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𝐶𝑗,𝑘)

k 1 2 3 … … 6 7

j

1 …

2

…

… …

… …

t-1

t …

k es el año en que se realiza el pago del siniestro

j es

el a

ño

en

el q

ue

los

sin

iest

ros

ocu

rrie

ron

0<

j<I+

1

SIN

IEST

RO

S A

CU

MU

LAD

OS 𝐶 , 𝐶 ,

𝐶 , 𝐶 ,

𝐶 ,

𝐶 ,

𝐶 ,

𝐶 , 𝐶 ,

𝐶 ,

𝐶 ,

𝐶𝑗𝑘

{𝐶 , , 𝐶 , , 𝐶 ,3, … 𝐶 , }, … , {𝐶𝑗, , 𝐶𝑗, , 𝐶𝑗,3, … 𝐶𝑗, } 𝑐𝑜𝑛 𝑗 ≠ 𝑘 𝑠𝑜𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑖𝑒𝑡𝑒𝑠 ( 𝑒𝑐. 1)

𝐶𝑗𝑘

𝐸[𝐶𝑗,𝑘+ | 𝐶𝑗, , 𝐶𝑗, , … 𝐶𝑗,𝑛 ] = 𝐶𝑗𝑘 𝑓𝑘 𝑑𝑜𝑛𝑑𝑒 1 ≤ 𝑗 ≤ 𝐼, 1 < 𝑘 ≤ 𝐼 − 1 (𝑒𝑐. 2)

𝑓𝑘

𝑓𝑘 =∑ 𝐶𝑗,𝑘+ 𝑘𝑗=

∑ 𝐶𝑗𝑘 𝑘𝑗=

𝐶𝑗, = 𝐶𝑗, + 𝑛 ∗ ∏ 𝑓𝑤

𝑤=𝑘+𝑛

𝐶𝑗𝑘

𝑉𝑎𝑟[𝐶𝑗,𝑘+ | 𝐶𝑗,0, 𝐶𝑗, , … 𝐶𝑗,𝑘 ] = 𝐶𝑗𝑘 ∝𝑘

∝𝑘 > 0

αk =

1

I − k − 1∑Cj,k (

Cj,k+

Cjk− fk)

I k

j=

𝜎𝑛 𝑓𝑗 ≈ 1 𝜎𝑗

= 0

𝜎0 , 𝜎

, 𝜎 , … . 𝜎𝑗

��𝑗

σj-32

σj-22=σj-22

σj-12 y .σj-3

2 >σj-22

σj = min (

σj 3

σj =

σj

σj , mi n ( . σj 3

, σj ))

𝐷 = {𝐶𝑗𝑘|𝑗 + 𝑘 ≤ 𝐼 + 1}

𝐸[𝐶𝑗, | 𝐷 ] = 𝐶𝑗, 𝑗 ∗ 𝑓 + 𝑗 ∗ … ∗ 𝑓

𝐶 ,

1 < 𝑗 < 𝐼 + 1

𝐸𝑗[𝑋 ] = 𝐸[𝑋| 𝐶𝑗, , … 𝐶𝑗, + 𝑗]

𝐸[𝐶𝑗, | 𝐷 ] = 𝐸[𝐶𝑗, ]

= 𝐸𝑗[𝐸(𝐶𝑗, |𝐶𝑗, , … , 𝐶𝑗, )]

= 𝐸𝑗[𝐶𝑗, ∗ 𝑓 ]

𝐶𝑗,

𝐸𝑗[𝐶𝑗, ] ∗ 𝑓 + 𝑗 ∗ …∗ 𝑓 = 𝐶𝑗, + 𝑗 ∗ 𝑓 + 𝑗 ∗ …∗ 𝑓

𝐶𝑗, 𝐸[𝐶𝑗, | 𝐷 ]

fk , 1 ≤ k ≤ I − 1

Bk[X ] = {Cji| i ≤ k, i + j ≤ I + 1}, 1 ≤ k < I

Bk[X ] Cji

fk 𝑓𝑘

𝐸[𝐶𝑗,𝑘+ | 𝐶𝑗, , 𝐶𝑗, , … 𝐶𝑗,𝑛 ] = 𝐶𝑗𝑘 𝑓𝑘 𝑑𝑜𝑛𝑑𝑒 1 ≤ 𝑗 ≤ 𝐼, 1 < 𝑘 ≤ 𝐼 − 1

Bk

E[Cj,k+ | Bk ] = E[Cj,k+ | Cj,I, … Cj,k] = Cj,k ∗ fk

fk 𝑓𝑘

E[fk ] = fk

𝐸[𝐶𝑗, | 𝐷 ] = 𝐸[𝐶𝑗, ] = 𝐶𝑗, + 𝑗 ∗ 𝑓 + 𝑗 ∗ … ∗ 𝑓

𝑓 + 𝑗 ∗ …∗ 𝑓 =𝐸[𝐶𝑗, | 𝐷 ]

