chain ladder excel caritat
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The chain ladder method
Calculation of the proportionality factors.
The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0
&')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)3 +'&* .'*$, /'))
4 +'** /',*+
5 .'$,6
Development year
Year of origin 0 1 2 3 4 5 60 $&/ $))* $))+ $)) $)).1 $&+ $)) $)). $))2 $&$ $)$) $))+3 $&*/ $)$,4 $&,5
6
01I!2)3 01I!2)3 )**., )**$ )**.&
$))+&+ $))$/ $))+,+ 1E!EL45**$+8 **$+8 **$+8 **&++8 **.*8 1E!EL4**$+8 ))))8 ))))8 )+&)8 )$.8 1E!EL4
The drun-off triangle
p:
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0 &')* +'&,)) +'+$$)) +'+)) +'+&.)) +'+./)) 4,456.00
1 &'&/, +'/.*)) +'/*/)) +',))) +',&))) 4,752.40 4,752.40
2 &',$ .'&+.)) .'&*)) .'+))) 5,40.07 5,455.7! 5,455.7! 3 +'&* .'*$,)) /')))) /')+/$. 0.00 0.00
4 +'** /',*+)) 0.00 0.00 0.00 0.00 0.00 5 .'$, 0.00 0.00 0.00 0.00 6
Development yearYear of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./ 4,456.00
1 &'&/, +'/.* +'/*/ +',) +',&) 4,752.40 4,752.40
2 &',$ .'&+. .'&* .'+) 5,40.07 5,455.7! 5,455.7!
3 +'&* .'*$, /')) 6,046.15 6,057.! 6,0!6.06 6,0!6.06
4 +'** /',*+ 0.00 0.00 0.00 0.00 0.00
5 .'$, 0.00 0.00 0.00 0.00 0.00 0.00
6
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $ "
1 &'&/, $'* &, + $) 22.40 "
2 &',$ $'+,+ .& 10.07 25.71 "
3 +'&* $'/, $)& 26.15 11.24 2!.6! "
4 +'** $'/. ### " " " "
5 .'$, (5,217.00) " " " " "6
Method 2 (using the proportionality factors p
1,i)
The incrementalrun-off triangle
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Development year
Year of origin 0 1 2 3 4 5 6
0 $&/ $))* $))+ $)) $)). $)))1 $&+ $)) $)). $)) $)). $)))
2 $&$ $)$) $))+ $)) $)). $)))
3 $&*/ $)$, $))+ - 01I!2)3 01I!2)3
4 $&, - 01I!2)3 01I!2)3 01I!2)3 01I!2)35 - 01I!2)3 01I!2)3 01I!2)3 01I!2)3 01I!2)36
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6ENT FACT4R7 6ENT RATI47
6ENT 5ERCENTA9E7
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The %old& chain ladder method
Here the proportionality factors are the mean of empirical proportionality factors.
Calculation of the proportionality factors.
The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))
4 +'** /',*+
5 .'$,6
Development year
Year of origin 0 1 2 3 4 5 6
The incrementalrun-off triangle
&'))) &'.)) +'))) +'.)) .'))) .'.))
+')))
+'.))
.')))
.'.))
/')))
/'.))
,')))
f(;# < $&$+*,&)$;f(;# < $+)&*.&); - *)&$),*$,*
+'))) +'.)) .'))) .'.)) /'))) /'.))
+'.)
+'+.)
+'/.)
+'.)
.').)
.'.)
.'+.)
.'/.)
.'.)
/').)
/'.)
f(;# < $)$$&.++;f(;# < $)+)+/$$*; - $+,/*,,&/&*,
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Completion of the cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&,)) +'+$$)) +'+)) +'+&.)) +'+./)) 4,456.00
1 &'&/, +'/.*)) +'/*/)) +',))) +',&))) 4,752.40 4,752.40
2 &',$ .'&+.)) .'&*)) .'+))) 5,40.0 5,455.74 5,455.74
3 +'&* .'*$,)) /')))) /')+/$, 6,057.5 6,0!6.0 6,0!6.0
4 +'** /',*+)) 0.00 0.00 0.00 0.00
5 .'$, 0.00 0.00 0.00 0.00 0.00 0.00 6
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./ 4,456.00
1 &'&/, +'/.* +'/*/ +',) +',&) 4,752.40 4,752.402 &',$ .'&+. .'&* .'+) 5,40.0 5,455.74 5,455.74
3 +'&* .'*$, /')) 6,046.17 6,057.5 6,0!6.0 6,0!6.0
4 +'** /',*+ 0.00 0.00 0.00 0.00 0.00
5 .'$, 0.00 0.00 0.00 0.00 0.00 0.00 6
Development yearYear of origin 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $ "
1 &'&/, $'* &, + $) 22.40 "
Method 1 (using the proportionality factors p i,i+1)
Method 2 (using the proportionality factors p1,i)
The incrementalrun-off triangle
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Development year
Year of origin 0 1 2 3 4 5 6
0 $&/ $))* $))+ $)) $)). $)))1 $&+ $)) $)). $)) $)). $)))
2 $&$ $)$) $))+ $)) $)). $)))
3 $&*/ $)$, $))+ $)) $)). $)))
4 $&, - 01I!2)3 01I!2)3 01I!2)3 01I!2)35 - 01I!2)3 01I!2)3 01I!2)3 01I!2)3 01I!2)36
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The %old& chain ladder method
Here the proportionality factors are the mean of empirical proportionality factors.
