FORECASTING VALUE—WEIGHTED REAL RETURNS OFTSE PORTFOLIOS USING DIVIDEND YIELDSbyJOSEPH WADE BLANCHARDB.Sc., Dalhousie University, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of StatisticsWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1993©Wade Blanchard, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of ^STAT sj )6 sThe University of British ColumbiaVancouver, CanadaDate ^'T^5 / 19`)3DE-6 (2/88)AbstractWe assess the ability of dividend yields denoted by DYt, to forecast value-weightedreal returns, denoted by Rt ,T of Toronto Stock Exchange (TSE) portfolios for followingreturn horizons, T: monthly, quarterly, and one to four year. Fama and French [4] appliedsimilar methods to the New York Stock Exchange and found the forecast power increasesas the return horizon increases. We find that the Fama and French methods generalize toTSE portfolios, however, it does not apply to all portfolios. We also determine that theFama and French approach may not lie on solid statistical ground, in that the residualvariance is not time invariant.With these drawbacks in mind we consider using the methods of Dynamic LinearModels as discussed in West and Harrison [13], which allow the model parameters tobe time varying. We conclude that for the majority of the portfolios, the two methodsagree, however, the regression DLM approach does slightly better in comparison withthe methods of Fama and French in terms of standarized forecast errors.1Table of ContentsAbstractList of TablesList of FiguresList of Financial VariablesAcknowledgements1 Introduction1.1 Data ^32 Fama and French^ 52.1 Basic Definitions ^52.2 Motivation ^62.3 Regressions ^72.3.1 Parameter Estimation ^82.3.2 Assumptions ^92.3.3 Corrections ^102.3.4 Out of Sample Forecasts ^ 112.3.5 Stationarity of Parameter Estimates ^ 132.4 Results ^ 142.4.1 Regressions ^182.4.2^Out of Sample Forecasts ^ 242.4.3^Stationarity of Parameter Estimates ^ 252.5 Conclusions ^ 283 Theory of Dynamic Linear Models 293.1 Notation & Preliminaries ^ 293.2 Dynamic Linear Models 333.3 Known Observational Variance Vt ^ 343.4 Unknown Constant Observational Variance V ^ 383.5 Discount Factors ^ 423.6 Discounted Variance Learning ^ 453.7 Reference Analysis ^ 483.7.1^Case of Wt Unknown^ 493.8 Model Assessment^ 503.9 Examples 523.9.1^Constant DLM ^ 523.9.2^Simple Regression 533.10 Computer Implementation ^ 544 Empirical Results of Applying DLM to Value—Weighted Real Returns 554.1 Constant DLM ^ 554.2 Simple Regression DLM ^ 604.3 Constant vs Regression DLM 654.3.1 Classical vs DLM ^ 674.4 Conclusions ^ 695 Conclusions 70iv5.1 Fama and French ^ 705.2 Dynamic Linear Model 71Bibliography^ 72Appendix 1 74Appendix 2^ 84STList of Tables1.1 The TSE Composite Index and its fourteen industry sectors ^42.1 Regression of real value-weighted TSE portfolio returns (Rt,T) on dividendyields (Dt/Pt), for differing return horizons T. ^192.1 Continued ^ 202.1 Continued 212.2 Out of Sample forecast power as measured by le ^ 264.1 Maximum Likelihood estimates of 6,, and KT for the constant DLM forreturn horizons T ^ 574.2 Maximum Likelihood estimates of 8T ^and KT for the Simple Regres-sion DLM for return horizons T ^ 624.3 Comparison of Constant DLM to the Regression DLM for each of the sixreturn horizons and each of the TSE portfolios using VLL^ 654.4 Comparison of Classical Regression to the Regression DLM for each of thesix return horizons and each of the TSE portfolios using RATIO.^ 68v IList of Figures2.1 Plots of TSE Composite portfolio {Rt,T} for return horizons, T. ^ 152.2 Plots of TSE Composite portfolio {DYt} for return horizons, T. ^ 162.3 Plots of TSE Composite portfolio {Rt,T} vs {DYt} for return horizons, T ^ 172.4 Plots of the time varying estimates of aT, 131' and 4 for the TSE Com-posite portfolio for monthly and quarterly return horizons. ^ 274.1 Plots of the actual returns {R,T} (points), the forecasted returns at timet (solid lines) and 95% forecast limits (dashed lines) for each of the sixreturn horizons for the TSE Composite portfolio for the Constant DLM.^594.2 Plots of the actual returns {Rt,r} (points), the forecasted returns at timet (solid lines) and 95% forecast limits (dashed lines) for each of the sixreturn horizons for the TSE Composite portfolio for the Regression DLM. 64List of Financial VariablesCPIt - A measure of inflation at time t based on the prices of produces the typicalconsumer purchases./tx - The continuously compounded inflation rate at time t for return horizon T.Dt - The dividends received in the time period t — 1 to t.Pt - The value of the portfolio at time t.DYt - The dividend yield at time t.rt,T - The continuously compounded nominal return at time t for return horizon T.C Dix - The accumulated dividends in the time period t to t + T.Rt,T - The continuously compounded real return at time t for return horizon TviiiAcknowledgementsI would like to thank Dr. Jian Liu, my supervisor, for his patience and generousityin helping me complete the manuscript. I would also like to thank Dr. Harry Joe for hiscomments and suggestions on improving the manuscript. Finally, I would like to thankmy employer, Dr. Chris Field of Dalhousie University, for his patience in my completingthe thesis while working at Dalhousie.ixChapter 1IntroductionDo stock market returns have predictable components? Several studies suggest that theanswer is, in fact, affirmative. However, the predictable components typically accountfor approximately 3% of return variances.Fama and Schwert [5] assess the predictability of one month U.S. treasury bill rateon the monthly return of the value—weighted portfolio of all New York Stock Exchange(NYSE) stocks for the period January 1953 to July 1971. Using regression techniques,Fama and Schwert [5] conclude that the one month treasury bill rate accounts for 3% ofthe stock market return variability.Keim and Stambaugh [8] using data for the period January 1928 to November 1978and the following variables:(a) (yuBBA — YTB) = difference in yields on long—term under—BAA—rated (low-grade)corporate bonds and short—term (approximately one month) U.S. treasury bills;(b) — log(SPt_i/SPt_i), where SPt_i is the real Standard and Poor's (S & P) Com-posite Index and SPt_i is the average of the year—end real index over the 45 yearsprior to the year containing month t — 1;(c) —log Pt where Pt is the share price, averaged equally across the quintile of firmswith the smallest market values on the NYSE.to predict stock market returns on firms of various sizes. Specifically,1(1) Q5 — common stocks making up the fifth quintile of firms ranked by size on theNYSE, i.e. the quintile containing the largest firms trading on the NYSE;(2) Q3 — common stocks making up the third quintile of size on the NYSE;(3) Q1 — common stocks making up the first quintile of size on the NYSE.Keim and Stambaugh [8] use weighted least squares, with weights corresponding to thevariance of the daily returns of the S & P. In addition, they consider two sub—periods:January 1928 to December 1952 and January 1952 to November 1978.Stock^(7/,,UBBA - YTB)adj R2-100 Pt_i I S P -t-i)^—log PQtadj R2^adj R2January 1928 — November 1978Q5 0.006 0.005 0.006Q3 0.002 0.002 0.008Q1 0.001 0.002 0.014January 1928 — December 1952Q5 -0.003 -0.001 -0.002Q3 0.001 0.000 0.005Q1 0.002 0.006 0.020January 1953 — November 1978Q5 0.000 -0.003 0.000Q3 -0.003 -0.003 0.001Q1 0.003 -0.003 -0.001As is evident from the Table, Keim and Stambaugh [8] conclude that the predictablecomponent of stock market returns accounts for less than 2% of the return variability.French, Schwert and Stambaugh [6] use daily stock market returns for the periodJanuary 1928 to December 1984 S & P index, to arrive at an estimate of the monthly2return volatility using autoregressive—integrated—moving average (ARIMA) models (Boxand Jenkins [2]). The estimates of volatility are then used as a predictor for the monthlyvalue—weighted returns of the NYSE. French, Schwert and Stambaugh conclude that thestock market volatility accounts for less than 2% of the return variance.Fama and French [4] use dividend yields to predict returns on the value-weightedportfolios of NYSE stocks for return horizons of one month, one quarter and one year tofour years. Fama and French [4] state the the amount of predictability in stock marketreturns increases as the return period increases.The purpose of this thesis is two—fold: firstly, we apply the methods of Fama andFrench [4] to the Toronto Stock Exchange (TSE) value—weighted portfolios; secondly, weextend the methods of Fama and French [4] by using Dynamic Linear Regression Modelsdescribed in West and Harrison [13].Chapter 2 will describe in more detail the method of Fama and French [4] and itsapplication to TSE portfolios. In chapter 3, we describe the basic theory of the DynamicLinear Model (DLM) as given in West and Harrison [13]. In chapter 4, we apply the Westand Harrison techniques to the TSE data. Finally in chapter 5, will give a summary ofthe major findings of the thesis.1.1 DataThe data consist of monthly Index Values and dividend yields of the Toronto StockExchange (TSE) 300 Composite and its 14 industry sectors (see Table 1.1). Appendix 1contains a description of the method of calculation of the Index and dividend yield for agiven component of the TSE Composite. Appendix 2 contains the Relative weights thateach of the 300 stocks have in determining the TSE Composite and its industry portfolioson the close of December 31, 1991.3Table 1.1: The TSE Com osite Index and its fourteen industry sectors.Series Begin End Percentagein Dec 1991SourceComposite Jan 1956 Dec 1992 TSE Review [11](1) Metals and Minerals Jan 1956 Dec 1992 7.70 TSE Review(2) Gold and Silver Jan 1956 Dec 1992 7.39 TSE Review(3) Oil and Gas Jan 1956 Dec 1992 6.71 TSE Review(4) Paper and Forest Products Jan 1956 Dec 1992 2.23 TSE Review(5) Consumer Products Jan 1956 Dec 1992 9.47 TSE Review(6) Industrial Products Jan 1956 Dec 1992 11.05 TSE Review(7) Real Estate and Construction Jan 1968 Dec 1992 0.91 TSE Review(8) Transportationand Environmental Services Jan 1956 Dec 1992 2.12 TSE Review(9) Pipelines Jan 1956 Dec 1992 2.06 TSE Review(10) Utilities Jan 1956 Dec 1992 13.72 TSE Review(11) Communications and Media Jan 1956 Dec 1992 4.74 TSE Review(12) Merchandising Jan 1956 Dec 1992 5.13 TSE Review(13) Financial Services Jan 1956 Dec 1992 20.98 TSE Review(14) Conglomerates Jan 1956 Dec 1992 5.76 TSE ReviewConsumer Price Index Jan 1956 Dec 1992 Bank ofCanada Review4Chapter 2Fama and French2.1 Basic DefinitionsThis section contains the basic definitions of the return rate of a portfolio, the rate ofinflation and dividend yields which will be used in the thesis.Definition 2.1 The continuously compounded inflation rate at time t, for return horizonT, is given byli,T = log(C Pli+T I C PIOwhere CPI is the consumer price index at time t.Definition 2.2 The dividend yield of a stock portfolio at time t is given as follows:DYi = A I Ptwhere Di is the dividend received in time period t — 1 to t and Pt is the value of theportfolio at time t.Definition 2.3 The continuously compounded nominal return of a stock portfolio at timet, for return horizon T, is given by1 (Pt+T + C Dt,T)rt,T = logPtwhere C Di,T are the accumulated dividends in the time period t to t + T.5Definition 2.4 The continuously compounded real return of a stock portfolio at time t,for return horizon T, is given byRt,T 1= rt,T -We can apply the above definitions to each of the portfolios given in Table 1.1 byidentifying the index value at time t with the price of the portfolio at time t.2.2 MotivationThis section will attempt to motivate the conjecture, proposed previously, that dividendyields predict returns. The following is taken from Fama and French [4].Consider a discrete-time deterministic model in which Dt, the dividend per share forthe time period from t to t +1, grows at the constant rate g, and the market interest ratethat relates the stream of future dividends to the stock price Pt at time t is the constantr. In this model the price, Pt isPt = Dt 11 + g + (1 + g)2 + . .1[1 + r (1 + r)2^]1 + g 1= Dtl+rl—liff"l+r1 + g= Dt r — gDt (r — g)Pt — 1 + gThe interest rate r is the discount rate for dividends and the period by period return onthe stock, this can be seen as follows:Dt _ r — g Dt+i _ r — gPt — 1 + g^Pt+i — 1 + g6however,Dt+i = (1 + g)Dt Pt-Fi = (1 + g)2 Dtr — gand the return r isr ==Pt-Fi + Dt-Fi - Pt Pt[(1 + g)2 1(r — g) + (1 + g)] -13t — Pt Pt[(1 + g)2 1(r — g) + (1 + g)Rr — 9)1(1 + Ali — Pt Pt(1 + r)Pt — PtPtThe transition form the deterministic model to a model that (a) allows uncertain futuredividends and discount rates and (b) shows the relationship between discount rates andtime—varying expected returns is difficult. The deterministic model given above, at thevery least, lends plausibility to the conjecture that dividend yields predict expectedreturns.2.3 RegressionsFollowing Fama and French [4], the following linear regression model is proposed to modelexpected returns:Rt,T = aT + NDYt -I- et,T t = 1,2, • • • , NT^(2.1)where• Rtx is the continuously compounded real return for return horizon T;• DYt is the dividend yield at time t;7• aT is the intercept for return horizon T;• OT is the slope for return horizon T;• etx , N[o, 41 is the error term;• NT is the number of observations in for return horizon T.This model will be applied to the value-weighted real returns for the TSE portfolios forreturn horizons of one month, one quarter, and one to four years. The monthly, quarterlyand annual returns are non-overlapping, the other return horizons are overlapping year—end values.2.3.1 Parameter EstimationThe usual regression estimators for equation (2.1) are given as follows:Eitv=Ti(Rt,T — RT)(DYt — DY)-I4T =EitV2i(DYt - DY)2^1aT = RT - -NM 1. ,1 NT2(TT = ^NT - 2 tE=1 ll'TandVar(o7i)Var(N)or, in matrix notation:---i- ( 1 ^DY2= crT —NT + EiNzi (Dyt _ Dy)2--i ^1. 0-,- Elt'17i(DYt — DY )2(aT , )3T) = (X1TXT)1rTRT8where1^DY1 R1,TXT=1^DY2and RT = R2,T1 DYNT RNT,T _andvar((aT, PT)) = 01(x'TxT)'for each return horizon T.2.3.2 AssumptionsThe estimates given above are valid provided the assumptions about the error sequence= (61,T 62,T, • • ' Er/TA are not violated i.e.Var(eT) = 4/(NT-T)x(NT-T)where /3.3 is the s X s identity matrix. That is, the errors are un-correlated and haveconstant variance.In order to check the assumption of lack of serial dependence, we define the following:Definition 2.5 The auto-covariance function^of a second order stationary time se-ries, {Z}, is given as follows:= COv(Zt, Zt+T)and the auto-correlation function pr is given by9A second order stationary time series is one in which the mean is independent of timeand the auto—covariance function depends only on the time difference T between any twotime points. The reader is referred to Box and Jenkins [2] for a more detailed discussionof stationarity.Definition 2.6 The sample analogue of 7,- , denoted by ir , based on a sample of size N,is given by1 N-Tir = i- E (zt -7)(zt+., -7)and hence the sample estimate of pr, denoted by Pr, is•;:.