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Unformatted text preview: Returns, Interest Rates, Volatility 0 Pt: asset price at time t. Rt 2 (Pt/Pt_1) — 1: return on asset between t — 1 and t. k—l
P
Rt(k) — t —1 = H(1+Rt_j)—1Z return over k periods.
.=0 — Pt—k 0 Time unit usually a year; multiyeQI returns annualized. k—l 1/16
Amfalized Rt(k) = { Ha + Rt_j)} — 1.
o Logarithmic (or contiguously compounded) return:
'rt 2 €n(Pt/Pt_1) 2 pt — pt_1, where pt 2 ZnPt.
6 As the length A of a period approaches 0,
rt 2 €n(Pt/Pt_1) = En(1 + Rt) i Rt.
0 When there is dividend payment Dt at time t,
Rt 2 (Pt + Dt)/Pt_1 — 1 and 1",; = €n(Pt + Di) — 21131. HSBC pﬁce level from Dec. 02, 91 to Mar. 25. 1998 price
50 100 150 200 250 0 200 400 600 am 1000 1200 1400
Elna HSBC percentage lettim from Dec. 03, 91 to Mar. 25, 1998 40 0 10 20 managelleium o In the case of a “risk—free” asset (e.g., 'Iieagﬁury bills, bonds)» the rate of return is called an interest Talia
— If interest rate is constant R and compounded once per
unit time, tilrem P; = P0(1 + R)t.
— If inteméét fate is continuously compounded at rate 7“, then
Pt 2 Fife” ($01 1‘ = £n(Pt/Pt_1)), i.e., dPt/Pt = rdt.
o A simple entension to risky assets is Geometric? Brownian Motion (GEM) 5%} = ,udt + adult (6 : volatility), Where {21175, t Z O} is Brownian motion:
(i) wt is independent of {103, s S t’} for t’ < t;
(ii) wt is normal with mean 0 and variance t; (iii) 7.0,; is continuous in t. o Discretetime Pt 2 €n(1 + Rt) independent and
identically distribﬁﬁ’éd With mean ,u and standard deviation
(volatility) or. Markowitz’s Portfolio Theory ( f 52)
0 Single—period portfolio of n assets With weights w;
(i) 2?;1'wi .2 1.
(ii) usually 1 2 wz 2 0 (but negative 1112 allowed in “short
selling”).
(iii) Betm of portfolio 7° = 23:11:51}. 0 Mean was: F a @613!) = Swim, where m : Em). o Volaﬁllity a of r : 3Q 2 E1S¢1jgnngWjGOVWiarjl . Left boundary of feasible set is . . set, the upper portion of Which gives, 1; a) conﬁgu
ration, called the eﬁicz’ent frontier.
0 Equations for efﬁcient portfolio given (Mi, Jig)1 SuﬁSn:
(i) Subject to Swim = u and 2w; = 1,
minimize Eijwimggij, Where aij = Cov('rz, 73). (ii) Method of ', QAP—M‘ an; Slam,ng i o Besidgs the n assume a riﬁefﬁee asset (with
varianee 0) yielding nonrandom return 1'*. Then the ef
ﬁcient frontier is a straight line tangent to the feasible
region oﬁﬁhe n risky assets at. some point M, so any ef ﬁcient portfolio $3. combination of M and the riskfree asset. (One Eund Theorem). . (.3. . 1.. 42.35% c CAPM (Capital Asset Pricing Model) assumes all investors
to be mean / volatility optimiaers and to adopt the same
meancovariance structure of the risky assets co—existing
with a risk—free asset. Then f0r the market to be in equi—
librium, the fund M in the One Fund Theorem must equal
the market portfolio (total sum of all assets in market) Whose return is denoted by r M, with mean u. M andwolatil— i" . M o
1ty 0M oreover, K): "b a I ht“ — 1"“ = ﬁiWM  7”“), Where Bi 2 Cov(ri,rM)/a%4. o for portfolio: fund such as S&P 506). o CAPM (Capital Asset Pricing Model) all investors
to be mean/ volatility Optimizers and no want the same
mean—covaniance structure of the riglﬁ co—existing
with a risig~fmee asset. Then for the to be in equi
librium, the ﬁund M in the One Fund must equal
the market portfolio (total sum of all in market) Whose return is denoted by r M, With mean and volatil ity 0 M. Moreover,
Mi — 74* = — ’I”*), where ,62‘ :T o Sharpe ratio = (,uz — r*)/oi (mean unit
of risk), 'to be commedz with benehmar'k. — 1“" GM
to aid investment?! Tents. 0 Proxy for market Index fund such as S&P 500. Multifactor Pricing Models ( g; Markowitzzs pogiifoﬁo selection idea for a period, one needs the value. of 72 means m, n variances and n(n — 1) / 2 covariances 0253 of the returns of the n
risky assets, in which n is usually large. Although one can
use historicak data to these parameters, how far
back in time. one should look at is a problem because of potential nonstationarity in certain parameters. Also some of these parameters may be estimated more accurately than others.
