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8/14/2019 Formulae and Tables for Actuarial Exams
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THEMAICLHOD R
xot fuct
x2 x3p()=e1+-+'"! 3
Ntr og
x log +x)=Il +x = x--+ . (-1
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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
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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
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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
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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
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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
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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
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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.
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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
DF:
MGF:Moe
The dbo con s abaed in he satsicalabe seion.
(X= X=
r re + )E(X ) = 2r ( r) ' r=246, .r -
2
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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
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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
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Unom sriution (connuous) a bParametes: ,b
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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
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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
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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
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2.4 TRUNCATE MOENTS
18
Nol disbuion
i he PDF o he ,2) ditibuion then
x ( =[
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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)
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8/14/2019 Formulae and Tables for Actuarial Exams
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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
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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)
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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
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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
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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-
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6 GEESED NEAR ODESxnna fay
or rndom vbl Y om h xponnti fy wh ntupt S n sc pmr
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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:
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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)]
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4 PD
casing/dcasig auiy fs
n
a; G v Da n Accumulan aco fo varal s as
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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