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Unformatted text preview: Quantilcs o if the (‘DF of X is continuous and strietly increasing;
lhen it has n iln'ersl.‘ l'nnelion F“ I [or r; lurtu'een [1 and 1. F‘Ltqi is caller] the r‘JLll quantile or 1UDqtll percentile Expectations and Variances The expectation of X is
+"\..
f .t‘f_\ [J7 hitr
— X
variance of X is (a := fl» — £‘{.\"J}3f_\lr}rl" = Et\'  Em}: ELY) := Useful lisrniuln: = ELY3] — {ElXilg standard deviation is the h't.llﬂl'l.‘ root of Ihe van'iauee: 1:{1’ — 1.‘{ X 1}? Il' X1 . . . . , A” is a siunple from a prob dist‘n. then a expertstimi estimated by sample mean a the \»';i.ri;i.nee estimated by sample variance Selim: — X)? .2
33:
'\ n—l Useful formulas: ox} : H(/\’Y)— h‘(;i.‘}H(y) UXi' = l“J‘ll/‘(— 1La‘lxllyl m = Hie" — Emu] a“ = til/Y1")ithXj=UorH(Y}=U Correlation coefﬁcient between X and Y:
,OXY I: (TXY/(TXUY for any (X.Y) it is true that
—1 S PXY S 1
After some algebra. (exercise) we find that
3i = HEY/Hi
and
in = Itilt} — 51 El X) = till") — (TU/ﬁx HEX} Thus. the lK‘Hi linear predictor of 1' is {32.10 + six 2 5(1'} + :2" {X — an}
X Another way to look at this: (we — an) (
on}. = W X — an) ‘7_\' X is observed — then expected squared prediction error is 2  2
01/11 — pXY)
Hypothesis Testing
Example: H”: ,u = 1 versus 11;: it 75 1 Rejection region: set of possil'ile samples that lead us
to reject H0. example: reject lln if a ll exceeds entol‘l' C.
Type I error: null hypothesis is true hill we reject il
Type II error: null hypothesis is [also and we accept. it “l'iias—em‘reeted" .\LH is the so—r:alled sample variance H 1 .2
n—lzai Y} i=1 : eXp(,u. + 02/2) — 1 > median (fact about
lognormals) .2 _
by" — sample skewness is T A 3
:97 1 Z rt — p.
T i=1 0
sample kurtosis is
T A 4
It I — E T11
T 0
i=1 “excess kurtosis” is K — 3
Three types of eliiciency weakform efﬁciency the information set. includes only
the history of priees or retm‘l'ls
semistrong efficiency the inlornuttion set inelmles all infm‘matiou that is 1':I1l'ilie:i.l_\' axailal'ﬂe strongform efficiency the inforlnation set inelur.l¢_.s all
infm‘matiou knrm‘n to any inru‘ket pru‘tieipaut p is the correlation function 7 Note that p(h) : [)(ihj Akaikes information criterion (AIC) is deﬁned as 7210g(L) + 2(1) + q)‘ covariance between Xi and XHh is denoted by 7(h) — L is the likelihood evaluated at the MLE is called the autocovariance function Note that 7(h) = 02p(h) and that 7(0) 2 02 since
{)(0) : 1 many financial time series not stationary * but the chnges in these time series may be
stationary
estimate autocovariance with n—h : n—1 WI) 3'21
estimate with , f 21.2....
,(m'1 ' 3ft — ,u : @(yH — e) + e
Properties of a stationary AR“) process \Yhen w 4:; 1 (statinmn‘ity}_ then Em?) = y W '? , . (I;
an) = V'srm} = .2 W
— w
nhﬂ”
1M}:(‘o\'{_r,r;.;,r,m} a " J.) W
1 — m nthl = Urnrm. m i h} = .th v‘r Only if o «1: l and only for ARU} processes
Also. for h > O 9“ . “G ' U2®h
N h : GOV E Et lab: E Q rd] : 6‘
i . ' ‘ +h’ ‘ s
/( ) I l r J 1 i (‘92
1:0 }:0
distinguish between 03 : variance of 61, E2. . . . and n{(0) : variance of y1.y2. . . .
