Basic chain ladder

In this sections, we will discuss the solution to estimate the IBNR reserves via the basic chain ladder.

Let’s recall the run-off triangle we have used in the previous sections.

	Development year
Origin year	0	1	2	3
2020	100	180	240	280
2021	120	220	300
2022	140	260
2023	160

Adjustment factors

From the above run-off triangle, we have already known that all accidents happened in year 2020 have been claimed in the next three development years, which means the IBNR reserve for year 2020 is zero.

For all other origin years, if we want to estimate the IBNR reserve, we should introduce two factors for adjustment:

• Development factor ($d$): is a ratio (usually $>$ 1) of the cumulative claims paid in the current development year to the cumulative claims paid in the previous development year, which is used as a multiplier to estimate the progression of claims from one development year to the next.

Example

For example, the development factor for year 2020 ($0 \rightarrow 1$) is $180/100 = 1.8$, and the development factor for year 2021 ($0 \rightarrow 1$) is $220/120 = 1.8333$.

• Grossing-up factor ($g$): is a ratio of the total claims paid in the previous year over the total claims paid in the current year.

Example

From the above example, the grossing-up factor in year 2021 for development year $1→2$ is $220/300 = 0.733$, and the grossing-up factor in year 2022 is $140/260 = 0.5385$.

Ultimate losses

Once we understand the above two factors, we could estimate the IBNR reserve. In this part, I will show you how to estimate the IBNR reserve by using the development factor.

In the beginning, we need to know that we are going to use the last available claims data for each origin year. For example, for year 2021, we have already known that the total claims paid in year 2023 (2021 + 2 development years) is 300.

	Development year
Origin year	0	1	2	3
2020	100	180	240	280
2021	120	220	300	?
2022	140	260
2023	160

If we assume that all claims will be settled in the next development year, we only need to use the total claims in developemt year 2 times the development factor from development year 2 to development year 3 to estimate the IBNR reserve. So we can have $? = 300 \times g_{2|3}$.

According to year 2020 (development year $2 → 3$), we can have $g_{2|3} = 280/240 = 1.1667$. Therefore, the ultimate loss for year 2021 in development year 3 is $300 \times 1.1667 = 350$.

	Development year
Origin year	0	1	2	3 (Ultimate Losses)
2020	100	180	240	280
2021	120	220	300	350
2022	140	260		?
2023	160

The next step is to estimate the ultimate losses for year 2022. The last available total paid claims in year 2022 with development year 1 is 260. According to the above formula, we should have $? = 260 \times g_{1|3}$. We use the last available total paid claims times the development factor from that development year to the ultimate development year.

In this case, the development factor has crossed two development years, so we can simply decompose it into two parts.

$$g_{1|3} = g_{1|2} \times g_{2|3}$$

From this equation, we can see that the development factor from development year $i$ to development year $j$ is the product of the development factor from development year $i$ to development year $i+1$ and the development factor from development year $i+1$ to development year $j$.

$$g_{i|j} = \prod_{k=i}^{j-1} g_{k|k+1}$$

We can use the above formula to create a development factor table for each origin year.

	Development year
Origin year	0 → 1	1 → 2	2 → 3
2020	1.80	1.33	1.17
2021	1.83	1.36
2022	1.86

From the calculation of year 2021, we have already known that $g_{2|3} = 1.1667$ and $g_{1|3} = g_{1|2} \times g_{2|3}$. If we want to calculate the aggregate development factor from development year 1 to development year 3, we need to calculate $g_{1|2}$.

Aggregate development factor

From the above development factor table, we can see that we have two $g_{1|2}$, which are 1.33 (origin year 2020) and 1.36 (origin year 2021).

Now we have two methods to aggregate two individual development factors into one:

Arithmetic average

The first method is to use the arithmetic average. We simply sum up all the development factors and divide by the number of development factors. In this case, the aggregate development factor is

$$g_{1|2} = \frac{(1.33 + 1.36)}{2} = 1.345$$

If we use the total paid claims to show the result, it will be:

$$g_{1|2} = (\frac{240}{180} + \frac{300}{220})/2 = 1.345$$

We can also extend the above calculation to the development factor from development year 0 to development year 1. It will include all three origin year 2020, 2021 and 2022.

$$\begin{array}{rl} g_{0|1} &= (\frac{180}{100} + \frac{220}{120} + \frac{260}{140})/3 \\ \\ &= (1.90 + 1.83 + 1.86)/3 = 1.830 \end{array}$$

In all, we can have the aggregate development factor from development year 0 to development year 3.

$$\begin{array}{rl} g_{2|3} &= 1.1667 \\ \\ g_{1|3} &= g_{1|2} \times g_{2|3} \\ \\ &= 1.345 \times 1.1667 = 1.57 \\ \\ g_{0|3} &= g_{0|1} \times g_{1|2} \times g_{2|3} \\ \\ &= 1.830 \times 1.345 \times 1.1667 = 2.68 \end{array}$$

Weighted average

On the other hand, we can also use the weighted average to estimate the aggregate development factor. The weight is the total paid claims in each origin year over the total paid claims in all origin years.