Calculation of the proportionality factors.
The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))
4 +'** /',*+
5 .'$,
6
$&)*& $)$$+& $))+&+ $))$/ $))+,+
Completion of the cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&,)) +'+$$)) +'+)) +'+&.)) +'+./)) 4,456.00
1 &'&/, +'/.*)) +'/*/)) +',))) +',&))) 4,752.40 4,752.402 &',$ .'&+.)) .'&*)) .'+))) 5,40.07 5,455.7! 5,455.7! 3 +'&* .'*$,)) /')))) 6,046.15 6,057.! 6,0!6.06 6,0!6.06 4 +'** /',*+)) 6,!71.67 6,$01.52 6,$14.4 6,$47.0! 6,$47.0! 5 . $, 7 204 7 2!6 6$ 7 1! 4 7 1 $4 7 66 66 7 66 66
&')))
.')))
+')))
+'.))
.')))
.'.))
/')))
/'.))
,')))
f(;#
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5RE6I>6
+.*$ +'+./))
+/, +',.+)
+/& .'+..,
.$,. /')/)/
./,& /'*+,)
/+&$ ,'&////
L477 RATI4 &.')/&**&$+).
$$$/.8
5RE6I>6) +'.*$ )) )
)))+,$& +'/, ) .)))/..* +'/& &$* &/))$).. .'$,. ./ /&)))&/ .'/,& $.) $+)) *$)* / +&$ $,/ / )*.
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The chain ladder method
Calculation of the proportionality factors.
The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))
4 +'** /',*+
5 .'$,6
Development year
Year of origin 0 1 2 3 4 5 6
0 $&/ $))* $))+ $)) $)).1 $&,& $)) $)). $))
2 $&,/ $))* $))+
3 $& $)$$
4 $&$
5
6
Development yearYear of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&.
1 &'&/, +'/.* +'/*/ +',) 4,727
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4
5
6
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&,
1 &'&/, 4,5!72
3
45
6
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* 0REF3 0REF3 0REF3 0REF3 0REF3 #%&'
1 &'&/, 0REF3 0REF3 0REF3 0REF3 #%&' #%&'
2 &',$ 0REF3 0REF3 0REF3 #%&' #%&' #%&' 3 +'&* 0REF3 0REF3 #%&' #%&' #%&' #%&' 4 +'** 0REF3 #%&' #%&' #%&' #%&' #%&'
5revious esti%ation (?ithout an@ correction for @ear )#Development year
Year of origin 0 1 2 3 4 5 6
0 &')* 0REF3 0REF3 0REF3 0REF3 #%&' #%&'
1 &'&/, 0REF3 0REF3 0REF3 #%&' #%&' #%&' 2 &',$ 0REF3 0REF3 #%&' #%&' #%&' #%&' 3 +'&* 0REF3 #%&' #%&' #%&' #%&' #%&' 4 +'** 6,!11 6,!71.$7 6,$02.$1 6,$1.!2 6,$1.!2 6,$1.!2
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Development year
Year of ori 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,6
Development yearYear of ori 0 1 2 3 4 5 6
0 &')* $'$/& &* $, ,
1 &'&/, $'* &, + 7
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4
5
6
Development year
Year of ori 0 1 2 3 4 5 6
0 &')* $'$/&
1 &'&/, 1,2202
3
45
6
0REF30REF30REF3 #%&'
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1ifference
&
(#
.)
$,
&,
1ifference
/
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For%ation Caritat' ))
The chain ladder method' *verage co!t
Number of claims
Development year
Year of origin 0 1 2 3 4 5 6
0 $')+&+ $')+.. $')+,. $')+,, $')+,, $')+,,
1 $')+&) $'),$ $'), $')* $'),
2 */.$ */,* */, *,)$
3 *,,) *+, */
4 $')**) $'$$.
