formulae and tables for actuarial exams

8/14/2019 Formulae and Tables for Actuarial Exams

1/45

THEMAICLHOD R

xot fuct

x2 x3p()=e1+-+'"! 3

Ntr og

x log +x)=Il +x = x--+ . (-1


2/45

AULS

ayr series (e vaiableh2fexh)=(x

hf'x -f"x

.2!ayr ser (wo rales)

fx h,y k=fx+(x f;(x

+ h2fxy)+hf(x,y)+k2;x,y)+ .

egra y pas

b d b f b duum=[uv] vm m mDoe iegras (changg te r integration

The domn of negron here s th se of ue x for whcax


3/45

13 SOLVING EQUATOS

4

ewton-Raphs med

If x is a suciently good approximaio t a roo of he equaiolex)= 0 he (provied cnvergec oc) a br aroxiioIS

* x) =--.{'

tegrang factors

The interatig ator or slng e dierenia uaiondy .+ P)y Q) S:

x (JP(d)

Secndoder dierence equaos

eeral solutn he ierne euaon

al2+bXn1+cXn=0is:

i b2 4ac > 0: Xn AAJ + BAisnt rea rs, A " 12)

f b2 4ac 0: i (A+ B)n(eua eal oos A = 1 A)

i b2-ac < 0: xn = Aos8+Bsn8)complex roos, 2 rei6)

were A and 12 a te roots he uadra euaiA2+b+CO


4/45

G CO

.5

Dio

r()

ftX-ledt, >O

opee

rex= r 1rn n n

BYS' FORMU

e > 2 ,Anbe a collecion omaly xs d exauiveevens w P 0, i , n

o any evenB uc a C) 0

P(AiIB)=(Ii)(AJ i=I,2 ,nLPBIAj)P(A

j=l


5/45

TC DISTR UONSNotaon

PF Probabli nctio,p(x)D Pobabiy densty nco f(D Disbut ncon F(x) Prbat grn ncG()MF Mme gera nc M(tNte ee fmulae hav e ted eow ts inices ha() tee s no sple foa r (b) he ncion oes ave nie vue (c he ncto quals zer.

DSC SUOS

Binomial disribution

ams:

F:

D

Mmns

Cecent

p (posive inteer 0 < p


6/45

BerouH distribion

he Beu disbutn e sme s he bnl sbnwh mete = I

Pisso isbo

Pamee:

PF

DF:

Mmen

Cecen

sewnes

>

exp(x)- xOI,2x

The dsibun ncn baed n he scbes secnG s

M e( )

EX X

7


7/45

8

Ngave mal sr

Paraetes:

PF

GF

MG

Momens

Coecen

k p ( positiv ntge, < p < 1 with q=1-p[X I)

k x-p{X)= P q , x=kklk+2..k-I

(s = lkl qs )M [ pet )k

-qe

E(X)!, v(X= kp

pof skewss: f


8/45

egativ binomia isn - Type ees

P

Moens :

Cocn

o skwnss:

k > 0 0 < < wi = - ) k q x= 012

GS= r

M rq

(X)

2-jGeerc distrbutionhgomericdstrun s thesae as the negave binomal

disibuonwhparamete k= 1.


9/45

Uniform dsrbutio (dicret)

Pameters: a b, h (a < b h> , b -a i a uipl of h )

P:

MF

Moments

hpx)= , x=a,a+h,a+2h . -,bba+h

G(s)=s sh [h a 1

ba+h sh-lh eCb+h)t ea

M(t) = -ba 1

X=2

a+b vX)=(b a(ba 2h)12

OUS STRBOS

10

Sandrd normal disrb - N(O,I)

Pameters one

PDF

DF:

MGF:Moe

The dbo con s abaed in he satsicalabe seion.

(X= X=

r re + )E(X ) = 2r ( r) ' r=246, .r -

2


10/45

Nom (Guan) itribo N,a2Paraeters: I,

2a>0)

PDF: 2 x f()= xp - -a5 a1 2 2

MGF M() =elt2c t

Moens E(=I vaX)=a2

xpi daaeer

D

D:

M

oens

oeento sewness

A A>O

fx= >O

x=l-eA

M(t) fT t


11/45

Gamma dstonarats a, A (a>O AO)

PDF' f a a x =x e x>na)

Wh 2a is itg obabiliis r th gaaistribto an fo using th rlatonsi

G

Mots

Coit

f kwss

EX I ' (X =

X ) 123

n

2Cie sibt

chisqa isribuio w ' gs o ro i th sa as

th gaa stribuio it paatrs

an = .2 2h istrbtio tio fo t iuar istibtio is ault i

th stastica tabs sco


12/45

Unom sriution (connuous) a bParametes: ,b


13/45


14/45

e dson ee pmeer eion)

aeters a A k (aO AO, kO)

PD

Moments E(X)=(a>1), var(X)k(k+-lA

(>2 )-I (a)(-2)

Web dsio

arametes c Y (cO YO)

D

DF: F =e

Y

Moment rl +./Y c

B isibion

Paeters a A Y (a 0 A0 Y 0

D

D:

Moment:

