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Forecasting value-weighted real returns of TSE portfolios using dividend yields 1993

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FORECASTING VALUE—WEIGHTED REAL RETURNS OF TSE PORTFOLIOS USING DIVIDEND YIELDS by JOSEPH WADE BLANCHARD B.Sc., Dalhousie University, 1990 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Statistics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1993 ©Wade Blanchard, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature Department of ^STAT sj )6 s The University of British Columbia Vancouver, Canada Date ^'T^5 / 19`)3 DE-6 (2/88) Abstract 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. 1Table of Contents Abstract List of Tables List of Figures List of Financial Variables Acknowledgements 1 Introduction 1.1 Data  ^3 2 Fama and French^ 5 2.1 Basic Definitions  ^5 2.2 Motivation  ^6 2.3 Regressions  ^7 2.3.1 Parameter Estimation  ^8 2.3.2 Assumptions  ^9 2.3.3 Corrections  ^10 2.3.4 Out of Sample Forecasts ^  11 2.3.5 Stationarity of Parameter Estimates ^  13 2.4 Results ^  14 2.4.1 Regressions  ^18 2.4.2^Out of Sample Forecasts ^ 24 2.4.3^Stationarity of Parameter Estimates ^ 25 2.5 Conclusions ^ 28 3 Theory of Dynamic Linear Models 29 3.1 Notation & Preliminaries ^ 29 3.2 Dynamic Linear Models 33 3.3 Known Observational Variance Vt ^ 34 3.4 Unknown Constant Observational Variance V ^ 38 3.5 Discount Factors ^ 42 3.6 Discounted Variance Learning ^ 45 3.7 Reference Analysis ^ 48 3.7.1^Case of Wt Unknown^ 49 3.8 Model Assessment^ 50 3.9 Examples 52 3.9.1^Constant DLM ^ 52 3.9.2^Simple Regression 53 3.10 Computer Implementation ^ 54 4 Empirical Results of Applying DLM to Value—Weighted Real Returns 55 4.1 Constant DLM ^  55 4.2 Simple Regression DLM ^  60 4.3 Constant vs Regression DLM  65 4.3.1 Classical vs DLM ^  67 4.4 Conclusions ^  69 5 Conclusions 70 iv 5.1 Fama and French ^  70 5.2 Dynamic Linear Model  71 Bibliography^ 72 Appendix 1 74 Appendix 2^ 84 ST List of Tables 1.1 The TSE Composite Index and its fourteen industry sectors ^4 2.1 Regression of real value-weighted TSE portfolio returns (Rt,T) on dividend yields (Dt/Pt), for differing return horizons T.  ^19 2.1 Continued ^  20 2.1 Continued  21 2.2 Out of Sample forecast power as measured by le ^ 26 4.1 Maximum Likelihood estimates of 6,, and KT for the constant DLM for return horizons T ^  57 4.2 Maximum Likelihood estimates of 8T ^and KT for the Simple Regres- sion DLM for return horizons T ^  62 4.3 Comparison of Constant DLM to the Regression DLM for each of the six return horizons and each of the TSE portfolios using VLL^ 65 4.4 Comparison of Classical Regression to the Regression DLM for each of the six return horizons and each of the TSE portfolios using RATIO.^ 68 v I List of Figures 2.1 Plots of TSE Composite portfolio {Rt,T} for return horizons, T. ^ 15 2.2 Plots of TSE Composite portfolio {DYt} for return horizons, T. ^ 16 2.3 Plots of TSE Composite portfolio {Rt,T} vs {DYt} for return horizons, T ^ 17 2.4 Plots of the time varying estimates of aT, 131' and 4 for the TSE Com- posite portfolio for monthly and quarterly return horizons. ^ 27 4.1 Plots of the actual returns {R,T} (points), the forecasted returns at time t (solid lines) and 95% forecast limits (dashed lines) for each of the six return horizons for the TSE Composite portfolio for the Constant DLM.^59 4.2 Plots of the actual returns {Rt,r} (points), the forecasted returns at time t (solid lines) and 95% forecast limits (dashed lines) for each of the six return horizons for the TSE Composite portfolio for the Regression DLM. 64 List of Financial Variables CPIt - A measure of inflation at time t based on the prices of produces the typical consumer 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 T viii Acknowledgements I would like to thank Dr. Jian Liu, my supervisor, for his patience and generousity in helping me complete the manuscript. I would also like to thank Dr. Harry Joe for his comments and suggestions on improving the manuscript. Finally, I would like to thank my employer, Dr. Chris Field of Dalhousie University, for his patience in my completing the thesis while working at Dalhousie. ix Chapter 1 Introduction Do stock market returns have predictable components? Several studies suggest that the answer is, in fact, affirmative. However, the predictable components typically account for approximately 3% of return variances. Fama and Schwert [5] assess the predictability of one month U.S. treasury bill rate on 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% of the stock market return variability. Keim and Stambaugh [8] using data for the period January 1928 to November 1978 and 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 years prior to the year containing month t — 1; (c) —log Pt where Pt is the share price, averaged equally across the quintile of firms with 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 the NYSE, 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 the variance 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 PQt adj R2^adj R2 January 1928 — November 1978 Q5 0.006 0.005 0.006 Q3 0.002 0.002 0.008 Q1 0.001 0.002 0.014 January 1928 — December 1952 Q5 -0.003 -0.001 -0.002 Q3 0.001 0.000 0.005 Q1 0.002 0.006 0.020 January 1953 — November 1978 Q5 0.000 -0.003 0.000 Q3 -0.003 -0.003 0.001 Q1 0.003 -0.003 -0.001 As is evident from the Table, Keim and Stambaugh [8] conclude that the predictable component 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 period January 1928 to December 1984 S & P index, to arrive at an estimate of the monthly 2 return volatility using autoregressive—integrated—moving average (ARIMA) models (Box and Jenkins [2]). The estimates of volatility are then used as a predictor for the monthly value—weighted returns of the NYSE. French, Schwert and Stambaugh conclude that the stock market volatility accounts for less than 2% of the return variance. Fama and French [4] use dividend yields to predict returns on the value-weighted portfolios of NYSE stocks for return horizons of one month, one quarter and one year to four years. Fama and French [4] state the the amount of predictability in stock market returns increases as the return period increases. The purpose of this thesis is two—fold: firstly, we apply the methods of Fama and French [4] to the Toronto Stock Exchange (TSE) value—weighted portfolios; secondly, we extend the methods of Fama and French [4] by using Dynamic Linear Regression Models described in West and Harrison [13]. Chapter 2 will describe in more detail the method of Fama and French [4] and its application to TSE portfolios. In chapter 3, we describe the basic theory of the Dynamic Linear Model (DLM) as given in West and Harrison [13]. In chapter 4, we apply the West and Harrison techniques to the TSE data. Finally in chapter 5, will give a summary of the major findings of the thesis. 1.1 Data The data consist of monthly Index Values and dividend yields of the Toronto Stock Exchange (TSE) 300 Composite and its 14 industry sectors (see Table 1.1). Appendix 1 contains a description of the method of calculation of the Index and dividend yield for a given component of the TSE Composite. Appendix 2 contains the Relative weights that each of the 300 stocks have in determining the TSE Composite and its industry portfolios on the close of December 31, 1991. 3 Table 1.1: The TSE Com osite Index and its fourteen industry sectors. Series Begin End Percentage in Dec 1991 Source Composite 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) Transportation and 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 Review Consumer Price Index Jan 1956 Dec 1992 Bank of Canada Review 4 Chapter 2 Fama and French 2.1 Basic Definitions This section contains the basic definitions of the return rate of a portfolio, the rate of inflation and dividend yields which will be used in the thesis. Definition 2.1 The continuously compounded inflation rate at time t, for return horizon T, is given by li,T = log(C Pli+T I C PIO where 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 Pt where Di is the dividend received in time period t — 1 to t and Pt is the value of the portfolio at time t. Definition 2.3 The continuously compounded nominal return of a stock portfolio at time t, for return horizon T, is given by 1 (Pt+T + C Dt,T)rt,T = log Pt where C Di,T are the accumulated dividends in the time period t to t + T. 5 Definition 2.4 The continuously compounded real return of a stock portfolio at time t, for return horizon T, is given by Rt,T 1= rt,T - We can apply the above definitions to each of the portfolios given in Table 1.1 by identifying the index value at time t with the price of the portfolio at time t. 2.2 Motivation This section will attempt to motivate the conjecture, proposed previously, that dividend yields predict returns. The following is taken from Fama and French [4]. Consider a discrete-time deterministic model in which Dt, the dividend per share for the time period from t to t +1, grows at the constant rate g, and the market interest rate that relates the stream of future dividends to the stock price Pt at time t is the constant r. In this model the price, Pt is Pt = Dt 11 + g + (1 + g)2 + . .1 [1 + r (1 + r)2^] 1 + g 1 = Dtl+rl—liff"l+r 1 + g = Dt r — g Dt (r — g) Pt — 1 + g The interest rate r is the discount rate for dividends and the period by period return on the stock, this can be seen as follows: Dt _ r — g Dt+i _ r — g Pt — 1 + g^Pt+i — 1 + g 6 however, Dt+i = (1 + g)Dt Pt-Fi = (1 + g)2  Dtr — g and the return r is r = = 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 — Pt Pt The transition form the deterministic model to a model that (a) allows uncertain future dividends and discount rates and (b) shows the relationship between discount rates and time—varying expected returns is difficult. The deterministic model given above, at the very least, lends plausibility to the conjecture that dividend yields predict expected returns. 2.3 Regressions Following Fama and French [4], the following linear regression model is proposed to model expected 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 for return horizons of one month, one quarter, and one to four years. The monthly, quarterly and annual returns are non-overlapping, the other return horizons are overlapping year— end values. 