𝐶𝑗,

fk

E[fk| 𝐵𝑘 ] =∑ 𝐸[𝐶𝑗,𝐼| 𝐵𝑘 ]𝐼−𝑘𝑗=1

∑ 𝐶𝑗𝑘𝐼−𝑘𝑗=1

=∑ 𝐶𝑗,𝑘+1𝐼−𝑘𝑗=1

∑ 𝐶𝑗𝑘𝐼−𝑘𝑗=1

= 𝑓𝑘

fk

𝑓𝑘 𝐵𝑘[𝑋 ] = {𝐶𝑗𝑖| 𝑖 ≤ 𝑘, 𝑖 + 𝑗 ≤

𝐼 + 1}, 1 ≤ 𝑘 < 𝐼 𝑖 ≤ 𝑘

𝐸[𝑓��𝑓�� ] = 𝐸[𝐸(𝑓��𝑓��| 𝐵𝑘) ] = 𝐸[𝑓��(𝑓��| 𝐵𝑘) ]

= 𝐸[𝑓��(𝑓𝑘) ] = 𝐸[𝑓��] ∗ 𝐸[𝑓��]

𝐶𝑗, − 𝐶𝑗,

𝐶𝑗,

𝐶𝑗,

𝑒. 𝑐.𝑚 (𝑅��) = 𝐸[( 𝐶𝑗, − 𝐶𝑗, ) | 𝐷]

𝐷 = {𝐶𝑗𝑘|𝑗 + 𝑘 ≤ 𝐼 + 1}

𝑅𝑗 𝑅��

𝑒. 𝑐.𝑚 (𝑅��) = 𝐸[( 𝑅�� − 𝑅𝑗) | 𝐷]

𝐶𝑗,

𝐶𝑗,

𝑒. 𝑐.𝑚 (𝑅��) = 𝐸[( 𝑅�� − 𝑅𝑗) | 𝐷] = 𝐸[(𝐶𝑗, − 𝐶𝑗, )|𝐷]

𝑒. 𝑐.𝑚 (𝑅��) = 𝑒. 𝑐.𝑚 (𝐶𝑗, )

𝐶𝑗,

𝐶𝑗,

𝑒. 𝑐.𝑚 (𝐶𝑗, ) = 𝐸 [(𝐶𝑗, − 𝐶𝑗, ) ] 𝑒𝑐. 2

𝑉𝐴𝑅(𝑋) = 𝐸(𝑋 ) − [𝐸(𝑋)]

𝑒. 𝑐.𝑚 (𝐶𝑗, ) = 𝐸 [(𝐶𝑗, − 𝐶𝑗, ) ] = 𝑉𝑎𝑟(𝐶𝑗, − 𝐶𝑗, ) + [𝐸(𝐶𝑗, − 𝐶𝑗, )]

𝐶𝑗,

𝑉𝐴𝑅(𝑋 + 𝑐) = 𝑉𝑎𝑟(𝑋), 𝑐 𝑒𝑠 𝑢𝑛𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒

𝑉𝑎𝑟(𝐶𝑗, − 𝐶𝑗, ) + [𝐸(𝐶𝑗, − 𝐶𝑗, )] = 𝑉𝑎𝑟(𝐶𝑗, ) + [𝐸(𝐶𝑗, − 𝐶𝑗, )]

𝑒𝑐. 3

𝐸[𝐶𝑗, | 𝐷 ] = 𝐸[𝐶𝑗, ]

𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌),

𝑒. 𝑐.𝑚 (𝐶𝑗, ) = 𝑉𝑎𝑟(𝐶𝑗, ) + [𝐸(𝐶𝑗, − 𝐶𝑗, )] = 𝑉𝑎𝑟(𝐶𝑗, |𝐷) + [𝐸(𝐶𝑗, |𝐷) − 𝐶𝑗, ]

𝑒. 𝑐.𝑚 (𝐶𝑗, ) = 𝑉𝑎𝑟(𝐶𝑗, |𝐷) + [𝐸(𝐶𝑗, |𝐷) − 𝐶𝑗, ] 𝑒𝑐. 4

𝐶𝑗,𝑘

𝛼 𝑘

𝑉𝑎𝑟[𝐶𝑗,𝑘+ | 𝐶𝑗,0, 𝐶𝑗, , … 𝐶𝑗,𝑘 ] = 𝐶𝑗𝑘 ∝𝑘

𝛼𝑘 =

1

𝐼 − 𝑘 − 1∑𝐶𝑗,𝑘 (

𝐶𝑗,𝑘+

𝐶𝑗,𝑘− 𝑓𝑘)

