Multiplicative methods of calculating RPNU
Multiplicative methods are based on the following assumptions:
1) the amount  the amount of payments in the financial period with the number j for insured events that occurred during the period with the number i ,  are independent random variables (not cumulative, accumulated ^ (we denote by capital letters), but made in a particular year yy (denoted with a lowercase letter))
2) the value can be represented in a multiplicative form:
(4.13)
where  the total cost of insurance payments that occurred in the period with the number i, expressed in constant monetary units;  the fraction of the amount paid during the fiscal year with the number  a measure of inflation and the impact of external factors on the size of payments in the period with the number ().
The estimation of the parameters of such a model can be made using the method of least squares, which consists in minimizing the following expression:
where  arbitrary weights, you can choose them equal to 1, and you can use them to take into account the importance, relevance, reliability of existing data.
The summation occurs over all elements of the development triangle.
Considering the model without inflation:
(4.14)
and equating to zero the partial derivatives of and we get the system:
(4.15)
(4.16)
It can be solved using the method of successive approximation.
Choosing some probabilities as initial values, it is iteratively calculated from the above formulas (4.15) and (4.16) and and, as a rule, the process converges very quickly, after several iterations, as a result of which, according to the estimates, and it is possible to calculate estimates of future payments according to the model formula (4.14):
Then the payment matrix of the insurance company is completed and the reserves are calculated by formulas (4.12) and (4.6) to ensure future payments.
One of the advantages of the method is that it is not necessary to have all the information included in the development triangle. If, for example, the insurance company's loss adjustment policy suddenly changes within one year, it is possible to ignore the previous information and analyze the triangle, excluding its first diagonals. The multiplicative method has the advantage that for the evaluation of inflation factors it uses all the information. The results obtained in this way are more stable with respect to small oscillations of the observed values.
The drawbacks of the method lie in the fact that the solution may not be unique, but also in the large number of parameters that need to be evaluated.
The multiplicative model and the chain ladder model are related by the following relationship:
where
The value is the total cost of insurance payments that occurred during the period with the number i, and the size, called delay factor , is the fraction of those payments that are made after the first j payment periods.
A multiplicative model with constant inflation is written in the form
where  the total cost of insurance payments for the period with the number r in the current price of money; its share paid after the j payment period.
Assuming
we get
Both methods give similar reserve values obtained with their use. Multiplicative methods have the advantage that they use all information to evaluate inflation factors, and the results obtained with their help are more stable with respect to small fluctuations of observed values.
Taylor's separation method
Another kind of multiplicative method is the method of separation, using a triangle not with elements , but with elements
where  the total number of insurance slips in the ith period;  average payments per one insured event in the jm period for losses of the ith period.
The model (4.13) is written as follows:
(4.17)
Taylor's method uses the following technique for estimating parameters (Table 4.12). Let the quantity be the sum of the expressions that are on a diagonal, i.e. all sums paid during the calendar period with the number to
, all the amounts ,(4.18)
Table 4.12
The Taylor dropout triangle
The periods of the occurrence of losses, i 
Loss payment periods, j 

1 
2 
3 
... 
N 

1 

2 

3 

... 

N 
Size  the sum of the values in the column with the number :
(4.19)
Then we get
(4.20)
Since by definition , we get
Given that , we get:
Next we find:
As a result, we get:
(4.21)
(4.22)
This makes it possible to calculate the lower triangle by (4.17):
Usually it is required to estimate the impact of future inflation and to find estimates using some forecasting method. As a rule, a simple linear trend often quite justifies itself.
Completing the matrix of average payments for one insured event in the year jm for losses of the yth year estimates (4.17) and assuming , that the amount of payments that must be made from the (N + 1) th period and until the losses of the ith period are fully settled, we estimate these quantities using the following formulas:
 for the 1st year; (4.23)
 in the following years. (4.24)
Therefore,
(4.25)
is an estimate for , and the method is completed in the same way as the chain ladder method  by formula (4.11), the expected total payments for losses of each year are calculated:
(4.26)
Then, using formula (4.12), we estimate the reserves of losses by the end of each ith period:
Finally, we find the estimate of the total RNPU using formula (4.6):
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