":fll = "(’3‘ 1.{h]=flifh >1. 0
1+ 92' {3(1J= Mb] 2 [I if lhl 2’ l Schwarz‘s Bayesian Criterion (SBC):
— deﬁned as —210s(L) +10s(‘n)(p + 9').
i n is the length of the time series — also called Bayesian Information Criterion (BIC) term “2(1) + q} in AlC or loghxﬂp + q) is a penalty on
having too many {Hirameters — llit_~1‘c_~.l.'t_n't_‘. AK". and 8150 try to lrmhjolf Ziyj+h — gxyj — 37) >‘< good [it to the data measured by L x the desire to use few 1J2I.I‘EIllLt.‘l.L‘.1'r§
which penalizes the most?
x log[NJ :> 2 if n .2 8
r'« most time series are much longer than 8
x so BBC Lwnalizes p  (1 111011) than AIC'
.:: 1‘lu“I'eful1‘n AIC will tend to Plll'n'lﬁl" I'm'u'le'ds With
more nammeters than SEC
two risky assets have returns R1 and R2
mix them in proportions w and 1 — w
the return is R : le + (1 — w)R2 expected return on the portfolio is
Eu?) 2 W1 + (1 — tum : #2 + aim1 — #2)
let p12 be the correlation so that 03132 : plgalag. the variance of the return 011 the portfolio is 0% : u12012+(1— Info; + 2w(l — w),012 0102. The covariance 012 can be estimated by sample
covariance 71 812 = 71—1 2(3111: * P1Xst * P2) t:1 correlation p12 can be estimated by the sample correlation
512 P12 2
8182 Finding the tangency portfolio Deﬁne
0%:m—w
0%:mim 0 V1 and V2 are called the “excess returns.’1 i l/lag — V2.012 (7102
V10; + V20? — (V1 + 1V2)}5'12C710"2I WT A = pig—11 = 1Tﬂ_l,u
B “To 1p.
C‘ : 1Tsz—‘1‘ Then (5} and (G) can he rewritten as
m1 +_41,\2
AM + (9‘3. NP 2
l : Let D = B(' — A? he the determinant of this sync111 of
linear eqlmtimm. The Hollltiml is —.r'l+(' 1',
Al: n I: ll. follows after Home algebra IllAI B—z’lﬂp and A2 = n win, =g+hron {T}
wl Ii‘I'l‘ l 1
BR 1 — A D— ,u
= 8
Q U  l l
and 1 I
h=LQ ,u—AQ 1' {U} D
Notice that g and h are fixed vettors. eiuee they depend
on the lle‘ll vettor p. and the lixerl matrix 0.. Also. tlu
scalars 31.1". and D are functions; of p mid 9 so they are
also fixed.
The target experted return. ﬂp. mil be varied over the
range _ minY ,1, l=l.. :3 m: 3 _Illmx\_;.',. J... of L‘lliL‘iL‘lll portfolios called the "ellieielit frontier." We can As pp varies over this range. we get u. llM'iIH w“, illnHtrzite the (‘lllt'lt‘lll frontier l1_\' the following algorithm _ \"+ir_\' m: along A grid. For (Pmll value of pp on this
grid. Compute (TN. by“:
(a) Computing wm, = g + hop {In} llli'll (olnpuling (rm, 2 wildr12pr '2. Plot the Willﬂ‘ﬁ (Noam) Then enlistinn.ng (I3) into (l0) new 2 r."p w. where
1”}: H [J'Jr (u M1ng ‘(u m1} :1};— and w=Q l{;r—;rﬁ). {l—l} {p — gr II} is the vector of "excess returns." that is.
the amount by which the expected relurns on the risky assets exceed the risk—free return. The excess returns measure how much the market pays for assuming risk. a is not quite a portfolio because these weights do “(ll 'Il(‘(‘(‘:"iﬁi'l'l'l.l_\' HI'II'Il tU (lllf'. The tmigeney portfolio lH a scalar Imlltiple of w:
w 17: {15} Lilr =
0.»; is a portfolio h'll'lCli‘ the \\'I_“1]_J"l1t are lltll‘li'lﬁl to sum to
one. The optimal weight V'E'C'tOl‘ us“? can be expressed in terms of the tangean portfolio as
w —r w—(‘(1Tw)w
pp — p — p I Therefore. C'p (1T?) tells us how much weight to put on the Lungeuey portfolio‘ Log; The amount of weight to put of the risk—free. asset is
=1— c‘p (GT1). Note that E and car do not depend on on.
Quadratic progrannning Quadrutie progrunmiing minimizes over :1: 1
EaJHa: + 1% i 16}
subject to
Am 5. b. {17"}
uml
Ama: = bra. (18} N is the dimension of the problem [number of
m I‘i :th PH} 2: and f are N ><  \'P!l"tt‘1]‘5i;'Il1l'l H is an X N matrix. ...
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 Spring '07
 ANDERSON

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