5 $'),/&6
$))&$/ $))$&* $)))* )****) $)))))
Development year
Year of origin 0 1 2 3 4 5 6
0 $')+& $')+..) $')+,.) $')+,,) $')+,,) $')+,,) 1,047.701 $')+& $'),$) $'),) $')*) $'),) 1,02!.70 1,02!.702 */. */,*) */,) *,)$) $70.01 $70.01 $70.013 *,, *+,) */) $!7.6! $!7.5! $!7.5! $!7.5! 4 $')** $'$$.) 1,120.06 1,121.05 1,120.$4 1,120.$4 1,120.$45 $'),/ 1,07$.70 1,0!1.20 1,0!2.16 1,0!2.06 1,0!2.06 1,0!2.066
Paid losses
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
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For%ation Caritat' ))
2 &',$ .'&+.)) .'&*)) .'+))) 5,40.07 5,455.7! 5,455.7!3 +'&* .'*$,)) /')))) 6,046.15 6,057.! 6,0!6.06 6,0!6.064 +'** /',*+)) 6,!71.67 6,$01.52 6,$14.4 6,$47.0! 6,$47.0!
5 .'$, 7,204. 7,2!6.6$ 7,1!.4 7,1.$4 7,66.66 7,66.666
Incurred losses
Development year
Year of origin 0 1 2 3 4 5 6
0 +'*,. +'/* +'+*, +'+,) +'+./ +'+./
1 .'$&. +'*+* +',& +',/) +',.)
2 .'/$ .'/&$ .'+* .'+,)3 /', /'$* /'$&$
4 ,'&/ ,'),
5 ,'&.&6
)*/*.. )*,/+/ )**.$& )**,+) $)))))
Development year
Year of origin 0 1 2 3 4 5 6
0 +'*,. +'/*)) +'+*,)) +'+,))) +'+./)) +'+./)) 4,456.001 .'$&. +'*+*)) +',&)) +',/))) +',.))) 4,750.00 4,750.002 .'/$ .'/&$)) .'+*)) .'+,))) 5,455.7! 5,455.7! 5,455.7!3 /', /'$*)) /'$&$)) 6,101.12 6,0!5.25 6,0!5.25 6,0!5.254 ,'&/ ,'),)) 6,$20.15 6,!!6.42 6,!6!.51 6,!6!.51 6,!6!.515 ,'&.& 7,12$.07 6,$61.2 6,$27.0 6,$0$.2$ 6,$0$.2$ 6,$0$.2$6
N>6BER A! C47T (I# A! C47T T4T C47 T4T C47 LATE7T RE7 (I#) $')+,,) +. +. ++./ ++./ +'+./ -$ $ ) ,) + / + / +,.) +,. + ,&) )
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For%ation Caritat' ))
Development year
Year of ori 0 1 2 3 4 5 6
0 &) +$ +$ +& +& +.
1 && +.+ +./ +.* +/)
AERA!E Paid losses
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2 +)$ .. .. ..* 5.60 5.62 5.62 3 +&+ /)$ /$) 6.12 6.1 6.16 6.16 4 ++ /), 6.14 6.16 6.17 6.20 6.20
5 +. 6.67 6.7 6.76 6.77 6.!0 6.!0 6
Development year
Year of ori 0 1 2 3 4 5 6
0 +,, ++& +* +, +. +.
1 +* + +/. +/& +/
2 .* . ./, ./+3 /+ /* /$
4 //, /&+
5 /&6
)*//$ )*,.&) )**+$ )**,.$ $)))))
Development yearYear of ori 0 1 2 3 4 5 6
0 +,, ++& +* +, +. +. 4.251 +* + +/. +/& +/ 4.62 4.62 2 .* . ./, ./+ 5.62 5.62 5.62 3 /+ /* /$ 6.1! 6.16 6.16 6.16 4 //, /&+ 6.1! 6.14 6.1 6.1 6.15 /& 6.60 6.44 6.40 6.! 6.! 6.! 6
RE7 (5# -
AERA!E Incurred losses
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The chain ladder method' +ariant "
Estimation of the linear least s"uare parameters
Development year
Year of origin 0 1 2 3 4 5
0 $&/ $))* $))+ $)) $)).
1 $&+ $)) $)). $))
2 $&$ $)$) $))+
3 $&*/ $)$,
4 $&,5
6
Method 1 (using &cel functions)
colu%n )= )))++
$&,$+
colu%n $= ))),$))/*
colu%n = ))))$
$))+
colu%n &= )))).
$))$/colu%n += $ (no least square regression nee"e"#
Method 2 (using for*ulas)
colu%n )= n .)