^irPT = —'YeNow, if the underlying process has pr = 0 for T > 0, then_^1Var(pr) = Tr.For a justification see Box and Jenkins [2].Thus we have a method of determining whether there is serial dependence presentin the residuals. Simply plot the sample auto—correlation function and determine if aninordinate number of sample auto-correlations fall outside the limits ±2 I 07 . We defineE to be the number of times the sample auto—correlation fails outside the rangeWhat about the homogeneity of variance assumption will be discussed in a latersection. The next section gives a correction applied to the variance of the estimatorswhen the returns are over—lapping.2.3.3 CorrectionsHansen and Hodrick [7] suggest a correction to the variances of the estimates givenSection 2.3.1 when the return horizon is larger than the sampling period. Specifically,10the correction should be applied to the 2 year, 3 year and 4 year return horizons. Hansenand Hodrick suggest the following modified covariance matrix for ((aT, PT))10 = —Var((aT,13T)) = 1 Rvivinoc-iNT NTT-10= ERR,= E(et,ret-Fr,T),= E(xitxt+T)where x't = (1, DYt) and etx are the regression errors for return horizon T.Now we must have estimates of Rrx and Re, for T = —T +1, ,T —1. The estimatorsare given by:ferC =^1 N 7' 7--E XliXt..F.,-NT t=1ik^ 1 NT-I-= E it ,Tet+ T,TNT t=iwhere, I are the residuals, which estimate the errors E.Thus, an estimator of e, denoted by -6, is—-1— _1e = ^fikociv TwhereT-1=2.3.4 Out of Sample ForecastsIt is well known in regression analysis that R2 tends to be overly optimistic (see Draperand Smith [3] or Weisberg [12]). To illustrate what is meant by this statement considerthe following:11• split the data randomly into two parts, a construction sample and a validationsample;• compute the parameter estimates using only the construction sample and computean Ronstruction for the construction sample;c2 • using the estimates of the parameters from the construction sample, compute pre-dicted values for the validation sample to form the following out of sample R2Evalidation(Yvalidation — Ypred)2 .Then R2c^will be, in general, larger than R2^provided the modelonstruction^ validationassumptions apply.It is the goal of this section to provide estimates of the forecast performance, i.e.R2, that are not totally model based. That is, we wish to have estimates of the forecastperformance of the model given in equation (2.1) when applied to future values of thereal returns. Consider the following strategy:(a) Choose a window length W which will provide the estimates of the parameters ofthe model (2.1).(b) For the next time point, W+ 1 compute the forecast using the parameter estimatesfrom (a) and the DYw+i to forecast Rw +1 x . Which results in a forecast errorew-Fix = RW-1-1,T - ii 147+1,T •(c) Move the window forward one point in time. Now we will base our parameterestimates on the data points 2, ... , W + 1 and then we repeat (b), using the nexttime point and continue with steps (b) and (c) until the end of the series is reached.Rv2alidation = 1 Evalidation(Yvalidation — Vvalidation)212^'Now we can compute the following measure of forecast performanceMSEOUt R2011t = 1e011twhereMSE _x--.NT ,2— Lat=W +1 9,T^1 2^v-,NT^f PPand s„t = L.t=w+ilitl,T — pout ) 2with^1 ^NT^ROLA = NT w^ E RtIT •t=W+12.3.5 Stationarity of Parameter EstimatesIs it reasonable to assume that the same model given in equation (2.1) should apply tothe entire sampling period. That is, should the parameters of the model (aT, /37, and4) be time varying. This section will discuss an ad hoc approach to this problem. Thenext chapter on Dynamic Linear Models will deal with this problem in a more rigorousfashion.We assume that the model (2.1) holds for some fixed time period, say L. We thenuse the following strategy:(a) Using the first L data points,i.e. t=1,...,L, to provide estimates &T, ST and F4of the parameters.(b) Move the data window forward, U time points.(c) Use the data provided at time points t=1-1-U,...,L+U to get new estimates forthe parameters.(d) Repeat steps (b) and (c) until we reach the end of the time series.One could then examine the evolution of the parameter values as we move through time.132.4 ResultsWe now apply the techniques discussed in the previous sections to value—weighted realreturns of TSE portfolios given in table 1.1. The real returns for return horizon T attime t are denoted by Rix, see definition 2.4. The availability of the returns are given inthe following table:Return Horizon NT Start End443(299) January 1956(1968) November 1992147(99) 1st Quarter 1956(1968) 3rd Quarter 19921 36(24) 1956(1968) 19912 35(23) 1956(1968) 19903 34(22) 1556(1968) 19894 33(21) 1956(1968) 1988The values in parentheses apply to the Real Estate and Construction portfolio, while theothers apply to remaining TSE portfolios.Figures 2.1, 2.2 and 2.3 display plots of the real returns {Rix}, dividend yields {DYt}and real returns {Rix} vs dividend yields {DYt} for of each of the six return horizons Tfor the TSE Composite portfolio.It is apparent by examining the three sets of plots that the dividends yields appearto track real returns more closely as the return horizon increases. This indicates, at leastfor the Composite portfolio, that the results of Fama and French [4] seem to apply.144)(10tolim•1090 1960 1900198019701970 19801980TimeT=Monthly^ T=QuarterlyTime^ TimeT=Yearly T=2 YearsT=3 Years^ T-4 Years 1960 1970 1980Time^ TimeFigure 2.1: Plots of TSE Composite portfolio {Rt,T} for return horizons, T.15T=Monthly T=QuarterlyTim.T=YearlyTimeT=2 Years1960 1970 1980 1090 1960 1990TimeT=3 Years1970^1980TimeT=4 Yearstr!v^ ■^ .1960 1970 1980Time1990 1960 1970 1980Time1960 1970 199019800.!Figure 2.2: Plots of TSE Composite portfolio {DYt} for return horizons, T.16173 4^6Dividend Yields3 4^6Dividend YieldsT=Monthly^ T=QuarteriyT=YearlyDividend YieldsT=3 Years.^.25^3.0^3.6^4.0^4.5Dividend Yields• ..4Cd9- 1.T-2 Years.^.^,3.0^3.5^4.0^4.6Dividend Yields25 5.0^5.6T=4 Years25 3.0^3.5^4.0^4.6Dividend Yields5.0^5.5Figure 2.3: Plots of TSE Composite portfolio {/it,T} vs {DYt} for return horizons, T.,. ••.....4 4A,4?ar•• "0.• .2.4.1 RegressionsTable 2.1 contains the results of applying the regression model (2.1) to the real returnsRt,T for the TSE portfolios. The following statistics are given in the table:_.,• the slope estimate, #T;• the standard error of the slope estimate se(13T) as well as the Hansen and Hodrick[7] correction given in parenthesis for the overlapping returns;• forecast performance, R2;• the residual variance (4;• the first four auto—correlation estimates -i., i32, -3 , i14 ;• standard error of the auto—correlation estimate assuming lack of serial dependence,se() = 1/NrN7-,;• the number of times the sample auto—correlation function exceeds 2 X se(ji). Thenumber of auto correlations assessed depends on the return horizon: monthly series— 40, quarterly — 20, annual — 10.18Table 2.1: Regression of real value-weighted TSE portfolio returns (Rt,T) on dividendyields (Da Pt), for differing return horizons T.Real Returns Rt,T Auto-correlationsT NT --)4T se(') R2^4 se() jil 132 1,3 /14 ECompositeM 443 0.006 0.003 0.008^0.002 0.048 0.104 -0.053 0.071 0.028 4Q 147 0.021 0.010 0.030^0.006 0.082 0.205 -0.014 -0.065 -0.077 11 36 0.081 0.038 0.118^0.022 0.167 0.020 -0.196 0.091 -0.013 02 35 0.132 0.046(0.053) 0.197^0.033 0.169 0.432 -0.108 -0.019 0.056 13 34 0.138 0.049(0.062) 0.200^0.036 0.171 0.616 0.199 -0.051 -0.047 14 33 0.178 0.051(0.071) 0.280^0.039 0.174 0.688 0.427 0.236 0.023 2Metals and MineralsM 443 0.001 0.003 0.000^0.004 0.048 0.044 -0.057 0.010 -0.040 5Q 147 0.004 0.009 0.001^0.011 0.082 0.077 -0.013 -0.105 -0.041 11 36 0.023 0.033 0.014^0.043 0.167 -0.111 -0.240 0.152 0.053 02 35 0.038 0.041(0.043) 0.025^0.066 0.169 0.310 -0.221 0.055 0.120 03 34 0.007 0.044(0.053) 0.001^0.074 0.171 0.550 0.179 -0.045 -0.026 14 33 0.034 0.050(0.066) 0.015^0.094 0.174 0.623 0.396 0.229 -0.137 2Gold and SilverM 443 0.004 0.003 0.003^0.009 0.048 0.014 -0.069 0.000 0.081 1Q 147 0.011 0.009 0.009^0.025 0.082 0.036 -0.070 -0.001 0.009 21 36 0.040 0.034 0.039^0.086 0.167 -0.186 -0.214 0.067 -0.190 02 35 0.092 0.042(0.043) 0.129^0.128 0.169 0.292 -0.284 -0.169 -0.237 13 34 0.102 0.044(0.047) 0.144^0.140 0.171 0.484 -0.020 -0.488 -0.350 44 33 0.108 0.048(0.048) 0.142^0.158 0.174 0.416 -0.004 -0.170 -0.383 3Oil and GasM 443 0.021 0.007 0.019^0.005 0.048 0.038 -0.009 0.096 -0.008 1Q 147 0.068 0.021 0.064^0.015 0.082 0.091 0.073 0.028 0.014 01 36 0.251 0.079 0.228^0.051 0.167 0.190 -0.190 -0.168 -0.002 02 35 0.461 0.102(0.116) 0.384^0.084 0.169 0.406 -0.145 -0.202 -0.031 13 34 0.658 0.094(0.116) 0.603^0.072 0.171 0.542 0.029 -0.017 -0.028 14 33 0.730 0.088(0.101) 0.691^0.062 0.174 0.406 0.275 0.019 0.217 1Paper and Forest ProductsM 443 0.000 0.002 0.000^0.004 0.048 0.141 -0.061 -0.036 0.032 6Q 147 0.003 0.007 0.001^0.013 0.082 0.157 0.009 -0.030 -0.079 11 36 0.003 0.025 0.000^0.046 0.167 -0.120 -0.217 0.010 -0.138 02 35 0.010 0.033(0.036) 0.003^0.077 0.169 0.327 -0.285 -0.233 -0.139 03 34 -0.010 0.036(0.042) 0.003^0.092 0.171 0.523 -0.004 -0.319 -0.266 14 33 0.020 0.039(0.046) 0.009^0.106 0.174 0.538 0.134 -0.178 -0.391 219Table 2.1: ContinuedReal Returns Rt,T Auto--correlationsT NT iiT se(14T) R2 4 se(x) ii1 -A. -fi3 -A EConsumer ProductsM 443 0.003 0.002 0.005 0.002 0.048 0.105 -0.053 0.042 0.080 3Q 147 0.012 0.007 0.019 0.007 0.082 0.166 0.091 -0.022 -0.043 11 36 0.047 0.029 0.071 0.029 0.167 -0.014 -0.160 0.225 -0.124 02 35 0.118 0.038(0.047) 0.225 0.045 0.169 0.460 0.041 0.113 0.033 13 34 0.149 0.042(0.060) 0.284 0.053 0.171 0.715 0.357 0.134 0.030 24 33 0.200 0.050(0.071) 0.345 0.069 0.174 0.693 0.484 0.372 0.093 4Industrial ProductsM 443 0.002 0.003 0.001 0.003 0.048 0.132 -0.010 0.004 -0.041 3Q 147 0.009 0.009 0.006 0.010 0.082 0.084 -0.145 -0.075 -0.100 01 36 0.034 0.029 0.039 0.024 0.167 -0.126 -0.231 0.207 0.052 02 35 0.045 0.037(0.039) 0.043 0.036 0.169 0.292 -0.218 0.127 0.203 03 34 0.032 0.037(0.050) 0.023 0.037 0.171 0.568 0.203 0.033 0.132 14 33 0.059 0.041(0.063) 0.061 0.046 0.174 0.630 0.440 0.334 0.067 2Real Estate and ConstructionM 299 -0.007 0.005 0.007 0.007 0.058 0.232 0.045 0.141 0.017 3Q 99 -0.020 0.018 0.013 0.031 0.101 0.230 0.102 -0.041 0.028 11 24 -0.015 0.090 0.001 0.164 0.204 0.279 0.109 -0.109 -0.111 02 23 0.104 0.149(0.180) 0.023 0.362 0.209 0.514 0.005 -0.214 -0.112 13 22 0.384 0.189(0.231) 0.171 0.474 0.213 0.476 0.084 -0.269 -0.251 14 21 0.498 0.186(0.233) 0.275 0.455 0.218 0.497 0.120 -0.255 -0.356 1Transportation and Environmental ServicesM 443 0.002 0.002 0.003 0.004 0.048 0.089 0.005 0.061 0.038 2Q 147 0.007 0.006 0.010 0.014 0.082 0.186 0.014 -0.046 -0.034 31 36 0.029 0.022 0.051 0.055 0.167 0.174 -0.147 0.050 -0.154 02 35 0.066 0.032(0.042) 0.111 0.115 0.169 0.527 -0.034 -0.135 -0.265 23 34 0.085 0.038(0.052) 0.133 0.146 0.171 0.661 0.156 -0.242 -0.405 34 33 0.103 0.043(0.057) 0.156 0.161 0.174 0.676 0.191 -0.163 -0.389 3PipelinesM 443 0.002 0.002 0.003 0.003 0.048 0.007 -0.015 0.095 0.050 1Q 147 0.008 0.005 0.016 0.009 0.082 0.104 -0.079 -0.115 -0.047 01 36 0.036 0.018 0.105 0.028 0.167 0.051 -0.150 0.049 -0.092 02 35 0.056 0.024(0.028) 0.144 0.046 0.169 0.431 -0.109 -0.090 0.012 13 34 0.064 0.026(0.036) 0.162 0.055 0.171 0.623 0.133 -0.138 -0.133 14 33 0.077 0.028(0.042) 0.201 0.063 0.174 0.670 0.350 0.056 -0.205 220Table 2.1: ContinuedReal Returns Rt ,T Auto-correlationsT NT 137'^se(r)2^- iR az' -se() iii -A 1:1 3 il4 EUtilitiesM 443 0.002 0.001 0.010^0.001 0.048 0.060 -0.072 -0.020 0.036 7Q 147 0.007 0.003 0.031^0.004 0.082 0.068 -0.081 -0.107 0.140 01 36 0.024 0.012 0.099^0.012 0.167 0.033 0.028 -0.077 -0.067 02 35 0.039 0.016(0.022) 0.148^0.022 0.169 0.529 0.003 -0.089 -0.123 23 34 0.054 0.019(0.030) 0.193^0.030 0.171 0.669 0.283 -0.102 -0.088 24 33 0.064 0.021(0.037) 0.226^0.036 0.174 0.750 0.413 0.134 -0.033 3Communications and MediaM 443 0.003 0.002 0.004^0.003 0.048 0.123 0.015 0.108 -0.013 2Q 147 0.008 0.006 0.013^0.009 0.082 0.173 -0.017 -0.012 -0.061 11 36 0.046 0.027 0.079^0.041 0.167 -0.047 -0.283 0.152 -0.068 02 35 0.094 0.035(0.038) 0.185^0.068 0.169 0.359 -0.232 -0.083 0.012 13 34 0.107 0.037(0.048) 0.210^0.076 0.171 0.604 0.069 -0.170 -0.143 14 33 0.139 0.039(0.055) 0.286^0.086 0.174 0.594 0.298 0.063 -0.299 2MerchandisingM 443 0.005 0.003 0.004^0.002 0.048 0.183 -0.005 0.049 0.090 7Q 147 0.017 0.011 0.014^0.009 0.082 0.191 0.017 0.029 -0.125 21 36 0.086 0.049 0.083^0.040 0.167 -0.036 -0.208 0.280 0.106 02 35 0.143 0.064(0.073) 0.132^0.065 0.169 0.358 -0.119 0.232 0.249 13 34 0.129 0.064(0.093) 0.113^0.065 0.171 0.621 0.261 0.157 0.205 24 33 0.185 0.072(0.119) 0.174^0.080 0.174 0.703 0.513 0.441 0.223 5Financial ServicesM 443 0.003 0.002 0.007^0.002 0.048 0.116 -0.048 0.045 0.063 4Q 147 0.012 0.006 0.028^0.007 0.082 0.209 -0.090 -0.156 -0.089 21 36 0.047 0.027 0.084^0.027 0.167 -0.120 -0.166 0.244 -0.018 02 35 0.058 0.032(0.033) 0.091^0.038 0.169 0.288 -0.212 0.109 0.104 03 34 0.051 0.033(0.044) 0.070^0.039 0.171 0.555 0.117 -0.124 -0.187 14 33 0.072 0.036(0.050) 0.114^0.047 0.174 0.561 0.306 0.058 -0.307 1ConglomeratesM 443 0.002 0.002 0.004^0.004 0.048 0.064 -0.031 0.074 0.000 1Q 147 0.006 0.005 0.008^0.012 0.082 0.136 -0.023 -0.001 -0.064 11 36 0.014 0.018 0.018^0.041 0.167 0.177 -0.219 0.011 0.010 02 35 0.019 0.028(0.033) 0.014^0.091 0.169 0.469 -0.141 -0.109 -0.070 13 34 0.005 0.032(0.041) 0.001^0.122 0.171 0.638 0.124 -0.214 -0.278 14 33 0.001 0.034(0.045) 0.000^0.139 0.174 0.680 0.267 -0.120 -0.382 221We make the following observations about Table 2.1:Composite - forecast power for all return horizons, R2 increases with return hori-zon;(1) Metals and Minerals - no forecast power, R2 does not increase with returnhorizon;(2) Gold and Silver - forecast power for the 2 to 4 year returns, R2 increases withreturn horizon;(3) Oil and Gas - forecast power for all return horizons, R2 increases with returnhorizon;(4) Paper and Forest Products - no forecast power, R2 does not increase withreturn horizon T;(5) Consumer Products - forecast power for 2 to 4 year returns, R2 increases withreturn horizon;(6) Industrial Products - no forecast power, R2 increases with return horizon T;(7) Real Estate and Construction - forecast power for 4 year returns, R2 increaseswith return horizon;(8) Transportation and Environmental Services - no forecast power, R2 increaseswith return horizon;(9) Pipelines - forecast power for 1 and 2 year returns, R2 increases with returnhorizon;(10) Utilities - forecast power for monthly, quarterly and annual returns, R2 increaseswith return horizon;22(11) Communications and Media — forecast power for 2 to 4 year returns, R2 in-creases with return horizon;(12) Merchandising — no forecast power, R2 increases with return horizon;(13) Financial Services — no forecast power, 1r does not increase with return horizon;(14) Conglomerates — no forecast power, Ir does not increase with return horizon T;where forecast power means the slope estimate, 14T, is more than two standard errorsfrom zero. For the overlapping annual returns the Hansen and Hodrick [7] corrections tothe standard errors are used.It is interesting to note that the results of Fama and French [4] do not apply to allTSE portfolios we saw above. That is, the forecast power, as measured by R2, increaseswith return horizon. Fama and French [4] give a two—part explanation as to why thishappens(a) If expected returns have strong positive auto-correlation, rational forecasts of one—year returns one to four years ahead are highly correlated. As a consequence, thevariance of the expected returns grows faster with the return horizon than thevariance of unexpected returns — the variation of expected returns becomes a largerfraction of the variation of returns.