On links between the different definitions of time in IBNR run-off triangles

1. On links between the different definitions of time in IBNR run-off triangles Ilja V. Erokhin, Andrey A. Kudryavtsev St.Petersburg, Russia 2007

2. Multiplicative Model for loss development: where is loss payments in the moment of u+d on accidents which happened inthe moment u; is a factor connecting to time moment u of accident origin; is a factor connecting to delay d of the development period; is a factor connecting to calendar time of payment which isdetermined as u+d; is a constant; is a dimension of loss development matrix.

3. So we have three time directions: d t = u+d u u - the time of origin of the accident insured; d - the time of development of loss settlement; t - the calendar time of loss settlement.

4. Time definition Notation Value the time of origin of the accident insured u 0 0 … 0 1 1 … 1 … n n … n the time of developm. (delay) of loss settlement d 0 1 … n 0 1 … n … 0 1 … n the calendar time of loss settlement t 0 1 … n 1 2 … n+1 … n n+1 … 2n Co-ordinates of elements of loss development matrix

5. u d t u 0 d 0 t Using the Principal Component Method, we have The variance-covariance matrix for time coordinates: After denotation: we can calculate eigenvalues and eigenvectors of this matrix.

6. If Y is eigenvalue and {X1, X2, X3} are eigenvector of the matrix then the following equations are stated: We have: where – i- th eigenvector, i {1, 2, 3}. – i-th eigenvalue, i {1, 2, 3},

7. No. 1 2 3 Eigenvalues Y(1) = 3 X Y(2) = X Y(3) = 0 The structure of eigen-vectors X1(1) = X2(1) X1(2) = – X2(2) X1(3)= X2(3) X2(1) = X1(1) X2(2) = – X1(2) X2(3)= X1(3) X3(1) = 2 X1(1) X3(2) = 0 X3(3) = – X1(3) It may be shown thateigenvalues and eigenvectors of thevariance-covariance matrix are: We can see that the elements of run-off matrix are characterized by two independent parameters regardless of its dimension.

8. The new coordinate system is derived from the existing system (according to the Principal Component Method) by equations: The additional feature of the procedure of rotating is As result the system of rotating equations is Principal Component (75% of system variation) see eigenvalues Calendar time

9. d LOSS DEVELOPMENT MATRIX D u U • Figure. Two variants of coordinate systems for loss development: {u, d} and {U, D}: • u – the time of origin of the accident insured; • d – the time of development (delay) of loss settlement; • U – the calendar time of loss settlement; • D – the orthogonal axe for time of loss settlement. The angle of rotation does not depend on the specific feature of the data investigated.

10. The forecasting efficiencies of different coordinate systems using in linear regression models are identical. For example: The following models can get equal prognoses: and (if the procedures of parameter estimation are identical for both of them)

11. Equivalent models must be compared on ratios of statistical significance of time factors (the basis of decision-making about an exception of non-significant factors). The theoretical arguments about properties of time definitions will be able to explain statistical values of the formulated criterion. As a result for any loss development model we can recommend the optimum coordinate system without subjectivity of statistical sample choice.

12. Conclusions: Overparametrization of the Multiplicative Model; 1. The Chain Ladder Model is got from the Multiplicative Model by exception from consideration the effect of payment time. However it carries 75% of loss payment size dependence on position in the system examined; 2. The interpretation of excluding the effect of payment time is the exception of inflation and dynamics of business. Such assumptions are unsuitable for modelling of Russian insurance market. 3.