2.3.1 Parameter Estimation The usual regression estimators for equation (2.1) are given as follows: Eitv=Ti(Rt,T — RT)(DYt — DY) -I4T = EitV2i(DYt - DY)2^1 aT = RT - -NM 1. , 1 NT2 (TT = ^NT - 2 tE=1 ll'T and Var(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)1rTRT 8 where 1^DY1 R1,T XT= 1^DY2 and RT = R2,T 1 DYNT RNT,T _ and var((aT, PT)) = 01(x'TxT)' for each return horizon T. 2.3.2 Assumptions The 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 have constant 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 by 9 A second order stationary time series is one in which the mean is independent of time and the auto—covariance function depends only on the time difference T between any two time points. The reader is referred to Box and Jenkins [2] for a more detailed discussion of stationarity. Definition 2.6 The sample analogue of 7,- , denoted by ir , based on a sample of size N, is given by 1 N-T ir = i- E (zt -7)(zt+., -7) and hence the sample estimate of pr, denoted by Pr, is •;:.^ir PT = — 'Ye Now, if the underlying process has pr = 0 for T > 0, then _^1 Var(pr) = Tr. For a justification see Box and Jenkins [2]. Thus we have a method of determining whether there is serial dependence present in the residuals. Simply plot the sample auto—correlation function and determine if an inordinate number of sample auto-correlations fall outside the limits ±2 I 07 . We define E to be the number of times the sample auto—correlation fails outside the range What about the homogeneity of variance assumption will be discussed in a later section. The next section gives a correction applied to the variance of the estimators when the returns are over—lapping. 2.3.3 Corrections Hansen and Hodrick [7] suggest a correction to the variances of the estimates given Section 2.3.1 when the return horizon is larger than the sampling period. Specifically, 10 the correction should be applied to the 2 year, 3 year and 4 year return horizons. Hansen and Hodrick suggest the following modified covariance matrix for ((aT, PT)) 1 0 = —Var((aT,13T)) = 1 Rvivinoc-i NT NT T-1 0= 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 estimators are given by: ferC =^1 N 7' 7--E XliXt..F.,- NT t=1 ik^ 1 NT-I- = E it ,Tet+ T,T NT t=i where, I are the residuals, which estimate the errors E. Thus, an estimator of e, denoted by -6, is —-1— _1e = ^fikoc iv T where T-1 = 2.3.4 Out of Sample Forecasts It is well known in regression analysis that R2 tends to be overly optimistic (see Draper and Smith [3] or Weisberg [12]). To illustrate what is meant by this statement consider the following: 11 • split the data randomly into two parts, a construction sample and a validation sample; • compute the parameter estimates using only the construction sample and compute an 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 R2 Evalidation(Yvalidation — Ypred)2  . Then R2c^will be, in general, larger than R2^provided the modelonstruction^ validation assumptions 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 forecast performance of the model given in equation (2.1) when applied to future values of the real returns. Consider the following strategy: (a) Choose a window length W which will provide the estimates of the parameters of the model (2.1). (b) For the next time point, W+ 1 compute the forecast using the parameter estimates from (a) and the DYw+i to forecast Rw +1 x . Which results in a forecast error ew-Fix = RW-1-1,T - ii 147+1,T • (c) Move the window forward one point in time. Now we will base our parameter estimates on the data points 2, ... , W + 1 and then we repeat (b), using the next time point and continue with steps (b) and (c) until the end of the series is reached. Rv2alidation = 1 Evalidation(Yvalidation — Vvalidation)2 12^ ' Now we can compute the following measure of forecast performance MSEOUt R2011t = 1 e011t where MSE _x--.NT ,2 — Lat=W +1 9,T^ 1 2^v-,NT^f PPand s„t = L.t=w+ilitl,T — pout ) 2 with ^1 ^NT ^ROLA = NT ŵ E RtIT • t=W+1 2.3.5 Stationarity of Parameter Estimates Is it reasonable to assume that the same model given in equation (2.1) should apply to the entire sampling period. That is, should the parameters of the model (aT, /37, and 4) be time varying. This section will discuss an ad hoc approach to this problem. The next chapter on Dynamic Linear Models will deal with this problem in a more rigorous fashion. We assume that the model (2.1) holds for some fixed time period, say L. We then use the following strategy: (a) Using the first L data points,i.e. t=1,...,L, to provide estimates &T, ST and F4 of 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 for the 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. 13 2.4 Results We now apply the techniques discussed in the previous sections to value—weighted real returns of TSE portfolios given in table 1.1. The real returns for return horizon T at time t are denoted by Rix, see definition 2.4. The availability of the returns are given in the following table: Return Horizon NT Start End 443(299) January 1956(1968) November 1992 147(99) 1st Quarter 1956(1968) 3rd Quarter 1992 1 36(24) 1956(1968) 1991 2 35(23) 1956(1968) 1990 3 34(22) 1556(1968) 1989 4 33(21) 1956(1968) 1988 The values in parentheses apply to the Real Estate and Construction portfolio, while the others 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 T for the TSE Composite portfolio. It is apparent by examining the three sets of plots that the dividends yields appear to track real returns more closely as the return horizon increases. This indicates, at least for the Composite portfolio, that the results of Fama and French [4] seem to apply. 14 4)(10tolim • 1090 1960 1900198019701970 19801980 Time T=Monthly^ T=Quarterly Time ^ Time T=Yearly T=2 Years T=3 Years ^ T-4 Years  1960 1970 1980 Time ^ Time Figure 2.1: Plots of TSE Composite portfolio {Rt,T} for return horizons, T. 15 T=Monthly T=Quarterly Tim. T=Yearly Time T=2 Years 1960 1970 1980 1090 1960 1990 Time T=3 Years 1970^1980 Time T=4 Years tr! v^ ■^ . 1960 1970 1980 Time 1990 1960 1970 1980 Time 1960 1970 19901980 0.! Figure 2.2: Plots of TSE Composite portfolio {DYt} for return horizons, T. 16 17 3 4^6 Dividend Yields 3 4^6 Dividend Yields T=Monthly^ T=Quarteriy T=Yearly Dividend Yields T=3 Years .^. 25^3.0^3.6^4.0^4.5 Dividend Yields • .. 4 Cd 9- 1. T-2 Years .^.^, 3.0^3.5^4.0^4.6 Dividend Yields 25 5.0^5.6 T=4 Years 25 3.0^3.5^4.0^4.6 Dividend Yields 5.0^5.5 Figure 2.3: Plots of TSE Composite portfolio {/it,T} vs {DYt} for return horizons, T. , . ••.....4 4A,4?ar•• "0. • . 2.4.1 Regressions Table 2.1 contains the results of applying the regression model (2.1) to the real returns Rt,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). The number of auto correlations assessed depends on the return horizon: monthly series — 40, quarterly — 20, annual — 10. 18 Table 2.1: Regression of real value-weighted TSE portfolio returns (Rt,T) on dividend yields (Da Pt), for differing return horizons T. Real Returns Rt,T Auto-correlations T NT --)4T se(') R2^4 se() jil 132 1,3 /14 E Composite M 443 0.006 0.003 0.008^0.002 0.048 0.104 -0.053 0.071 0.028 4 Q 147 0.021 0.010 0.030^0.006 0.082 0.205 -0.014 -0.065 -0.077 1 1 36 0.081 0.038 0.118^0.022 0.167 0.020 -0.196 0.091 -0.013 0 2 35 0.132 0.046(0.053) 0.197^0.033 0.169 0.432 -0.108 -0.019 0.056 1 3 34 0.138 0.049(0.062) 0.200^0.036 0.171 0.616 0.199 -0.051 -0.047 1 4 33 0.178 0.051(0.071) 0.280^0.039 0.174 0.688 0.427 0.236 0.023 2 Metals and Minerals M 443 0.001 0.003 0.000^0.004 0.048 0.044 -0.057 0.010 -0.040 5 Q 147 0.004 0.009 0.001^0.011 0.082 0.077 -0.013 -0.105 -0.041 1 1 36 0.023 0.033 0.014^0.043 0.167 -0.111 -0.240 0.152 0.053 0 2 35 0.038 0.041(0.043) 0.025^0.066 0.169 0.310 -0.221 0.055 0.120 0 3 34 0.007 0.044(0.053) 0.001^0.074 0.171 0.550 0.179 -0.045 -0.026 1 4 33 0.034 0.050(0.066) 0.015^0.094 0.174 0.623 0.396 0.229 -0.137 2 Gold and Silver M 443 0.004 0.003 0.003^0.009 0.048 0.014 -0.069 0.000 0.081 1 Q 147 0.011 0.009 0.009^0.025 0.082 0.036 -0.070 -0.001 0.009 2 1 36 0.040 0.034 0.039^0.086 0.167 -0.186 -0.214 0.067 -0.190 0 2 35 0.092 0.042(0.043) 0.129^0.128 0.169 0.292 -0.284 -0.169 -0.237 1 3 34 0.102 0.044(0.047) 0.144^0.140 0.171 0.484 -0.020 -0.488 -0.350 4 4 33 0.108 0.048(0.048) 0.142^0.158 0.174 0.416 -0.004 -0.170 -0.383 3 Oil and Gas M 443 0.021 0.007 0.019^0.005 0.048 0.038 -0.009 0.096 -0.008 1 Q 147 0.068 0.021 0.064^0.015 0.082 0.091 0.073 0.028 0.014 0 1 36 0.251 0.079 0.228^0.051 0.167 0.190 -0.190 -0.168 -0.002 0 2 35 0.461 0.102(0.116) 0.384^0.084 0.169 0.406 -0.145 -0.202 -0.031 1 3 34 0.658 0.094(0.116) 0.603^0.072 0.171 0.542 0.029 -0.017 -0.028 1 4 33 0.730 0.088(0.101) 0.691^0.062 0.174 0.406 0.275 0.019 0.217 1 Paper and Forest Products M 443 0.000 0.002 0.000^0.004 0.048 0.141 -0.061 -0.036 0.032 6 Q 147 0.003 0.007 0.001^0.013 0.082 0.157 0.009 -0.030 -0.079 1 1 36 0.003 0.025 0.000^0.046 0.167 -0.120 -0.217 0.010 -0.138 0 2 35 0.010 0.033(0.036) 0.003^0.077 0.169 0.327 -0.285 -0.233 -0.139 0 3 34 -0.010 0.036(0.042) 0.003^0.092 0.171 0.523 -0.004 -0.319 -0.266 1 4 33 0.020 0.039(0.046) 0.009^0.106 0.174 0.538 0.134 -0.178 -0.391 2 19 Table 2.1: Continued Real Returns Rt,T Auto--correlations T NT iiT se(14T) R2 4 se(x) ii1 -A. -fi3 -A E Consumer Products M 443 0.003 0.002 0.005 0.002 0.048 0.105 -0.053 0.042 0.080 3 Q 147 0.012 0.007 0.019 0.007 0.082 0.166 0.091 -0.022 -0.043 1 1 36 0.047 0.029 0.071 0.029 0.167 -0.014 -0.160 0.225 -0.124 0 2 35 0.118 0.038(0.047) 0.225 0.045 0.169 0.460 0.041 0.113 0.033 1 3 34 0.149 0.042(0.060) 0.284 0.053 0.171 0.715 0.357 0.134 0.030 2 4 33 0.200 0.050(0.071) 0.345 0.069 0.174 0.693 0.484 0.372 0.093 4 Industrial Products M 443 0.002 0.003 0.001 0.003 0.048 0.132 -0.010 0.004 -0.041 3 Q 147 0.009 0.009 0.006 0.010 0.082 0.084 -0.145 -0.075 -0.100 0 1 36 0.034 0.029 0.039 0.024 0.167 -0.126 -0.231 0.207 0.052 0 2 35 0.045 0.037(0.039) 0.043 0.036 0.169 0.292 -0.218 0.127 0.203 0 3 34 0.032 0.037(0.050) 0.023 0.037 0.171 0.568 0.203 0.033 0.132 1 4 33 0.059 0.041(0.063) 0.061 0.046 0.174 0.630 0.440 0.334 0.067 2 Real Estate and Construction M 299 -0.007 0.005 0.007 0.007 0.058 0.232 0.045 0.141 0.017 3 Q 99 -0.020 0.018 0.013 0.031 0.101 0.230 0.102 -0.041 0.028 1 1 24 -0.015 0.090 0.001 0.164 0.204 0.279 0.109 -0.109 -0.111 0 2 23 0.104 0.149(0.180) 0.023 0.362 0.209 0.514 0.005 -0.214 -0.112 1 3 22 0.384 0.189(0.231) 0.171 0.474 0.213 0.476 0.084 -0.269 -0.251 1 4 21 0.498 0.186(0.233) 0.275 0.455 0.218 0.497 0.120 -0.255 -0.356 1 Transportation and Environmental Services M 443 0.002 0.002 0.003 0.004 0.048 0.089 0.005 0.061 0.038 2 Q 147 0.007 0.006 0.010 0.014 0.082 0.186 0.014 -0.046 -0.034 3 1 36 0.029 0.022 0.051 0.055 0.167 0.174 -0.147 0.050 -0.154 