(b) Fama and French [4] claim that residual variances for regressions of returns onyields increase less than in proportion to the return horizon. They base theirexplanation on the so called discount—rate effect, which simply put states that theoffsetting adjustment of current prices triggered by shocks to discount rates andexpected returns. They find that estimated shocks to expected returns are indeedassociated with opposite shocks to prices. The cumulative price effect of these23shocks is roughly zero; on average, the expected future price increases implied byhigher expected returns are offset by the immediate decline in the current price.The corrections, given in Hansen and Hodrick [7] seem to do the correct thing, i.e.increases the estimate of the standard error whenever the residual auto-correlations arelarge. We would expect E, the number of exceedances of the auto—correlation function1) 1 - above twice its standard error, to be about two for the monthly series, one for thequarterly series, and about one half for the annual series. Assuming a null hypothesisof no auto—correlation then we would expect 5% of the values to fall outside the ±2 x1/077: limits. In many cases, there seem to be an inordinate number of auto-correlationsexceeding these limits. This indicates the model given in 2.1 may be inadequate for thesedata. The next chapter discusses an alternative approach.2.4.2 Out of Sample ForecastsAs was mentioned in Section 2.4.2 the R2 values can be over stated when based on asingle sample. This section, applies the method of Section 2.4.2 to the value-weightedreal returns of TSE portfolios except for the Real Estate and Construction portfolio, dueto its late inclusion in the TSE composite (see Table 1.1).It was decided to choose a sliding window of 20 years to base the estimates on, whichwould lead to out of sample forecasts using the next available data point. The 20 yearperiod was chosen to closely resemble the 30 year period chosen by Fama and French[4]. Specifically the period 20 year period January 1956 to December 1975. Thus we areproviding out of sample forecasts for the period January 1976 to the end of the series. Thefollowing table shows the number of in sample and out of sample observations availablefor each return horizon T.24T Nin sample Nout sampleM^240^203Q^80^671 20^162^20^153^20^144^20^13Table 2.2 contains the out of sample forecast performance measured by R2. It is quiteevident that we don't do very well when forecasting out of the range of the data for themajority of the portfolios. For example, consider the composite portfolio, the in—sampleR2 are as follows: 0.008, 0.030, 0.118, 0.197, 0.200, 0.280 for the monthly, quarterly, andone to four year return horizons respectively. These values are quite inflated comparedto the ones given in Table 2.2. As a further example, consider the consumer productsportfolio, the in—sample R2 are 0.005, 0.019, 0.071, 0.225, 0.284 and 0.345 for the sixreturn horizons while in Table 2.2 the R2 are all negative. This due to the fact the thedividends are changing are low over the latter part of the out-of-sample period comparedto the remaining period. Thus, negative R2 values could alert us to changing modelcharacteristics.2.4.3 Stationarity of Parameter EstimatesIn this section we determine whether the regression model (2.1) applies for entire sampleperiod. We will attempt to answer this question by examining the two largest samples,namely the monthly returns and the quarterly returns. It was decided quite arbitrarilyto choose a window length of 5 years. That is, we are assuming that the parametersremain approximately constant over a five year period. Some sort of validation of this25Table 2.2: Out of Sample forecast power as measured by R2Portfolio Monthly Quarterly Yearly 2 Year 3 Year 4 YearComposite -0.006 -0.028 0.022 0.084 0.081 0.060Metals -0.016 -0.053 -0.186 -0.248 -0.326 -0.673Gold -0.015 -0.044 -0.177 -0.232 -0.223 -0.091Oil 0.007 0.027 0.168 0.297 0.588 0.585Paper -0.010 -0.028 -0.111 -0.226 -0.218 -0.341Consumer -0.011 -0.031 -0.117 -0.342 -0.971 -1.061Industrial -0.016 -0.059 -0.195 0.046 0.268 0.387Transportation -0.019 -0.071 -0.203 -0.198 -0.208 -0.334Pipelines 0.008 0.025 0.098 0.213 0.275 0.249Utilities -0.004 -0.014 0.088 0.061 0.264 0.296Communications -0.003 0.000 -0.023 0.022 0.085 0.194Merchandising -0.035 -0.098 -0.209 -0.475 -0.439 -0.372Financial -0.021 -0.073 -0.001 -0.299 -0.286 -0.030Conglomerates -0.010 -0.036 -0.156 -0.255 -0.226 -0.197value should be done, however, we did not pursue this avenue any further, since the topicof stationarity of parameter estimates is discussed in more generality in the next chapter.We also, decided that we would slide the window along by one time period.Note that these assumptions provide a sliding window of 60 data points for themonthly series and 20 data points for the quarterly series. Figure 2.4 contains plots(solid lines) of the estimated intercept, estimated slope and estimated variance at time t-and return horizon T denoted by atT, fitT and erh, respectively for the TSE Compositeportfolio. Also, plotted are eitT ± 2 x se(EitT) and AT ± 2 x se(AT) (dotted lines). Notethat when examining Figure 2.4 that successive estimates of aT, ,ST and 4 are not dis-tinct, in that each successive estimate contains data used to form the previous estimate.With this in mind, it is evident that the parameter values do not simply fluctuate aboutsome mean level which would support the hypothesis of stationary or constant parametervalues. The behavior of the parameter estimates seem to exhibit some of the propertiesof a random walk.26Monthly QuarterlyMonthly QuarterlyITim.MonthlyT1m0QuarterlyTim.1^ v^ I1970 19130 1990TimI^ v^ 11960^1970 1080 1990Tim.1900^1970^1980^1990Tim.IFigure 2.4: Plots of the time varying estimates of aT, OT and 4 for the TSE Compositeportfolio for monthly and quarterly return horizons.272.5 ConclusionsThe results of Fama and French [4] which state that dividend yields show increasedforecast power to predict real returns for increasing return horizons do not extend toall portfolios of the Toronto Stock Exchange. The Fama and French results only applyto the Composite portfolio and the Oil and Gas portfolio where there is forecast powerfor all return horizons as shown in Fama and French. However, the result of increasingforecast performance also apply to the following portfolios: Gold and Silver, ConsumerProducts, Industrial Products, Real Estate and Construction, Transportation, Pipelines,Utilities, Communications, and Merchandising.However, the model suffers from some problems, most notably:• residual auto—correlation• dramatic decreases in out—of—sample R2 indicating lack of stationarity• Model 2.1 may not apply for all time periods as suggested in Fama and French[4] as evidenced by the changing residual variance results given in Section 2.4.3.It should be noted however, that Fama and French [4] mention the fact that thereturn variances are not constant throughout their sampling period. They presentresults for various sub periods of interest. We did not pursue this option since wehad only a limited amount of data available.For the reasons mentioned above, we should look for a more acceptable method ofanalysis which incorporates both of the deficiencies of the Fama and French [4] approach.This is the subject of the next chapter, namely Dynamic Linear Models.28Chapter 3Theory of Dynamic Linear Models3.1 Notation & PreliminariesThis section will introduce the notations, basic definitions and theorems used in thesequel.Vectors and matrices will be denoted in bold face. For example if X is a p-dimensionalvector and E is an n x p dimensional matrix, we denote them as followsX = (Xi, X2, • • • ,i, )'andan 012 • • • 0-1p021 022 •• • 0-2pE =0-711 an2 •• • crnpThe density of a random vector will be denoted by f(X). The joint density of tworandom vectors X and Y is f(X, Y) and f(XIY) denotes the conditional density of Xgiven Y. Let E denote the expectation operator and let Var denote the variance operator.Theorem 3.1 (Bayes) Let X1 be a pi-dimensional random vector and let X2 be aP2-dimensional random vector and suppose we are given the conditional distributionf(X2IX1). Then the conditional distribution f(X1(X2) is given as follows:f (X21X1) f (Xi) f (XilX2) =ff. f (X2IZ1)f (Zi)dzi29f (x2Ixi)f (xi) f(X2)oc f (xi ,x2)Definition 3.1 A p-dimensional random vector X is said to have a Multivariate NormalDistribution, with mean p, and covariance matrix E, denoted by X -, N[A,Z] if11(X) = (2r), P/21Eiii.2 exp( — (X — AYE-1(X — p)/2)provide E is positive definite.Definition 3.2 A random variable 0 > 0 is said to have a Gamma distribution withparameters, n > 0 and d> 0, denoted by 0 e.s., G[n,d] ifdn f&) = r (n) 0n-1 exp(-0).where r(n) is the gamma functionr(n) = I xn-1 exp(—x)dx.oNote that E[0] = n I d and Var[0] = E[0]2 In = n/&.Definition 3.3 A p-dimensional random vector X is said to have a Multivariate Stu-dent's T Distribution with n degrees of freedom, mode A and scale matrix E, denoted byX es, Tn[p, E] ifff ( < oo, i =,,^nni2r((n + /3)/2) (714.(x_ttrz-i(x_11))-(n+p)/2 _00 < X,A.) = rior(n/2)1E11/2 k 1 kTheorem 3.2 Consider the following Multivariate Normal Distribution:X ^-, N[p, EJ30X = ( xi ),it = (111 ) , and, E =)E21 E22X2^/1'2which we can partition as follows:Then we have the following conditional distribution:whereand(X1 1X2) - NEtli (X2), Ei(X2 )]p1 (X2) = p i + E 12Eg(x2 - 112)E1(X2) = Ell - E l2E221 E21.Theorem 3.3 Suppose that0 Ps, G[n 12, d12],and that the p-dimensional random vector X is normally distributed conditional on 0 i.e.(x10) - MA, E0-11.Here the p-vector p and the (p x p) matrix E are known. Then(a) (01X) "d G[n*/2, d*/2]wheren* = n + p and d* = d + (X — 1.1)'E-1(X — /./)(b) X has a (marginal) multivariate Student's T distribution in p dimensions with ndegrees of freedom, mode p and scale matrix R = E(dIn) = E I E(0) denoted byX ,-, Tn[p,11.1.31Proof: By Bayes Theorem (Theorem 3.1) we have:f(0X) x f (x,oc f (x10) f (0)0„,2oc (270p/21E1i/2 exp( —0(X AY E-1 (X 14)12) ,2 idn 2exp(-0d/2)2n/ 11(71/2) e/2-1oc 0(n+P)/2-1 exp(-0{(X —^(X — it)) d}/2)thus,(01X) G[n*/2, d*/2]where n* = n p and d* = d + (X — E -1 (X — it). This proves assertion (a).Now to prove the assertion (b) consider the following:f(X) =^46)/i(OIX)op/2^exp(-0/2(X — itrE-1(X — A 22r(n/2)))^dn/2(2-w/2 1E11/2^ cp/2-1exp( —012)n 0+(x-p),E-1(X-it))(n+P)12 0(n+P)/2-1 exp(-0(d + (X — A)'E-1(X — tt))/2)2(n+p)/2r((n+p)/2)op/2^ P/2 d,n/2-1(27)P/21E11/2 2'. 21—/--'(,--42) r(d1-(X-p)/E-1(X-10)(n+P)/2 0(n÷P)/2-12(n+P)/2“(n+P)/2)(d (X — AY E 1(X — 11))-(n+P)/2OC (d (X — it)'11.-1(dIn)(X — 1.1)) - (n+p)/2 letting R = dlnEa (n + (X — AY 11-1 (X — ii))-(n+P)/2 factoring out d nFinally we havef(X) Tnwhere R = (d /n) =^Ekkj. That is, X has a student multivariate student T distri-bution with n degrees of freedom, mode it and scale matrix R. This proves assertion (b).0323.2 Dynamic Linear ModelsBayesian methods, not surprisingly, use Bayes theorem to update our knowledge aboutthe current state of nature 0 once we have observed some data Y. Let us assume thatour initial state of knowledge can be expressed in terms of a prior distribution, f(0).Let us also assume that we have a model which we assume generates our data Y, givenby f(YI0). Note that once we observe Y, our model can be interpreted as a likelihoodfor the various values of 0 and Bayes theorem tells us how to combine the prior and thelikelihood to form the posterior:f(91Y) 0( i(Y10)f(0)posterior cx likelihood x priorNote that the posterior updates our knowledge of the current state of nature 0, in lightof the new data Y we have observed. That is, Bayes theorem gives us a sequential wayof incorporating new information in our beliefs about the state of nature 0.Dynamic models, as the name implies, are ones in which change is the driving force.Now applied to a time series, f1iatN_1, dynamic models are models which change or adaptas time progresses. Now combining dynamic models with Bayesian methods gives onea unified approach to the problem of forecasting a time series {Yt}, since the dynamicmodel tells us how the time series changes and the Bayesian approach tells us how theinformation we received about the time series can be incorporated into our state ofknowledge.Specifically, the approach given in West and Harrison [13] for Bayesian forecastingand dynamic modeling is,(i) a sequential model definition;(ii) structuring using parametric models with meaningful parameterization;33(iii) probabilistic representation of information about parameters;(iv) forecasts derived as probability distributions.Now, let Di represent our state of knowledge at time t. The sequential approach basesour statements about Yt, made at time t —1, on the information set available Di_1. Theparametric model at time t is f (1'dt:it, Dt_1), which represents our belief as to how thedata at time t is derived. As mentioned above we need a prior distribution about ourparameters, i.e. f (0 ilDt_i). Now after we observe the data at time t we have a posterior,which in effect becomes the prior for the next time period. Thus, the prior—posterior paireffectively store the information about the defining parameter Ot as we move throughtime.The following sections introduce various aspects of the Dynamic Linear Model andtheir Bayesian analysis.3.3 Known Observational Variance VtThis section will describe the most basic dynamic linear model, in which one assumes,that all defining parameters are known. The basic definition is as follows:Definition 3.4 The Univariate Normal Dynamic Linear Model is defined by:observation equation:Yt = FOt + lit, lit ^., N[0, li],system equation:Ot = Gt Ot-i + wt, wt r's N[0, Wd,initial prior:(90 'Do) ,-, N[mo, Co],34where:(a) Ft is a known (p x 1) matrix;(b) Gt is a known (p x p) matrix;(c) Vt is a known constant;(d) Wt is a known (p x p) matrix;for some prior moments mo and Co. The observational and evolution error sequencesare assumed to be independent and mutually independent, and are independent of (OolDo) •Now we will describe the basic elements of the dynamic linear model:• Ft plays the role of the regression matrix,• Ot the vector of regression parameters also called the state vector,• vt is the observational error with variance Vt,• Gt is the evolution transfer matrix,• Cat is the evolution error with known covariance matrix W.We assume that the information set at time t is updated as follows:Dt = {Y, D1}The key result is given in the following theorem:Theorem 3.4 For the DLM of Definition 3.4, one-step forecast and posterior distribu-tions are given, for each t, as follows:(a) Posterior at t —1:(0 t-ilDt-i) ", N[mt_i, Ci-1].For some mean mt_i and variance matrix Ct_1,35(b) Prior at t:(0t1Dt-i) — N[at, Re],where at = Gimt-i and Rt = GiCt_iq + Wt(c) One-step forecast:(ft1-Dt-i) - N[ft,Qt]where ft = Fiat and Qt = rtRtFt + Vt(d) Posterior at t:(0t1Dt) ,-,, N[mt, Cd,with mt = at + Atet and Ct = Rt — -NAM,where At = RtRIQT1 and et = Yt — ft.Proof: By induction on t. Assume a) holds, i.e.:(Ot_ilDt_i) ^, Ar[mi- i, Ct-1].And from the system equation we have:Ot = GtOt-i + wtwhere: Ot_i ,-., N[mt_i, Ct_il and Wt^N[0, Wt] and Ot-i and cot are independent ofeach other.This implies thatOt .