0 The number of parameters is substantially «reduced in a regression model of the form
(*) Ti=ai+ﬂ$f+€i Where f 2 (f1, . ., a ,Jifk)‘T is a vector of factors and (51, . . . ,en)
has mean 0 is uncorrelated With f. An example is
CAPM, with 0:1, : 'r* and a single factor f 2 MM — 'r*,
Where MM is the mean return of the market portfolio and r*
is the return of the riskfr asset. The efﬁcient frontier is
a straight line jointing (0,7") and (0M, [1, M). An alternative to CAPM is the Arbitrage Pﬁicing Theory (APT) of Ross, which allows for multiple factors determining mean returns. lio, leading to the regression model 0 Although is considerably more general than CAPM,
the theory does, not provide insights into the identity and the
number kaéfﬁf fact‘brs, which can be portfolios and / or macroe
conomje variables (*such as GDP and bond yields). In its
most general form, APT provides anﬁapproximate relation ship for large n in the absence of arbitrage and leaves the . . . . 2» £3455 . 0.” g4”
fact rs unidentiﬁed. Pr1nc1pal component analySIS ( 
an.» ’9': m.. ———~ ﬁ) can be used to determine the factors.
a," we Applications of Linear Regression o APT Once the faetors are speciﬁed in APT, we can perform esti mation and. testing of the regression parameters; see Section; o The Markowitz theory, CAPM and APT, are singleperiod
theories, andrtherefore the parameters relate only to that
period. Using data from other periods to estimate the pa—
rameters make the implicit assumption of stationarity, which may not be appropriate if one takes historical data too far back into the past. 0 Tests for CAPM. ( @303'9‘9 Suppose one has data on the returns of K assets (e.g.
IBM, GE, Pﬁzer) and a proxy of the market portfolio (e.g.