𝑘

𝑗=

, 1 ≤ 𝑘 ≤ 𝐼 − 1

𝑒. 𝑐.𝑚 (𝑅��)

𝑒. 𝑐.𝑚 (𝑅��) = 𝐶𝑗

∑𝛼𝑘

𝑓𝑘

𝑘= + 𝑗

(1

𝐶𝑗,𝑘 +

1

∑ 𝐶𝑥𝑘 𝑘𝑥=

)

𝐸𝑗 = 𝐸(𝑋|𝐶𝑗, ,….,𝐶𝑗, + 𝑗 )

𝑉𝑎𝑟𝑗 = 𝑉𝑎𝑟(𝐶𝑗,𝑘+ |𝐶𝑗, ,….,𝐶𝑗, + 𝑗 )

𝑒. 𝑐.𝑚 (𝐶𝑗, ) = 𝑉𝑎𝑟(𝐶𝑗, |𝐷) + [𝐸(𝐶𝑗, |𝐷) − 𝐶𝑗, ]

𝐸𝑗= = 𝐸(𝑋|𝐶 , ,….,𝐶 , + 𝑗 ) = 𝐶 ,

𝑉𝑎𝑟𝑗= = 𝑉𝑎𝑟(𝐶 , |𝐶 , ,….,𝐶 , ) = 0

𝐶 ,

𝐶 ,

𝑉𝑎𝑟(𝐶𝑗, |𝐷) = 𝑉𝑎𝑟(𝐶𝑗, )

𝑉𝑎𝑟(𝐶𝑗, ) = 𝐸𝑗[𝑉𝑎𝑟(𝐶𝑗, )] + 𝑉𝑎𝑟𝑗[𝐸(𝐶𝑗, ,|𝐶𝑗, ,….,𝐶𝑗, )]

𝛼𝑘

𝑉𝑎𝑟(𝐶𝑗, ) = 𝑉𝑎𝑟(𝐶 , ) = 𝛼 𝐶 ,

𝐶 ,

𝐸(𝐶𝑗, ,|𝐶𝑗, ,….,𝐶𝑗, ) = 𝐸(𝐶 , ,|𝐶 , ,….,𝐶 , ) = 𝑓 𝑗+ ∗ 𝐶 , + 𝑗, = 𝑓 + ∗ 𝐶 , + ,

𝑉𝑎𝑟𝑗[𝐸(𝐶𝑗, ,|𝐶𝑗, ,….,𝐶𝑗, )] = 𝑉𝑎𝑟𝑗[𝑓 + ∗ 𝐶 , + ,]

𝑉𝑎𝑟𝑗[𝐸(𝐶𝑗, ,|𝐶𝑗, ,….,𝐶𝑗, )] = 𝑓 + ∗ 𝛼 𝐶 ,

= 𝐸𝑗[𝐶𝑗, ] ∗ 𝛼 + 𝑉𝑎𝑟𝑗[𝐶𝑗, ]𝑓

= 𝐸𝑗[𝐶𝑗, ] ∗ 𝑓 ∗ 𝛼

+ 𝐸𝑗[𝐶𝑗, ] ∗ 𝑓 ∗ 𝛼

+ 𝑉𝑎𝑟𝑗[𝐶𝑗, ]𝑓

𝐶𝑗,𝑘 𝑓𝑘

Cj, I+ +j* ∑ fI+ -j

I-

k=I+ -j

* …. * fk- *αk * fk+

* …. * fI-

{𝐸[𝐶𝑗, | 𝐷] − 𝐶𝑗, } = 𝐶𝑗, + 𝑗

∗ [(𝑓 𝑗 ∗ … .∗ 𝑓 ) − ( 𝑓 𝑗 ∗…∗ 𝑓 )]

𝒇𝒌 𝑪𝒋,𝒌

𝐶𝑗, 𝑓��

{𝐸[𝐶𝑗, | 𝐷] − 𝐶𝑗, } = 𝐶𝑗, + 𝑗

∗ [(𝑓 𝑗 ∗ … .∗ 𝑓 ) − ( 𝑓 𝑗 ∗…∗ 𝑓 )]

𝐶𝑗, + 𝑗 ∗ [(𝑓 𝑗 ∗ … .∗ 𝑓 ) − ( 𝑓 𝑗 ∗…∗ 𝑓 )]