10.0
The d-triangle
a)
)
a$
$
a
a&
&
xi
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1.0
)
$)$)
)))).$))$/
colu%n += $))+, (no least square regression nee"e"#
Completion of the d-triangle
Method 1 (using &cel functions)
Development yearYear of origin 0 1 2 3 4 5
0 $&/+ $))* $))&* $))$/ $))+, $))))1 $&&, $)),* $)).$ $))$ 1.0047 $))))2 $&) $))** $))+$ 1.0027 1.0047 $))))3 $&*. $)$,+ 1.0046 1.002 1.0047 $))))4 $&,+ 1.0047 1.007 1.0047 $))))5 1.$4 1.0206 1.004! 1.004 1.0047 $))))
6
Method 2 (using for*ulas)
Development year
Year of origin 0 1 2 3 4 5
0 $&/+ $))* $))&* $))$/ $))+, $))))1 $&&, $)),* $)).$ $))$ 1.0047 $))))2 $&) $))** $))+$ 1.0027 1.0047 $))))
3 $&*. $)$,+ 1.0046 1.002 1.0047 $))))4 $&,+ 1.017$ 1.0047 1.007 1.0047 $))))5 1.$4 1.0206 1.004! 1.004 1.0047 $))))6
xi
yi
xiyi(x
i)
a&
&
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6
Estimated Reserves
Year of origin #e!erve
) )$ +)& ,/+ -/',*+. '&)/
Total$ -4'350
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6
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6
6
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The chain ladder method' +ariant ""
Development year
Year of origin 0 1 2 3 4 5 6
0 )*&) )*,$ )**+ )**, $)))
1 )*/+ )*// )**. )**
2 )**$ )*,. )**/
3 )* )**4 )*/,5
6
t-: 5
: )i
) $ )*&). )*&)+.$ )*/& $*,../ & )**$ *,&.*/& + )* &*.)/+ . )*/,+ +&/
)*,+,.&
t-: .
: $
i
Calculation of the #eighted average of each column $#i%&$ i'%'()*)
The d-triangle
alculation d0
?i)
"i)
?i)"
i)
")
alculation d1
? " ? "
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i) + )**/* &*,+,$ . )**,* +**+*/
)**,++$
t-: : +
i) . $)))) .
$
Completion of the d-triangle
)*,+
)*,,
)**.
)**,+
$))))
". $))))
Development year
Year of origin 0 1 2 3 4 5 6
0 )*&) )*,$ )**+ )**, $))) $)))
1 )*/+ )*// )**. )** 1.000 $)))
2 )**$ )*,. )**/ 0.$$7 1.000 $)))
3 )* )** " 0.$$7 1.000 $)))
4 )*/, 0.$7! " 0.$$7 1.000 $)))
5 0.$75 0.$7! 0.$$7 1.000 $)))6
?i&
"i&
?i&"
i&
"&
alculation d4
?i+
"i+
?i$"
i+
"+
")
"$
"
"&
"+
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6
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Estimated Reserves
Year of origin #e!erve
) )$ ) -$+& -/'))+ -/',*+. -.'$,
Total$ -1('045
F ti B k I
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The chain ladder method' +ariant """
The logarithm of the cumulative run-off triangle.The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))
4 +'** /',*+5 .'$,6
ln%,umulative triangle&
Development year
Year of origin 0 1 2 3 4 5 6
0 ), & &* +) +) +)
1 $ +. +. +/ +/2 / . .* /)3 &. /* ,)4 .) 5 ./6
Estimation of the lineair least s"uare parameters
Method 1 (using &cel functions)colu%n )= )$)+/+*&
).)$*,)*
colu%n $= ) $$)//,&&
0
a)
1
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Cost level ad%ustment of the incremental run-off triangle
The incremental run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,
6 -
n-%i& (n < /#Development year
Year of origin 0 1 2 3 4 5 6
0 .)) +)) &)) )) $)) )))1 +)) &)) )) $)) )))2 &)) )) $)) )))3 )) $)) )))4 $)) )))5 )))6
The /-run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 .'+$/$+ $'),* .&) )$ ,+, Err=.)+1 .'$$,** $')&$ +.. /.. $)))2 .'**/ $'++& .*)) ))3 .'/, $',/** $)&))4 . +, *, $ /. ))
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colu*n 1
$n-: .
i
) $'),* &/+$.,$ & $')&$ .++*+ + $'++& ,&,,**& . $',/** *&+*+*+ / $'/.)) $$$*). , ))) )
, &,)$,.