0 2 35 0.066 0.032(0.042) 0.111 0.115 0.169 0.527 -0.034 -0.135 -0.265 2 3 34 0.085 0.038(0.052) 0.133 0.146 0.171 0.661 0.156 -0.242 -0.405 3 4 33 0.103 0.043(0.057) 0.156 0.161 0.174 0.676 0.191 -0.163 -0.389 3 Pipelines M 443 0.002 0.002 0.003 0.003 0.048 0.007 -0.015 0.095 0.050 1 Q 147 0.008 0.005 0.016 0.009 0.082 0.104 -0.079 -0.115 -0.047 0 1 36 0.036 0.018 0.105 0.028 0.167 0.051 -0.150 0.049 -0.092 0 2 35 0.056 0.024(0.028) 0.144 0.046 0.169 0.431 -0.109 -0.090 0.012 1 3 34 0.064 0.026(0.036) 0.162 0.055 0.171 0.623 0.133 -0.138 -0.133 1 4 33 0.077 0.028(0.042) 0.201 0.063 0.174 0.670 0.350 0.056 -0.205 2 20 Table 2.1: Continued Real Returns Rt ,T Auto-correlations T NT 137'^se(r)2^- iR az' -se() iii -A 1:1 3 il4 E Utilities M 443 0.002 0.001 0.010^0.001 0.048 0.060 -0.072 -0.020 0.036 7 Q 147 0.007 0.003 0.031^0.004 0.082 0.068 -0.081 -0.107 0.140 0 1 36 0.024 0.012 0.099^0.012 0.167 0.033 0.028 -0.077 -0.067 0 2 35 0.039 0.016(0.022) 0.148^0.022 0.169 0.529 0.003 -0.089 -0.123 2 3 34 0.054 0.019(0.030) 0.193^0.030 0.171 0.669 0.283 -0.102 -0.088 2 4 33 0.064 0.021(0.037) 0.226^0.036 0.174 0.750 0.413 0.134 -0.033 3 Communications and Media M 443 0.003 0.002 0.004^0.003 0.048 0.123 0.015 0.108 -0.013 2 Q 147 0.008 0.006 0.013^0.009 0.082 0.173 -0.017 -0.012 -0.061 1 1 36 0.046 0.027 0.079^0.041 0.167 -0.047 -0.283 0.152 -0.068 0 2 35 0.094 0.035(0.038) 0.185^0.068 0.169 0.359 -0.232 -0.083 0.012 1 3 34 0.107 0.037(0.048) 0.210^0.076 0.171 0.604 0.069 -0.170 -0.143 1 4 33 0.139 0.039(0.055) 0.286^0.086 0.174 0.594 0.298 0.063 -0.299 2 Merchandising M 443 0.005 0.003 0.004^0.002 0.048 0.183 -0.005 0.049 0.090 7 Q 147 0.017 0.011 0.014^0.009 0.082 0.191 0.017 0.029 -0.125 2 1 36 0.086 0.049 0.083^0.040 0.167 -0.036 -0.208 0.280 0.106 0 2 35 0.143 0.064(0.073) 0.132^0.065 0.169 0.358 -0.119 0.232 0.249 1 3 34 0.129 0.064(0.093) 0.113^0.065 0.171 0.621 0.261 0.157 0.205 2 4 33 0.185 0.072(0.119) 0.174^0.080 0.174 0.703 0.513 0.441 0.223 5 Financial Services M 443 0.003 0.002 0.007^0.002 0.048 0.116 -0.048 0.045 0.063 4 Q 147 0.012 0.006 0.028^0.007 0.082 0.209 -0.090 -0.156 -0.089 2 1 36 0.047 0.027 0.084^0.027 0.167 -0.120 -0.166 0.244 -0.018 0 2 35 0.058 0.032(0.033) 0.091^0.038 0.169 0.288 -0.212 0.109 0.104 0 3 34 0.051 0.033(0.044) 0.070^0.039 0.171 0.555 0.117 -0.124 -0.187 1 4 33 0.072 0.036(0.050) 0.114^0.047 0.174 0.561 0.306 0.058 -0.307 1 Conglomerates M 443 0.002 0.002 0.004^0.004 0.048 0.064 -0.031 0.074 0.000 1 Q 147 0.006 0.005 0.008^0.012 0.082 0.136 -0.023 -0.001 -0.064 1 1 36 0.014 0.018 0.018^0.041 0.167 0.177 -0.219 0.011 0.010 0 2 35 0.019 0.028(0.033) 0.014^0.091 0.169 0.469 -0.141 -0.109 -0.070 1 3 34 0.005 0.032(0.041) 0.001^0.122 0.171 0.638 0.124 -0.214 -0.278 1 4 33 0.001 0.034(0.045) 0.000^0.139 0.174 0.680 0.267 -0.120 -0.382 2 21 We 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 return horizon; (2) Gold and Silver - forecast power for the 2 to 4 year returns, R2 increases with return horizon; (3) Oil and Gas - forecast power for all return horizons, R2 increases with return horizon; (4) Paper and Forest Products - no forecast power, R2 does not increase with return horizon T; (5) Consumer Products - forecast power for 2 to 4 year returns, R2 increases with return 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 increases with return horizon; (8) Transportation and Environmental Services - no forecast power, R2 increases with return horizon; (9) Pipelines - forecast power for 1 and 2 year returns, R2 increases with return horizon; (10) Utilities - forecast power for monthly, quarterly and annual returns, R2 increases with 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 errors from zero. For the overlapping annual returns the Hansen and Hodrick [7] corrections to the standard errors are used. It is interesting to note that the results of Fama and French [4] do not apply to all TSE portfolios we saw above. That is, the forecast power, as measured by R2, increases with return horizon. Fama and French [4] give a two—part explanation as to why this happens (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, the variance of the expected returns grows faster with the return horizon than the variance of unexpected returns — the variation of expected returns becomes a larger fraction of the variation of returns. (b) Fama and French [4] claim that residual variances for regressions of returns on yields increase less than in proportion to the return horizon. They base their explanation on the so called discount—rate effect, which simply put states that the offsetting adjustment of current prices triggered by shocks to discount rates and expected returns. They find that estimated shocks to expected returns are indeed associated with opposite shocks to prices. The cumulative price effect of these 23 shocks is roughly zero; on average, the expected future price increases implied by higher 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 are large. We would expect E, the number of exceedances of the auto—correlation function 1) 1 - above twice its standard error, to be about two for the monthly series, one for the quarterly series, and about one half for the annual series. Assuming a null hypothesis of no auto—correlation then we would expect 5% of the values to fall outside the ±2 x 1/077: limits. In many cases, there seem to be an inordinate number of auto-correlations exceeding these limits. This indicates the model given in 2.1 may be inadequate for these data. The next chapter discusses an alternative approach. 2.4.2 Out of Sample Forecasts As was mentioned in Section 2.4.2 the R2 values can be over stated when based on a single sample. This section, applies the method of Section 2.4.2 to the value-weighted real returns of TSE portfolios except for the Real Estate and Construction portfolio, due to 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, which would lead to out of sample forecasts using the next available data point. The 20 year period 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 are providing out of sample forecasts for the period January 1976 to the end of the series. The following table shows the number of in sample and out of sample observations available for each return horizon T. 24 T Nin sample Nout sample M^240 ^ 203 Q^80 ^ 67 1 20 ^ 16 2^20 ^ 15 3^20 ^ 14 4^20 ^ 13 Table 2.2 contains the out of sample forecast performance measured by R2. It is quite evident that we don't do very well when forecasting out of the range of the data for the majority of the portfolios. For example, consider the composite portfolio, the in—sample R2 are as follows: 0.008, 0.030, 0.118, 0.197, 0.200, 0.280 for the monthly, quarterly, and one to four year return horizons respectively. These values are quite inflated compared to the ones given in Table 2.2. As a further example, consider the consumer products portfolio, the in—sample R2 are 0.005, 0.019, 0.071, 0.225, 0.284 and 0.345 for the six return horizons while in Table 2.2 the R2 are all negative. This due to the fact the the dividends are changing are low over the latter part of the out-of-sample period compared to the remaining period. Thus, negative R2 values could alert us to changing model characteristics. 2.4.3 Stationarity of Parameter Estimates In this section we determine whether the regression model (2.1) applies for entire sample period. 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 arbitrarily to choose a window length of 5 years. That is, we are assuming that the parameters remain approximately constant over a five year period. Some sort of validation of this 25 Table 2.2: Out of Sample forecast power as measured by R2 Portfolio Monthly Quarterly Yearly 2 Year 3 Year 4 Year Composite -0.006 -0.028 0.022 0.084 0.081 0.060 Metals -0.016 -0.053 -0.186 -0.248 -0.326 -0.673 Gold -0.015 -0.044 -0.177 -0.232 -0.223 -0.091 Oil 0.007 0.027 0.168 0.297 0.588 0.585 Paper -0.010 -0.028 -0.111 -0.226 -0.218 -0.341 Consumer -0.011 -0.031 -0.117 -0.342 -0.971 -1.061 Industrial -0.016 -0.059 -0.195 0.046 0.268 0.387 Transportation -0.019 -0.071 -0.203 -0.198 -0.208 -0.334 Pipelines 0.008 0.025 0.098 0.213 0.275 0.249 Utilities -0.004 -0.014 0.088 0.061 0.264 0.296 Communications -0.003 0.000 -0.023 0.022 0.085 0.194 Merchandising -0.035 -0.098 -0.209 -0.475 -0.439 -0.372 Financial -0.021 -0.073 -0.001 -0.299 -0.286 -0.030 Conglomerates -0.010 -0.036 -0.156 -0.255 -0.226 -0.197 value should be done, however, we did not pursue this avenue any further, since the topic of 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 the monthly 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 Composite portfolio. Also, plotted are eitT ± 2 x se(EitT) and AT ± 2 x se(AT) (dotted lines). Note that 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 about some mean level which would support the hypothesis of stationary or constant parameter values. The behavior of the parameter estimates seem to exhibit some of the properties of a random walk. 26 Monthly Quarterly Monthly Quarterly I Tim. Monthly T1m0 Quarterly Tim. 1^ v^ I 1970 19130 1990 Tim I^ v^ 1 1960 ^ 1970 1080 1990 Tim. 1900 ^ 1970 ^ 1980 ^ 1990 Tim. I Figure 2.4: Plots of the time varying estimates of aT, OT and 4 for the TSE Composite portfolio for monthly and quarterly return horizons. 27 2.5 Conclusions The results of Fama and French [4] which state that dividend yields show increased forecast power to predict real returns for increasing return horizons do not extend to all portfolios of the Toronto Stock Exchange. The Fama and French results only apply to the Composite portfolio and the Oil and Gas portfolio where there is forecast power for all return horizons as shown in Fama and French. However, the result of increasing forecast performance also apply to the following portfolios: Gold and Silver, Consumer Products, 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 the return variances are not constant throughout their sampling period. They present results for various sub periods of interest. We did not pursue this option since we had only a limited amount of data available. For the reasons mentioned above, we should look for a more acceptable method of analysis 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. 