= 𝐶𝑗, + +𝑗 ∗ ∑ 𝑓 + 𝑗

𝑘= + 𝑗

∗ … .∗ 𝑓𝑘 ∗ 𝛼𝑘 ∗ 𝑓𝑘+

∗ … .∗ 𝑓

= 𝐶𝑗, + +𝑗 ∗ ∑ 𝑓 + 𝑗

𝑘= + 𝑗

∗ … .∗ 𝑓𝑘 ∗ 𝛼𝑘 ∗ 𝑓𝑘+

∗ … .∗ 𝑓

= 𝐶𝑗, ∗ ∑𝛼𝑘 𝑓𝑘

⁄

= 𝐶𝑗, + +𝑗

𝑘= + 𝑗

𝐹 = [ 𝑓 + 𝑗 ∗ … .∗ 𝑓 ] − [ 𝑓 + 𝑗 ∗ … ∗ 𝑓 ]

𝑆𝑘 = 𝑓 + 𝑗 ∗ … ∗ 𝑓 = ∏ 𝑓𝑥

𝑥= + 𝑘

2 ≤ 𝑘 ≤ 𝐼 + 1

𝐹 = [ 𝑓 + 𝑗 ∗ … .∗ 𝑓 ] − [𝑆𝑘] = ∑ 𝑆𝑤

𝑤= + 𝑗

𝐹 = [ ∑ 𝑆𝑤

𝑤= + 𝑗

]

= ∑ 𝑆𝑤

𝑤= + 𝑗

+ 2 ∗ ∑ 𝑆𝑥𝑆𝑤

𝑥<𝑤

𝑆𝑤 = 𝐸[𝑆𝑤

| 𝐵𝑤] 𝑦 𝑡𝑎𝑚𝑏𝑖é𝑛 𝑆𝑥𝑆𝑤 = 𝐸[𝑆𝑥𝑆𝑤| 𝐵𝑤]

𝑆𝑤 𝑆𝑥𝑆𝑤

E [(fk − fk)|Bk] = Var [fk|Bk] =∑Var [Ci,k+1|Bk]

(∑ Ct,kI−kt=1 )

2 =αk2

∑ Ct,kI−kt=1

I−k

i=1

𝐸[ 𝑆𝑘 |𝐵𝑘] =

[(𝑓 𝑗 ∗ … .∗ 𝑓 ) ∗ ( 𝑓 𝑗 ∗…∗ 𝑓 )]

∑ Ct,kI kt=

𝐹 = [∑ 𝑆𝑤 𝑤= + 𝑗 ]