$&,$)/+colu*n 2
n-: +
i
) & .&) $/$&*.&$ + +.. $&&**/ . .*)) *+**+,& / $)&)) /$+ , ))) )
. $.,,*
.)&$$.$colu*n
&n-: &
i ) + )$ &&&*,$ . /.. $&,/& / )) $&
i'01 ;Di'$ i'0 /i'1
;D$
i'2
;Di'
i'2
/i'2
;D
i'3;D
i'& i'3 /i'3
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colu*n 6
/n-: )
i
) , ))) ), )
)
The completed +,-run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 .'+$/$+ $'),* .&) )$ ,+, Err=.)+ )))1 .'$$,** $')&$ +.. /.. $))) &rr504 0.00 2 .'**/ $'++& .*)) )) 5.41 &rr504 0.00 3 .'/, $',/** $)&)) 15.!2 5.41 &rr504 0.00 4 .'+,*, $'/.)) 50.1 15.!2 5.41 &rr504 0.00 5 .'$,)) 1,71.06 50.1 15.!2 5.41 &rr504 0.00 6
Reversion of the cost level ad%ustmenti-n
Development year
Year of origin 0 1 2 3 4 5 6
0 -.)) -+)) -&)) -)) -$)) )))1 -+)) -&)) -)) -$)) ))) $))2 -&)) -)) -$)) ))) $)) ))3 -)) -$)) ))) $)) )) &))
4 -$)) ))) $)) )) &)) +))5 ))) $)) )) &)) +)) .))6
i'6 ;Di'/ i'6 /i'6
;D/
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Estimated Reserves
Year of origin #e!erve
) Err=.)+
$ Err=.)+ Err=.)+& Err=.)++ Err=.)+. Err=.)+
Total$ rr$504
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Corporate Actuarial Control
The chain ladder method' +ariant "+
The logarithm of the cumulative run-off triangle.The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))
4 +'** /',*+5 .'$,6
ln%,umulative triangle&
Development year
Year of origin 0 1 2 3 4 5 6
0 ), & &* +) +) +)
1 $ +. +. +/ +/2 / . .* /)3 &. /* ,)4 .) 5 ./6
Estimation of the lineair least s"uare parameters
Method 1 (using &cel functions)colu%n )= )$)+/+*&
).)$*,)*
l $ ) $$)//,&&
a)
)
a
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Corporate Actuarial Control
5 ) Err=.)+su% .. Err=.)+
)$)+/+* Err=.)+ ,,.*)
)$$)//, Err=.)+ $,*,+
)$),,. Err=.)+ $))$/*)$)$),&* Err=.)+ ++.+&+
))/+&*,/ Err=.)+ $)//.$/
Err=.)+ Err=.)+ Err=.)+su% Err=.)+
Cost level ad%ustment of the incremental run-off triangle
The incremental run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./1 &'&/, +'/.* +'/*/ +',) +',&)2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))4 +'** /',*+5 .'$,6
n-%i& (n < .#Development year
Year of origin 0 1 2 3 4 5 6
0 . + & $ )1 + & $ )2 & $ )3 $ )
?.
5
0
'0
?)01
'1
?$1
2
'2
?2
3
3?
&3
4
'4
?+4
5
'5
?.5
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Corporate Actuarial Control
colu*n 0
)n-: /
i
) $ Err=.)+ Err=.)+$ Err=.)+ Err=.)+ & Err=.)+ Err=.)+& + Err=.)+ Err=.)++ . Err=.)+ Err=.)+. / Err=.)+ Err=.)+/ , Err=.)+ Err=.)+
Err=.)+Err=.)+
colu*n 1
$n-: .
i
) Err=.)+ Err=.)+$ & Err=.)+ Err=.)+ + Err=.)+ Err=.)+& . Err=.)+ Err=.)++ / Err=.)+ Err=.)+. , Err=.)+ Err=.)+
, Err=.)+
Err=.)+colu*n 2
n-: +
i
) & Err=.)+ Err=.)+
i'0 ;Di') i'0 /i'0
;D)
i'01
;Di'$
i'0
/i'1
;D$
i'2
;Di'
i'2
/i'2
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Corporate Actuarial Control
+n-:
i
) . Err=.)+ Err=.)+$ / Err=.)+ Err=.)+ , Err=.)+ Err=.)+
$ Err=.)+
Err=.)+colu*n 5
.n-: $
i
) / Err=.)+ Err=.)+$ , Err=.)+ Err=.)+
$& Err=.)+
Err=.)+colu*n 6
/
n-: )i
) , 0!AL>E3 0!AL>E3, 0!AL>E3
0!AL>E3
The completed +,-run-off triangle
Development year
Year of origin 0 1 2 3 4 5 60 Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ 0!AL>E31 Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ #-/&2 Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ &rr504 #-/&
i'4
;Di'+
i'4
/i'4
;D+
i'5 ;Di'. i'5 /i'5
;D.
i'6
;Di'/
i'6
/i'6
;D/
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Corporate Actuarial Control
Development year
Year of origin 0 1 2 3 4 5 6
0 Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ 0!AL>E3
1 Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ #-/&2 Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ &rr504 #-/&3 Err=.)+ Err=.)+ Err=.)+ Err=.)+ &rr504 &rr504 #-/& 4 Err=.)+ Err=.)+ Err=.)+ &rr504 &rr504 &rr504 #-/& 5 Err=.)+ Err=.)+ &rr504 &rr504 &rr504 &rr504 #-/& 6 Err=.)+ &rr504 &rr504 &rr504 &rr504 &rr504 #-/&
The completed cumulative run-off triangle
Development yearYear of origin 0 1 2 3 4 5 6
0 ')/)) Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+1 ')&$)) Err=.)+ Err=.)+ Err=.)+ Err=.)+ Err=.)+ &rr5042 '$/+)) Err=.)+ Err=.)+ Err=.)+ Err=.)+ &rr504 &rr5043 '&))) Err=.)+ Err=.)+ Err=.)+ &rr504 &rr504 &rr5044 '+/)) Err=.)+ Err=.)+ &rr504 &rr504 &rr504 &rr5045 '/.$)) Err=.)+ &rr504 &rr504 &rr504 &rr504 &rr504
6 &')+)) &rr504 &rr504 &rr504 &rr504 &rr504 &rr504
Estimated Reserves
Year of origin #e!erve
) ))$ Err=.)+ Err=.)+& Err=.)++ Err=.)+. Err=.)+/ Err=.)+
Total$ rr$504
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Corporate Actuarial Control
The chain ladder method
Calculation of the proportionality factors.
