Unformatted text preview: leastsquares 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 — (mmn“ “ 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+ . — 
il {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 _ illX! Where :3, in {Lil is the some in; in (5}. (1+r)sl _ 32
SE _ “alumni mm Let P: Im the wipeend return on the ﬁt]: scramit}: 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 hemeon 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 that1} 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 {SMLl. 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' {markn1. 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‘lJlK‘iq to 3113' “5501: Mid ﬁﬂl'li tllﬂt. Ms“ hr I.
. , ,.
Thus, lmger Inﬁdel}, havc' ii. — ii! = gillin — Ifrl.  typically E5 is negative beenww.‘ money Inui lintn
— 1955 bios (300d) The total risk of the jth asset is thUl
— 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 titk) 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‘ = IslI {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'qi — a“ is the standard deviation of RM 6‘er = ﬂjﬁjio}! [01‘ j 59 j" Danger: lle:i\'_\' tll‘)l'llll('11(:(‘ ull msumpriuns ...
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 Spring '07
 ANDERSON

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