∑ 𝐸[ 𝑆𝑘

|𝐵𝑘] 𝑤= + 𝑗

𝑓𝑘 𝛼𝑘

𝐹

𝐹 = 𝑓 + 𝑗 ∗ … .∗ 𝑓𝑘 ∗ 𝛼𝑘

∗ 𝑓𝑘+ ∗ … .∗ 𝑓

∗∗ ∑

αk

fk

⁄

∑ Ct,kI kt=

𝑤= + 𝑗

𝑒. 𝑐.𝑚 (��) = [𝑒. 𝑐.𝑚. (��)] + 𝑐𝑗, ∑ 𝑐𝑖,

𝑖=𝑗+

∗ ∑

αk

fk

⁄

∑ Ct,kI kt=

𝑤= + 𝑗

𝒇𝒌 ��𝒌

𝜶𝒌 𝒚 𝑪𝒋,𝒌

k 1 2 3 … … 6 7

j

1 …

2

…

… …

… …

t-1

t …

k es el año en que se realiza el pago del siniestro

SIN

IEST

RO

S A

CU

MU

LAD

OS

j es

el a

ño

en

el q

ue

los

sin

iest

ros

ocu

rrie

ron

0<

j<I+

1

𝐶 , 𝐶 ,

𝐶 , 𝐶 ,

𝐶 ,

𝐶3,

𝐶 ,

𝐶 , 𝐶 ,

𝐶 ,

𝐶3,

𝐶 ,

𝒇, 𝒇, 𝒇, 𝒇, 𝒌 + 𝒇 𝒌 𝒇,

𝐶𝑗𝑘 =∑𝑆𝑗,

𝑘

=

k 1 2 3 … … I

pago

j sin

1 …

2

…

…

…

t-1

t

SIN

IEST

RO

S P

AG

AD

OS

𝑆 , 𝑆 ,

𝑆 ,

𝑆 ,

𝑆 ,

k 1 2 3 4 5 6 7

pago 2011 2012 2013 2014 2015 2016 2017

i sin

1 2011 120 50 30 20 15 10 2

2 2012 150 100 80 70 20 15

3 2013 200 180 170 100 50

4 2014 305 245 182 121

5 2015 380 220 150

6 2016 400 350

7 2017 500

REC

LAM

AC

ION

ES

k 1 2 3 … … I

pago 2011 2012 2013 2014 2015 2016 2017

j sin

1 2011 …

2 2012

… 2013

… 2014

… 2015

t-1 2016

t 2017 SIN

IEST

RO

S A

CU

MU

LAD

OS

𝐶 , 𝐶 ,

𝐶 ,

𝐶 ,

𝐶 ,

𝑓 𝑓 𝑓

𝐶𝑗𝑘 =∑𝑆𝑗,

𝑘

=

𝐶 =∑𝑆 , = 𝑆 , +𝑆 ,

=

= 120 + 50 = 170

𝐶4,3 =∑𝑆4, = 𝑆4, +𝑆4, +𝑆4,3

3

=

= 305 + 245 + 182 = 732

𝑓𝑘

𝑓𝑘 = ∑ 𝐶𝑗,𝑘+ 𝑘𝑗=

∑ 𝐶𝑗,𝑘 𝑘𝑗=

k 1 2 3 4 5 6 7

pago 2011 2012 2013 2014 2015 2016 2017

j

1 120 170 200 220 235 245 247

2 150 250 330 400 420 435

3 200 380 550 650 700

4 305 550 732 853

5 380 600 750

6 400 750

7 500 SIN

IEST

RO

S A

CU

MU

LAD

OS

k 1 2 3 … … I

j

1 …

2

…

…

…

t-1

t

calc

ulo

de

fact

ore

s d

e

des

arro

llo

año de pago k

año

de

ocu

rren

cia

del

sin

iest

ro j

𝐶 , 𝐶 ,

𝐶 ,

𝐶 ,

𝐶 ,

𝑓, 𝑓, 𝑓,

𝑓 =∑ 𝐶𝑗,1+17−1=6𝑗=1

∑ 𝐶𝑗,17−1=6𝑗=1

=𝐶1,2+𝐶2,2+𝐶3,2+𝐶4,2+𝐶5,2+𝐶6,2

𝐶1,1+𝐶1,2+𝐶1,3+𝐶1,4+𝐶1,5+𝐶1,6= 0+ 50+380+550+ 00+ 50

0+ 50+ 00+305+380+400= 1.74

𝑓 =∑ 𝐶𝑗,1+17−1=6𝑗=1

∑ 𝐶𝑗,17−1=6𝑗=1

=𝐶1,3+𝐶2,3+𝐶3,3+𝐶4,3+𝐶5,3

𝐶1,2+𝐶2,2+𝐶2,3+𝐶2,4+𝐶2,5= 00+330+350+ 3 + 50

0+ 50+380+550+ 00= 1.31

𝑓3 =∑ 𝐶𝑗,1+17−1=6𝑗=1

∑ 𝐶𝑗,17−1=6𝑗=1

=𝐶1,4+𝐶2,4+𝐶3,4+𝐶4,4

𝐶1,3+𝐶2,3+𝐶3,3+𝐶4,3= 0+400+ 50+853

00+330+550+ 3 = 1.17

𝑓4 = ∑ 𝐶𝑗,4+17−4=3𝑗=1

∑ 𝐶𝑗,47−4=3𝑗=1

= 𝐶1,5+𝐶2,5+𝐶2,5

𝐶1,4+𝐶1,4+𝐶1,4= 35+4 0+ 00

0+400+ 50= 1.07

𝑓5 = ∑ 𝐶𝑗,4+17−4=3𝑗=1

∑ 𝐶𝑗,47−4=3𝑗=1

= 𝐶1,6+𝐶2,6

𝐶1,5+𝐶2,5= 45+435

35+4 0= 1.04

𝑓 = ∑ 𝐶𝑗,4+17−4=3𝑗=1

∑ 𝐶𝑗,47−4=3𝑗=1

= 𝐶1,7

𝐶1,6= 4

45= 1.01

𝐸[𝐶𝑗,𝑘+ | 𝐶𝑗, , 𝐶𝑗, , … 𝐶𝑗,𝑛 ] = 𝐶𝑗𝑘 𝑓𝑘

k 1 2 3 4 5 6 7