The final lo!! run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 +'*,. +'/* +'+*, +'+,) +'+./ +'+./
1 .'$&. +'*+* +',& +',/) +',.)
2 .'/$ .'/&$ .'+* .'+,)
3 /', /'$* /'$&$4 ,'&/ ,'),
5 ,'&.&6
$)&$+ $)+$ $))+* $))/ $))))
)*/*.. )*,/+/ )**.$& )**,+) $))))) 1E!EL45
Completion of the cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 +'*,. +'/*)) +'+*,)) +'+,))) +'+./)) +'+./)) 4,456.00
1 .'$&. +'*+*)) +',&)) +',/))) +',.))) 4,750.00 4,750.00
2 .'/$ .'/&$)) .'+*)) .'+,))) 5,455.7! 5,455.7! 5,455.7!
3 /', /'$*)) /'$&$)) /'$)$$ 6,0!5.25 6,0!5.25 6,0!5.25 4 ,'&/ ,'),)) 6,$20.15 6,!!6.42 6,!6!.51 6,!6!.51 6,!6!.515 ,'&.& 7,12$.07 6,$61.2 6,$27.0 6,$0$.2$ 6,$0$.2$ 6,$0$.2$6
p:
Method 1 (using the proportionality factors pi,i+1
)
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Corporate Actuarial Control
Calculation of the proportionality factors.
The final lo!! run-off triangle
Development year
Year of ori 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))4 +'** /',*+
5 .'$,6
),+$ )*, )**., )**$ )**.&
6ENT FAC $&)*& $)$$+& $))+&+ $))$/ $))+,+ 1E!EL45
Completion of the cumulative run-off triangle
Development year
Year of ori 0 1 2 3 4 5 6
0 &')* +'&,)) +'+$$)) +'+)) +'+&.)) +'+./)) 4,456.00
1 &'&/, +'/.*)) +'/*/)) +',))) +',&))) 4,752.40 4,752.40
2 &',$ .'&+.)) .'&*)) .'+))) 5,40.07 5,455.7! 5,455.7!
3 +'&* .'*$,)) /')))) 6,046.15 6,057.! 6,0!6.06 6,0!6.064 +'** /',*+)) 6,!71.67 6,$01.52 6,$14.4 6,$47.0! 6,$47.0!5 .'$, 7,204. 7,2!6.6$ 7,1!.4 7,1.$4 7,66.66 7,66.66 6
p:
Method 1 (using the proportionality factors pi,i+1
)
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Corporate Actuarial Control
6ENT FACT4R7
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Corporate Actuarial Control
The +ylder! ea!t uare ethod.
Calculation of the proportionality factors.
The cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./
1 &'&/, +'/.* +'/*/ +',) +',&)
2 &',$ .'&+. .'&* .'+)
3 +'&* .'*$, /'))4 +'** /',*+
5 .'$,6
Development year
Year of origin 0 1 2 3 4 5 60 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,6
The regression "ataset ) $ & line colu%n L) L$ L L&
& )* ) ) $ ) ) )
The incrementalrun-off triangle
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p
tatistiues de la r3gression
Coefficient "e )**..+)&.
Coefficient " )**$$)).+)&Coefficient " )*)$))/
Erreur-t@pe +&*,+&,.+
4servations $
ANALG7E 1E !ARIANCE
egr3 de liert3 **e des carryenne des car ' aleur critiue de '
RHgression $) //*$&$ //*$&$ $$$&//&, /,+*E-))*
RHsi"us $) .*.&+,. .*.&+,.Total ) //+&///
oefficients &rreur"type tatistiue t roailit3 e pour seuil de c pour seuil de
Constante $/.. &,,../* ++/.+.,/ )))$). ++/+.&+$, .,))+//
L) -$//+. $))* -.,,/$/ ))))$,), -&)/,.&*,+ -$)*/)$
L$ -$/).. $))* -..,&/* ))))&/$& -+,,.&*,+ -*/&*/)$&
L -$+&. *)/,), -+*&*$. ))))/+/$ -),$+,/ -,,/$/,&
L& -$$/*. */&).,*$ -&*+)&/ ))),&* -$&))&.++ -.)*/$+..,L+ -/)., &)/)./ -$*//*, )),).,+ -$*&&$//. $*$//.