-,., N[at,114]where at = Gtmt_i and Rt = GiCt-iGit + W.Since the sum of two independent normal random variables is again a normal randomvariable with mean equaling the sum of the means and the variance equaling the sum ofthe variances. Thus (b) is established, i.e.f (9 ilDt-i) = N[at,Itt]36Now using the observation equation Yt = F'tOt vt we have the following conditionaldensity:f(Yt 10t, Dt-i) = N[ft, Qt]where ft = Fiat and Qt = FiRtFt Vt since, Yt is the sum of two independent normals.Thus (c) is established.Now combining (b) f(OtIDt_i) and (c) f (YtiOt,Dt-1) we have that f(Yt,OilDt_i) isbivariate normal with covariance given as follows:Cov(Yt,OtiDt- i ) = Cov (COt + vt , Ot IDt-i)= rtCov(0t, OtiDt_i) + Cov(vt, Ot IDt-i)= FWar(Ot, Ot1Dt-i) + 0FRtThus,Dt_i) N[(^t )]tfi^Q CRNow we want to find the conditional density of (OtlYt,Dt_1) which is what we wantin part (d) since Dt = {Yt, Dt_1}. Note that since (Of, YtiDt_i) is multivariate normalTheorem 3.2 applies. Thus if we label X1 = Ot and X2 = Yt we havewhereand(OtlYt, Dt-1) N[mt, Ct]mt = at + FiRtQT1[17t — ft]Ct = Rt — FtRtQTirtRtNow if we define et = Yt — ft, At = RtFtQt-1, we have mt = at + Atet and Ct =Rt — AtQtA't. The induction is complete since by definition (001D0) N[mo, Co].^0at^FtRt Rt37The vector At is known as the adaptive coefficient vector, since it changes the forecasterrors et and at into the new estimate of the state vector mtNow there are several stumbling blocks in the basic definition of the univariate DLM.Firstly, we do not know the observational variance, Vt, this problem will be solved intwo parts. Section 3.4 will deal with constant observational variance case (Vt = V) andSection 3.6 will deal with a slowly varying observational variance V. Secondly, we do notknow the evolutional variance Wt, this problem will be dealt with in Section 3.5. Thirdly,we may not have any "reasonable" priors, this is dealt with in Section 3.7. Section 3.8will describe the model assessment strategy and finally Section 3.9 will give two simpleexamples to illustrate the theory.It should be noted that the model given in Definition 3.4 is closely related to theKalman Filter (see Preistley [10] and references therein) with the exception of the initialprior distributional assumptions.3.4 Unknown Constant Observational Variance VThis section generalizes the previous section to the case of constant unknown observa-tional variance, that is we consider Vt = V for all times. Consider modeling the precision0 = 1/V, if for convenience we choose to apply a Gamma prior to 0, and combiningit with the new information expressed in terms of normal densities, then the posteriordistribution also turns out to be a Gamma distribution with a simple updating of theparameters. Also a the expected value of a Gamma with parameters no and do has ex-pected value no/do = 1/S0 where So is the prior estimate of the observational varianceV. Using these facts we generalize the definition and theorem of the previous sectionto incorporate the unknown observational variance in the form of a new definition andtheorem discussed presently.38Definition 3.5 The General Univariate Dynamic Linear Model with unknown constantvariance is defined by: observation equation:Yt = Ftet + vt, ii ^N[0,17],system equation:Ot = GO_ i + Wt, cot — N[0, Wt],initial priors:(001Do, 0) e- N[rno, VCOI,(01130) -, G[no/2, do/2],where 0 = V-1.Theorem 3.5 In the univariate DLM of Definition , one-step forecast and posteriordistributions are given, for each t, as follows:(a) Conditional on V:(Ot_i IDt_i, V) --i N[rrit-i, -MI-1],(0t1Dt_i,V) — N[at,VR7],(YtiDt-i,V) ^-, N[ft,VQ7],(0t1Dt, V) , N[mt, VC71,with at = Gtmt_i, and RI = Gtq_iG't +WI, ft = gat and Q*;and1 + Ftlt;Ft,mt = at + Atetq = 11; — AtAMwhere et = Yt — ft and At = R;Ft/C27 •39(b) For precision q= V-1(01Dt-i) ^-, G[nt_i/2, dt-i /21,(01Dt) ^d G[ntI2, dt/2],where nt = nti +1 and dt = dt_1+ 41Q;(c) Unconditional on V:(Ot_iiDt-i) ''''(0t1Dt-i)(yot_i) -(otipt) -Tnt_,[mt-i, Ct-d,Tnt_, [at, Rd,Tnt_t [ft, Qt],Tnt [Mt ) C ti ,where Ct-1 = St-iq_i, Rt = St-inI, Qt = St-iQt* and Ct = Stq,dt_iInt-i and St = dant •(d) Operational definition of updating equations:mt -= at + Atet,Ct = (St/St-i)[Rt — AtA'tQt],St -= dtInt,andwith St_1lit = nt-i + 1dt = dt-i + St-ie?/Qtwhere Qt = St_1+ FtittFt and At = RtFt/Qt•Proof: Part (a) follows directly from Theorem 3.4. The proof of remainder is again byinduction on t. In order to prove part (b), assume(01Dt_1) ,-,, G[nt_l /2, dt_i/21,40and from (a) we have(ftIpt-1,0) - N[ft, Q7/0],Hence, by Theorem 3.3 (a) we have(01Dt-i,Yt)= (01Dt)"d G[nt12,di12] since Di = {D-1,Y}wherezit= ni_i+ 1 since Yt is univariate i.e. p =1anddi = dt-i+ (Yt — ft)21q;= dt-i+ 41q;and (b) is proved.The results of (c) can be proved in the following manner For example, in order toprove (Yt1Dt_i) , Tnt_i[ft,Qt] we proceed as follows:we know from part (a) that(Ytipt-1,0"d N[ft,VQ1and from (b) we have(01Dt-i) ,--d G[nt--112,dt-112]thus by Theorem 3.3 (b) we have(17t ipt-i ) ''" Tnt-i [ft,Qt]where Qt = St_iQt* with St-1 = dt_i/nt_i the other results can be shown similarly.Finally (d) follows from (c) as follows: for example,Ct = St C't'41= St[R; — AtAM]= St[Rt/St-i — AtiOt/St-i]= (St/St_i)[Rt — AtAitQt]The theorem is now proved by induction since the results hold for at t = 0 and the initialpriors given in Definition 3.5.^ 0Note that St = dtInt estimates V = 1/0.3.5 Discount FactorsThis section will deal with the problem of specifying the state evolution variance Wt,which controls the amount of stochastic variation in the state vector and hence controlsthe model stability over time. Consider the following,Var(Ot_i I Dt-i ) -- -= Ct-ii.e. the prior variance of the current state vector. Now using the updating equations,the posterior variance is given byVar(et-ilDt-i) = Pt + Wt = R.where Pt = GiCt_iGit. Thus, we see that Wt has the effect of increasing the variance ofthe state vector, which can also be seen as a loss of information about the state vector,Ot, from time t — 1 to time t.Ameen and Harrison [1], suggest that a multiplicative rate of information decay isappropriate, since the relative magnitudes of Wt and Pt are important. That is, considermodeling, the information decay byRt = Pt/452.42where 0 < St < 1. The dependence of 8 on time allows the possibility of a differentrate of information decay at different times. Thus, from time t — 1 to time t the loss ofinformation is 100(1 — Et)/6t% andRt = Pt + Wt and Wt = Pt(1 — 8)/8tNote that this implies that for all components of the state vector, the information decaysat the same rate.A more general approach to discounting is the method of component discountingdiscussed in West and Harrison [13]. This approach involves partitioning the state vectorOt into h components, such that p = p1 -I- p2 +... + ph as follows:0; = At, • • • , ° 'ht),rt = (Fit, • • • , rht),Gt = block diag[Git, • • • , Ght]_Glt o-o0 G2t^. . .^0=0 0^. . .^Ghtwithand,Wt = block diag[Wit, • • • , Wht]43Wit 0 .^00^W2t . . .^00^0^. . . WhtNow, as before Pt represents the variance before the addition of the evolution noise term.Form a block diagonal matrix corresponding to the partition of Ot as follows:Pit = Var(Giteit IDt_i) (i = 1, . . . , h).Now, if we discount each component separately, we need discount factors 0 < (St', ... Sth <1, which correspond to the partitions of Ot. Then, the evolution variance, Wt is given asfollows:P lt (1 - 8t1)18t1^00^P2t(1 - 6t2)/6t2 • • •00wt =0^0^• • • Pht(1 — 45th)gth _and finally,Rt = Pt + Wt•This approach allows the various components of Ot to suffer information loses at differingrates. For example, some components of the state vector may be more robust to infor-mation loss and would hence have a discount factor near 1, and other components maychange quite rapidly and hence would have a smaller discount factor.Note that in most practical situations we can remove the dependence of the discountfactors on t.443.6 Discounted Variance LearningIn the previous sections we have assumed that the unknown observational variance, Vt,is constant, namely V. Ameen and Harrison [1] and West and Harrison [13] discuss arelaxation in this assumption by allowing for stochastic changes in V. The key idea is tointroduce a random walk component to the precision estimate (At = 1/Vs. Now, at timet - 1, the precision has posterior,(Ot- i IDt- i) ^-, G[nt_1/2, dt_1/2].Now modeling the stochastic variation as a random walk, we haveOt = Ot-i. -I- Iktwhere ikt is un-correlated with Ot_ilDt_i. Lettkt fsd [O, U]denote a distribution for tki, of form unspecified, with mean 0 and variance U. Now weknow the following,E[Ot-ilDt-i] = nt-i/dt-i = 1/St-1andVar[g6t_i IDt_i] = 2nt_i /dt2_1 = 2/ (nt_i St2_1).Now using the updating equation we see that the mean is unchanged and the varianceof Ot increases toVar[Ot I Dt-i] = Ut + 2/(nt_1.5_1).However, as in the previous section (Section 3.5) on Discount factors, it is practicallyuseful to think of variance increases in a multiplicative sense; thusVar[Otipt-i] = 2/(Kint-iSt2-1).45Where Kt, (0 < Kt < 1), is implicitly defined viaUt = Varkki_ilDt_illicT1 — 1).Ameen and Harrison [1] recommend constraining Kt to the range (.95,1). They alsorecommend removing the dependence on t, that is, assume that Kt = K for all t. Thusas in the section on Discount factors, K represents the amount of information loss aboutthe precision estimate Ot moving from time t — 1 to t.The following definition and theorem show how the discount factor is incorporatedinto the analysis.Definition 3.6 The General Univariate Dynamic Linear Model with unknown stochasticobservational variance, V, is defined by: observation equation:Yt = Ftet + vt, vt "a MO, Vtl,system equation:Ot = Gtat- I + wt, wt ,-,, N[O,Wt],initial priors:(001Do,0o) "a Nkno, VoCo4d,(OolDo) ^a G[no/2,d0/2],where 00 = V0-1 .Theorem 3.6 In the above definition (3.6) , one-step forecast and posterior distributionsare given, for each t, as follows:(0t-ilDt-i) "a G[nt_i/2,dt_i/2],(OtIpt-i) "a G[Ktnt--1/2,Ktdt_1/2],(OA) "a G[nt/2,dt/2],46(0t-ilDt-1) ^." Tnt_i [rnt-i, C_],(0t1Dt_1) ,--, Tnt_i [at, Rti,(Ytipt-i) ^- Ltnt_i [ft, Qt],(0t1Dt) -, Tnt[mt, Ct],where,at . Gtmt-i,Rt = GtCt_iG; + Wt,ft = rtat,Qt = St-i + Ft'RtFt,At = RtFt/Qtet = Yt — ft,nt = Ktnt_i +1,dt = Ktdt-i + St-iet2 IQt,St = dtInt,nit = at + Atet,ci = (St/St-1 )[Rt — At-A;(2d •Proof: The proof proceeds along the same lines as the proof of Theorem 3.5 and willnot be presented here.^ 0Ameen and Harrison [1] note that we can in most practical situations we can removethe dependence of K on t.473.7 Reference AnalysisWhat if the forecaster is uncertain about an initial prior for an analysis or if one wishesto have a baseline for comparison of a given prior with the "non-informative" prior.This section gives some results based on the "non-informative" prior distribution of theparameters 00 and cb. Thus if the forecaster is unable or unwilling to specify the initialpriors then reference analysis gives a data based alternative.Theorem 3.7 For the model defined in (3.5) let the initial prior information be repre-sented byf (0 t, OlDt_i) oc 17-1Then the joint prior and posterior distributions of the state vector and the observationvariance at time t = 1,2, ... are given byf (0 t, V IDt_i) oc V-'4."'-'/2 exp{-1/2V-1(0ith0t — 20itht + At)}f (0 t, V IDt_i) oc V-14)42 exp{-1/217-1(0Kt0t — 20'tkt + St)}whereHt = wq-i _ w;.