pago 2011 2012 2013 2014 2015 2016 2017

j

1 120 170 200 220 235 245 247

2 150 250 330 400 420 435

3 200 380 550 650 700

4 305 550 732 853

5 380 600 750

6 400 750

7 500

1.74 1.31 1.17 1.07 1.04 1.01

CA

LCU

LO D

E FA

CTO

RES

DE

DES

AR

RO

LLO

𝒇, 𝒇, 𝒇 𝒇, 𝒇, 𝒇,

Cj,k = Cj,I j+ ∏ fx

I

x=I k+

Rj = CJi − CI,I j+

𝐶 , = 𝐶 , + ∏ 𝑓𝑥

𝑥= +

= 𝐶 , ∗ 𝑓 = 435 ∗ 1.01 = 439

𝐶 ,4 = 𝐶 , + ∏ 𝑓𝑥

𝑥= +

= 𝐶 , ∗ 𝑓 = 500 ∗ 1.74 ∗ 1.31 ∗ 1.17 = 1,332

k 1 2 3 … … 6 7

pago 2011 2012 2013 2014 2015 2016 2017

j sin

1 2011 …

2 2012

… 2013

… 2014 …

… 2015 …

t-1 2016

t 2017 …

SIN

IEST

RO

S A

CU

MU

LAD

OS 𝐶 , 𝐶 ,

𝐶 , 𝐶 ,

𝐶 ,

𝐶 ,

𝐶 ,

𝐶 , 𝐶 ,

𝐶 ,

𝐶 ,

k 1 2 3 4 5 6 7

pago 2011 2012 2013 2014 2015 2016 2017

j

1 120 170 200 220 235 245 247

2 150 250 330 400 420 435

3 200 380 550 650 700

4 305 550 732 853

5 380 600 750

6 400 750

7 500

1.74 1.31 1.17 1.07 1.04 1.01

PA

GO

S ES

TIM

AD

OS

𝒇, 𝒇, 𝒇 𝒇, 𝒇, 𝒇,

𝐶 ,4

𝐶 ,

Rt = Ct,I − Ct,I t+ = 735 − 700 = 35

R3 = C3, − C3,5 = 735 − 700 = 35

R = C , − C3, = 1289 − 750 = 539

VAR(Cj, | … . , Cj,k) = Cj,Iαk

𝛼𝑘 =

1

𝐼 − 𝑘 − 1∑𝐶𝑗,𝑘

𝑘

𝑗=

(𝐶𝑗,𝑘+

𝐶𝑗,𝑘− 𝑓𝑘)

, 1 ≤ 𝑘 ≤ 𝐼 − 1

𝛼 =

1

7 − 2 − 1∑ 𝐶𝑗, (

𝐶𝑗, +

𝐶𝑗, − 𝑓 )

𝑘= =5

𝑗=

=1

5∑𝐶𝑗, (

𝐶𝑗,3

𝐶𝑗, − 𝑓 )

5

𝑗=

k 1 2 3 4 5 6 7

pago 2011 2012 2013 2014 2015 2016 2017

j

1 120 170 200 220 235 245 247

2 150 250 330 400 420 435 439

3 200 380 550 650 700 728 735

4 305 550 732 853 912 948 957

5 380 600 750 877 938 975 984

6 400 750 982 1,148 1,228 1277 1289

7 500 870 1,139 1,332 1,425 1482 1496

1.74 1.31 1.17 1.07 1.04 1.01

PA

GO

S ES

TIM

AD

OS

𝒇, 𝒇, 𝒇 𝒇, 𝒇, 𝒇,

k 1 2 3 4 5 6 7

D 1.74 1.31 1.17 1.07 1.04 1.01

i

1 120 170 200 220 235 245 247 247 247 0

2 150 250 330 400 420 435 439 439 435 4

3 200 380 550 650 700 728 735 735 700 35

4 305 550 732 853 912 948 957 957 853 104

5 380 600 750 877 938 975 984 984 750 234

6 400 750 982 1,148 1,228 1277 1289 1,289 750 539

7 500 870 1,139 1,332 1,425 1482 1496 1,496 500 996

suma 6,147 4,235 1,912

año de pago k

año

del

sin

iest

ro j

reserva

estimada

siniestros

pagados a la

fecha

siniestros

totales

estimados

𝑠𝑖 𝑗 = 1 → 𝐶 , (𝐶 ,3𝐶 , − 𝑓 ) = 170 ∗ (

200

170− 1.31)

= 0.01

𝑠𝑖 𝑗 = 2 → 𝐶 , (𝐶 ,3𝐶 , − 𝑓 ) = 250 ∗ (

330

250− 1.31)

= 5.63

𝑠𝑖 𝑗 = 3 → 𝐶3, (𝐶3,3𝐶3, − 𝑓 ) = 380 ∗ (

550

380− 1.31)

= 29.23

𝑠𝑖 𝑗 = 4 → 𝐶4, (𝐶4,3𝐶4, − 𝑓 ) = 550 ∗ (

732

550− 1.31)