8&1?» 500). CAPM is a (ﬁlinear) hypothesis on E73 and Er M
of the form "Hog: ai = 0 (j = 1, . . . , J) in the linear model 771:3 :aj—IﬂjftMFétj = 1,...,T), where ftj Empirical evidence: Early empirical work was largely positive in favor of CAPM, but in the past 30 years US stock—
market data have provided some statistical evidence against
CAPM. In spite of this, CAPM and the associated Sharpe
ratios still remain widely used tools in ﬁnance. There is con—
troversyaconcenning how the evidence against CAPM should
be interpreted. Some argue that evidence against CAPM is
overstated because of sample selection bias, misspeciﬁcation
of the market portfolio, and violation of the assumptions
underlying the tests used. Others argue that CAPM is too
simple to accomodate the anomalies of the stock market and that richer and more ﬂexible multifacth models should be used. c The linear predictor a + b1X1 + ngg + . . . + prp of Y,
a ﬁnancial variable of interest (such as stock price or mort
gage rate‘), using other ﬁnancial variables X1, . . . ,Xp, arises
in many modeling and hedging problems in ﬁnance. The co
efﬁcients b1, . . . ,bp are sometimes required to satisfy the con—
gtraint g b, = 1 in the case of portfolio weights), under the. predictor can be rewritten (with bp 2 1 — b1 —. . .— :é“Irrit’bii'(X1 *Xp)+. . .+b _1(Xp—1 ~Xp)+Xp, 0r equa , = a+b1(X1—X_p)+. . Q+b _1(X,,_1—Xp)+e. .Part Statistical Methods
1:1 Linear Regression LELinear prediction of random variable Y from X1, . . . , XP. 0 Linear predictor l7 .= a + b1X1 + . . . + quq 0 To minimize S = — (a+b1X1 +. . .+quq)]2}, solve the “normal equations” 2—: = 0, 3—1;: = 0 (j. 21,. . . , q), leading to a = EY — (blEXl + . . . + prXp) 0000/, X1)
= (00v(Xi,XJ))'1 131',qu Egg? 00,00]: Xq)
. CG'EQX, Y) = ngX—sXxY—Em = EXY—(EX)(EY). 0 Choose a, by». . . ,. big to minimize the residual sum of squares 117i
R. Z — (a + blxﬂ + bq$tq)}2 Setting to 0 the partial derivative of RSS w.r.t. a, b1, . . . ,bq yields 0 Projection and AN OVA It is convenient to write a1 as a1  1 = boat“), Where b0 = a1
and zeta = 1. Let p = q + 1 and re—label the indices 0,1,. . . ,q
as 1, . . . my. We can think of the method of least squares ge
ometrically in terms of projection of Y 6 R" into the linear space £(X1,. . . spanned by X1, . . . ,Xp, Where A X = (x1, . . . , xp), and ﬁt = ,BTxt is the “ﬁtted value”, yt — ﬁt
is the “residual”. Then 3? is orthogonal to Y — Y, and by Pythagoras theorem, we have the ANOVA decomposition: HYII2 = IIYH2+ llYYHZ, (1)
73.6., Total SS 2 SS(regression) + RSS.
(For x, y E R”, x2 = 23;;1 x3, x y if 232:1 xiyz = 0.)
«LS estimate:
is = (XTX)—1XTY. (2)
0 Projection formulas (i) Projection $1 of y into £(x) is (xTy/xTx)x = xxTy/xTx. "Eastiwlar, for x = 1, $7 = yl, Where y = (23?:1 /n and (ii) Projection of Y into £(X1,. ._._ ,Xp) = PY, where P = X(XTX)‘1XT and X 2 (X1, . . . ,Xp), an n X p matrix. (iii) An n X 72 matrix P is a projection matrix if it is symmetric (PT 2 P) and idempotent (P2 = P). (iv) If P is associated with projection into a linear space L
with dimension p, then I — P is associated with projection into
£i (=linear space of vectors orthogonal to E) with dimension n — p. Moreover, there exists an n X n orthogonal matrix Q such that 1,, 0 0 0
QTPQ : y — = 7 O '0 0 In_p
in which I? d‘éﬁ’éﬁapx p identity matrix. (An n X 71 matrix Q is an orthogonal maﬁﬁc @T 2 Q71. Thus, its column (row) . vectors are orthogonalﬁﬁé eaéli and. have unit length.) 5 o N onnegative deﬁnite matrices (i) V = X TX is symmetric, nonnegative deﬁnite (i.e. xTVx 2 0 for all x). (ii) A symmetric matrix V has real eigenvalues (i.e., solutions
to the polynomial equation det()\I — V) = O are all real). The
eigenvalues are nonnegative (resp. positive) if V is nonneg—
ative (resp. positive) deﬁnite. In this case, there exists an
orthogonal matrix Q such that V = QDQT, where D is a
diagonal matrix Whose elements are eigenvalues of V. Q can be formedby eigenvectors (Vx = Ax) normalized to have unit length. (iii) In the @ﬁtive deﬁnite case (xTVx > 0 for all x aé 0), V is inveItiBle. Gtherwise V‘1 is taken to be a generalized inverse, (not uniquely) deﬁned by property VV‘IV = V. For
an n X p matrix X, V = X TX is positive deﬁnite if X is
of full rank p(§ When X = (X1, . . . ,Xp) has rank q < p, £(X1, . . . ,Xp) can be spanned by q linearly independent column vectors of X. 1.14. Statistical properties of LS estimates
Assumptions: yt = Blastl + . . . + 31933,, + 6,: (t = 1,. . . ,n) (A) mtj nonrandom constants (to be relaxed later in time series regression). (B) 6,; unobserved random disturbances With Eet = 0.