-iGt p t- 1 G t/ w 4; -1Pt = C4-WriGt + Kt--1ht = WriGtPTiktKt = Ht + Ftllkt = ht + FtYt.7t = 7t-i + 1At — 8t-i —14_iPt-lkt-i8t = At + Yt248with initial values H1 = 0, h1 = 0, Ai = 0 and 71 = 0. Provided that W; are non-singular and known for each time.Proof: See Pole and West [9].Now, following Pole and West [9], we revert to the usual updating equations oncesufficient observations have been processed to give rise to a proper posterior distribution.In the general dynamic linear model (3.5) this happens after p+1 observations have beenprocessed, where p is the dimension of the parameter vector 0. The following theoremfrom Pole and West proves this result.Theorem 3.8 For t = p +1 the posterior distribution is(1941)t) "d Tnt[mt,Ct](V-11Dt) ,-,, G[nt12,dt12]withCt = StICT1 and mt = KTiktwhere nt = 1, and St = dt = eVQI.Proof: See Pole and West [9].3.7.1 Case of Wt UnknownThis section will present a result which frees us from the assumption of known Wt,the system equation variance. Ameen and Harrison [1] have a avoided the problem ofspecification of Wt, by the use of discount techniques. However, these methods do notapply in the reference analysis for t < p+ 1 because the posterior covariances do not yetexist. The proposed method is to assume that Wt = 0 for t = 1, 2, ... ,p + 1The rationale behind this approach is as follows. In a reference analysis with p + 1parameters, we need p + 1 observations to obtain a fully specified proper joint posterior49distribution for Ot and V. At time p + 1 we have one observation per parameter. NowW allows for changes in the parameter estimates, however, it is not possible to estimateany changes over the first p + 1 time points so setting Wt = 0 for t = 1, 2, ... ,p + 1,results in no loss of information. At time p + 1 we can revert to the usual updating aswas shown in Theorem (3.8).Theorem 3.9 In the framework of Theorem (3.7) with Wt = 0 the prior and posteriordistributions of Ot and V have the forms of Theorem (3.7) with the recursions modifiedas follows:Ht = GTliKt_iGt-1ht = GT1'kt_1Kt = Ht + Ft}1kt = ht -I- FtYtlit = -yt_i + 1At = at-ibt = At + Yt2with initial values H1 = 0, h1 = 0, Ai = 0 and -yi = 0.Proof: See Pole and West [9].3.8 Model AssessmentHow do we determine whether our model does an adequate job in terms of forecastperformance. We concentrate on overall measures of forecast performance as opposed tothe approaches of West [14] and West and Harrison [15]. Their approach is as follows:Assess the performance of the model at each time point and determine if (a) a change50in the model is necessary or (b) an observation should be considered an outlier. Theirapproach is based on cumulative Bayes factors.We will instead concentrate on overall model forecast performance, recognizing thatthe continual assessment techniques mentioned above will provide better results. Therationale behind our approach is to have an objective method of comparison with thestandard regression methods discussed in chapter 2.The measures considered are the following:Definition 3.7 The Mean Square Prediction Error, denoted by M S E, isMSE = E 411VDefinition 3.8 The Mean Absolute Prediction Error, denoted by MAD, isMAD = Et..Definition 3.9 The observed predictive densityt=NPD f(YN,YN-1, • • • , Y.91D0) = H f (YtIt -1)t=st=N dry /2rt^11191"^-LH^1I2r(df 12)QV2 (df (yt ft)2 I Q i)—(df +10til=s 7 where df =^Thus PD is the product of the sequence of one—step forecastdensities evaluated at the actual observation. Note that all products start at s, if areference analysis was done, then s = (p+1)+1, where p is the dimension of 0, otherwises = 1.Since the discount factors, 81, • • • , 8h and K are assumed to be part of the initialinformation set Do. We can consider the predictive density PD as a likelihood for thediscount factors. That is, we have the following definition,51Definition 3.10 The Log Likelihood for a parameter n = ((51,...,sh,K), denoted byLL (i1), isN_um= log PD = E log( f (YilDt-i))t=,This suggests a method of finding, "optimal" values for the discount factors in themodel. Evaluate the predictive density on a grid of discount (Si, ... , Sh, K) values andchoose the combination of values which maximizes the predictive density or likelihood.3.9 ExamplesThis section will present two examples: the Constant DLM and second the Simple Re-gression DLM. These models are simple, yet illustrate all of the basic concepts of theDLM.3.9.1 Constant DLMThe constant model is obtained from 3.6 by making the following simplifications,Ft = 1 for all t,Ot = lit 1G= 1.Thus the observation equation becomes:Yt = fit + vt, vt — N [0 , Vt] ,and the system equation is:tit = Pt-i + wt cot ''s N[0, Hit]52with initial priors:(Po Po, (ko) ^-, N[0, VoC},(OolDo) '-', G[no/2,d0/2],where 00 = VC'.Since there is only one element to the state vector we only need one discount factor,80 and if we use variance learning (section 3.6) we need a second discount factor, K inorder to specify the system variance W.3.9.2 Simple RegressionThe simple regression model is obtained from 3.6 by making the following simplifications,F't = (1,x),O't = (atogt),[ 1 0 IGt = 0 1,where, xt is the independent variable observed at the same time as Y. That is, given thevalue of xt we hope to be able to say something about the value of Y.Thus the observation equation becomes:Yt = at + Otxt + vt, vt .--, N[0, Vt],and the system equation is:at( A ) = ( at-i +^)A-1 ) (WitW2t53where(Wit)N[0,W t,T}W2twith initial priors:(00ID0, 00) ^- N[mo, Voq],(000) — G [no/2,c10/2],where 00 = V0-1.Now if we wish to perform component discounting on the simple regression model,we will need two discount factors 8,, and 60 to apply to the constant term and the sloperespectively to specify the system variance W. Since we are using variance learning(Section 3.6) we will need a third discount factor, namely K.In chapter 4 the Constant DLM and the Simple Regression DLM will be applied tothe the problem of forecasting value-weighted real returns of TSE portfolios discussed inChapter 2.3.10 Computer ImplementationThis section briefly describes the computer implementation of the Unvariate DLM. Theauthor programmed the dynamic linear model in Fortran 77 using NAG subroutines(Numerical Algorithms Group). We then used the dyn.load features of SPLUS 3.1 toprovide a user friendly interface.54Chapter 4Empirical Results of Applying DLM to Value—Weighted Real ReturnsIn this chapter we apply the methods of Bayesian forecasting using Dynamic LinearModels to the problem of forecasting value—weighted real returns, Rtx , of TSE portfoliosusing dividend yields, DYE, for different return horizons, T. Specifically, we will usethe two models discussed in Section 3.9, namely the Constant DLM and the SimpleRegression DLM.4.1 Constant DLMIn this section we deal with the problem of specifying a model for the mean level ofthe real return series Rtx . That is, are the returns made up of a slowly varying meancomponent plus observational noise with a time varying variance? Specifically, the modelfor return horizon T is given by:observation equation:and system equation:with initial priors:Rtx = Pt,T + Vt,T , Vt rsa N[0,14,71,lit,T = P (t-1),T + Wt,T Wt '''' N[0, Wt,T](110,TID0, 00,T) r's N{MO,T, VO,TC0*,11,(00,TIDO) '''' G[n0,712, do,r/2],55where 00,T . V0721.Notice that we have different parameter values for each return horizon T. Now, inorder to specify the system variance WiT we need a discount factor 5T• We shall alsoemploy the methods of variance learning, thus we need a second discount factor K. Arewe justified in assuming that the precision Ot follows a random walk? Figure 2.4 shedssome light on this assumption; the estimated residual variance doesn't seem to varyingrandomly about constant value. This graphical evidence seems to suggest that some timevarying variance is appropriate. However, there is no guarantee that the random walkapproach of variance learning is the "best" approach to take.The initial priors are based on the "non-informative" priors and are discussed in thesection on reference analysis (see Section 3.7).We employ the methods of Section 3.8 to choose the "optimal" values of the dis-count factors. That is, we evaluate the predictive density at a grid of values for SAT -=0.01, 0.02, ... , 1.00 and KT = 0.95, 0.96, ... ,1.00 and choose the pair that maximizesthe predictive density or equivalently the log likelihood, namely 31,7, and RT. Table 4.1presents the maximizing values.56Table 4.1: Maximum Likelihood estimates of 64,7, and KT for the constant DLM for returnhorizons T.Monthly Quarterly Yearly 2 Year 3 Year 4 YearPortfolio -gar -IZ7' ;-50.7. 11T --(-5aT KT -gaT 1-.4 LT 1-4 0a7 icTComposite 1.00 0.95 1.00 0.95 1.00 1.00 1.00 0.95 0.24 1.00 0.38 0.95Metals 1.00 0.95 1.00 0.95 1.00 1.00 1.00 0.97 0.37 0.96 0.41 1.00Gold 1.00 0.95 1.00 0.95 1.00 0.96 1.00 0.95 0.01 1.00 0.19 0.95Oil 1.00 0.95 1.00 0.95 1.00 1.00 0.02 1.00 0.08 0.95 0.20 1.00Paper 1.00 0.95 1.00 0.95 1.00 1.00 1.00 0.99 0.15 1.00 0.32 0.95Consumer 1.00 0.95 1.00 0.97 1.00 1.00 0.68 0.96 0.06 0.95 0.25 1.00Industrial 1.00 0.96 1.00 1.00 1.00 1.00 0.91 1.00 0.07 1.00 0.50 1.00Real Estate 0.92 0.95 0.84 0.95 0.58 1.00 0.36 1.00 0.27 1.00 0.21 1.00Transportation 1.00 0.97 1.00 0.97 1.00 1.00 0.16 1.00 0.12 1.00 0.18 0.95Pipelines 1.00 0.95 1.00 0.95 1.00 0.96 1.00 0.95 0.22 1.00 0.35 0.95Utilities 1.00 0.95 1.00 0.95 1.00 1.00 0.23 1.00 0.23 1.00 0.26 1.00Communications 1.00 0.95 1.00 0.98 1.00 1.00 0.95 0.95 0.08 0.95 0.25 1.00Merchandising 1.00 0.95 1.00 1.00 1.00 1.00 0.93 1.00 0.01 1.00 0.33 1.00Financial 1.00 0.95 1.00 0.98 1.00 1.00 1.00 0.95 0.22 1.00 0.38 1.00Conglomerates 1.00 0.95 1.00 1.00 1.00 0.99 0.16 1.00 0.01 0.97 0.05 1.00Table 4.1 indicates that for most portfolios and return horizons the static model isnot appropriate, that is, either the variance is changing indicated by 1£ being less thanone or the mean is time varying indicated by 807, being less than one. There appears tobe a shift from models with changing observational variance and static mean componentto models with a highly adaptive mean component and constant observational varianceas the return horizon increases. That is, the models are changing from becoming lessadaptive with changing observational variance to models where the mean component ischanging rapidly and the observational variance is more or less constant. This change inmodel character as we move from monthly to four year returns may be due to the factthat the constant model does not apply, hence the rapidly changing mean estimates.Figure 4.1 shows plots of the real returns, {Rix}, forecasted returns and %95 forecastlimits for the TSE composite portfolio using the constant DLM. The figure reflects thepattern shown in Table 4.1 for the Composite series. For example, for the monthly return57horizon we have higher predictive density by not varying the level of the forecast, but wesuffer a time varying forecast variance. On the other hand, for the 4 year return horizon,we maximize the predictive density by having the forecasts adapt or change very quicklyin response to the changing returns, and the forecast variance is relatively constant.The reason for discussing the Constant DLM model is to have a reference modelin which to compare the forecasts generated with the Simple Regression DLM to bediscussed in the following section. That is, if the dividend yields, DYt , are to be usefulin forecasting the value-weighted real returns of the TSE portfolios then the SimpleRegression DLM should do better than the Constant DLM.581,199 -........ ................................^.........• • ••••• .......... . ......^..-es..• ............ ..... ................... .............. .....^Composite^CompositeMonthly Quarterly1970TimeComposite2 YearTim*Composite1 Year1960^1970^1980^1990^1960^1970^1980^1990Tim.^TimeComposite Composite3 Year^4 YearThnoFigure 4.1: Plots of the actual returns {Rt,T} (points), the forecasted returns at time t(solid lines) and 95% forecast limits (dashed lines) for each of the six return horizons forthe TSE Composite portfolio for the Constant DLM.59a(t -1),T )^ wt,T/30-1),T^wt,Tsystem equation:wherewith initial priors:(wti'T^N[O,Wt,T12Wt,T4.2 Simple Regression DLMThis section applies the simple regression DLM (see section 3.9.2), to problem of fore-casting the value-weighted real returns of the TSE portfolios, Rix, using dividend yields,DYt for varying return horizons, T. The advantage of using the DLM approach over (2.1)is that the the parameters of the model can be time varying. Specifically, the SimpleRegression DLM is given by:observation equation:Rt,T = at,T i3t,TDYt vt,T, vt N[0, Vt,Tb^(0 0,TI Do 0o,T)^N[mo,T Vo,TCO],^(0o,TID0)^G[no,T 12, do,T 12],where 00,7,As in the previous section we employ reference analysis to choose the initial priors(00,TID0, 00,T) and (00,TID0). In addition, we must specify the the evolution variancematrix Wtx. To accomplish this we employ the method of component discounting.That is, we must choose discount factors 8„7, for aT and Sp, for (3T, which representthe amount of information decay in the parameter estimates in the time period t — 1 tot. We also employ the method of variance learning, which allows the variance to be a60slowly varying function of time, thus we need a third discount factor KT to account forthe changing variance estimates. A graphical justification for assuming a random walkfor the parameter estimates is discussed in the previous section (see Figure 2.4).As for the Constant DLM we evaluate the predictive density or equivalently thelog likelihood over a grid of the discount values in order to find the "optimal" ormaximum likelihood values of the discount factors. Specifically, the grid consists of8,2T = 0.01,0.02, ... ,1.00, 8,3T0.01, 0.02, ... ,1.00, and icT = 0.95,0.96, ... ,1.00. Themaximum likelihood estimates of the discount factors, are given in Table 4.2 for the sixreturn horizons, T, and for each of the fifteen portfolios of the TSE.61Table 4.2: Maximum Likelihood estimates of 8„,,, SoT and KT for the Simple RegressionDLM for return horizons T.Monthly Quarterly YearlyPortfolio oar 15/3T 2T '-gaT -6,8T jCT -LT -I-5/3T 27'Composite 1.00 1.00 0.95 0.96 0.89 0.95 1.00 1.00 1.00Metals 1.00 1.00 0.95 1.00 1.00 0.95 1.00 1.00 1.00Gold 1.00 1.00 0.95 1.00 1.00 0.95 1.00 1.00 0.95Oil 1.00 0.99 0.95 1.00 1.00 0.95 1.00 1.00 0.95Paper 1.00 1.00 0.95 1.00 1.00 0.95 1.00 1.00 1.00Consumer 1.00 1.00 0.95 0.99 1.00 0.96 1.00 1.00 1.00Industrial 1.00 1.00 0.96 1.00 1.00 1.00 1.00 0.99 1.00Real Estate 0.99 0.96 0.95 1.00 0.95 0.95 0.93 0.89 1.00Transportation 1.00 1.00 0.97 1.00 1.00 0.97 1.00 1.00 0.95Pipelines 1.00 0.99 0.95 1.00 0.97 0.95 1.00 0.93 0.97Utilities 1.00 1.00 0.95 1.00 1.00 0.95 0.97 0.95 1.00Communications 1.00 1.00 0.95 1.00 1.00 0.98 1.00 1.00 1.00Merchandising 0.97 0.99 0.95 1.00 1.00 0.95 1.00 0.99 1.00Financial 1.00 1.00 0.95 0.94 0.89 1.00 1.00 0.97 1.00Conglomerates 1.00 1.00 0.95 1.00 1.00 1.00 1.00 1.00 1.002 Year 3 Year 4 YearPortfolio O. -gpT ICT L T LT le T LT -(5/3T 2TComposite 0.68 0.96 1.00 0.80 0.95 1.00 0.80 1.00 1.00Metals 0.98 1.00 0.95 0.90 1.00 0.95 0.90 0.75 1.00Gold 0.55 0.72 0.95 0.45 0.75 1.00 0.60 0.65 1.00Oil 1.00 1.00 0.95 1.00 0.95 0.95 0.95 1.00 0.97Paper 1.00 1.00 0.97 0.85 0.50 1.00 0.95 1.00 0.95Consumer 0.87 0.93 1.00 0.80 0.90 1.00 0.90 0.95 1.00Industrial 1.00 0.96 1.00 0.60 1.00 0.97 0.65 1.00 1.00Real Estate 0.74 0.81 1.00 0.70 0.70 1.00 0.65 0.75 1.00Transportation 0.86 0.55 1.00 0.90 0.45 0.95 0.90 0.45 0.95Pipelines 0.77 1.00 1.00 0.85 1.00 1.00 0.80 1.00 0.95Utilities 0.95 1.00 1.00 0.95 1.00 0.95 1.00 0.95 1.00Communications 0.95 0.55 0.95 0.35 0.95 0.95 0.85 0.80 1.00Merchandising 0.84 0.83 1.00 0.55 0.95 0.95 0.75 0.95 1.00Financial 0.97 0.57 1.00 0.90 0.95 1.00 0.95 0.90 0.98Conglomerates 0.55 0.91 1.00 0.95 0.25 0.95 0.55 0.85 1.0062It is apparent from examining Table 4.2 that the assumption of non—time varyingparameters is suspect for the majority of the portfolios and return horizons. The pa-rameters a and # for Fama and French model (2.1) are time varying for the majorityof the portfolios for the two to four year return horizons as evidenced by the maximumlikelihood estimates for Sa, and 8,37, being less than one. However, in most instances forthe two to four year return horizons the observational variance is constant as indicated bythe KT estimate being one. For the monthly and quarterly return horizons the oppositeappears to happen, that is the parameters a and # are constant and the observationalvariance is time varying. And finally, for the yearly returns, the model with no timevarying parameters is "best" for most return horizons.Figure 4.1 shows plots of the real returns, {Rix}, forecasted returns and %95 fore-cast limits for the TSE composite portfolio using the simple regression DLM. Figure 4.2depicts a similar pattern to that of Figure 4.1, that is, for short return horizons the domi-nate feature is the changing forecast variance and for longer return horizons the adaptionis more pronounced, indicating that dividend yields actually forecast real returns. Ofcourse, this refers only to the Composite portfolio.In the following two sections we compare the simple regression DLM with the constantregression DLM to determine if dividend yields contribute exhibit any forecast power.Secondly, we compare the simple regression DLM with the classical regression approachdiscussed in chapter 2.631960.