= 14.24

𝛼 =1

4 (0.01 + 5.63 + 29.23 + 14.24) = 12.27

k 1 2 3 4 5 6 7

j

1 120 170 200 220 235 245 247

2 150 250 330 400 420 435 439

3 200 380 550 650 700 728 735

4 305 550 732 853 912 948 957

5 380 600 750 877 938 975 984

6 400 750 982 1,148 1,228 1277 1289

7 500 870 1,139 1,332 1,425 1482 1496

1.74 1.31 1.17 1.07 1.04 1.01

PA

GO

S ES

TIM

AD

OS

añ

o d

e si

nie

stro

año de pago k

𝒇, 𝒇, 𝒇 𝒇, 𝒇, 𝒇,

k 1 2 3 4 5 6 7

j 1 2 3 4 5 6 7

I-1-j 5 4 3 2 1

i sin

1 2011 1.37 0.01 0.18 0.17 0.25 0.00

2 2012 19.08 5.63 6.67 0.04

3 2013 69.62 29.23 6.88

4 2014 74.21 14.24

5 2015 27.49

6 2016

7 2017

38.35344 12.2768 4.574101 0.107364 0.016327 0

𝜶

𝜶

(𝑠. 𝑒 (𝑅𝑗))

= 𝐶𝑗 ∑

𝛼𝑘

𝑓𝑘

𝑘= + 𝑗

(1

𝐶𝑗𝑘 +

1

∑ 𝐶𝑥𝑘 𝑘𝑥=

) ,

(𝑠. 𝑒 (𝑅 )) = 𝐶

∑𝛼𝑘

𝑓𝑘

𝑘= +

(1

𝐶 𝑘 +

1

∑ 𝐶𝑥𝑘 𝑘𝑥=

) = 𝐶 , ∑

𝛼𝑘

𝑓𝑘

𝑘=

(1

𝐶 𝑘 +

1

∑ 𝐶𝑥𝑘 𝑘𝑥=

)

= 𝐶 , ∗𝛼

𝑓 ∗ (

1

𝐶 , +

1

𝐶 , ) = 439 ∗

0.0163

1.01∗ (1

435+1

245) = 41.7374

𝑒. 𝑐.𝑚(𝑅𝑇) = ∑

{

(𝑠. 𝑒 (𝑅𝑗))

+ 𝐶𝑗 ∗ ( ∑ 𝐶𝑥

𝑥=𝑗+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦= + 𝑗)

}

𝑗=

𝑒. 𝑐.𝑚(𝑅𝑇) =∑

{

(𝑠. 𝑒 (𝑅𝑗))

+ 𝐶𝑗, ∗ ( ∑ 𝐶𝑥,

𝑥=𝑗+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦=8 𝑗)

}

𝑗=

1 0.0000

2 41.7374

3 33.2034

4 19.5886

5 0.6658

6 0.1391

7 0.0000ESTI

MA

CIÓ

N D

EL E

RR

OR

CU

AD

RÁ

TIC

O M

EDIO

PO

R

AÑ

O

j

(𝑠. 𝑒 (𝑅 )) + 𝐶 , ∗ ( ∑ 𝐶𝑥,

𝑥=𝑗+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦= )

= (𝑠. 𝑒 (𝑅 )) + 𝐶 , ∗ (𝐶3, + 𝐶4, + 𝐶5, + 𝐶 , + 𝐶 , ) ∗ (

2𝛼

𝑓 ⁄

𝐶 , + 𝐶 , + 𝐶3, + 𝐶4, + 𝐶5, )

= 41.73 + 439 ∗ (735 + 957 + 984 + 1,289 + 1,496) ∗ (

2(0)1.02⁄

245 + 435 + 728 + 948 + 975) = 41.73

(𝑠. 𝑒 (𝑅3)) + 𝐶3, ∗ ( ∑ 𝐶𝑥,

𝑥=3+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦=5)

= (𝑠. 𝑒 (𝑅 )) + 𝐶3, ∗ (𝐶4, + 𝐶5, + 𝐶 , + 𝐶 , ) ∗ (

2𝛼5

𝑓5⁄

𝐶 ,5 + 𝐶 ,5 + 𝐶3,5 + 𝐶4,5+

2𝛼

𝑓 ⁄

𝐶 , + 𝐶 , + 𝐶3, + 𝐶4, )

= 33.20 + 735 ∗ (957 + 984 + 1,289 + 1,496) ∗ (

2(0.1163)1.08⁄

235 + 420 + 700 + 912+

2(0)1.02⁄

245 + 435 + 728 + 948)

