Average cost per claim

In the last section, we use the development factor to estimate the ultimate losses of the chain and calculate the IBNR reserves. However, using the total paid claims to estimate the ultimate losses may not be accurate. In the real world, the number of claims is also an important factor to estimate the ultimate losses.

In this section, we will discuss another method to estimate the IBNR reserve with introducing the number of claims and groosing-up factors.

From previous sectios, we know the total paid claims for different accident years. Then, we can add the cumulative total number of claims for each accident year in the same table. The run-off triangle will be like this:

	Development year
Origin year	0		1		2		3
	C	N	C	N	C	N	C	N
2020	100	12	180	16	240	22	280	30
2021	120	14	220	18	300	25
2022	140	16	260	20
2023	160	10

In the above table, the values in the cells represent the cumulative cost (C) and the number of claims (N) for each origin year and development year. We can estimate the IBNR reserves by using the following steps.

Step 1: Calculate the average cost per claim

In the first step, we need to calculate the average cost per claim for each cell. The average cost per claim is calculated by dividing the cumulative cost (C) by the number of claims (N). We can get the table below:

	Development year
Origin year	0	1	2	3
2020	8.333	11.250	10.909	9.333
2021	8.571	12.222	12.000
2022	8.750	13.000
2023	16.000

Step 2: Calculate the adjustment factor

In the previous section, we use the development factor to estimate the ultimate losses. In this section, we will try to use grossing-up factors ($g_{i|j}$) to estimate the ultimate losses.

The grossing-up factor is similar to the development factor, but it is calculated by dividing the underlying average per claim cost in development year $i$ by the average per claim cost in development year $j$. For example, the grossing-up factor from development year 2 to development year 3 for origin year 2020 is calculated as follows:

$$g_{2|3} = \frac{A_{2020,2}}{A_{2020,3}} = \frac{10.909}{9.333} = 1.169$$

The grossing-up factor can be calculated for each cell in the run-off triangle. We can also apply the weighted average method or arithmetic average method to calculate the grossing-up factor for each development year.

In this section, we will use the arithmetic average method to calculate the grossing-up factor. The formula is as follows:

$$\begin{array}{rl} g_{2|3} & = 1.169 \\ \\ g_{1|2} & = (\frac{11.25}{10.909} + \frac{12.222}{12})/2 = 1.025 \\ \\ g_{0|1} & = (\frac{8.333}{11.25} + \frac{8.571}{12.222} + \frac{8.75}{13})/3 = 0.705\\ \\ \end{array}$$

Then, the cumulative grossing-up factor for each cell can be calculated by multiplying the grossing-up factors from the origin year to the development year.

$$\begin{array}{rl} g_{2|3} & = 1.169 \\ \\ g_{1|3} & = g_{1|2} \times g_{2|3} = 1.025 \times 1.169 = 1.198 \\ \\ g_{0|3} & = g_{0|1} \times g_{1|2} \times g_{2|3} = 0.705 \times 1.025 \times 1.169 = 0.846\\ \\ \end{array}$$

Once we get the grossing-up factors, we can calculate the ultimate average cost per claim, dividing the average cost per claim in the latest available development year by the cumulative grossing-up factor.

For example, the ultimate average cost per claim for origin year 2021 is calculated by $\frac{12}{1.169} = 10.265$

	Development year
Origin year	0	1	2	3	Ultimate average per claim
2020	8.333	11.250	10.909	9.333	9.333
2021	8.571	12.222	12.000		10.265
2022	8.750	13.000			10.851
2023	16.000				18.913
g_{i|3}	0.846	1.198	1.169	1

From the above table, we have estimated the average cost per claim and the grossing-up factor for each cell.

Step 3: Calculate the ultimate total number of claims

In the third step, we can estimate the ultimate total number of claims for each origin year with the same method as the above calculation. First, let’s display the table below:

	Development year
Origin year	0	1	2	3
2020	12	16	22	30
2021	14	18	25
2022	16	20
2023	10

Then, we use the arithmetic average method to calculate the grossing-up factor for each development year. The formula is as follows:

$$\begin{array}{rl} g_{2|3} & = \frac{22}{30} = 0.733 \\ \\ g_{1|2} & = (\frac{16}{22} + \frac{18}{25})/2 = 0.724 \\ \\ g_{0|1} & = (\frac{12}{16} + \frac{14}{18} + \frac{16}{20})/3 = 0.776\\ \\ \end{array}$$

The cumulative grossing-up factors are:

$$\begin{array}{rl} g_{2|3} & = \frac{22}{30} = 0.733 \\ \\ g_{1|3} & = 0.724 \times 0.733 = 0.531 \\ \\ g_{0|1} & = 0.776 \times 0.724 \times 0.733 = 0.412\\ \\ \end{array}$$

We can fill in the table below:

	Development year
Origin year	0	1	2	3	Ultimate total claims
2020	12	16	22	30	30
2021	14	18	25		34.1
2022	16	20			37.66
2023	10				24.27
g_i|3	0.412	0.531	0.733	1

Step 4: Calculate the ultimate losses and IBNR reserves

In the last step, we can sum up the ultimate total claims for each origin year by multiplying the ultimate average cost per claim and the ultimate total number of claims.

$$\begin{array}{ll} \text{Ultimate Losses} & = \sum(\text{Ultimate Average Cost per Claim} \times \text{Ultimate Total Number of Claims}) \\ \\ & = 9.333 \times 30 + 10.265 \times 34.1 + 10.851 \times 37.66 + 18.913 \times 24.27 \\ \\ & = 1497.694 \\ \\ \text{Total Paid claims} & = 280 + 300 + 260 + 160 = 1000 \\ \\ \text{IBNR Reserves} & = \text{Ultimate Losses} - \text{Total Paid Claims} \\ \\ & = 1497.694 - 1000 = 497.694 \end{array}$$

Conclusion

In this section, we introduce another method to estimate the IBNR reserves by using the number of claims and grossing-up factors.

The calculation can be summarized as follows:

1. Calculate the average cost per claim for each cell in the run-off triangle.
2. Calculate the ultimate average cost per claim with grossing-up factors or development factors.
3. Calculate the ultimate total number of claims with grossing-up factors or development factors.
4. Estimate the ultimate losses by multiplying the ultimate average cost per claim and the ultimate total number of claims.
5. Estimate IBNR reserves