Let’s focus on the development factor from development year 1 to development year 2 (since there is only one available development factor from development year 2 to development year 3).

From the total paid claim table, we can know that the total paid claims in origin year 2020 and 2021 at development year 1 are 180 and 220, respectively. The total paid claims in all origin years are $180 + 220 = 400$. The weight for origin year 2020 is $180/400 = 0.45$, and the weight for origin year 2021 is $220/400 = 0.55$. The aggregate development factor $g_{1|2}$ is

$$\begin{array}{rl} g_{1|2} &= \frac{180}{400} \times g_{1|2, 2020} + \frac{220}{400} \times g_{1|2, 2021} \\ \\ &= \frac{180}{400} \times \frac{240}{180} + \frac{220}{400} \times \frac{300}{220} \\ \\ \end{array}$$

The result can be simply rewritten as:

$$\begin{array}{rl} g_{1|2} &= \frac{240}{400} + \frac{300}{400} \\ \\ &= \frac{240 + 300}{400} \\ \\ &= \frac{240 + 300}{180 + 220} = 1.35 \end{array}$$

From the above formula, we can see that the weighted aggregate development factor $g_{1|2}$ is the sum of total paid claims in development year 2 over the sum of total paid claims in development year 1 for all origin years.

Similary, we can calculate the weighted aggregate development factor from development year 0 to development year 1 by using the total paid claims directly.

$$\begin{array}{rl} g_{0|1} = \frac{180 + 220 + 260}{100 + 120 + 140} = 1.833 \end{array}$$

The weighted aggregate development factor from development year 0 to development year 3 is

$$\begin{array}{rl} g_{2|3} &= 1.1667 \\ \\ g_{1|3} &= g_{1|2} \times g_{2|3} \\ \\ &= 1.35 \times 1.1667 = 1.57 \\ \\ g_{0|3} &= g_{0|1} \times g_{1|2} \times g_{2|3} \\ \\ &= 1.833 \times 1.35 \times 1.1667 = 2.887 \end{array}$$

Compare the above two methods, we can see that the arithmetic average is slightly different from the weighted average.

Estimate Ultimate losses

Once we have the aggregate development factor, we can estimate the ultimate losses for each origin year.

In here, we assume that we want to use the weighted aggregate development factors to estimate the ultimate losses.

For origin year 2022, we calculate the ultimate losses by using the total paid claims in development year 1 times the development factor from development year 1 to development year 3, $260 \times g_{1|3} = 260 \times 1.35 = 351$.

Also, the ultimate loss for year 2023 is $160 \times g_{0|3} = 160 \times 2.887 = 461$.

	Development year
Origin year	0	1	2	3 (Ultimate Losses)
2020	100	180	240	280
2021	120	220	300	350
2022	140	260		351
2023	160			462

Estimate IBNR reserve

Finally, we can estimate the IBNR reserve by using the ultimate losses. The IBNR reserve is the difference between the ultimate losses and the total paid claims.

$$\begin{array}{rl} \text{IBNR reserve} &= \text{Ultimate losses} - \text{Total paid claims} \\ \\ &= (280 + 350 + 351 + 462) - (280 + 300 + 260 + 160) \\ \\ & = 1443 - 1000 = 443 \end{array}$$

Because all claims in year 2020 have been settled, the IBNR reserve for year 2020 is zero. When we calculate the result, we can simply ignore the total paid claims in year 2020. The result will only include the total paid claims in year 2021, 2022 and 2023.

$$\begin{array}{rl} \text{IBNR reserve} &= \text{Ultimate losses} - \text{Total paid claims} \\ \\ &= (350 + 351 + 462) - (300 + 260 + 160) \\ \\ & = 1163 - 720 = 443 \end{array}$$

Conclusion

In this section, we have discussed how to estimate the IBNR reserve by using the basic chain ladder. We have introduced two factors, the development factor and the grossing-up factor, to estimate the ultimate losses. We have also discussed how to aggregate the development factors by using the arithmetic average and the weighted average. Finally, we have shown how to estimate the IBNR reserve by using the ultimate losses.

It can be separated into the following steps:

1. Calculate the development factor for each development year.
2. Aggregate the development factors by using the arithmetic average or the weighted average.
3. Estimate the ultimate losses for each origin year.
4. Estimate the IBNR reserve by using the difference between the ultimate losses and the total paid claims.

In the next section, we will discuss how to estimate the IBNR reserve by using the Average cost per claim method.