C) &.&$$,. $))* $./,/, &*&.E-)), *+/)$. +$,&$)&**
C$ $$).,. $))* &,)+&.+ )))&&&+*+ +/)/+/)$.,, $,++.)&**
C -$/$,. *)/,), -)../+/* ).*)$+, -)*+),,/$* +.*),,/
C& -$)) */&).,*$ -)&&,+*$ ),+,.$* -,/)$)++,. ./)$)++&
C+ -+ &)/)./ -)$&/)*.*+ )*+++/$ -,*/$//+ /+./$//
ANALG7E 1E7 R7I1>7 R5ARTITI4N 1E7 5R4BABILIT7
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The incured proected run-off triangle
Development year
Year of origin 0 1 2 3 4 5 60 &'.. $'$+ ($+$# (,*# ($# $
1 &'/$$ $'$& (# ()# & )
2 &',*& $'&/. $)) $/ ) /
3 +')+, $'/$* &.+ +$/ +,+ .$/
4 +'/$$ '$& *$ *) $')& $'))
5 .'$, ', $'.+ $'./ $'/++ $'//6
&'.&$ $$).,. -$/$,. ($))# (+# -
Estimated Reserves
Year of origin #e!erve
) )$ ) +&+ +)$,. *
Total$
#e!idual!
Development year
Year of origin 0 1 2 3 4 5 6
0 (&+ &* $) */ -1 (++# $)* $$* ++ (#2 , $)* (+,# ($+)#3 $* .* (.$#
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+ ) $ & + .L+ C) C$ C C& C+ C.
) $ ) ) ) ) )
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our seuil de c our seuil de confiance $5,08
++/+.&+ .,))+//
-&)/,.&** -$)*/)$
-+,,.&** -*/&*/)$&
-),$+,, -,,/$/,&
-$&))&.++ -.)*/$+..,-$*&&$//& $*$//.
*+/)$ +$,&$)&**
+/)/+/)$& $,++.)&**
-)*+),,/ +.*),,/
-,/)$)++& ./)$)++&
-,*/$// /+./$//
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($'//.#
($'/)/#
($'++#
($'$,)#
(/)/#
-
$'//
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The +ylder! ea!t uare ethod.
Estimation of the parameters
The triangle of the model
Development yearYear of origin 0 1 2 3 4 5 6
0
1
2
3
4
5
6
The incremental run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,6
"teration
1. ).)))) )&)))) ))))) )$)))) )).))) ))+))) +*+ .&)& /)*) /*., *+2. )./,& )*&) ))$)&& )))&+ )))$// )))+$ ..,+ .* /,/, ,+.$ /$)3. ).,)/, )$$, )))*&+ )))&+. )))$+* )))&,, ../ .*) /,) ,+/& //4. ).,), )$,, )))*&& )))&++ )))$+* )))&,/ ../ .*)& /,) ,+/& //5. ).,), )$,,, )))*&& )))&++ )))$+* )))&,/ ../ .*)& /,) ,+/& //
))/.,re!caled ),),/, ),))) ))$$./ )))+, )))$+ )))+// +.). .+/ /)$* /*./ /+&.
The resulting parameters
) ,),/, + .). +,.$
p)7
)p
$7
)p
7
)p
&7
)p
+7
)p
.7
)p
/7
)
p)7
$p
$7
$p
7
$p
&7
$p
+7
$p
.7
$
p)7
p
$7
p
7
p
&7
p
+7
p)7
&p
$7
&p
7
&p
&7
&
p)7
+p
$7
+p
7
+
p)7
.p
$7
.
p)7
/
p0
p1
p2
p3
p4
p5
70
71
72
73
74
su% p:
p r 7 r
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),))) .'+/+,
))$$./ /')$*$.
)))+, /'*./)&)
)))$+ /'+&.&.&
)))+// ,'&,$&
The completed incremental run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &'$&, $'$/+/ .$) $*+ &) $))1 &'/*/ $'+,/+, /&+ &&. $)), 25.4$2 +'.*.. $'/.$. /*/$ .,) 11.0$ 2!.06 3 +'*/, $',$. )++ 2$.70 12.!1 2.42 4 +'..+) $',&,. 74.42 27.4! 11.!5 0.00 5 .'$,)) 1,$$0.45 !5.25 1.4! 1.5! 4.6 6
The completed cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &'$&, +'+)+& +'+./*+ +'+,/$ +'+++ +'.).+1 &'/*/ .'&+/&& .'+)*., .'+&* .'++** 5,46!.4!2 +'.*.. .'+,$ .'*.+&$ .'*)) 5,$$1.10 6,01$.163 +'*/, /')) /'$/ 6,$10.$7 6,$2.7! 6,$56.204 +'..+) /'*$/) 6,66.02 6,$.50 6,405.6 6,45.5 5 .'$,)) 7,207.45 7,2$2.70 7,24.1! 7,7.76 7,72.12 6
p1
r 71
r
p2
r 72
r
p3
r 73
r
p4
r 74
r
p5r 75
r
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The linear *rithmetic eparation ethod
Estimation of the parameters
The triangle of the model
Development year
Year of origin 0 1 2 3 4 5
0
1
2
3
4
5
6
The incremental run-off triangle
Development year
Year of origin 0 1 2 3 4 50 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,6
,'&))
In this e;a%ple the seperation %etho" starts fro% the non-cu%ulative triangle e assu%e therefotriangle that ?as given al?a@s has an esti%ate of one acci"ent at the en" of @ear )
r00
r11
r22
r33
r44
r55
r01
r12
r23
r34
r45
r02
r13
r24
r35
r03
r14
r25
r04
r15
r05
alculation of r5and
5
5
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Estimation of the parameters corresponding to the future calender years.