28 Chapter 3 Theory of Dynamic Linear Models 3.1 Notation & Preliminaries This section will introduce the notations, basic definitions and theorems used in the sequel. Vectors and matrices will be denoted in bold face. For example if X is a p-dimensional vector and E is an n x p dimensional matrix, we denote them as follows X = (Xi, X2, • • • ,i, )' and an 012 • • • 0-1p 021 022 •• • 0-2p E = 0-711 an2 •• • crnp The density of a random vector will be denoted by f(X). The joint density of two random vectors X and Y is f(X, Y) and f(XIY) denotes the conditional density of X given 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 a P2-dimensional random vector and suppose we are given the conditional distribution f(X2IX1). Then the conditional distribution f(X1(X2) is given as follows: f (X21X1) f (Xi) f (XilX2) = ff. f (X2IZ1)f (Zi)dzi 29 f (x2Ixi)f (xi) f(X2) oc f (xi ,x2) Definition 3.1 A p-dimensional random vector X is said to have a Multivariate Normal Distribution, with mean p, and covariance matrix E, denoted by X -, N[A,Z] if 11(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 with parameters, n > 0 and d> 0, denoted by 0 e.s., G[n,d] if dn f&) = r (n) 0n-1 exp(-0). where r(n) is the gamma function r(n) = I xn-1 exp(—x)dx. o Note 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 by X es, Tn[p, E] iff f ( < 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 k Theorem 3.2 Consider the following Multivariate Normal Distribution: X ^-, N[p, EJ 30 X = ( xi ) , it = (111 ) , and, E = ) E21 E22X2^/1'2 which we can partition as follows: Then we have the following conditional distribution: where and (X1 1X2) - NEtli (X2), Ei(X2 )] p1 (X2) = p i + E 12Eg(x2 - 112) E1(X2) = Ell - E l2E221 E21. Theorem 3.3 Suppose that 0 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] where n* = 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 n degrees of freedom, mode p and scale matrix R = E(dIn) = E I E(0) denoted by X ,-, Tn[p,11.1. 31 Proof: By Bayes Theorem (Theorem 3.1) we have: f(0X) x f (x, oc f (x10) f (0) 0„,2 oc (270p/21E1i/2 exp( —0(X AY E-1 (X 14)12) ,2 i dn 2 exp(-0d/2) 2n/ 11(71/2) e/2-1 oc 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/2̂  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)/2-1 2(n+P)/2“(n+P)/2) (d (X — AY E 1(X — 11))-(n+P)/2 OC (d (X — it)'11.-1(dIn)(X — 1.1)) - (n+p)/2 letting R = dlnE a (n + (X — AY 11-1 (X — ii))-(n+P)/2 factoring out d n Finally we have f(X) Tn where 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). 0 32 3.2 Dynamic Linear Models Bayesian methods, not surprisingly, use Bayes theorem to update our knowledge about the current state of nature 0 once we have observed some data Y. Let us assume that our 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, given by f(YI0). Note that once we observe Y, our model can be interpreted as a likelihood for the various values of 0 and Bayes theorem tells us how to combine the prior and the likelihood to form the posterior: f(91Y) 0( i(Y10)f(0) posterior cx likelihood x prior Note that the posterior updates our knowledge of the current state of nature 0, in light of the new data Y we have observed. That is, Bayes theorem gives us a sequential way of 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 adapt as time progresses. Now combining dynamic models with Bayesian methods gives one a unified approach to the problem of forecasting a time series {Yt}, since the dynamic model tells us how the time series changes and the Bayesian approach tells us how the information we received about the time series can be incorporated into our state of knowledge. Specifically, the approach given in West and Harrison [13] for Bayesian forecasting and 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 bases our statements about Yt, made at time t —1, on the information set available Di_1. The parametric model at time t is f (1'dt:it, Dt_1), which represents our belief as to how the data at time t is derived. As mentioned above we need a prior distribution about our parameters, 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 pair effectively store the information about the defining parameter Ot as we move through time. The following sections introduce various aspects of the Dynamic Linear Model and their Bayesian analysis. 3.3 Known Observational Variance Vt This 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], 34 where: (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 sequences are 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 + wt where: Ot_i ,-., N[mt_i, Ct_il and Wt ^N[0, Wt] and Ot-i and cot are independent of each other. This implies that Ot .-,., 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 random variable with mean equaling the sum of the means and the variance equaling the sum of the variances. Thus (b) is established, i.e. f (9 ilDt-i) = N[at,Itt] 36 Now using the observation equation Yt = F'tOt vt we have the following conditional density: 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) is bivariate 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) + 0 FRt Thus, Dt_i) N[(^ t )]tfi^Q CR Now we want to find the conditional density of (OtlYt,Dt_1) which is what we want in part (d) since Dt = {Yt, Dt_1}. Note that since (Of, YtiDt_i) is multivariate normal Theorem 3.2 applies. Thus if we label X1 = Ot and X2 = Yt we have where and (OtlYt, Dt-1) N[mt, Ct] mt = at + FiRtQT1[17t — ft] Ct = Rt — FtRtQTirtRt Now 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].^0 at^FtRt Rt 37 The vector At is known as the adaptive coefficient vector, since it changes the forecast errors et and at into the new estimate of the state vector mt Now 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 in two parts. Section 3.4 will deal with constant observational variance case (Vt = V) and Section 3.6 will deal with a slowly varying observational variance V. Secondly, we do not know 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 3.8 will describe the model assessment strategy and finally Section 3.9 will give two simple examples to illustrate the theory. It should be noted that the model given in Definition 3.4 is closely related to the Kalman Filter (see Preistley [10] and references therein) with the exception of the initial prior distributional assumptions. 3.4 Unknown Constant Observational Variance V This 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 precision 0 = 1/V, if for convenience we choose to apply a Gamma prior to 0, and combining it with the new information expressed in terms of normal densities, then the posterior distribution also turns out to be a Gamma distribution with a simple updating of the parameters. 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 variance V. Using these facts we generalize the definition and theorem of the previous section to incorporate the unknown observational variance in the form of a new definition and theorem discussed presently. 38 Definition 3.5 The General Univariate Dynamic Linear Model with unknown constant variance 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 posterior distributions 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*; and 1 + Ftlt;Ft, mt = at + Atet q = 11; — AtAM where 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, and with St_1 lit = nt-i + 1 dt = dt-i + St-ie?/Qt where Qt = St_1+ FtittFt and At = RtFt/Qt• Proof: Part (a) follows directly from Theorem 3.4. The proof of remainder is again by induction on t. In order to prove part (b), assume (01Dt_1) ,-,, G[nt_l /2, dt_i/21, 40 and 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} where zit= ni_i+ 1 since Yt is univariate i.e. p =1 and di = 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 to prove (Yt1Dt_i) , Tnt_i[ft,Qt] we proceed as follows: we know from part (a) that (Ytipt-1,0"d N[ft,VQ1 and 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 initial priors given in Definition 3.5.^ 0 Note that St = dtInt estimates V = 1/0. 3.5 Discount Factors This 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 controls the model stability over time. Consider the following, Var(Ot_i I Dt-i ) -- -= Ct-i i.e. the prior variance of the current state vector. Now using the updating equations, the posterior variance is given by Var(et-ilDt-i) = Pt + Wt = R. where Pt = GiCt_iGit. Thus, we see that Wt has the effect of increasing the variance of the 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 is appropriate, since the relative magnitudes of Wt and Pt are important. That is, consider modeling, the information decay by Rt = Pt/452. 42 where 0 < St < 1. The dependence of 8 on time allows the possibility of a different rate of information decay at different times. Thus, from time t — 1 to time t the loss of information is 100(1 — Et)/6t% and Rt = Pt + Wt and Wt = Pt(1 — 8)/8t Note that this implies that for all components of the state vector, the information decays at the same rate. A more general approach to discounting is the method of component discounting discussed in West and Harrison [13]. This approach involves partitioning the state vector Ot 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 - o 0 G2t^. . .^0= 0 0^. . .^Ght with and , Wt = block diag[Wit, • • • , Wht] 43 Wit 0 .^0 0^W2t . . .^0 0^0^. . . Wht Now, 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 as follows: P lt (1 - 8t1)18t1^0 0^P2t(1 - 6t2)/6t2 • • • 0 0 wt = 0 ^ 0^• • • Pht(1 — 45th)gth _ and finally, Rt = Pt + Wt• This approach allows the various components of Ot to suffer information loses at differing rates. 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 may change quite rapidly and hence would have a smaller discount factor. Note that in most practical situations we can remove the dependence of the discount factors on t. 44 3.6 Discounted Variance Learning In 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 a relaxation in this assumption by allowing for stochastic changes in V. The key idea is to introduce a random walk component to the precision estimate (At = 1/Vs. Now, at time t - 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 have Ot = Ot-i. -I- Ikt where ikt is un-correlated with Ot_ilDt_i. Let tkt fsd [O, U] denote a distribution for tki, of form unspecified, with mean 0 and variance U. Now we know the following, E[Ot-ilDt-i] = nt-i/dt-i = 1/St-1 and Var[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 variance of Ot increases to Var[Ot I Dt-i] = Ut + 2/(nt_1.5_1). However, as in the previous section (Section 3.5) on Discount factors, it is practically useful to think of variance increases in a multiplicative sense; thus Var[Otipt-i] = 2/(Kint-iSt2-1). 