Prelim 2 Formulas

Prelim 2 Formulas - least-squares estimate finds be and b...

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Unformatted text preview: least-squares estimate. finds be and b. to minimizes “50- define How do “1' Iind this Inngiml Vfllllt‘ 0f ‘1? . In“ 'M " ii} = —;2 . (5) I q must satisfy Zed- — (mm-n“ “ 1 ,, 4‘. s i=| From the theory of best linear lmliction: rhfla} + {1 - qlfl‘lll‘ = II {31 _ 1 j } ‘ F 2‘ ‘3' l' using calculus, one can show that — ,8; is the slope of the best linear predictor of the 1 ‘ I Substituting fi2] = 0 und “3} = s: — K into jth security‘s return hm] upon the return ofthe _ _ eqmnuu and wiring [or q: market portfolio. _ BLIP/I - VIEX: - Ti 1 — n — 2 1+ -. — - i-l {X5 _ X) — The best linear predictor of R1 based on Rm is *l' = w+laa and R! = 11.,- + 31'3“ ‘ (G) From previous slide: b” = Y _ ill-X! Where :3,- in {Lil is the some in; in (5}. (1+r)sl _ 32 SE _ “alumni mm Let P: Im the wipe-end return on the fit]: scram-it}: q = 3 i . __ 83 — ‘ 2 — This is estinmtfll thutlnnl deviation of the [emit Th7" J"; _ ll! ii“ ”1" "1'in Dl'i'lllilllll>" _ _ _ I We wimt q to he hem-eon I] and 1. winner estunntor and tells In; the prlx'lsmn of that Using CAPM it. can be shown that Swim?“ i m d r 3 (s) a From (5) one can see that-1} 5 q 5 1 if An 1's 0 var once, , an Masts p" M __a1(p_” ML 82 E (1 +1151 5 53‘ mm SH = E“? _ T)“. This equution is called the seeln'ity market line Then “1 undo j.- [-1 {SM-Ll. I The “line of the up! ion is In {8) (SJ )5 the vnriable to the hnenr equation {U} = mpl—r6£}{qul2j+ 1}+{1_ mum}. I't‘ull'h'filull 55 = gin—1713. Wl'll‘l'l' Pi = ii. + a.“ in] The slope is p“ - i1}. where """'"'t""'1“1'l"‘ 5" = Z”; ‘filu- Comparison of the CML with the SM'L o The Ill'liitrngeIlulrrluinml w i. ll] . . . .rar _ . tulnl SE) = mgreaslou 35+ l'esltlnnl error 55. . The CML applies only to tilt! return R of an 9', = ‘ ’1- "b‘ I“) _ nwwinu SS _ _ residual error 35 efficient portfolio and ‘im-‘S that- 5-,”. :52; - total 55 _ mm] SS ' I The LIlLlllllt‘r of shame of Hlllck to be holding in 0' . _ The mean sum of mumres {MS} fnr an)r fiOllfl‘O is its p}; — it! 2 (Si) (mu — pf). o, = —fl2’ 4’ 1: - “2'” = [mim- mm sum 01 squamxl divided In- its ilngnuts ni' {mark-n1. M D “3’“ _ “3' h 1 I: . . ’ more the nnmunt of capitol ro o d in the rigs —fi'ee The residual MS is an unbiimi estimator of a2. ' '1 h" SML “F-‘lJl-K‘i-q to 3113' “5501: Mid fifll'li t-llflt. Ms“ hr I. . , ,. Thus, lmger Infidel}, havc' ii.- — ii! = gill-in — If-rl. - typically E5 is negative beenww.‘ money Inui lint-n — 1955 bios (300d) The total risk of the jth asset is thU-l- — more variability {bad} 2 2 Slim: portfolio n1)l.i:.-ntes option. {U} = sin: + {-3. aj=1fflfcru+o¢ ‘1' _ Suppose there are 3-! pmlietors. bomb“- — a" is the estimate of 0? using all or them. The risk has two components: "I = {ml ‘ ‘5!"4‘1‘ (r) ll“, changes in \‘nlne tu r.-“"{_fU} — ojsj} alter Ulll'.‘ — SSEUJ} is the sum of sqimreo for error for a model _ 536% is called till: market 01. systematic time tit-k) with only p g M pmlimnrs. component of IE and TI h by r A h ‘ i r r ‘I l n is the sample size. 2 If‘ pro ll fly :1 am} [ml is [11131 1|(.. n in h n nug — a}, is culled the unique, nonmarket, or the path. The" C” is unsvstematic eomooneut of risk. Cuwlill'im: “Mel [W's 10'- SSE . .‘ . . _ (“p 2 32(9) _ N + 2(1) +1} The ehaiaetenstle line Black Scholes formula " c = Md. )5“ — ordain expl Hr'f') R}: = i1}: + leflm — .flfcl + Ene 1 (X3 _ X) 2 where d? is the stnlulutd normal CDF. HE. _ : + implies that. = [UH[S[|{I\'l + [:.+6.2}J2fl‘ g, n (n _ 1)5§ a} = .5353! + of}, if. ”if? untl d2 = d. - off. RS’l‘UDEN'l‘ = Isl-I {Si—ii \/1——Hu-} ”’3" = fii'fli’”i" m J 7‘ j! (“midi") and that The CIVIL is 2 m: — w 6“" = 830“" M? = n“! ‘l‘ —00u R~ Using CAPM in portfolio analysis Suppose we have esthnated: Where - R is the return on a given elfieient portfolio " be“ and a? {01' mall MW in a Portfolio RM is the return on the market portfolio ' ”it - M? 2 Elm ' .tw ’ if." = Elfin} Then since n; is also known we can compute the _ 1”! is the rate of return on the risk—Elm asset expectations. variances. and cmmisnoes of all asset — an is the standard deviation of R ““1““ ii; = M + gill!“ — w}- 2_ . 2 2 or:- — life” + o'q-i — a“ is the standard deviation of RM 6‘er = fljfijio}! [01‘ j 59 j" Danger: l-le:i\'_\' tll‘|)l'llll('11(:(‘ ull msumpriuns ...
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