Deriving the Chain-Ladder Projection from Link Ratios

Why Successive Link Ratios Produce a Projection

A cumulative loss triangle records amounts at successive development ages. A chain-ladder projection extends a cumulative amount from its latest observed age to a later age by applying the selected age-to-age factors for the intervening transitions. The key is to include each required transition once, and only once.

Actual development ages are often stated in months, such as 12, 24, and 36 months. The factors, however, are indexed by the transition between consecutive ages, not by the number of months. To keep the formula unambiguous, this tutorial uses ordinal indices $j=1,2,3,\ldots$ for the ordered development ages. For example, 12, 24, and 36 months can be indexed as $t_1,t_2,t_3$. The value $f_2$ is then the selected factor from $t_2$ to $t_3$, or from 24 to 36 months in this example.

Definition: Let $C_{i,j}$ be the cumulative loss for accident period $i$ at development age $t_j$, where $j$ is an ordinal index. Let $f_j$ be the selected age-to-age factor for the transition from $t_j$ to $t_{j+1}$. If the latest observed cumulative loss is $C_{i,k}$, the chain-ladder projection to a later development age $t_m$, with $m>k$, is

\widehat{C}_{i,m}=C_{i,k}\prod_{j=k}^{m-1} f_j =C_{i,k} f_k f_{k+1}\cdots f_{m-1}.

Here the hat indicates that the amount at age $t_m$ is projected rather than observed. The factors are the selected factors for the transitions after the latest observed age and up to the target age. In particular, the factor from $t_{k-1}$ to $t_k$ is not included: that transition has already been reflected in $C_{i,k}$.

Why the Projection Is a Product

The multiplication follows directly from the meaning of each link ratio. For a single transition, a factor $f_k$ takes a cumulative amount at age $t_k$ to its projected amount at age $t_{k+1}$. To project one more transition, apply $f_{k+1}$ to that new amount. Repeating this step gives the complete projection.

Start with the latest observed amount.
At age $t_k$, the cumulative amount is $C_{i,k}$.

Project one transition.
At age $t_{k+1}$, the projected amount is $C_{i,k}f_k$.

Project the next transition.
At age $t_{k+2}$, apply $f_{k+1}$ to the amount from Step 2, giving $C_{i,k}f_kf_{k+1}$.

Continue to the target age.
Each transition contributes one factor, so the final expression includes factors $f_k$ through $f_{m-1}$, inclusive.

More formally, define each projected step by $\widehat{C}_{i,j+1}=\widehat{C}_{i,j}f_j$, starting with $\widehat{C}_{i,k}=C_{i,k}$. Applying the recurrence once gives $\widehat{C}_{i,k+1}=C_{i,k}f_k$. If after $r$ transitions the amount is $C_{i,k}\prod_{j=k}^{k+r-1}f_j$, one further transition multiplies it by $f_{k+r}$, giving $C_{i,k}\prod_{j=k}^{k+r}f_j$. Thus, by repeated application, after $m-k$ transitions the projection is $C_{i,k}\prod_{j=k}^{m-1}f_j$, as stated.

This derivation describes the mechanics of the chain-ladder projection. It does not establish that the selected factors will accurately predict future experience. Their selection and suitability depend on the data and assumptions used in the reserving analysis.

Worked Projections

Worked Example: Projecting Across Two Future Transitions

A hypothetical paid-loss triangle has a latest observed cumulative amount of $480,000 at 24 months. The development ages are 12, 24, 36, and 48 months, indexed respectively by $t_1,t_2,t_3,t_4$. The selected factors are $f_1=1.16$, $f_2=1.25$, and $f_3=1.12$. Project the cumulative amount to 48 months.

Method. The latest observed age is $t_2$, and the target is $t_4$. The required transitions are $t_2$ to $t_3$ and $t_3$ to $t_4$, so use $f_2$ and $f_3$. The 12-to-24-month factor $f_1$ is not needed.

\widehat{C}_{i,4}=480{,}000(1.25)(1.12).

First project from 24 to 36 months:

$$ 480{,}000(1.25)=600{,}000. $$

Then project from 36 to 48 months:

$$ 600{,}000(1.12)=672{,}000. $$

The projected cumulative amount at 48 months is $672,000. As a check, the combined multiplier is $1.25(1.12)=1.40$, and $480{,}000(1.40)=672{,}000$. A common mistake is to multiply by $f_1$ as well, even though the 12-to-24 transition is already reflected in the observed 24-month amount.

Worked Example: Keeping Actual Ages and Indices Separate

A hypothetical liability portfolio is valued at development ages 6, 18, 30, and 48 months. These ages are indexed as $t_1,t_2,t_3,t_4$. For one accident period, the latest observed cumulative loss is $275,000 at 18 months. The selected factors are $f_1=1.20$, $f_2=1.10$, and $f_3=1.05$. Project to 48 months.

Method. The latest observed age is $t_2$, and the target age is $t_4$. There are two future transitions: from 18 to 30 months, represented by $f_2$, and from 30 to 48 months, represented by $f_3$. The fact that the intervals have different lengths does not change the indexing: $f_j$ refers to the transition from $t_j$ to $t_{j+1}$.