45 Where Kt, (0 < Kt < 1), is implicitly defined via Ut = Varkki_ilDt_illicT1 — 1). Ameen and Harrison [1] recommend constraining Kt to the range (.95,1). They also recommend removing the dependence on t, that is, assume that Kt = K for all t. Thus as in the section on Discount factors, K represents the amount of information loss about the precision estimate Ot moving from time t — 1 to t. The following definition and theorem show how the discount factor is incorporated into the analysis. Definition 3.6 The General Univariate Dynamic Linear Model with unknown stochastic observational 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 distributions are 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/Qt et = 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 will not be presented here.^ 0 Ameen and Harrison [1] note that we can in most practical situations we can remove the dependence of K on t. 47 3.7 Reference Analysis What if the forecaster is uncertain about an initial prior for an analysis or if one wishes to 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 the parameters 00 and cb. Thus if the forecaster is unable or unwilling to specify the initial priors 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 by f (0 t, OlDt_i) oc 17-1 Then the joint prior and posterior distributions of the state vector and the observation variance at time t = 1,2, ... are given by f (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)} where Ht = wq-i _ w;.-iGt p t- 1 G t/ w 4; -1 Pt = C4-WriGt + Kt--1 ht = WriGtPTikt Kt = Ht + Ftll kt = ht + FtYt .7t = 7t-i + 1 At — 8t-i —14_iPt-lkt-i 8t = At + Yt2 48 with 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 once sufficient 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 been processed, where p is the dimension of the parameter vector 0. The following theorem from 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] with Ct = StICT1 and mt = KTikt where nt = 1, and St = dt = eVQI. Proof: See Pole and West [9]. 3.7.1 Case of Wt Unknown This 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 of specification of Wt, by the use of discount techniques. However, these methods do not apply in the reference analysis for t < p+ 1 because the posterior covariances do not yet exist. The proposed method is to assume that Wt = 0 for t = 1, 2, ... ,p + 1 The rationale behind this approach is as follows. In a reference analysis with p + 1 parameters, we need p + 1 observations to obtain a fully specified proper joint posterior 49 distribution for Ot and V. At time p + 1 we have one observation per parameter. Now W allows for changes in the parameter estimates, however, it is not possible to estimate any 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 as was shown in Theorem (3.8). Theorem 3.9 In the framework of Theorem (3.7) with Wt = 0 the prior and posterior distributions of Ot and V have the forms of Theorem (3.7) with the recursions modified as follows: Ht = GTliKt_iGt-1 ht = GT1'kt_1 Kt = Ht + Ft}1 kt = ht -I- FtYt lit = -yt_i + 1 At = at-i bt = At + Yt2 with initial values H1 = 0, h1 = 0, Ai = 0 and -yi = 0. Proof: See Pole and West [9]. 3.8 Model Assessment How do we determine whether our model does an adequate job in terms of forecast performance. We concentrate on overall measures of forecast performance as opposed to the 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 change 50 in the model is necessary or (b) an observation should be considered an outlier. Their approach is based on cumulative Bayes factors. We will instead concentrate on overall model forecast performance, recognizing that the continual assessment techniques mentioned above will provide better results. The rationale behind our approach is to have an objective method of comparison with the standard 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, is MSE = E 411V Definition 3.8 The Mean Absolute Prediction Error, denoted by MAD, is MAD = E t.. Definition 3.9 The observed predictive density t=N PD f(YN,YN-1, • • • , Y.91D0) = H f (YtIt -1) t=s t=N dry /2rt^11191 "^-LH ^1I2r(df 12)QV2 (df (yt ft)2 I Q i)—(df +10 til=s 7 where df =^Thus PD is the product of the sequence of one—step forecast densities evaluated at the actual observation. Note that all products start at s, if a reference analysis was done, then s = (p+1)+1, where p is the dimension of 0, otherwise s = 1. Since the discount factors, 81, • • • , 8h and K are assumed to be part of the initial information set Do. We can consider the predictive density PD as a likelihood for the discount factors. That is, we have the following definition, 51 Definition 3.10 The Log Likelihood for a parameter n = ((51,...,sh,K), denoted by LL (i1), is N _um= log PD = E log( f (YilDt-i)) t=, This suggests a method of finding, "optimal" values for the discount factors in the model. Evaluate the predictive density on a grid of discount (Si, ... , Sh, K) values and choose the combination of values which maximizes the predictive density or likelihood. 3.9 Examples This 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 the DLM. 3.9.1 Constant DLM The constant model is obtained from 3.6 by making the following simplifications, Ft = 1 for all t, Ot = lit 1 G= 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] 52 with 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 in order to specify the system variance W. 3.9.2 Simple Regression The simple regression model is obtained from 3.6 by making the following simplifications, F't = (1,x), O't = (atogt), [ 1 0 I Gt = 0 1 , where, xt is the independent variable observed at the same time as Y. That is, given the value 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 ) (WitW2t 53 where (Wit) N[0,W t,T} W2t with 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 slope respectively 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 to the the problem of forecasting value-weighted real returns of TSE portfolios discussed in Chapter 2. 3.10 Computer Implementation This section briefly describes the computer implementation of the Unvariate DLM. The author 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 to provide a user friendly interface. 54 Chapter 4 Empirical Results of Applying DLM to Value—Weighted Real Returns In this chapter we apply the methods of Bayesian forecasting using Dynamic Linear Models to the problem of forecasting value—weighted real returns, Rtx , of TSE portfolios using dividend yields, DYE, for different return horizons, T. Specifically, we will use the two models discussed in Section 3.9, namely the Constant DLM and the Simple Regression DLM. 4.1 Constant DLM In this section we deal with the problem of specifying a model for the mean level of the real return series Rtx . That is, are the returns made up of a slowly varying mean component plus observational noise with a time varying variance? Specifically, the model for 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], 55 where 00,T . V0721. Notice that we have different parameter values for each return horizon T. Now, in order to specify the system variance WiT we need a discount factor 5T• We shall also employ the methods of variance learning, thus we need a second discount factor K. Are we justified in assuming that the precision Ot follows a random walk? Figure 2.4 sheds some light on this assumption; the estimated residual variance doesn't seem to varying randomly about constant value. This graphical evidence seems to suggest that some time varying variance is appropriate. However, there is no guarantee that the random walk approach of variance learning is the "best" approach to take. The initial priors are based on the "non-informative" priors and are discussed in the section 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 maximizes the predictive density or equivalently the log likelihood, namely 31,7, and RT. Table 4.1 presents the maximizing values. 56 Table 4.1: Maximum Likelihood estimates of 64,7, and KT for the constant DLM for return horizons T. Monthly Quarterly Yearly 2 Year 3 Year 4 Year Portfolio -gar -IZ7' ;-50.7. 11T --(-5aT KT -gaT 1-.4 LT 1-4 0a7 icT Composite 1.00 0.95 1.00 0.95 1.00 1.00 1.00 0.95 0.24 1.00 0.38 0.95 Metals 1.00 0.95 1.00 0.95 1.00 1.00 1.00 0.97 0.37 0.96 0.41 1.00 Gold 1.00 0.95 1.00 0.95 1.00 0.96 1.00 0.95 0.01 1.00 0.19 0.95 Oil 1.00 0.95 1.00 0.95 1.00 1.00 0.02 1.00 0.08 0.95 0.20 1.00 Paper 1.00 0.95 1.00 0.95 1.00 1.00 1.00 0.99 0.15 1.00 0.32 0.95 Consumer 1.00 0.95 1.00 0.97 1.00 1.00 0.68 0.96 0.06 0.95 0.25 1.00 Industrial 1.00 0.96 1.00 1.00 1.00 1.00 0.91 1.00 0.07 1.00 0.50 1.00 Real Estate 0.92 0.95 0.84 0.95 0.58 1.00 0.36 1.00 0.27 1.00 0.21 1.00 Transportation 1.00 0.97 1.00 0.97 1.00 1.00 0.16 1.00 0.12 1.00 0.18 0.95 Pipelines 1.00 0.95 1.00 0.95 1.00 0.96 1.00 0.95 0.22 1.00 0.35 0.95 Utilities 1.00 0.95 1.00 0.95 1.00 1.00 0.23 1.00 0.23 1.00 0.26 1.00 Communications 1.00 0.95 1.00 0.98 1.00 1.00 0.95 0.95 0.08 0.95 0.25 1.00 Merchandising 1.00 0.95 1.00 1.00 1.00 1.00 0.93 1.00 0.01 1.00 0.33 1.00 Financial 1.00 0.95 1.00 0.98 1.00 1.00 1.00 0.95 0.22 1.00 0.38 1.00 Conglomerates 1.00 0.95 1.00 1.00 1.00 0.99 0.16 1.00 0.01 0.97 0.05 1.00 Table 4.1 indicates that for most portfolios and return horizons the static model is not appropriate, that is, either the variance is changing indicated by 1£ being less than one or the mean is time varying indicated by 807, being less than one. There appears to be a shift from models with changing observational variance and static mean component to models with a highly adaptive mean component and constant observational variance as the return horizon increases. That is, the models are changing from becoming less adaptive with changing observational variance to models where the mean component is changing rapidly and the observational variance is more or less constant. This change in model character as we move from monthly to four year returns may be due to the fact that 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 forecast limits for the TSE composite portfolio using the constant DLM. The figure reflects the pattern shown in Table 4.1 for the Composite series. For example, for the monthly return 57 horizon we have higher predictive density by not varying the level of the forecast, but we suffer 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 quickly in 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 model in which to compare the forecasts generated with the Simple Regression DLM to be discussed in the following section. That is, if the dividend yields, DYt , are to be useful in forecasting the value-weighted real returns of the TSE portfolios then the Simple Regression DLM should do better than the Constant DLM. 58 1,1 9 9 - ........ ......................... .......^......... • • ••••• .......... . ......^.. -es ..• ............ ..... ......... .......... .............. ..... ^ Composite ^ Composite Monthly Quarterly 1970 Time Composite 2 Year Tim* Composite 1 Year 1960^1970^1980^1990 ^ 1960 ^ 1970^1980 ^ 1990 Tim. ^ Time Composite Composite 3 Year ^ 4 Year Thno Figure 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 for the TSE Composite portfolio for the Constant DLM. 59 a(t -1),T )^ wt,T /30-1),T^wt,T system equation: where with initial priors: (wti ' T^ N[O,Wt,T12 Wt,T 4.2 Simple Regression DLM This 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 Simple Regression 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 variance matrix 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 represent the amount of information decay in the parameter estimates in the time period t — 1 to t. We also employ the method of variance learning, which allows the variance to be a 60 slowly varying function of time, thus we need a third discount factor KT to account for the changing variance estimates. A graphical justification for assuming a random walk for 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 the log likelihood over a grid of the discount values in order to find the "optimal" or maximum likelihood values of the discount factors. Specifically, the grid consists of 8,2T = 0.01,0.02, ... ,1.00, 8,3T0.01, 0.02, ... ,1.00, and icT = 0.95,0.96, ... ,1.00. The maximum likelihood estimates of the discount factors, are given in Table 4.2 for the six return horizons, T, and for each of the fifteen portfolios of the TSE. 61 Table 4.2: Maximum Likelihood estimates of 8„,,, SoT and KT for the Simple Regression DLM for return horizons T. Monthly Quarterly Yearly Portfolio 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 1.00 Metals 1.00 1.00 0.95 1.00 1.00 0.95 1.00 1.00 1.00 Gold 1.00 1.00 0.95 1.00 1.00 0.95 1.00 1.00 0.95 Oil 1.00 0.99 0.95 1.00 1.00 0.95 1.00 1.00 0.95 Paper 1.00 1.00 0.95 1.00 1.00 0.95 1.00 1.00 1.00 Consumer 1.00 1.00 0.95 0.99 1.00 0.96 1.00 1.00 1.00 Industrial 1.00 1.00 0.96 1.00 1.00 1.00 1.00 0.99 1.00 Real Estate 0.99 0.96 0.95 1.00 0.95 0.95 0.93 0.89 1.00 Transportation 1.00 1.00 0.97 1.00 1.00 0.97 1.00 1.00 0.95 Pipelines 1.00 0.99 0.95 1.00 0.97 0.95 1.00 0.93 0.97 Utilities 1.00 1.00 0.95 1.00 1.00 0.95 0.97 0.95 1.00 Communications 1.00 1.00 0.95 1.00 1.00 0.98 1.00 1.00 1.00 Merchandising 0.97 0.99 0.95 1.00 1.00 0.95 1.00 0.99 1.00 Financial 1.00 1.00 0.95 0.94 0.89 1.00 1.00 0.97 1.00 Conglomerates 1.00 1.00 0.95 1.00 1.00 1.00 1.00 1.00 1.00 2 Year 3 Year 4 Year Portfolio O. -gpT ICT L T LT le T LT -(5/3T 2T Composite 0.68 0.96 1.00 0.80 0.95 1.00 0.80 1.00 1.00 Metals 0.98 1.00 0.95 0.90 1.00 0.95 0.90 0.75 1.00 Gold 0.55 0.72 0.95 0.45 0.75 1.00 0.60 0.65 1.00 Oil 1.00 1.00 0.95 1.00 0.95 0.95 0.95 1.00 0.97 Paper 1.00 1.00 0.97 0.85 0.50 1.00 0.95 1.00 0.95 Consumer 0.87 0.93 1.00 0.80 0.90 1.00 0.90 0.95 1.00 Industrial 1.00 0.96 1.00 0.60 1.00 0.97 0.65 1.00 1.00 Real Estate 0.74 0.81 1.00 0.70 0.70 1.00 0.65 0.75 1.00 Transportation 0.86 0.55 1.00 0.90 0.45 0.95 0.90 0.45 0.95 Pipelines 0.77 1.00 1.00 0.85 1.00 1.00 0.80 1.00 0.95 Utilities 0.95 1.00 1.00 0.95 1.00 0.95 1.00 0.95 1.00 Communications 0.95 0.55 0.95 0.35 0.95 0.95 0.85 0.80 1.00 Merchandising 0.84 0.83 1.00 0.55 0.95 0.95 0.75 0.95 1.00 Financial 0.97 0.57 1.00 0.90 0.95 1.00 0.95 0.90 0.98 Conglomerates 0.55 0.91 1.00 0.95 0.25 0.95 0.55 0.85 1.00 62 It is apparent from examining Table 4.2 that the assumption of non—time varying parameters 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 majority of the portfolios for the two to four year return horizons as evidenced by the maximum likelihood estimates for Sa, and 8,37, being less than one. However, in most instances for the two to four year return horizons the observational variance is constant as indicated by the KT estimate being one. For the monthly and quarterly return horizons the opposite appears to happen, that is the parameters a and # are constant and the observational variance is time varying. And finally, for the yearly returns, the model with no time varying 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 4.2 depicts 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 adaption is more pronounced, indicating that dividend yields actually forecast real returns. Of course, this refers only to the Composite portfolio. In the following two sections we compare the simple regression DLM with the constant regression DLM to determine if dividend yields contribute exhibit any forecast power. Secondly, we compare the simple regression DLM with the classical regression approach discussed in chapter 2. 63 1960 .^• 1980 1e70 Thee I v 1900 1980^1970^1080^1990 Ti me ^ Composite ^ Composite Monthly Quarterly Composite ^ Composite 1 Year 2 Year 1980^1970^1960^1980 Time ^ rim. Composite Composite 3 Year ^ 4 Year Time Time Figure 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 for the TSE Composite portfolio for the Regression DLM. 64 4.3 Constant vs Regression DLM We compare the predictive densities of the Constant DLM and the Simple Regression DLM to determine if dividend yields, DYt, have any ability to forecast value-weighted real returns, Rt,T, of TSE portfolios. A model is better if it has a higher predictive density or 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 each of the TSE portfolios. Note that there is no guarantee that if we add a regression variable that the Predictive Density will increase. Table 4.3: Comparison of Constant DLM to the Regression DLM for each of the six return horizons and each of the TSE portfolios using V'LL. Portfolio Monthly Quarterly 1 Year 2 Year 3 Year 4 Year Composite -3.481 -4.422 -0.283 4.614 3.082 7.287 Metals and Minerals -5.601 -5.832 -1.931 -1.612 -2.435 0.430 Gold and Silver -4.078 -3.731 -0.940 6.249 4.930 5.662 Oil and Gas 2.306 1.833 0.715 5.918 8.507 9.377 Paper and Forest -5.690 -4.043 -2.568 -3.024 -2.910 -4.776 Consumer -3.787 -1.174 -0.667 7.190 1.773 6.196 Industrial -5.287 -4.627 -2.284 1.052 2.019 5.466 Real Estate -2.624 -3.286 -3.041 -1.031 3.075 2.542 Transportation -5.833 -6.025 -2.299 -0.434 0.745 2.234 Pipelines -2.377 0.715 1.247 6.754 7.669 13.680 Utilities -6.321 -5.711 -0.444 7.784 8.659 10.050 Communications -6.184 -4.204 -0.568 4.370 -0.308 3.272 Merchandising -0.257 -2.570 0.199 9.747 5.064 9.814 Financial Services -5.293 -5.712 -0.505 8.586 4.621 6.867 Conglomerates -7.678 -2.905 0.518 2.088 -1.541 -1.626 Examining 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 all portfolios 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 the pattern of increased predictability of real returns as the return horizon increases using dividend yields. The only portfolios, where this is not the case are the following: Paper and Forest, Communications, and Conglomerates. This lends support to the hypothesis of Fama and French [4], that is, as the return horizon increases the predictive ability dividend yields increases. The above results should be interpreted with some caution, since we do not have a method 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 not always monotonic in nature. 66 4.3.1 Classical vs DLM In order to make a fair comparison between the classical regression approach and the regression DLM approach we would like to have a measure which behaves in the following manner It should penalize a forecast which is far from the observed value whenever our forecast variance is small i.e. we have a lot of information available. It should not penalize as heavily a forecast which is far from the observed value if the forecast variance is large i.e. we don't have a lot of information about the next observation. So, we can summarize the aspects of the measure in the following table: forecast error forecast variance measure small small large large small large small large acceptable acceptable unacceptable acceptable With these critera in mind we define the following measure for a forecast a time t Et= forecast error/Vforecast variance which is then used as follows: SDMSE = EV In order to make a fair comparison we compare the out—of—sample forecasts (see Section 2.2) generated by Fama and French [4] model with the DLM approach forecasts for the same time period. We then examine the ratio of the two standardized mean squared errors as follows: RATIO = MSEDLm/MSEciassical Note that we did not carry out the comparison for the Real Estate and Construction portfolio due to the limited number of observations. It should also be stressed that for 67 Table 4.4: Comparison of Classical Regression to the Regression DLM for each of the six return horizons and each of the TSE portfolios using RATIO. Portfolio Monthly Quarterly 1 Year 2 Year 3 Year 4 Year Composite 0.918 0.905 0.945 1.479 1.806 0.521 Metals and Minerals 0.716 0.893 0.943 0.486 0.438 0.395 Gold and Silver 0.795 0.756 1.077 0.942 0.908 0.960 Oil and Gas 0.905 0.949 1.579 2.585 1.936 0.941 Paper and Forest 0.924 0.959 1.187 1.351 1.161 1.354 Consumer 0.863 0.910 1.000 0.046 0.042 0.006 Industrial 1.013 1.035 0.916 1.508 1.861 1.813 Transportation 0.751 2.401 0.874 1.154 1.262 1.153 Pipelines 0.900 0.897 0.868 1.046 0.313 0.481 Utilities 0.882 1.051 0.946 0.013 0.009 0.041 Communications 1.088 1.382 1.086 1.180 1.223 1.448 Merchandising 1.097 0.993 1.260 0.805 0.423 0.156 Financial Services 0.978 0.875 1.040 1.032 2.448 2.018 Conglomerates 1.116 1.320 1.192 0.043 0.045 0.008 the one to four year return horizons, the out of sample mean squared errors for the Classical 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 14 cases; • Quarterly Return Horizon - Regression DLM beats classical regression in 9 of 14 cases; • One Year Return Horizon - Regression DLM beats classical regression in 6 of 14 cases; • Two Year Return Horizon - Regression DLM beats classical regression in 6 of 14 cases; • Three Year Return Horizon - Regression DLM beats classical regression in 7 of 14 cases; 68 • Four Year Return Horizon — Regression DLM beats classical regression in 9 of 14 cases; It is interesting to note that when the classical regression beats the Regression DLM the greatest margin of victory is 2.6, however, in the some cases the regression DLM the margin of victor is as much as 170 times better than the classical. 