F 1 aAX ) /y X )=r a- r 1+ - , =123, . . r


15/45

COMPUND DISTRBUONS

6

Conditonal expectatin and vaiE(Y E[(Y IXI]va ) vr Y I X)] [ Y I X)]

Mmts f a opud dstrbution X1,X2, . are lD rndom ibles with MF MxCta idependet oegti iteg-vlued d vaable hS = + .. + Xl i S 0 wen 0 as th flwgpopis:Mean: E S NE(Varian va(S N)v + v [ X]G:

Compound Piss dston

en AmVariance AmThird centra

w A = N) a m r


16/45

ecsve oa o ege-vaed disios(ab,O) cla

g p r) r 0,12 an Ij PX= =13.f P N r wr =a+P-Ir=23 tn

rb

)go=Po and gr =La 1ijg-Jr=12,3,..j

und ossondistrbtiof N as a osson istiution wt n a a b a

A goe' an gr=LIig-' =23r


17/45

2.4 TRUNCATE MOENTS

18

Nol disbuion

i he PDF o he ,2) ditibuion then

x ( =[


18/45

No

RLAIONSHPSBETWEEN

B

DSC

LX,p '

=

u \

B

Pn.p

minX

LX

(mp

,/=np

-

LXi1

po:sn

.

k=-.

I.r=

L eXlogX'X-

/

C

N

r

n

( l-

p)

.-y

.

10 r=/ ="

/ +(X

'

\

xr

t

r =

I

La+bX)

LX

XVI

a=1v

z

'

j

2

a

/apL,

V;

k=

lv\

k.

. F Iae

eto

- "

v

ralk

r

a

(af2+P+I)


19/45


20/45

3 STATISTCL METHOD

3.1 AME M D VAAEhe random sample !, . ,xn ha the following ampe

momens:

ampe ma

ap varce s2 = 1 xf } 1 i132 PTC FN (O MO)

One sample

F a ingle sampe of sze ne he omal moe N I,c2): -! t ! d nl/F

n

Two samples

For wo ndepeet ames o szs m ad ue the oalmoes X N and Y I:

FmlnSy y


21/45

Under the addonal assumption that O = O :

x y

lm2

S -p

wr S = {(m S1 + (lSf} s the pood saple+2

vaanc.

MAXIU LIKELHOOD ESIMATORSAsymptoc ditibuio

If s te maxum keoo stmator of a arameter e bsd ona sae X ten s aymtocay noay dstrbtd wth eane and vaance equa to te Cramr-Rao ower bound

Likelhood rao tes

ere p = ax 10gLo

nd e p ma 0gLHHj

appoxatey (under Ho

s the mamm og-keood for te

od under Ho (n whch there are ee arameters)

s t mamum og-kehoo for the

mode under u H (in wc thereare q e pareers)


22/45

34 LINE R REGSIO DEL WITH ORMAL EOR

4

Model

Intermedia cacuaons

n )2 2 -2sx xi - X = Xj -nxi=l ;=1

2 2 2S = Yi - Y =Y - ny=

Sxy = L(X X(Y - y) =LYi - nx=1 i=

aame smes

Disibuon of


23/45

Vaiae of edied ea esose

An aiioa c2 mu be added o oai e arian of preie inivia repoe.Tesig he oeao oe

sr= xSXSy

rIf=Oen - -2 1- Fse ansfomaio

z

N Zp,_l approximaeyere z = arIOg(l +r ad z tanp =O/1 + .lr lpSum of squaes eaiosi

nYi y2 L(Yi v;)2 Le y2i ;=

5


24/45

5 ALYSI O VE

6

Sgle facor nma me

2Y N + i,O = 1,2,..k j 2 ;

k kwhere >i' wh I;'; 0;=1 ;=1eredae calcos (sms sqares)

k ni k ni 2al SST= Yij -y.)2=IyJ-Yi=lj1 =1 n

k k2

2B SS (- -)2 Yi Yn reame: B= Y;. -Y.. k,i= ;=

Residual RS SBVarance esmae

Sasa es

Under he rorte nul yohes

SB /SR Fk-1 n-k-l n-


25/45

6 GEESED NEAR ODESxnna fay

or rndom vbl Y om h xponnti fy wh ntupt S n sc pmr


26/45

3.7 BAYSAN EODS

28

eltiop eeen teo n p dtibtion

oseroro PorX Lkhoo

Te osero sbon I. o paaee s eae oe o disbo ia e keoo ncion

D

orml nrm moeIf . is a anom sape o sze o a N( ,2) sion,e

2 is kon an e pio sbuon o e paaee ! sN(!5) en poseo sbuon o s:


27/45


28/45

9 EMRCAL BAYES CREB - MOD

0

ata requirements

{ 2 N,}=1} {P, ,, N}=12,} epeen e aggege am n e } yea m e i k; e oepondng kvoumenrmda aaons

l2j' J=,p*=_12 -n N N -j

j i P

Yi XI'= i , X= I p. . p .. Pj1 i i=j=1Paraeer estimaio

Quanti

E[m 8]

va[(8)]

Estimator

( N n 2 { I I 2})

- - 'PX i -X) -IIp(x -X )P * Nn I ) N n l I= J= =1 JCrdb aor

2Pi

Z n s 8)2 p - -j=1 I var[m(8)]


29/45

4 PD

casing/dcasig auiy fs

n

a; G v Da n Accumulan aco fo varal s as


30/45

5 SURVIVAL MODES

OTA "AS

Sv patis

Gpz

k's

_ gcX(c-l) -Ax = A+B (P where =eGtzk fr

Te Gmprtz-Makeh grdutin fula, ene by rsstts tha

wher t na fncion ox nd olY and o aeymas of egree s reecvey.