After the 18-to-30-month transition:

$$ 275{,}000(1.10)=302{,}500. $$

After the 30-to-48-month transition:

$$ 302{,}500(1.05)=317{,}625. $$

Equivalently, $1.10(1.05)=1.155$, and $275{,}000(1.155)=317{,}625$. The projected cumulative loss at 48 months is $317,625. Including $f_1=1.20$ would incorrectly apply the already completed 6-to-18-month transition.

Worked Example: Starting From a Later Observed Age

A hypothetical accident period has observed cumulative losses of $420,000 at 12 months and $504,000 at 24 months. The selected factors for the next transitions are $f_2=1.10$ from 24 to 36 months and $f_3=1.05$ from 36 to 48 months. Project the 48-month cumulative loss.

Method. The observed 12-to-24-month link ratio is $504{,}000/420{,}000=1.20$. This confirms that the observed amount at 24 months already includes development through the first transition. The projection to 48 months therefore starts with $504,000 and uses only $f_2$ and $f_3$.

First project from 24 to 36 months:

$$ 504{,}000(1.10)=554{,}400. $$

Then project from 36 to 48 months:

$$ 554{,}400(1.05)=582{,}120. $$

The projected cumulative amount at 48 months is $582,120. The check is $1.10(1.05)=1.155$, and $504{,}000(1.155)=582{,}120$. The common error is to apply the observed 12-to-24 link ratio again; doing so would restart the projection from an amount that is already at 24 months.

Reading the Formula Correctly

The lower and upper limits of the product encode the starting and ending ages. If a projection starts at index $k$ and ends at index $m$, the first factor is $f_k$, the factor for the transition leaving the starting age. The last is $f_{m-1}$, the factor for the transition arriving at the target age. There is no $f_m$ in the product, because that factor would move beyond $t_m$.

Projection	Factors used	Reason
$t_2$ to $t_3$	$f_2$	One transition, from index 2 to index 3
$t_2$ to $t_4$	$f_2,f_3$	Two transitions, ending at index 4
$t_1$ to $t_4$	$f_1,f_2,f_3$	Three transitions, starting at index 1

For ordinary positive cumulative amounts and positive selected factors, each multiplication preserves positivity. If a factor is greater than 1, it increases the projected amount for that transition; if it is between 0 and 1, it reduces it. The formula itself does not require every selected factor to exceed 1. It requires that the factors be defined for the relevant transitions and that the chain-ladder multiplicative projection be the method being applied.

This formula projects a cumulative amount to a specified future development age. It is not itself a calculation of an unpaid reserve: that requires comparing projected and observed amounts, a separate step. Nor should a factor beyond the last selected transition be inserted simply because the target age is stated in months. The index range, rather than the month number, determines which factors belong in the product.

Common Exam Trap

Suppose the latest observation is at $t_3$ and the requested projection is to $t_5$. The relevant product is $f_3f_4$, because those factors represent $t_3$ to $t_4$ and $t_4$ to $t_5$. A tempting but incorrect calculation is $C_{i,3}f_2f_3f_4$. It includes $f_2$, the transition from $t_2$ to $t_3$, even though $C_{i,3}$ is already the cumulative amount at $t_3$.

Key takeaway: Begin with the cumulative amount at the latest observed development index $k$. Multiply by exactly the selected factors $f_k$ through $f_{m-1}$ to project to index $m$. Keep actual development ages, such as months, separate from the ordinal indices used to identify transitions.

Check Your Understanding

For each hypothetical projection, identify the required transitions before multiplying. Answers are provided at the end of each question.

A cumulative amount of $240,000 is observed at $t_2$. The selected factor for the transition from $t_2$ to $t_3$ is $f_2=1.15$. What is the projection at $t_3$? (Answer: $276,000.)
A cumulative amount of $90,000 is observed at $t_1$. The factors are $f_1=1.20$ and $f_2=1.10$. Project to $t_3$. (Answer: $118,800.)
Development ages are 12, 24, 36, and 48 months, indexed $t_1$ through $t_4$. A cumulative amount of $640,000 is observed at 36 months, and $f_3=1.075$. Project to 48 months. (Answer: $688,000.)
A cumulative amount of $500,000 is observed at $t_2$. The target is $t_5$, and the applicable factors are $f_2=1.08$, $f_3=1.04$, and $f_4=1.02$. Calculate the projection sequentially. (Answer: $572,832.)
Development ages are 6, 18, 30, and 42 months, indexed $t_1$ through $t_4$. A cumulative amount of $315,000 is observed at 18 months. The factors are $f_1=1.20$, $f_2=1.12$, and $f_3=1.06$. Project to 42 months without applying a factor for an already observed transition. (Answer: $373,968.)

Projection	Factors used	Reason
\(t_2\) to \(t_3\)	\(f_2\)	One transition, from index 2 to index 3
\(t_2\) to \(t_4\)	\(f_2,f_3\)	Two transitions, ending at index 4
\(t_1\) to \(t_4\)	\(f_1,f_2,f_3\)	Three transitions, starting at index 1

What You'll Learn