4.4 Conclusions In all but two cases the Fama and French [4] results hold true in that the regression DLM with dividend yields does better than the constant DLM for all return portfolios except for the Paper and Forest, Communications and Conglomerates as measured by the change in predictive density from the regression DLM to the constant DLM. When comparing the regression DLM approach with the classical approach discussed in chapter 2, there is not clear cut winner. However, the regression DLM approach does beat the classical case in the majority of cases, 55% in fact. 69 Chapter 5 Conclusions This concluding chapter summarizes the findings of the thesis. 5.1 Fama and French The results of Fama and French [4] which state that dividend yields show increased forecast power to predict real returns for increasing return horizons do not extend to all portfolios of the Toronto Stock Exchange. The Fama and French results only apply to the Composite portfolio and the Oil and Gas portfolio where there is forecast power for all return horizons as shown in Fama and French. However, the result of increasing forecast performance also apply to the following portfolios: Gold and Silver, Consumer Products, 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 the return variances are not constant throughout their sampling period. They present 70 results for various sub periods of interest. We did not pursue this option since we had only a limited amount of data available. 5.2 Dynamic Linear Model In all but two cases the Fama and French [4] results hold true in that the regression DLM with dividend yields does better than the constant DLM for all return portfolios except for the Paper and Forest, Communications and Conglomerates as measured by the change in predictive density from the regression DLM to the constant DLM. When comparing the regression DLM approach with the classical approach discussed in chapter 2, there is not clear cut winner. However, the regression DLM approach does beat the classical case in the majority of cases 55%. 71 Bibliography [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. and Smith, 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 of Financial Economics, 5, 115-146. [6] French,K.R., Schwert,G.W. and Stambaugh,R.F. (1987), "Expected Stock Returns and 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 Political Economy, 88, no 5, 829-853. [8] Keim, D.B. and Stambaugh, R.F. (1986), "Predicting Returns in the Stock and Bond 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 Statistical Society 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. 73 Appendix 1 1 Index Formula and Rules Each of the Toronto Stock Exchange indices measure the current aggregate market value (i.e. number of presently outstanding shares x current price) of the stocks included in the index as a proportion of an average base aggregate market value (number of base outstanding shares x average base price ± changes proportional to changes made in the current aggregate market value figure) for such stocks. The starting level of the base value has been set equal to 1000. Expressed more briefly this is: INDEX = ^ Current aggregate market value x1000 Adjusted average base aggregate market value Essentially, 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 of the market. Following is a detailed description of how The Toronto Stock Exchange indices are produced. The following formula is the basis for initial calculation of each of the indices 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 the index. QA,QB, • . . Q N: the numbers of currently outstanding shares of each stock in 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 each stock in the index during the base period QAB,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 blocks of 20% or less. 74 The base period is 1975. Calculation of the 1975 average base aggregate market 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 number of shares (share weight) of each stock outstanding at the beginning of the base period i.e. January 1, 1975 less any individual and/or control blocks of 20% or more. The current aggregate market value is determined using closing prices for each period for which the index is calculated multiplied by the number of shares then outstanding, less any individual and/or related control blocks of 20% or more, as at that period. As an example of these calculations, assume there are only two stocks in a hypothetical index. The problem is to calculate the level of the index as of January 31, 1975. Company 1 The current price (January 31, 1975) is $10 and the number of shares currently outstanding is 18,000. The average base aggregate market value in 1975 is $162,000. Company 2 The current price (January 31, 1975) is $25 and the number of shares currently outstanding is 30,000. The average base aggregate market value in 1975 is $690,000. Computation of the index would be as follows: 10 x 18, 000) + (25 x 30, 000) INDEX —^ x 1000 162, 000 + 690, 000 930' 000 INDEX =^x 1000 852,000 INDEX = 1091.55 ADJUSTMENT TO INDEX To calculate the indices subsequent to the establishment of the average base aggregate market value, recurring capital changes must be taken into account. Adjustments to the indices resulting from these changes 75 must normally be introduced without altering the level of the in- dex(see Bankruptcy Rule (7) for exception). In other words, continuity of the index must be preserved. To accomplish this, certain procedures are followed. These vary according to whether the adjustments result from: (1) the issuance of additional shares of a stock in the indices; or the addition to, 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) an asset 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 in Number of Stocks Two steps are necessary to make adjustments for additions or withdrawals of shares to or from the index calculations: (1) Updating the current aggregate market value of the index. If additional shares of an index stock are issued, the current aggregate market value of the stocks in that index will be accordingly higher. Likewise, if a new stock is added to the index, or if a stock is removed, the current aggregate market value of that stock will be added to, or subtracted from the current aggregate market value of the other stocks in that index. (2) Adjusting the average base aggregate market value of the index proportional to the change in the current aggregate market value so that the index level will remain the same. The first step, therefore, towards making an adjustment is to calculate the new current aggregate market value as indicated in (1) above. The second step is to calculate the new average base aggregate market value. Expressed as a formula the second step would be as follows: Let the old average base aggregate market value = A. Let the un—adjusted current aggregate market value = C. Let the current aggregate market value of the capital to be added or withdrawn = D. The current adjusted aggregate market 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) C 76 To calculate the index on the new base, the formula for the hypothetical example given above would be: INDEX= (C ± D) x 1000B Continuing the example above, assume that Company 1 issued 2,000 new shares. 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 current aggregate 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 be changed proportionately. Here the formula B = A x ig1311 is used. B (930, 000 + 20, 000)= 852 000,^x 930,00 B = 870,323 The index level remains unchanged as shown below: INDEX = 950,000 x 1000 870,323 INDEX = 1091.55 1.2 Stock Rights The day the stock sells ex—rights, the additional shares resulting from the rights 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 accordingly aggregate market value, discounts the rights. The formula to calculate the new base aggregate market value following subscription to stock rights would be: 77 C + C B = S x C + D — S where S = the total capital subscribed for the newly issued shares. A concrete example of how a stock rights issue is incorporated into the index is the December 5, 1975 Bank of Nova Scotia offer. The Bank, with an outstanding capital of 18,562,500 shares, offered the shareholders of record at the close of business on December 5, 1975 rights to buy one new share at $36 per share of each 9 shares held. As a result, 2,062,500 new shares were 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 index was 2,062,500 shares times the current price (theoretically, at this opening on the "ex" date, $41frac38 adjusted for the value of the right) amounting to $85,335,938. Actual subscription price was 2,062,500 shares times $36, amounting to $74,250,000. Calculations for the proportionately adjusting the base were as follows: Bank Index Un—adjusted current aggregate market value:. $4,193,109,375 Un—adjusted base aggregate market value:. $1,337,840,000 New current aggregate market value after allowing for rights: (4,194,109,375 + 85,335,938) = $4,278,445,313 New base aggregate market value after allowing for rights offering: 4,278,445,313 1,337,840,000 x ^  = $1,361,467,499 4,278,445,313 — 74, 250,000 . As at the close on the day prior to the ex—date. Adjustments are made after the close and before the market opens the following day. Bank of Nova Scotia closed at $42 on December 2, 1975. 1.3 Stock Dividends, Splits, and Consolidations On 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 the stock should drop by the extent of the worth of the dividend. The current aggregate market value, therefore, will not change. Hence the base figure is not adjusted. Similarly, in the case of share splits, the increased number of shares times the lower price should equal the old number of shares times 78 the higher price. Thus, the current aggregate market value is theoretically unchanged, and the base figure is not adjusted. The same reasoning holds in the case of stock consolidations, except that the higher price time the smaller number of shares leaves the current aggregate market value unchanged. 1.4 Liquidation of A Company Effective January, 1979, where a capital distribution is announced as being a liquidation of a company whose stock is included in the index, that stock will be removed from the index effective the ex—distribution date. 1.5 Asset Spin—off Effective January, 1979, adjustments necessary to leave the level of an index unchanged when a stock in that index has its per share value decreased through an asset spin—off are made at the opening of the ex—distribution day or as soon thereafter as the value of the asset being spun—off is known by the Exchange staff. Thus the staff may have to recalculate index values if a stock trades "ex—asset spin—off" without the index being stabilized. 