31/45

52 EC SIMAONGeewood's fo fo e ve of he Kan Meeeso

- [ , jvar[ F] = \ F(t)151njn-dj )

Vae f he Nelson-aen esae o he inegae azad

5.3 ITY SSMS

54

sso/ q1 1- tqx x i n inge 0 t 1)

GENER L MAKOV ODE

Kgv oa diferential eqaona gh_ ( jh . h h )t I x - L - +rh

3


32/45

55 GRADUAnON SS

Grupng f signs s

ere are n, otve gn nd n2 negatve g and G denoe eobeed nmber o oitive then :

Cra vaue or e gropng o ig e ae ablaed eaa able eon o a vae o , ad n2' Fo argevae o l and 2 e nra aoximaion an be ed

Srial crran s

m. L (zi-Zi+j m } _,r " I) m-LZi_:)2

m

X O axiaey .Varanc adjsm far

4 1ir _i__x Li1

1 mee Li

m!

ee 1i e propoon o ve age wo ave exaty ipoie


33/45

56 MUIPLE DCMNT ESFo a ll cren abl w re crnt , an y,ach fo vr ya of age x, + 1 n t nl centabl, t

57 OUO PROJECTI MDLS

ogsc odel

dP p_ kP a gnal oltn pe ppe d Cpe-Pi +k

whr C i a conat

35


34/45


35/45

63 EMMS AD SVS

64

remm onvesion eionship eeen annies anssurnes

Siiar raionip od o ndowent arane po ia ) .X:ne prem eseve

a - tt X

av 1- t x -

axiia oa od or o aane poii itate and ) .XnI t:n IH'S DA QOWhoe fe ssne

Siiar ora od or or ype o poiiee sae mode

3


36/45


37/45

tg s

y = n { : Bs + } wr > a < 0 n;o

Onsnlc pcss

7.3 O CAO ODS

xll frla

If UI an U a npnn rano vaab o h U(O)

rbuon

ar ipndn anar noa varaba t

an 2 ar npndn ran varab fm -)buon d S J2 + vl n conna on < S 1

ar npnn ana noa varab

Poao vau o h U O,) rbuon an h N(Oisbuon ar ncu n t sascal ab con

39


38/45


39/45

aa aocoao nco o 1For e proe I = el el_

< k+!1 - , k=,2,3...l +

8 ME SERES RQUN OAISpca dnsy ncon

j' ) _ 1 iwc-1 "e Yk, 1


40/45

8 I SS BOXJNNS OOOGY

4

jn an Bo pomaeau es o he resias or a, moe

m 2rk n(n+2) Xm-(p+q)k=! n

hee r (k1,2 . . te ette ue the kthtet eet the resdual n te nmbe dt vales e in he AR() ee.rnn oin es

eqece n epenet ranom vabes the umbe ftng pont i uch that:

ET=3n-2 ad v=16n-2990


41/45

9 ECONOMIC DELS

9.1 TILTY THORY

y fucn

Exponental: U(w)=- - , a> 0

ogaithmic: U(w)= gw

Poer

Quadaic

Msres o rsk avrso

Abse sk avesion:

Rlatv s vson

A(w)= _ "()

U'(W)

R w) w(w)9.2 CAPIA ASSET PCING MODL CA

Scry mr

Cpia mr line (fr ci foos

pp=)M

43


42/45


43/45

10 FCL DERVTSNote. otnuously-paabe)ivenrae.

0 1 PCE F A FO O T COFor an asset with fxed income ofpresent value I:

F o - J)eT

For an asst with diidends:

F- Sr-qT- oe

10.2 A PRCING " skna pas

t dUp-s proabiity e - ,

u-

d

5


44/45

10 3 STOCHAS FF QAOSGalsd Wi poces

=a+bz

where a an b a cons an i he incn for a Wiproe (tanar Brownian otin

t c

axt bx

's lea f a fcin Gx, t)

G ab +bz( dG 1 2cPG aG ] aGa 2 a2 a ad f th sho at rt

HLee: 8(

uWte r [OCt a ]t Vaek r (b +

CxIneR a r

104 BAKSCOLES O FOR OPEA OPOS

46

oic owna moo md f a tock pic St

BackSc aia dal qation

a a 2 2a-qS S at as s?


45/45

GaaKoae fouae fo e e al ad ut ot

h d iog(St/K)+(r-q+IO2T)w ere

nd

O T/ 2d - iog(St!K)+ rq IO Tt

2 r; 1 0" -\T- PT PRIT LAHP

K-T-t S q(t)Ct e -t + te

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