) +'&*$ +'$./
$ +'/)/ +',.*
.'+) .'&/$
& .',*$ .'*/+
+ /',$) /'./,. ,'& ,'$,)
/ 7,77
, !,75
!,$7!
* $,5!1
$) 10,1!4
$$ 10,7!7$ 11,$0
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6
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The completed cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5
0 &')* +'&, +'+$$ +'+ +'+&. +'+./1 &'&/, +'/.* +'/*/ +',) +',&) 4,752 &',$ .'&+. .'&* .'+) 5,42$ 5,4543 +'&* .'*$, /')) 6,045 6,055 6,0!14 +'** /',*+ 6,!66 6,!$ 6,$04 6,$2 5 .'$, 7,1!0 7,25! 7,2!6 7,2$! 7,2! 6
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6
6
re that the
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,&)+*...$
)*,./
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6
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The linear *rithmetic eparation ethod
Estimation of the parameters
The triangle of the model
Development year
Year of origin 0 1 2 3 4 5 6
0
12
3
4
5
6
The incremental run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,6
In this e;a%ple the seperation %etho" starts fro% the non-cu%ulative triangle e assu%e therefore thatthe triangle that ?as given al?a@s has an esti%ate of one acci"ent at the en" of @ear )
r00
r11
r22
r33
r44
r55
r01 r12 r23 r34 r45r02
r13
r24
r35
r03
r14
r25
r04
r15
r05
alculation of r and
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E ti ti f th t di t th f t l d
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Estimation of the parameters corresponding to the future calender years.
) +'&*$ +//,).$&
$ +'/)/ +,+/$,,/
.'+) .,*/)*
& .',*$ .,*/*$.
+ /',$) /.&&))$&,
. ,'& ,/,$$$.
/ !0!.$4$2, !$$2.446
1000.05$$
* 11127.251
$) 1277.7!4!
$$ 176!.!5$2
$ 1516.26$1
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The completed cumulative run-off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./1 &'&/, +'/.* +'/*/ +',) +',&) 4,752 &',$ .'&+. .'&* .'+) 5,40 5,456 3 +'&* .'*$, /')) 6,046 6,057 6,0!6 4 +'** /',*+ 6,!6$ 6,!$! 6,$10 6,$42 5 .'$, 7,25$ 7,42 7,74 7,!! 7,4246
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The inear 9eometric eparation ethod
Estimation of the parameters
The triangle of the model
Development year
Year of origin 0 1 2 3 4 5 6
0
1
2
3
4
5
6
The incremental run-off triangle
Development yearYear of origin 0 1 2 3 4 5 6
0 &')* $'$/& &* $, , $
1 &'&/, $'* &, + $)
2 &',$ $'+,+ .&
3 +'&* $'/, $)&
4 +'** $'/.
5 .'$,
6
In this e;a%ple the seperation %etho" starts fro% the non-cu%ulative triangle e assu%e therefore that
the triangle that ?as given al?a@s has an esti%ate of one acci"ent at the en" of @ear )
r00
r11
r22
r33
r44
r55
r01
r12
r23
r34
r45
r02 r13 r24 r35r03
r14
r25
r04
r15
r05
alculation of r5and
5
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Estimation of the parameters corresponding to the future calender years
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Estimation of the parameters corresponding to the future calender years.
) /$ .*/
$ ,$. ,$
,,& *
& . *+.
+ $)&, $)/$. $*$ $$,,
/ 12$.4
, 141.0
152.6
* 164.
$) 175.$
$$ 1!7.5
$ 1$$.2
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6
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The completed cumulative run-off triangle
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The completed cumulative run off triangle
Development year
Year of origin 0 1 2 3 4 5 6
0 &')* +'&, +'+$$ +'+ +'+&. +'+./1 &'&/, +'/.* +'/*/ +',) +',&) 4,7512 &',$ .'&+. .'&* .'+) 5,42$ 5,452 3 +'&* .'*$, /')) 6,046 6,056 6,0!14 +'** /',*+ 6,!65 6,!$4 6,$05 6,$15 .'$, 7,21 7,$$ 7,42$ 7,441 7,470 6
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The completed incremental run-off triangle
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