lec1 - Returns, Interest Rates, Volatility 0 Pt: asset...

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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) — 2113-1. 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, til-rem 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 Discrete-time Pt 2 €n(1 + Rt) independent and identically distribfifi’éd With mean ,u and standard deviation (volatility) or. Markowitz’s Portfolio Theory ( f 5-2) 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 Sufi-Sn: (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 $3. combination of M and the risk-free asset. (One Eund Theorem). . (.3. . 1.. 42.35% c CAPM (Capital Asset Pricing Model) assumes all investors to be mean / volatility optimiaers and to adopt the same mean-covariance structure of the risky assets co—existing with a risk—free asset. Then f0r the market to be in equi— librium, the fund M in the One Fund Theorem must equal the market portfolio (total sum of all assets in market) Whose return is denoted by r M, with mean u. M andwolatil— i" . M o 1ty 0M oreover, K): "b a I ht“ — 1"“ = fiiWM - 7”“), Where Bi 2 Cov(ri,rM)/a%4. o for portfolio: fund such as S&P 506). o CAPM (Capital Asset Pricing Model) all investors to be mean/ volatility Optimizers and no want the same mean—covaniance structure of the riglfi co—existing with a risig~fmee asset. Then for the to be in equi- librium, the fiund M in the One Fund must equal the market portfolio (total sum of all in market) Whose return is denoted by r M, With mean and volatil- ity 0 M. Moreover, Mi — 74* = — ’I”*), where ,62‘ :T o Sharpe ratio = (,uz- — r*)/oi (mean unit of risk), 'to be commedz with benehmar'k. — 1“" GM to aid investment?! Tents. 0 Proxy for market Index fund such as S&P 500. Multifactor Pricing Models ( g; Markowitzzs pogiifofio selection idea for a period, one needs the value. of 72 means m, n variances and n(n — 1) / 2 covariances 0253- of the returns of the n risky assets, in which n is usually large. Although one can use historicak data to these parameters, how far back in time. one should look at is a problem because of potential non-stationarity in certain parameters. Also some of these parameters may be estimated more accurately than others. 0 The number of parameters is substantially «reduced in a regression model of the form (*) Ti=ai+fl$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 risk-fr 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 Markowit-z theory, CAPM and APT, are single-period 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—I-fljftM-Fétj = 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“Irr-it’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+ llY-YHZ, (1) 73.6., Total SS 2 SS(regression) + RSS. (For x, y E R”, ||x||2 = 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 Gauss-Markov 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 chi-square 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 t-distribution 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 . 11 ...
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