence is $(298.08-276)+(224.25-195)=22.08+29.25=51.33$ thousand dollars.

Check. The starting cells have calendar index 2, the first projected cells index 3, and the second projected cells index 4. The two second-step factors are $f_3$ and $f_2$, because the current ages have advanced.

Common Mistake: Reusing the Previous Factor

A frequent error is to keep using the factor that was used for a row’s previous projection. In the third example, a wrong second projection for accident period 1 would be $276(1.15)=317.4$, reusing $f_2$. But the cell is now at age 3, so the correct transition is age 3 to age 4 and requires $f_3$: $276(1.08)=298.08$. The wrong cell would also overstate second-period emergence for that accident period: $317.4-276=41.4$, rather than the correct $22.08$.

Another related error is to report the sum of the new cumulative cells as the diagonal’s emergence. In the first example, the first projected cumulative cells sum to $110+192+250=552$ thousand dollars, but emergence is $92$ thousand dollars. Emergence is the sum of the increases from the preceding diagonal, not the sum of cumulative amounts.

Putting the Projection Into Practice

A reliable calculation follows the same sequence for every future diagonal: identify the current development age of each cell, apply its matching selected factor, record the new cumulative cell, and then move to the next age before projecting again. When calculating emergence, compare each new cell with the corresponding cumulative cell on the preceding diagonal.

Locate the latest diagonal.
Record each accident period’s latest cumulative amount and its development age.

Project one age at a time.
For a cell currently at age $j$, multiply by $f_j$ to project to age $j+1$.

Re-index before the next projection.
After moving to age $j+1$, use $f_{j+1}$ for the following transition, if that factor is supplied.

Group by calendar diagonal and calculate emergence.
Cells with the same $i+j-1$ belong to the same calendar period. Subtract each corresponding prior-diagonal amount and add the differences.

Projected cells are useful for seeing how a triangle develops across future calendar periods, not only for finding a projected endpoint. This makes the indexing important: a correct overall method can still give incorrect diagonal values if the wrong age factor is used. These calculations use only the factors supplied for the required transitions; any projection beyond their range requires additional input.

Key takeaway: A selected link ratio is indexed by the development age being left. Projected cumulative cells belong to future diagonals, while emergence is the change from the preceding diagonal. Re-index the development age after every projection.

Check Your Understanding

For each problem, apply the factor for the current development age and distinguish cumulative cell values from emergence.

A latest-diagonal cell is 80 at development age 2. If $f_2=1.25$, what is its projected age-3 cumulative value and its increment? (Answer: cumulative value 100; increment 20.)
The latest diagonal has accident period 1 at age 3 with 140 and accident period 2 at age 2 with 120. Given $f_3=1.10$ and $f_2=1.20$, find the next diagonal’s cells and total emergence. (Answer: cells 154 and 144; emergence 38.)
The latest diagonal has cells 300 at age 3 and 250 at age 2. Use $f_3=1.04$ and $f_2=1.12$ to find the next diagonal and its total emergence. (Answer: cells 312 and 280; emergence 42.)
A latest diagonal has four cells: 500 at age 4, 300 at age 3, 200 at age 2, and 100 at age 1. Given $f_4=1.02$, $f_3=1.10$, $f_2=1.20$, and $f_1=1.30$, find the next diagonal and its total emergence. (Answer: cells 510, 330, 240, and 130; emergence 110.)
The latest diagonal has accident period 1 at age 2 with 180 and accident period 2 at age 1 with 125. Given $f_2=1.10$, $f_1=1.20$, and $f_3=1.05$, project two diagonals and calculate emergence on the second. (Answer: first diagonal cells 198 and 150; second diagonal cells 207.9 and 165; second-diagonal emergence 24.9.)

What You'll Learn