^•1980 1e70TheeIv1900 1980^1970^1080^1990Ti me^Composite^CompositeMonthly QuarterlyComposite^Composite1 Year 2 Year1980^1970^1960^1980Time^ rim.Composite Composite3 Year^4 YearTime TimeFigure 4.2: Plots of the actual returns {Rt,T} (points), the forecasted returns at time t(solid lines) and 95% forecast limits (dashed lines) for each of the six return horizons forthe TSE Composite portfolio for the Regression DLM.644.3 Constant vs Regression DLMWe compare the predictive densities of the Constant DLM and the Simple RegressionDLM to determine if dividend yields, DYt, have any ability to forecast value-weightedreal returns, Rt,T, of TSE portfolios. A model is better if it has a higher predictive densityor higher log likelihood. We define the following difference of log likelihoods:VLL = LLRegression - LLConstant •The following table gives values of VLL for each of the six return horizons and for eachof the TSE portfolios. Note that there is no guarantee that if we add a regression variablethat the Predictive Density will increase.Table 4.3: Comparison of Constant DLM to the Regression DLM for each of the sixreturn horizons and each of the TSE portfolios using V'LL.Portfolio Monthly Quarterly 1 Year 2 Year 3 Year 4 YearComposite -3.481 -4.422 -0.283 4.614 3.082 7.287Metals and Minerals -5.601 -5.832 -1.931 -1.612 -2.435 0.430Gold and Silver -4.078 -3.731 -0.940 6.249 4.930 5.662Oil and Gas 2.306 1.833 0.715 5.918 8.507 9.377Paper and Forest -5.690 -4.043 -2.568 -3.024 -2.910 -4.776Consumer -3.787 -1.174 -0.667 7.190 1.773 6.196Industrial -5.287 -4.627 -2.284 1.052 2.019 5.466Real Estate -2.624 -3.286 -3.041 -1.031 3.075 2.542Transportation -5.833 -6.025 -2.299 -0.434 0.745 2.234Pipelines -2.377 0.715 1.247 6.754 7.669 13.680Utilities -6.321 -5.711 -0.444 7.784 8.659 10.050Communications -6.184 -4.204 -0.568 4.370 -0.308 3.272Merchandising -0.257 -2.570 0.199 9.747 5.064 9.814Financial Services -5.293 -5.712 -0.505 8.586 4.621 6.867Conglomerates -7.678 -2.905 0.518 2.088 -1.541 -1.626Examining Table 4.3 we can make the following observations:• Monthly return horizon - constant DLM outperforms regression DLM for all port-folios except for the Oil and Gas portfolio;65• Quarterly return horizon — constant DLM outperforms regression DLM for all port-folios except the Oil and Gas portfolio and the Pipelines portfolio;• One Year return horizon — constant DLM outperforms regression DLM for all port-folios except for the Oil and Gas portfolio, the Pipelines portfolio, the Merchandis-ing portfolio and the Conglomerates portfolio;• Two Year return horizon — regression DLM outperforms constant DLM for all port-folios except for the Metals and Minerals portfolio, the Paper and Forest portfolio,the Real Estate and Construction portfolio, and the Transportation portfolio.• Three Year return horizon — regression DLM outperforms constant DLM for allportfolios except for the Metals and Minerals portfolio, the Paper and Forest port-folio, and the Communications portfolio.• Four Year return horizon — regression DLM outperforms constant DLM for all port-folios except for the Paper and Forest portfolio, and the Conglomerates portfolio.It is interesting to note that, generally speaking, all portfolios seem to follow thepattern of increased predictability of real returns as the return horizon increases usingdividend yields. The only portfolios, where this is not the case are the following: Paperand Forest, Communications, and Conglomerates. This lends support to the hypothesisof Fama and French [4], that is, as the return horizon increases the predictive abilitydividend yields increases.The above results should be interpreted with some caution, since we do not have amethod of determining how large a difference in predictive density is meaningful. Also,even when there is a change in predictive density with return horizon the increase in notalways monotonic in nature.664.3.1 Classical vs DLMIn order to make a fair comparison between the classical regression approach and theregression DLM approach we would like to have a measure which behaves in the followingmanner It should penalize a forecast which is far from the observed value whenever ourforecast variance is small i.e. we have a lot of information available. It should notpenalize as heavily a forecast which is far from the observed value if the forecast varianceis large i.e. we don't have a lot of information about the next observation. So, we cansummarize the aspects of the measure in the following table:forecast error forecast variance measuresmallsmalllargelargesmalllargesmalllargeacceptableacceptableunacceptableacceptableWith these critera in mind we define the following measure for a forecast a time tEt= forecast error/Vforecast variancewhich is then used as follows:SDMSE = EVIn order to make a fair comparison we compare the out—of—sample forecasts (see Section2.2) generated by Fama and French [4] model with the DLM approach forecasts for thesame time period. We then examine the ratio of the two standardized mean squarederrors as follows:RATIO = MSEDLm/MSEciassicalNote that we did not carry out the comparison for the Real Estate and Constructionportfolio due to the limited number of observations. It should also be stressed that for67Table 4.4: Comparison of Classical Regression to the Regression DLM for each of the sixreturn horizons and each of the TSE portfolios using RATIO.Portfolio Monthly Quarterly 1 Year 2 Year 3 Year 4 YearComposite 0.918 0.905 0.945 1.479 1.806 0.521Metals and Minerals 0.716 0.893 0.943 0.486 0.438 0.395Gold and Silver 0.795 0.756 1.077 0.942 0.908 0.960Oil and Gas 0.905 0.949 1.579 2.585 1.936 0.941Paper and Forest 0.924 0.959 1.187 1.351 1.161 1.354Consumer 0.863 0.910 1.000 0.046 0.042 0.006Industrial 1.013 1.035 0.916 1.508 1.861 1.813Transportation 0.751 2.401 0.874 1.154 1.262 1.153Pipelines 0.900 0.897 0.868 1.046 0.313 0.481Utilities 0.882 1.051 0.946 0.013 0.009 0.041Communications 1.088 1.382 1.086 1.180 1.223 1.448Merchandising 1.097 0.993 1.260 0.805 0.423 0.156Financial Services 0.978 0.875 1.040 1.032 2.448 2.018Conglomerates 1.116 1.320 1.192 0.043 0.045 0.008the one to four year return horizons, the out of sample mean squared errors for theClassical approach are based on a small number of observations.After examining Table 4.4 we make the following observations:• Monthly Return Horizon - Regression DLM beats classical regression in 10 of 14cases;• Quarterly Return Horizon - Regression DLM beats classical regression in 9 of 14cases;• One Year Return Horizon - Regression DLM beats classical regression in 6 of 14cases;• Two Year Return Horizon - Regression DLM beats classical regression in 6 of 14cases;• Three Year Return Horizon - Regression DLM beats classical regression in 7 of 14cases;68• Four Year Return Horizon — Regression DLM beats classical regression in 9 of 14cases;It is interesting to note that when the classical regression beats the Regression DLM thegreatest margin of victory is 2.6, however, in the some cases the regression DLM themargin of victor is as much as 170 times better than the classical.4.4 ConclusionsIn all but two cases the Fama and French [4] results hold true in that the regressionDLM with dividend yields does better than the constant DLM for all return portfoliosexcept for the Paper and Forest, Communications and Conglomerates as measured bythe change in predictive density from the regression DLM to the constant DLM.When comparing the regression DLM approach with the classical approach discussedin chapter 2, there is not clear cut winner. However, the regression DLM approach doesbeat the classical case in the majority of cases, 55% in fact.69Chapter 5ConclusionsThis concluding chapter summarizes the findings of the thesis.5.1 Fama and FrenchThe results of Fama and French [4] which state that dividend yields show increasedforecast power to predict real returns for increasing return horizons do not extend toall portfolios of the Toronto Stock Exchange. The Fama and French results only applyto the Composite portfolio and the Oil and Gas portfolio where there is forecast powerfor all return horizons as shown in Fama and French. However, the result of increasingforecast performance also apply to the following portfolios: Gold and Silver, ConsumerProducts, Industrial Products, Real Estate and Construction, Transportation, Pipelines,Utilities, Communications, and Merchandising.However, the model suffers from some problems, most notably:• residual auto—correlation• dramatic decreases in out—of—sample R2 indicating lack of stationarity• Model 2.1 may not apply for all time periods as suggested in Fama and French[4] as evidenced by the changing residual variance results given in Section 2.4.3.It should be noted however, that Fama and French [4] mention the fact that thereturn variances are not constant throughout their sampling period. They present70results for various sub periods of interest. We did not pursue this option since wehad only a limited amount of data available.5.2 Dynamic Linear ModelIn all but two cases the Fama and French [4] results hold true in that the regressionDLM with dividend yields does better than the constant DLM for all return portfoliosexcept for the Paper and Forest, Communications and Conglomerates as measured bythe change in predictive density from the regression DLM to the constant DLM.When comparing the regression DLM approach with the classical approach discussedin chapter 2, there is not clear cut winner. However, the regression DLM approach doesbeat the classical case in the majority of cases 55%.71Bibliography[1] Ameen, J.R.M. and Harrison, P.J. (1985), "Normal Discount Bayesian Models",Bayesian Statistics 2, eds Bernardo, J.M., De Groot, M.H.„ Lindley, D.V. andSmith, A.F.M., North-Holland,Amsterdam, 271-298.[2] Box,G.E.P. and Jenkins,G.M., (1976), Time Series Analysis: Forecasting and Con-trol, Rev. Ed., Holden—Day, Oakland[3] Draper, N.R., and Smith, H. (1981), Applied Regression Analysis, 2d ed., Wiley,New York.[4] Fama, E.F. and French, K.R. (1988), "Dividend Yields and Expected Returns",Journal of Financial Economics, 22, 3-25.[5] Fama, E.F. and Schwert, R.F. (1977), "Asset Returns and Inflation", Journal ofFinancial Economics, 5, 115-146.[6] French,K.R., Schwert,G.W. and Stambaugh,R.F. (1987), "Expected Stock Returnsand Volatility", Journal of Financial Economics, 19, 3-29.[7] Hansen, L.R. and Hodrick, R.J. (1980), "Forward Exchange Rates as Optimal Pre-dictors of Future Spot Rates: An Econometric Analysis", Journal of the PoliticalEconomy, 88, no 5, 829-853.[8] Keim, D.B. and Stambaugh, R.F. (1986), "Predicting Returns in the Stock andBond Markets", Journal of Financial Economics, 17, 357-390.[9] Pole, A. and West,M. (1989), "Reference Analysis of the Dynamic Linear Model",Journal of Time Series Analysis, 10,no 2,131-147.[10] Priestley, M.B. (1981), "Spectral Analysis and Time Series", Academic Press,Toronto.[11] Traynor, P., ed The Toronto Stock Exchange Review, The Toronto Stock Exchange.[12] Weisberg,S (1985), Applied Linear Regression, 2d ed., Wiley, New York.[13] West, M. and Harrison, P.J. (1989), Bayesian Forecasting and Dynamic Models,Springer-Verlag, New York.72[14] West, M. (1986), "Bayesian Model Monitoring", Journal of the Royal StatisticalSociety Series B, 48, no 1,70-78.[15] West, M. and Harrison, P.J. (1986), "Monitoring and Adaptation in Bayesian Fore-casting Models", Journal of the American Statistical Association, 81, no 395,741-750.73Appendix 11 Index Formula and RulesEach of the Toronto Stock Exchange indices measure the current aggregatemarket value (i.e. number of presently outstanding shares x current price) ofthe stocks included in the index as a proportion of an average base aggregatemarket value (number of base outstanding shares x average base price ±changes proportional to changes made in the current aggregate market valuefigure) for such stocks. The starting level of the base value has been set equalto 1000. Expressed more briefly this is:INDEX = ^Current aggregate market valuex1000Adjusted average base aggregate market valueEssentially, there are two stages in the production of indices: (1) es-tablishment of an initial base and initial calculation of the indices; and (2)subsequent calculation of the indices taking into account recurring shifts ofthe market. Following is a detailed description of how The Toronto StockExchange indices are produced.The following formula is the basis for initial calculation of each of theindices of The Toronto Stock Exchange:INDEX=(PA x QA) + (PB x QB) + ... + (PN x QN)^x1000___^ ,(P AB X Q AB) + (P BBBB X Q BB) +... + (P NB X Q NB )A,B, ... N: the various stocks in the index portfolio.PA,PB, ... PN: the current board—lot market prices of each stock in theindex.QA,QB, • . . Q N: the numbers of currently outstanding shares of each stockin the index less any individual and/or related control blocks of 20% or more.PAB,PBB, ... FN: the trade weighted average board—lot prices of eachstock in the index during the base periodQAB,QBB, ... QNB: the number of shares of each stock in the index out-standing in the base period less any individual and/or related control blocksof 20% or less.74The base period is 1975. Calculation of the 1975 average base aggregatemarket value i.e.