Prelim 1 Formulas - Quantilcs o if the (‘DF of X is...

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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 1UDqt-ll percentile Expectations and Variances The expectation of X is +"-\.. f .t‘f_\- [J7 hit-r — X variance of X is (a := fl» — £‘{.\"J}3f_\-l-r}rl-" = Et\' - Em}: ELY) := Useful lisrniuln: = ELY-3] — {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 l-ln 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 weak-form efficiency the information set. includes only the history of priees or retm‘l'ls semi-strong efficiency the inlornuttion set inelmles all infm‘matiou that is 1':I1l'ilie:i.|l_\-' axailal'fle strong-form efficiency the inforlnation set inelur.l¢_-.s all infm‘matiou knrm-‘n to any inr-u‘ket pr-u‘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 (statinm-n‘ity}_ then Em?) = y W '? , . (I; an) = V's-rm} = .2 W — w nhfl” 1M}:(‘o\'{_r,r;.;,r,m} a " J.) W 1 — m- nth-l = Urn-rm. 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]=flif|h| >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 + aim-1 — #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 sync-111 of linear eqlmtimm. The Hollltiml is —.r'l+(' 1', Al: n I: ll. follows after Home algebra Ill-AI 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 vet-tors. eiuee they depend on the lle‘ll vet-tor 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 _ min-Y ,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“, illnHtrz-ite the (‘lllt'lt‘lll- frontier l1_\' the following algorithm |_ \-"+ir_\' m: along A grid. For (Pm-ll value of pp on this grid. Compute (TN. by“: (a) Computing wm, = g + hop {In} l-lli'll (-olnpuling (rm, 2 wild-r12pr '2. Plot the Willfl‘fi (Noam) Then enlist-inn.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 rel-urns 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 t-U (lllf'. The tmigeney portfolio lH a scalar Imlltiple of w: w 17: {1-5} Lil-r = 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't-Ol‘ 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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Prelim 1 Formulas - Quantilcs o if the (‘DF of X is...

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