An actuary wishes to analyse the amounts paid by a group of insurers on their respective portfolios of com$-$

mercial property insurance policies using the models of Empirical Bayes Credibility Theory.

The actuary obtains the following information about the amounts of claim payments made and the number of

policies sold for each of three different insurers. The data obtained are as follows.

Insurer A $14\mathrm{.}2$m $15\mathrm{.}8$m $22\mathrm{.}7$m $19\mathrm{.}0$m

$163189252199$

Insurer B $58\mathrm{.}6$m $63\mathrm{.}1$m $81\mathrm{.}0$m $64\mathrm{.}2$m

$4,4354,7615,5764,581$

Insurer C $123$m $132$m $161$m $133$m

$16,18417,44320,10218,000$

$($i$)$ Analyse the data using EBCT Model $1,$ and calculate the expected total claim payment

to be made by Insurer B in the coming year.

$($ii$)$ Analyse the data using EBCT Model $2,$ and again calculate the expected payout amount

for Insurer B in the coming year, assuming that the expected number of policies sold for

the coming year for Insurer B is $4,800.$ You may use the summary statistics given below,

which have been calculated using the formulae and notation given in the Tables, again

working in millions of pounds. Subscripts $1,2$ and $3$ refer to Insurers A$,$ B and C

respectively.

$$

$2$

$1$

$0\mathrm{.}014667$P x x i i $$

$2$

$$P x x ij j $15\mathrm{.}106461$

$$

$2$

$$P x x $222$ j j $0\mathrm{.}006103$

$2$

$$P x x $22$ j j $0\mathrm{.}006408$

$$

$2$

$$P x x $333$ j j $0\mathrm{.}003979$

$2$

$$P x x $33$ j j $0\mathrm{.}292641$

$($iii$)$ Comment on your results.$($i$)$ Analyse the data using EBCT Model $1,$ and calculate the expected total claim payment

to be made by Insurer B in the coming year.

$($ii$)$ Analyse the data using EBCT Model $2,$ and again calculate the expected payout amount

for Insurer B in the coming year, assuming that the expected number of policies sold for

the coming year for Insurer B is $4,800.$ You may use the summary statistics given below,

which have been calculated using the formulae and notation given in the Tables, again

working in millions of pounds. Subscripts $1,2$ and $3$ refer to Insurers A$,$ B and C

respectively.

${\displaystyle \underset{?}{\overset{?}{}}}{P}_i({\stackrel{}{x}}_1-{\stackrel{}{x}}_i{)}^2=0\mathrm{.}014667,{\displaystyle \underset{?}{\overset{?}{}}}{P}_{ij}(x_{1j}-(\stackrel{}{x}){)}^2=5\mathrm{.}106461$

${\displaystyle \underset{?}{\overset{?}{}}}{P}_{2j}(x_{2j}-{\stackrel{}{x}}_2{)}^2=0\mathrm{.}006103,{\displaystyle \underset{?}{\overset{?}{}}}{P}_{2j}(x_{2j}-(\stackrel{}{x}){)}^2=0\mathrm{.}006408$

${\displaystyle \underset{?}{\overset{?}{}}}{P}_{3j}(x_{3j}-{\stackrel{}{x}}_3{)}^2=0\mathrm{.}003979,{\displaystyle \underset{?}{\overset{?}{}}}{P}_{3j}(x_{3j}-(\stackrel{}{x}){)}^2=0\mathrm{.}292641$

$($iii$)$ Comment on your results.