(75AB X QAB) + (TIBB X QBB) + • • • + (T3NB X QNB)for each index was accomplished by multiplying the trade-weighted aver-age board—lot price for each stock for the 1975 base period by the numberof shares (share weight) of each stock outstanding at the beginning of thebase period i.e. January 1, 1975 less any individual and/or control blocksof 20% or more. The current aggregate market value is determined usingclosing prices for each period for which the index is calculated multiplied bythe number of shares then outstanding, less any individual and/or relatedcontrol blocks of 20% or more, as at that period.As an example of these calculations, assume there are only two stocks ina hypothetical index. The problem is to calculate the level of the index as ofJanuary 31, 1975.Company 1The current price (January 31, 1975) is $10 and the number of sharescurrently outstanding is 18,000. The average base aggregate market value in1975 is $162,000.Company 2 The current price (January 31, 1975) is $25 and the numberof shares currently outstanding is 30,000. The average base aggregate marketvalue in 1975 is $690,000.Computation of the index would be as follows:10 x 18, 000) + (25 x 30, 000) INDEX —^ x 1000162, 000 + 690, 000930'000 INDEX =^x 1000852,000INDEX = 1091.55ADJUSTMENT TO INDEXTo calculate the indices subsequent to the establishment of the averagebase aggregate market value, recurring capital changes must be taken intoaccount. Adjustments to the indices resulting from these changes75must normally be introduced without altering the level of the in-dex(see Bankruptcy Rule (7) for exception). In other words, continuityof the index must be preserved. To accomplish this, certain procedures arefollowed. These vary according to whether the adjustments result from: (1)the issuance of additional shares of a stock in the indices; or the additionto, withdrawal from, or substitution of stocks in the indices; (2) stock rights;(3) stock dividends and stock splits; (4) a liquidation of the company; (5) anasset spin-off; (6) takeover bid, amalgamation or merger; (7) a bankruptcy;or (8) a control block adjustment.1.1 Addition or Withdrawal of Shares or Changes inNumber of StocksTwo steps are necessary to make adjustments for additions or withdrawalsof shares to or from the index calculations:(1) Updating the current aggregate market value of the index. Ifadditional shares of an index stock are issued, the current aggregate marketvalue of the stocks in that index will be accordingly higher. Likewise, if a newstock is added to the index, or if a stock is removed, the current aggregatemarket value of that stock will be added to, or subtracted from the currentaggregate market value of the other stocks in that index.(2) Adjusting the average base aggregate market value of theindex proportional to the change in the current aggregate market value sothat the index level will remain the same.The first step, therefore, towards making an adjustment is to calculatethe new current aggregate market value as indicated in (1) above.The second step is to calculate the new average base aggregate marketvalue. Expressed as a formula the second step would be as follows:Let the old average base aggregate market value = A. Let the un—adjustedcurrent aggregate market value = C. Let the current aggregate market valueof the capital to be added or withdrawn = D. The current adjusted aggregatemarket value will equal C ± D.Therefore, to establish a new average base aggregate market value (B)for an index that formula is:B=Ax(C ± D) C76To calculate the index on the new base, the formula for the hypotheticalexample given above would be:INDEX= (C ± D) x 1000BContinuing the example above, assume that Company 1 issued 2,000 newshares. This required an addition of $20,000 ($10 x 2,000) to the aggre-gate market value of the stocks in the index and therefore the new currentaggregate market value resulting from the change is: 930,000 + 20,000 =950,000.The average base aggregate market value of the index also has to bechanged proportionately.Here the formula B = A x ig1311 is used.B (930, 000 + 20, 000)= 852 000,^x930,00B = 870,323The index level remains unchanged as shown below:INDEX = 950,000 x 1000870,323INDEX = 1091.551.2 Stock RightsThe day the stock sells ex—rights, the additional shares resulting from therights are included in the calculations to establish the current aggregate mar-ket value of the indices. The average base aggregate market value, however,is adjusted by taking into account both the market price and the subscrip-tion price because on ex—rights day the current market price, and accordinglyaggregate market value, discounts the rights.The formula to calculate the new base aggregate market value followingsubscription to stock rights would be:77C + C B = S x C + D — Swhere S = the total capital subscribed for the newly issued shares.A concrete example of how a stock rights issue is incorporated into theindex is the December 5, 1975 Bank of Nova Scotia offer. The Bank, with anoutstanding capital of 18,562,500 shares, offered the shareholders of recordat the close of business on December 5, 1975 rights to buy one new shareat $36 per share of each 9 shares held. As a result, 2,062,500 new shareswere issued. Ex—rights date was December 3, 1975 and from that date ad-ditional capitalization for the Bank of Nova Scotia used in the bank indexwas 2,062,500 shares times the current price (theoretically, at this openingon the "ex" date, $41frac38 adjusted for the value of the right) amountingto $85,335,938. Actual subscription price was 2,062,500 shares times $36,amounting to $74,250,000. Calculations for the proportionately adjustingthe base were as follows:Bank IndexUn—adjusted current aggregate market value:. $4,193,109,375 Un—adjustedbase aggregate market value:. $1,337,840,000 New current aggregate marketvalue after allowing for rights: (4,194,109,375 + 85,335,938) = $4,278,445,313New base aggregate market value after allowing for rights offering:4,278,445,3131,337,840,000 x ^ = $1,361,467,4994,278,445,313 — 74, 250,000. As at the close on the day prior to the ex—date. Adjustments are madeafter the close and before the market opens the following day. Bank of NovaScotia closed at $42 on December 2, 1975.1.3 Stock Dividends, Splits, and ConsolidationsOn the ex—dividend day the outstanding share total is increased by the num-ber of shares issued in the form of dividends. Theoretically, the price of thestock should drop by the extent of the worth of the dividend. The currentaggregate market value, therefore, will not change. Hence the base figureis not adjusted. Similarly, in the case of share splits, the increased numberof shares times the lower price should equal the old number of shares times78the higher price. Thus, the current aggregate market value is theoreticallyunchanged, and the base figure is not adjusted. The same reasoning holds inthe case of stock consolidations, except that the higher price time the smallernumber of shares leaves the current aggregate market value unchanged.1.4 Liquidation of A CompanyEffective January, 1979, where a capital distribution is announced as beinga liquidation of a company whose stock is included in the index, that stockwill be removed from the index effective the ex—distribution date.1.5 Asset Spin—offEffective January, 1979, adjustments necessary to leave the level of an indexunchanged when a stock in that index has its per share value decreasedthrough an asset spin—off are made at the opening of the ex—distribution dayor as soon thereafter as the value of the asset being spun—off is known bythe Exchange staff. Thus the staff may have to recalculate index values if astock trades "ex—asset spin—off" without the index being stabilized.1.6 Takeover Bid, Amalgamation or MergerEffective January, 1979, changes in share weight or control blocks resultingfrom takeover bids, amalgamations or mergers are incorporated into the indexas soon as is administratively possible after the fact. This procedure replacesthe former procedure of incorporating such changes at the next quarterlyupdate made just after the end of the calendar quarter to which they relate.1.7 Bankruptcy of Stock in Index SystemIf and when any company, whose stock is included within the TSE "300"indices, has made an assignment in bankruptcy or been placed in receivership,its stock will be removed as soon as possible at the lowest possible price pershare (one—half cent under the present computer programmes) rather thanat the last board—lot price before trading was suspended. If, as, and whenthe company recovers in any form, it will only be eligible to be included in79the index system again after fully complying with and meeting all criteria;that is, after qualifying in the normal fashion.1.8 Control Blocks(a) All known individual and related control blocks equal to 20% or moreof the share capital of any stock included in the indices is removed in orderto reflect, as nearly as may be practical, the market float or stock normallyavailable to portfolio investors.(b) If at any time more than 90% of the outstanding shares whichare included in the TSE 300 index is held by a controlling group; as definedby the methods of computing control group holdings for index weightingpurposes, or if the shares in public hands of the same class are so reducedthat the value calculated by multiplying the most recent share price by thenumber of shares held by parties other than the control group is insufficientto meet the market capitalization criterion for admission to the index, theneach such class of equity security shall be removed from the index as soon asis conveniently practicable.(c) If an individual control block of 20% or more, or a related group ofcontrol blocks which in aggregate total 20% or more of the relevant sharesoutstanding, are initially removed from the total of such shares then out-standing for purposes of computing the share weight of the stock in theindex portfolio, and (1) the holder or holders of such stock subsequently sellstock from their position to reduce the amount of such stock holding(s) below20%, then the holding(s) will be added back to the float at the first practicaltime subsequent to such sale; (2) if the 20% or more block(s) subsequentlyfalls below 20% as a result of an increase or increases in the total of suchshare capital outstanding, then such block(s) will not be added back to theshare weight until such time as the holding falls or is reduced to 15% or lessand as soon thereafter as is practical for it to be added back.1.9 Frequency of Adjusting the IndexStock rights, stock dividends, splits, consolidations, and liquidations are re-flected in the calculations of the indices immediately as they become affective,i.e. on the "ex" date. Asset spin—offs are reflected effective the "ex" date or80as soon thereafter as the value of the asset being spun-off is known by the Ex-change staff. Takeovers, amalgamations and mergers are reflected as soon aspossible after the fact. Bankruptcy and receivership situations are reflectedas soon as possible after they are announced. Any changes resulting fromthe annual post—year—end revision as noted in the section entitled "StockEligibility Criteria" are made at the end of the first calendar quarter. Otherchanges (such as those related to control blocks or to addition or withdrawalof shares) are usually made on a quarterly basis. Additions or deletions ofstocks are usually made on a quarterly basis but may be necessary at othertimes due to delistings caused by takeovers, amalgamations, or mergers or tonormal delistings.2 Dividends, Current Indicated Annual Yields,and Dividends Adjusted to Index2.1 GeneralEach time an index value or price is computed, a "current indicated annualdividend yield" (in percent terms) and a corresponding "dividends adjustedto index amount" (in dollars and cents) may also be produced for that index.The Exchange has computed and included both of these dividend valuesin this book for each of the historical index values contained in the tableswhether on a closing monthly, weekly, or daily basis.2.2 Compilation Assumptions2.2.1 Historical Series (1956 through 1976)In general, the historical dividend per share series for each stock in the indexsystem have been compiled on the following basis:• the original dividend payments per share against payment date of thedividend;• if it could be determined that the payment in question formed part of aregular annual dividend policy of a company (whether paid quarterly,semi-annually, or annually) then the payment was multiplied by the81appropriate factor e.g. 4 in the case of a quarterly amount to forecastthe current indicated annual rate at that point in time. In addition,any extra paid within the last twelve months was then added to theregular annual rate to become the current annual rate for that stock atthat point in time.• In cases where the regular periodic amount was not level, as in the casewhere a company has a policy of paying $1 per share per annum payable20 cents in the first quarter, 30 cents in the second quarter, 20 centsin the third quarter and 30 cents in the fourth quarter, no forecastingfrom the periodic amount was carried out; instead the annual amountwas used directly in making payout calculations.• Irregular payments in terms of either time or amount were treated assuch and included on payment date and were carried forward in yieldcalculations for twelve months, after which time they were removedfrom the calculations.2.2.2 Current Series (1977 forward)Updating of the dividend per share data takes place daily as new dividendreports are received by the Exchange, subject to computer cut-off require-ments;In general, the same compilation rules as were used for the historical seriesare used for the daily up-dates, except that updating is on an "as reported"basis as noted above rather than on the payment date.2.2.3 Translation of Foreign Currency AmountsIf dividends per share are stated in a foreign currency, then Exchange stafftranslate the amounts into Canadian currency on the following basis:(a) Historical (January 1956 through December 1976):Dividends originally expressed in U.S. funds were converted to Canadianfunds using monthly average noon spot rate applicable to the month duringwhich the dividend was paid. The annual rate for companies paying dividendsin U.S. funds was the sum of all dividends (including extra dividends) paidwithin the past 12 months, as converted by using the monthly average noonspot rate applicable to the month during which each dividend was paid.82Therefore the annual rate in Canadian funds may included four differentexchange rates.