(i) [a is unbiased estimate of s (i.e., E3} = {9).
(ii) cov([i)(= (owtﬁiﬁjnlgm) = 02(XTX)—1 if in addi—
tion (C) Var/“(cg = 02 and the et’s are uncorrelated 7 (i.e., Cov(ei, 6]) = 0 for z' 75 j).
(A); (3),. constitute the GaussMarkov model. In this case, an unbiased estimate of 02 is
s2 = :(yt — 3702 / (n — p) (= RSS/ degrees of freedom). (4)
t: Suppose (C) is replaced by the stronger assumption
(C*) 61, . . . ,6“ are independent N (O, 02). Then the following distributional properties hold: [‘3 ~ N03, 02(XTX)—1), (5) «gt—ma? ~ (6) [:3 and 52 are independent, (7) Bj — ’Bj N t where C =   — (XTX)_1 (8)
8 ij n—p: 2] 19,1510   (iv) If we replaee (0*) by the weaker assumption that 6,; be independent with (53‘ 331d variance 0'2 (Without normality), 8 but include additional assumptions on the xtj and boundedness
of higher moments of 6t so that the Central Limit Theorem is
agﬁlicable, then (5)—(8) still hold asymptotically as n ——> 00. (v) Under (C*), the LS estimates are also maximum likelihood estimates. Some basic distribution theory and its applications Cov(Z) = E {(z — EZ)(Z — EZ)T}
o For nonrandom k x m matrix A and k X 1 vector 0,
E(AZ + c) = AEZ + c, @W(14E«Z~=Eo) : Cov(AZ) = ACov(Z)AT. 9 Application to LS: Y = X ﬂ + 6 with E5 = 0, 009(5) 2 021, 6 = (61, . . . ,6n)T. Ea?) = (XTX)‘1XTE(Y) = (XTX)‘1XTXﬂ = a mutivariate normal distribution if it is of the form Y 2 pl, + AZ, Where Z has i.i.d. N (O, 1) components and [1,, A are nonrandom m X 1 vector and m x m matrix, respectively. 0 Since EZ = 0 and 0012(2) = 021, Z = p. + AZ has mean
u and covariance matrix V = AAT. If V is nonsingular,
then a change of variables applied to the density function {1:11 6f Z shows that the density function of Y 10 is f( )— —l——6
y — (mm/2W ’
We write Y N N(p,, V). 0 Deﬁnition: If Z1, . . . , Z, are independent N (O, 1) variables, then U = Z? + . . . + Z3, is said to have the chisquare distri bution with n degrees of freedom, written U ~ xi.
E(U) = n, Var(U) = 2n. 0 Deﬁnition: IfZ ~ N(O, 1), U ~ xi, and Z and U are inde pendent, then T = Z / ‘/U / n is said to have the tdistribution with n degrees of freedom, written T N tn. 0 Deﬁnition: If U N xfn, W ~ xi, and U and W are in—
dependent, then F : (U/m)/ is said to have the F— distribution with m and 71 degrees of freedom, written F .
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