= 79.46

(𝑠. 𝑒 (𝑅4)) + 𝐶4, ∗ ( ∑ 𝐶𝑥,

𝑥=4+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦=4)

k 1 2 3 4 5 6 7

3.0276 1.7161 1.3689 1.1449 1.0816 1.0201

38.3534 12.2768 4.5741 0.1074 0.0163 0.0000

1 k=1 120 170 200 220 235 245 247 0.00

2 k=2 150 250 330 400 420 435 439 41.74

3 k=3 200 380 550 650 700 728 735 79.46

4 k=4 305 550 732 853 912 948 957 632.62

5 k=5 380 600 750 877 938 975 984 35,510.77

6 k=6 400 750 982 1,148 1,228 1,277 1,289 228,622.45

7 k=7 500 870 1,139 1,332 1,425 1,482 1,496 0.00

TOTALj

𝜶 𝒇

= (𝑠. 𝑒 (𝑅4)) + 𝐶4, ∗ (𝐶5, + 𝐶 , + 𝐶 , ) ∗ (

2𝛼4

𝑓4⁄

𝐶 ,4 + 𝐶 ,4 + 𝐶3,4+

2𝛼5

𝑓5⁄

𝐶 ,5 + 𝐶 ,5 + 𝐶3,5+

2𝛼

𝑓 ⁄

𝐶 , + 𝐶 , + 𝐶3, )

= 33.20 + 735 ∗ (957 + 984 + 1,289 + 1,496)

∗ (

2(0.1073)1.14⁄

220 + 400 + 650+

2(0.1163)1.08⁄

235 + 420 + 700+

2(0)1.2⁄

245 + 435 + 728) = 632.62

(𝑠. 𝑒 (𝑅4)) + 𝐶4, ∗ ( ∑ 𝐶𝑥,

𝑥=4+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦=4)

= (𝑠. 𝑒 (𝑅5)) + 𝐶5, ∗ (𝐶 , + 𝐶 , ) ∗ (

2𝛼3

𝑓3⁄

𝐶 ,3 + 𝐶 ,3+

2𝛼4

𝑓4⁄

𝐶 ,4 + 𝐶 ,4+

2𝛼5

𝑓5⁄

𝐶 ,5 + 𝐶 ,5+

2𝛼

𝑓 ⁄

𝐶 , + 𝐶 , )

= 19.58 + 957 ∗ (984 + 1,289 + 1,496)

∗ (

2(4.57)1.36⁄

200 + 330+

2(0.1073)1.14⁄

220 + 400+

2(0.1163)1.08⁄

235 + 420+

2(0)1.02⁄

245 + 435) = 35,511.04

(𝑠. 𝑒 (𝑅5)) + 𝐶5, ∗ ( ∑ 𝐶𝑥,

𝑥=5+

) ∗

(

∑

2𝛼𝑦

𝑓𝑦⁄

∑ 𝐶𝑤,𝑦 𝑦𝑤=

𝑦=3)

= (𝑠. 𝑒 (𝑅 )) + 𝐶 , ∗ (𝐶 , ) ∗ (

2𝛼

𝑓 ⁄

𝐶 , +

2𝛼3

𝑓3⁄

𝐶 ,3+

2𝛼4

𝑓4⁄

𝐶 ,4+

2𝛼5

𝑓5⁄

𝐶 ,5+

2𝛼

𝑓 ⁄

𝐶 , )

= 0.6658 + 984 ∗ (1,289 + 1,496)

∗ (

2(12.27)1.71⁄

170+

2(4.57)1.36⁄

200+

2(0.1073)1.14⁄

220+

2(0.1163)1.08⁄

235+

2(0)1.02⁄

245)

= 35,511.04

(𝑠. 𝑒 (𝑅𝑇))

(𝑠. 𝑒 (𝑅𝑇)) = 264,887.03

X~Lognormal(μ,σ2)

μ≈μ=Ln(RT)-σ2

2 σ2 ≈σ2 =Ln(1+

(s.e (RT))2

(RT)2)

𝜎 ≈ 𝜎 = 𝐿𝑛 (1 +264,887.03

(1912) ) = 0.07

𝜇 ≈ �� = 𝐿𝑛(𝑅𝑇) −𝜎

2 = 𝐿𝑛(1912) −

0.07

2= 7.52

𝑋~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙(𝜇 = 0.07, 𝜎 = 7.52)

1 − β

𝑥 𝛽% = 𝑒𝑥𝑝(�� + �� ∗ Φ(1 − 𝛽%))

Φ(1 − 𝛽%) 1 − 𝛽%

1 0.0000

2 41.7374

3 79.4614

4 632.6164

5 35510.7687

6 228622.4471

7 0.0000

TOTAL 264887.03

ESTI

MA

CIÓ

N D

EL E

RR

OR

CU

AD

RÁ

TIC

O T

OTA

L j EC

Φ(1 − 𝛽%) = Φ(99.5%) = 2.576

𝑥99.5% = 𝑒𝑥𝑝(�� + �� ∗ Φ(0.995)) = 𝑒𝑥𝑝(0.07 + √7.52 ∗ 2.576) = 2,208.04