(b) Current (January 1977 to present):Dividends originally expressed in U.S. funds are converted to Canadian fundsusing the closing foreign exchange rate as reported in the current Globeand Mail Report on Business for the day prior to the date of receipt of thedividend declaration.3 Calculation Procedure and ProductThe computation procedure to produce a "current indicated dividend yield"on an index and the corresponding "dividends adjusted to index" amount indollars and cents is straightforward and analogous to that on an individualstock. The current indicated annual dividend rate for each stock within anindex is multiplied by the share weight of that stock within the index as atthat computation date. The resulting dividend payout figures for the stockswithin the index are then summed. This aggregate pool of dividend dollars isthen divided by the aggregate current market value (numerator of the index)as at that same point in time for the stocks within that index. The resultantfigure is the current indicated annual yield expressed as a decimal; it may bemultiplied by 100 to express the amount as a percent.This current indicated annual yield on an index, stated as a decimal maythen be multiplied by the index for that same computation date to producethe "Dividends Adjusted to Index" amount in dollars and cents. This amountmay then be used in market valuation formulae along with other assumptionsto forecast the expected level of the index based on these assumptions.83Appendix 2TSE 300 Composite IndexRELATIVE WEIGHTSThe following list (containing 300 stocks, fourteen Group Indices, andforty-three Sub-Group Indices) provides the weight which individual stocks,Group Indices, and Sub-Group Indices bear on the Toronto Stock Exchange300 Composite Index. Computations are as of the close of December 31, 1991.Percentages in brackets indicate available float on which relative weights onComposite Index are calculated. Numbers in brackets after each Index nameindicate the number of stocks within that Index. TOTALS MAY NOT ADDDUE TO ROUNDING. All stocks in the TSE 300 are common stock or inter—convertible pairs of common stocks (shown as A, B or A,B,C) unless otherwisenoted.StockSymbol Company NameRelative Weight onComposite %1.0 METALS & MINERALS (16) 7.701.1 Integrated Mines (6) 6.87AL^Alcan Aluminum 3.00BMS^Brunswick Mining & Smelt (15%) 0.02CLT^Cominco (56%) 0.55HBM.S^Hudson Bay Min. & Smelt S (52%) 0.02N^Inco 2.15NOR^Noranda (55 %) 1.131.2 Metal Mines (8) 0.33CCH^Campbell Resources (51%) 0.01COR^Cominco Resources (35%) 0.02KER^Kerr Addison (51%) 0.08MLM^Metal Mining (41%) 0.11MVA^Minnova (50%) 0.07NGX^Northgate Exploration (75%) 0.01PMC^Princetown Mining 0.02WMI^Estmin Resources (26%) 0.021.4 Non-Base Metal Mining (2) 0.49POT^Potash Corp. of Saskatchewan (62%) 0.29ROM^Rio Algom (49%) 0.20842.0 GOLD & SILVER (28)^7.392.1 Gold Sz Silver Mines (25)^7.30ABX^American Barrick (79%) 2.04AGE^Agnico-Eagle^0.08AUR^Aur Resources (79%)^0.05BGO^Berma Gold 0.03BWR^Breakwater Resources (73%)^0.01CBJ^Cambior (79%)^0.13DML.A Dickenson Mines CL A (39%)^0.01ECO^Echo Bay Mines 0.52EN^Euro-Nevada (80%)^0.11FN^Franco Nevada (77%) 0.12GKR^Golden Knight Resources (56%) 0.04GLC^Galactic Resources (66%)^0.01GXL^Granges Inc (49%)^0.01HEM^Hemlo (45%) 0.28ICR^Intl Corona Corp (70%)^0.18LAC^LAC Minerals^0.70MVG^Minven Gold (55%)^0.00PDG^Placer Dome 1.72PGU^Pegasus Gold 0.23RAY^Rayrock Yellowknife (75%)^0.02RY0^Royal Oak Mines^0.05TEK.B Teck Corporation CL B^0.80TVX^TVX Gold (24%) 0.07VOY^Viceroy Resources 0.05WFR^Wharf Resources (49%)^0.032.2 Precious Metal Funds (3) 0.10BPT.A BGR Precious Metals CL A^0.03CEF.A Central Fund of Canada CL A 0.04G^Goldcorp^ 0.033.0 OIL Sc GAS (38) 6.713.1 Integrated Oils (4) 2.04CCT^Canadian Turbo^0.02IMO^Imperial Oil (30%) 1.36SEC^Shell Canada A (22%)^0.54TPN^Total Petroleum N.A. (47%)^0.12853.2 Oil & Gas Producers (34)^4.66AEC^Alberta Energy (63%) 0.31AXL^Anderson Exploration (33%)^0.04BPC^B.P. Canada (43%)^0.15BVI^Bow Valley Industries (67%)^0.27CBE^Cabre Exploration 0.06CEX.B Conwest Exploration^0.09CGH^Computalog Ltd 0.01CHA^Chauvco Resources (38%)^0.10CID^Chieftain International (45%)^0.04CIR^Cimarron Petroleum 0.05CNQ^Canadian Natural Resources (81%) 0.080011^Coho Resources (63%)^0.01CSW^Canadian Southern Petroleum^0.03CXY^Canadian Occidental (52%) 0.54ECR^Encor Inc^ 0.02GOU^Gulf Canada Resources (26%)^0.14LMO^Lasmo Canada (66%)^0.06MHI^Morgan Hydrocarbons 0.07MKC^Mark Resources (36%)^0.04MRP^Morrison Petroleums (77%)^0.08NCN^Norcen Energy (63%) 0.24NCN.A Norcen Energy CL A (70%)^0.29NCO^North Canadian Oils (48%) 0.11NMC^Numac Oil & Gas (53%)^0.05NWS^Nowsco Well Service 0.10PCP^PanCanadian Petroleum (13%)^0.23PNN^Pinnacle Resources^0.05POC^Poco Petroleums (65%) 0.13RES^Renaissance Energy 0.49RGO^Ranger Oil 0.49RRL^Ranchmen's Resources (44%)^0.02SKO^Saskatchewan Oil and Gas (76%)^0.18SRL^Sceptre Resources (55%)^0.05ULP^Ulster Petroleums 0.05864.0 PAPER & FOREST PRODUCTS (18)^2.234.1 Paper & Forest Products (18) 2.23A^Abitibi-Price (18%)^ 0.10CAS^Cascades (41%) 0.08CFI^Crestbrook Forest (46%) 0.04CFP^Canfor (55%)^ 0.21DHC.B Donohue CL B (46%)^0.10DOM.B Doman Industries CL B (74%) 0.03DTC^Domtar (57%) 0.25FCC.A Fletcher Challenge Canada CL A (28%)^0.16IFP.A^International Forest Products CL A (76%) 0.07MB^MacMillan Bloedel (51%)^0.61NF^Noranda Forest (18%) 0.12PFP^Canadian Pacific Forest (20%)^0.12RPP^Repap Enterprises (62%) 0.04SPL^Scott Paper (50%)^ 0.08TBC.A Tembec CL A (59%) 0.05WFT^West Fraser Timber (58%)^0.08WGL^Westar Group (70%) 0.04WLW^Weldwood of Canada (15%) 0.035.0 CONSUMER & PRODUCTS (25)^ 9.475.1 Food Processing (5)^ 0.69BCS.A B.C. Sugar Refinery CL A,B 0.12CFL^Corporate Foods (33%) 0.07CMG^Canada Malting (60%)^0.10FPL^FPI Ltd^ 0.06MFL^Maple Leaf Foods (44%) 0.355.2 Tabacco (2) 1.56IMS^Imasco (60%)^ 1.49ROC^Rothmans (29%) 0.075.3 Distilleries (2) 4.56CDL.A Corby Distilleries CL A (49%)^0.08VO^Seagram (62%)^ 4.485.4 Breweries St Beverages (4) 1.65KOC^Coca-Cola Beverages (51%)^0.09LBT^Labatt, John (62%)^ 0.71MOL.A Molson CL A 0.71MOL.B Molson CL B (50%) 0.13875.5 Household Goods (4)^0.28COC^Camco Inc (29%) 0.03CRW^Cinram Ltd (62%) 0.05DTX^Dominion Textile^0.15NMA.A Noma industries CL A (71%)^0.055.6 Autos & Parts (4) 0.50FMC^Ford Motor of Canada (6%)^0.04HAY^Hayes-Dana (43%)^0.05MG.A^Magna International CL A^0.36UAP.A UAP Inc CL A (68%) 0.055.7 Packaging Products (4)^0.23CCQ.B CCL Industries CL B (73%)^0.12CGC^Consumers Packaging (25%) 0.01INO^International Innnopac (63%)^0.01LMP.A Lawson Mardon CL A^0.096.0 INDUSTRIAL & PRODUCTS (38) 11.056.1 Steel (5)^ 1.42CE!^Co-Steel (84%)^0.17DFS^Dofasco 0.75ISP^Ipsco^ 0.19IVA.A^Ivaco CL A 0.03STE.A Stelco CL A,B^ 0.296.2 Metal Fabricators (5) 0.32DRE.A Dreco Energy Services CI A^0.04DRL^Derlan Industries^0.05HLY^Haley Industries (20%)^0.01SHL.A Shaw Industries (73%) 0.09SNU.A SNC Group CL A 0.136.3 Machinery (1)^ 0.07UDI^United Dominion Industries (44%) 0.076.4 Transportation Equipment (3)^1.00BBD.A Bombardier CL A (20%) 0.07BBD.B Bombardier CL B^0.88HSC^Hawker Siddeley Canada (41%)^0.056.5 Electrical/Electronic (10)^4.29AAZ^Archer Communications (63%)^0.01CAE^CAE Industries 0.41CMW^Canadian Marconi (48%)^0.10GAN^Gandalf Technologies 0.02KGL.B Kaufel Group CL B 0.0388MLT^Mite! (49%)^ 0.02NNC^Newbridge Networks (58%)^0.10NTL^Northern Telecom (47%) 3.43SHK^SHL Systemhouse (33%) 0.06SPZ Spar Aerospace^ 0.116.6 Cement/Concrete Products (3)^0.26GYP^CGC Inc (24%) 0.03LCI.PR.E Lafarge Canada E^ 0.12ST.A^St. Lawrence Cement CL A (63%)^0.116.7 Chemicals (6)^ 2.03ANG^Alberta Natural Gas (50%)^0.08CCL^Celanese Canada (44%) 0.14DUP.A^Du Pont Canada CL A (23%)^0.17NVA^Nova Corp^ 1.36OIL Ocelot Industries 0.12SE^Sherritt Gordon 0.156.9 Business Services (5)^ 1.65COS^Corel Systems (53%) 0.07CSN^Cognos Inc. (61%) 0.06IIT.A^Intera Information CL A (72%)^0.03MCL^Moore Corp^ 1.40XXC.B^Xerox Canada CL B^0.117.0 REAL ESTATE & CONSTRUCTION (11)^0.917.1 Developers & Contractors (4) 0.18COT^Coscan (43%)^ 0.04CXA.A^Consolidated HCI Holdings CL A (42%) 0.01ITW^Intrawest Development^0.10RLG^Royal Lepage (42%) 0.037.2 Property Managers (7) 0.74BCD^Bramalea (30%)^ 0.08CBG^Cambridge Shopping Centres (75%)^0.25CDN^Carena Development (33%)^0.10MKP^Markborough Properties (19%) 0.06RPC^Revenue Properties (29%) 0.02TZC.A^Trizec CL A (28%)^ 0.15TZC.B^Trizec CL B (21%) 0.08898.0 TRANSPORTATION & ENVIRONMENTAL SERVICES (8) 2.128.1 Transportation & Environmental Services (8)^2.12AC^Air Canada^ 0.34ALC^Algoma Central (59%)^0.01GEL^Greyhound Lines of Canada (31%)^0.05LDM.A Laidlaw Inc Cl A (53%) 0.15LDM.B Laidlaw Ind Cl B^ 1.20PEN^Philip Environmental (64%)^0.09PWA^PWA Corporation 0.16TMA^Trimac (70%)^0.129.0 PIPELINES (4) 2.069.1 Oil Pipelines (2) 0.29IPL^Interprovincial Pipe Line (36%)^0.27TMP^Transmountain Pipelines (33%) 0.039.2 Gas Pipelines (2)^1.77TRP^TransCanada Pipelines (78%)^1.34W^Westcoast Energy (63%) 0.4310.0 UTILITIES (17)^13.7210.1 Gas Utilities (3) 0.54BCG^BC Gas Inc 0.38GWT^GW Utilities (11%)^0.04UEI^Union Energy (39%) 0.1210.2 Electrical Utilities (5) 1.77ACO.X Atco CL I (80%)^0.14CU^Canadian Utilities CL A (60%)^0.26CU.X^Canadian Utilities CL B (35%) 0.10FTS^Fortis Inc^ 0.14TAU^TransAlta Utilities^ 1.1210.3 Telephone Utilities (9) 11.41AGT^Telus Corp 1.26B^BCE Inc^ 8.51BCT^British Columbia Telephone (50%)^0.70BCX^BCE Mobile Communications (26%) 0.30BRR^Bruncor (67%) 0.14MTT^Maritime Telephone & Telegraph (66%)^0.22NEL^Newtel Enterprises (44%)^0.08QT^Quebec Telephone (49%) 0.09TGO^Teleglobe Inc (49%) 0.129011.0 COMMUNICATIONS & MEDIA (19)^4.7411.1 Broadcasting (5)^ 0.31BNB^Baton Broadcasting (47%)^0.05CF^CFCF (71%) 0.03CHM.B CHUM CL B^ 0.12TM.B^Tele-Metropole CL B (73%)^0.03WIC.B WIC Western CL B (67%) 0.0811.2 Cable Se Entertainment (6)^0.79CPX Cineplex Odeon (43%) 0.04RCI.A Rogers Communications CL A (9%) 0.04RCI.B Rogers Communications CL B (35%) 0.52SAT Canadian Satellite (26%) 0.02SCL.B Shaw Cablesystems CL B (78%)^0.10VDO^Le Groupe Videotron (29%)^0.0611.3 Publishing Se Printing (8) 3.64HLG^Hollinger (32%)^ 0.12MHP^Maclean Hunter (80%)^1.01QBR.A Quebeccor CL A (44%) 0.09QBR.B Quebeccor CL B 0.14STM^Southam (76%)^ 0.45TOC^Thompson Corporation (30%)^1.57TSP^Toronto Sun Publishing (36%) 0.07TS.B^Torstar CL B (43%)^0.2012.0 MERCHANDISING (31) 5.1312.1 Wholesale Distributors (5)^0.37ACK^Acklands (76%)^ 0.03EML^Emco Ltd (51%) 0.02FTT^Finning Ltd 0.26UWB^United Westburne (28%)^0.04WJX.A Wajax CL A,B (46%) 0.0212.2 Food Stores (6)^ 1.46EMP.A Empire CL A 0.12L^Loblaw Companies (24%)^0.19OSH.A Oshawa Group CL A 0.47PGI^Provigo (48%)^ 0.23UGO.B Unigesco CL B 0.03WN^Weston, George (42%)^0.4212.3 Department Stores (3) 0.49HBC^Hudson's Bay Co (27%)^0.26SCC^Sears Canada (37%) 0.20WDS^Woodward's Ltd (49%) 0.039112.4 Clothing Stores (3)^ 0.17DLX.A^Dylex CL A (76%) 0.10GFG.A^Grafton Group CL A 0.00RET.A^Reitman's CL A^ 0.0712.5 Specialty Stores (5) 1.38CTR.A^Canadian Tire CL A 1.12GDS.A^Gendis CL A,B (43%)^0.09PCJ.A^Peoples Jewellers CL A 0.01PJC.A^Jean Coutu Group CL A 0.14TOG^TOG International (41%)12.6 Health Hospitality (9)^1.27BCH^Biochem (IAF) (61%) 0.30CAO.A^Cara Operations CL A (63%)^0.12CAO^Cara Operations (38%)^0.07DEP^Deprenyl Research (68%) 0.15FSH^Four Seasons Hotel 0.15LWN^Loewen Group (77%)^0.23MHG.A MDS Health Group CL A (42%)^0.03MHG.B MDS Health Group CL B^0.18QLT^Quadra Logic (67%) 0.0413.0 FINANCIAL SERVICES (35) 20.9813.1 Banks (7)^ 18.12BMO^Bank of Montreal^3.01BNS^Bank of Nova Scotia 2.51CM Cdn Imperial Bank of Commerce^3.62LB^Laurentian Bank of Canada (38%)^0.07NA National Bank of Canada^0.86RY^Royal Bank of Canada 4.88TD Toronto-Dominion Bank 3.1713.2 Trust, Savings & Loan (6)^0.60CEH^Central Capital (32%) 0.00CEH.A^Central Capital CL A (78%)^0.00CGA^Central Guaranty Trustco (12%)^0.00NT National Trustco (53%) 0.24RYL^Royal Trustco (47%)^0.33TTG^General Trustco (35%) 0.0213.3 Investment Co's & Funds (6)^0.61CGI^Canadian General Investments (52%)^0.05CNN.UN Canada Trust Income Investments^0.02FMS.A^First Marathon CL A^0.12IGI^Investors Group (25%) 0.15MKF^Mackenzie Financial Corp^0.21UNC^United Corporations (50%) 0.069213.5 Insurance (6)^ 0.49CRX^Crownx (22%) 0.00CRX.A Crownx CL A 0.01ELF^E-L Financial (64%)^0.08FFH^Fairfax Financial 0.05GWO^Great West Lifeco (14%)^0.09LON^London Insurance Group (43%)13.6 Financial Management Co's (10)^1.15CXE^Consol. Canadian Express (57%)^0.01CXS^Counsel Corp (53%)^0.03DBC.A Dundee Bancorp CL A 0.04FC^FCA International (74%)^0.02GLZ^Great Lakes Group (8%) 0.05HIL^Hees International Bancorp (37%)^0.25LGC.B Laurentian Group CL B (31%)^0.03PGC.A Pagurian CL A (65%)^0.18PWF^Power Financial Corp (31%)^0.30TFC.A Trilon Financial CL A (42%)14.0 CONGLOMERATES (12)^5.7614.1 Conglomerates (12) 5.76AGR.B Agra Industries CL B (74%)^0.05BL.A^Brascan CL A (49%)^0.45CAM.A Canam Manac CL A (31%)^0.01CP^Canadian Pacific 3.31FIL.A^Federal Industries CL A,B^0.14HSM^Horsham^ 0.49ISE^International Semi-Tech (65%)^0.07JN^Jannock (73%) 0.19OCX^Onex Corporation (75%)^0.07POW^Power Corp of Canada (61%)^0.60SRC^Scott's Hospitality 0.33SRC.0 Scott's Hospitality CL C^0.06TSE Composite^100.0093
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Forecasting value-weighted real returns of TSE portfolios using dividend yields Blanchard, Joseph W. 1993
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Title | Forecasting value-weighted real returns of TSE portfolios using dividend yields |
Creator |
Blanchard, Joseph W. |
Date Issued | 1993 |
Description | We assess the ability of dividend yields denoted by DYt, to forecast value-weighted real returns, denoted by Rt ,T of Toronto Stock Exchange (TSE) portfolios for following return horizons, T: monthly, quarterly, and one to four year. Fama and French [4] applied similar methods to the New York Stock Exchange and found the forecast power increases as the return horizon increases. We find that the Fama and French methods generalize to TSE portfolios, however, it does not apply to all portfolios. We also determine that the Fama and French approach may not lie on solid statistical ground, in that the residual variance is not time invariant. With these drawbacks in mind we consider using the methods of Dynamic Linear Models as discussed in West and Harrison [13], which allow the model parameters to be time varying. We conclude that for the majority of the portfolios, the two methods agree, however, the regression DLM approach does slightly better in comparison with the methods of Fama and French in terms of standarized forecast errors. |
Extent | 4286891 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0086206 |
URI | http://hdl.handle.net/2429/2276 |
Degree |
Master of Science - MSc |
Program |
Statistics |
Affiliation |
Science, Faculty of Statistics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-11 |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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