1.6 Takeover Bid, Amalgamation or Merger Effective January, 1979, changes in share weight or control blocks resulting from takeover bids, amalgamations or mergers are incorporated into the index as soon as is administratively possible after the fact. This procedure replaces the former procedure of incorporating such changes at the next quarterly update made just after the end of the calendar quarter to which they relate. 1.7 Bankruptcy of Stock in Index System If 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 per share (one—half cent under the present computer programmes) rather than at the last board—lot price before trading was suspended. If, as, and when the company recovers in any form, it will only be eligible to be included in 79 the 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 more of the share capital of any stock included in the indices is removed in order to reflect, as nearly as may be practical, the market float or stock normally available to portfolio investors. (b) If at any time more than 90% of the outstanding shares which are included in the TSE 300 index is held by a controlling group; as defined by the methods of computing control group holdings for index weighting purposes, or if the shares in public hands of the same class are so reduced that the value calculated by multiplying the most recent share price by the number of shares held by parties other than the control group is insufficient to meet the market capitalization criterion for admission to the index, then each such class of equity security shall be removed from the index as soon as is conveniently practicable. (c) If an individual control block of 20% or more, or a related group of control blocks which in aggregate total 20% or more of the relevant shares outstanding, are initially removed from the total of such shares then out- standing for purposes of computing the share weight of the stock in the index portfolio, and (1) the holder or holders of such stock subsequently sell stock from their position to reduce the amount of such stock holding(s) below 20%, then the holding(s) will be added back to the float at the first practical time subsequent to such sale; (2) if the 20% or more block(s) subsequently falls below 20% as a result of an increase or increases in the total of such share capital outstanding, then such block(s) will not be added back to the share weight until such time as the holding falls or is reduced to 15% or less and as soon thereafter as is practical for it to be added back. 1.9 Frequency of Adjusting the Index Stock 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 or 80 as 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 as possible after the fact. Bankruptcy and receivership situations are reflected as soon as possible after they are announced. Any changes resulting from the annual post—year—end revision as noted in the section entitled "Stock Eligibility Criteria" are made at the end of the first calendar quarter. Other changes (such as those related to control blocks or to addition or withdrawal of shares) are usually made on a quarterly basis. Additions or deletions of stocks are usually made on a quarterly basis but may be necessary at other times due to delistings caused by takeovers, amalgamations, or mergers or to normal delistings. 2 Dividends, Current Indicated Annual Yields, and Dividends Adjusted to Index 2.1 General Each time an index value or price is computed, a "current indicated annual dividend yield" (in percent terms) and a corresponding "dividends adjusted to index amount" (in dollars and cents) may also be produced for that index. The Exchange has computed and included both of these dividend values in this book for each of the historical index values contained in the tables whether on a closing monthly, weekly, or daily basis. 2.2 Compilation Assumptions 2.2.1 Historical Series (1956 through 1976) In general, the historical dividend per share series for each stock in the index system have been compiled on the following basis: • the original dividend payments per share against payment date of the dividend; • if it could be determined that the payment in question formed part of a regular annual dividend policy of a company (whether paid quarterly, semi-annually, or annually) then the payment was multiplied by the 81 appropriate factor e.g. 4 in the case of a quarterly amount to forecast the current indicated annual rate at that point in time. In addition, any extra paid within the last twelve months was then added to the regular annual rate to become the current annual rate for that stock at that point in time. • In cases where the regular periodic amount was not level, as in the case where a company has a policy of paying $1 per share per annum payable 20 cents in the first quarter, 30 cents in the second quarter, 20 cents in the third quarter and 30 cents in the fourth quarter, no forecasting from the periodic amount was carried out; instead the annual amount was used directly in making payout calculations. • Irregular payments in terms of either time or amount were treated as such and included on payment date and were carried forward in yield calculations for twelve months, after which time they were removed from the calculations. 2.2.2 Current Series (1977 forward) Updating of the dividend per share data takes place daily as new dividend reports are received by the Exchange, subject to computer cut-off require- ments; In general, the same compilation rules as were used for the historical series are 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 Amounts If dividends per share are stated in a foreign currency, then Exchange staff translate 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 Canadian funds using monthly average noon spot rate applicable to the month during which the dividend was paid. The annual rate for companies paying dividends in U.S. funds was the sum of all dividends (including extra dividends) paid within the past 12 months, as converted by using the monthly average noon spot rate applicable to the month during which each dividend was paid. 82 Therefore the annual rate in Canadian funds may included four different exchange rates. (b) Current (January 1977 to present): Dividends originally expressed in U.S. funds are converted to Canadian funds using the closing foreign exchange rate as reported in the current Globe and Mail Report on Business for the day prior to the date of receipt of the dividend declaration. 3 Calculation Procedure and Product The computation procedure to produce a "current indicated dividend yield" on an index and the corresponding "dividends adjusted to index" amount in dollars and cents is straightforward and analogous to that on an individual stock. The current indicated annual dividend rate for each stock within an index is multiplied by the share weight of that stock within the index as at that computation date. The resulting dividend payout figures for the stocks within the index are then summed. This aggregate pool of dividend dollars is then 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 resultant figure is the current indicated annual yield expressed as a decimal; it may be multiplied by 100 to express the amount as a percent. This current indicated annual yield on an index, stated as a decimal may then be multiplied by the index for that same computation date to produce the "Dividends Adjusted to Index" amount in dollars and cents. This amount may then be used in market valuation formulae along with other assumptions to forecast the expected level of the index based on these assumptions. 83 Appendix 2 TSE 300 Composite Index RELATIVE WEIGHTS The following list (containing 300 stocks, fourteen Group Indices, and forty-three Sub-Group Indices) provides the weight which individual stocks, Group Indices, and Sub-Group Indices bear on the Toronto Stock Exchange 300 Composite Index. Computations are as of the close of December 31, 1991. Percentages in brackets indicate available float on which relative weights on Composite Index are calculated. Numbers in brackets after each Index name indicate the number of stocks within that Index. TOTALS MAY NOT ADD DUE 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 otherwise noted. Stock Symbol Company Name Relative Weight on Composite % 1.0 METALS & MINERALS (16) 7.70 1.1 Integrated Mines (6) 6.87 AL^Alcan Aluminum 3.00 BMS^Brunswick Mining & Smelt (15%) 0.02 CLT^Cominco (56%) 0.55 HBM.S^Hudson Bay Min. & Smelt S (52%) 0.02 N^Inco 2.15 NOR^Noranda (55 %) 1.13 1.2 Metal Mines (8) 0.33 CCH^Campbell Resources (51%) 0.01 COR^Cominco Resources (35%) 0.02 KER^Kerr Addison (51%) 0.08 MLM^Metal Mining (41%) 0.11 MVA^Minnova (50%) 0.07 NGX^Northgate Exploration (75%) 0.01 PMC^Princetown Mining 0.02 WMI^Estmin Resources (26%) 0.02 1.4 Non-Base Metal Mining (2) 0.49 POT^Potash Corp. of Saskatchewan (62%) 0.29 ROM^Rio Algom (49%) 0.20 84 2.0 GOLD & SILVER (28)^7.39 2.1 Gold Sz Silver Mines (25)^7.30 ABX^American Barrick (79%) 2.04 AGE^Agnico-Eagle^0.08 AUR^Aur Resources (79%)^0.05 BGO^Berma Gold 0.03 BWR^Breakwater Resources (73%)^0.01 CBJ^Cambior (79%)^0.13 DML.A Dickenson Mines CL A (39%)^0.01 ECO^Echo Bay Mines 0.52 EN^Euro-Nevada (80%)^0.11 FN^Franco Nevada (77%) 0.12 GKR^Golden Knight Resources (56%) 0.04 GLC^Galactic Resources (66%)^0.01 GXL^Granges Inc (49%)^0.01 HEM^Hemlo (45%) 0.28 ICR^Intl Corona Corp (70%)^0.18 LAC^LAC Minerals^0.70 MVG^Minven Gold (55%)^0.00 PDG^Placer Dome 1.72 PGU^Pegasus Gold 0.23 RAY^Rayrock Yellowknife (75%)^0.02 RY0^Royal Oak Mines^0.05 TEK.B Teck Corporation CL B^0.80 TVX^TVX Gold (24%) 0.07 VOY^Viceroy Resources 0.05 WFR^Wharf Resources (49%)^0.03 2.2 Precious Metal Funds (3) 0.10 BPT.A BGR Precious Metals CL A^0.03 CEF.A Central Fund of Canada CL A 0.04 G^Goldcorp^ 0.03 3.0 OIL Sc GAS (38) 6.71 3.1 Integrated Oils (4) 2.04 CCT^Canadian Turbo^0.02 IMO^Imperial Oil (30%) 1.36 SEC^Shell Canada A (22%)^0.54 TPN^Total Petroleum N.A. (47%)^0.12 85 3.2 Oil & Gas Producers (34)^4.66 AEC^Alberta Energy (63%) 0.31 AXL^Anderson Exploration (33%)^0.04 BPC^B.P. Canada (43%)^0.15 BVI^Bow Valley Industries (67%)^0.27 CBE^Cabre Exploration 0.06 CEX.B Conwest Exploration^0.09 CGH^Computalog Ltd 0.01 CHA^Chauvco Resources (38%)^0.10 CID^Chieftain International (45%)^0.04 CIR^Cimarron Petroleum 0.05 CNQ^Canadian Natural Resources (81%) 0.08 0011^Coho Resources (63%)^0.01 CSW^Canadian Southern Petroleum^0.03 CXY^Canadian Occidental (52%) 0.54 ECR^Encor Inc^ 0.02 GOU^Gulf Canada Resources (26%)^0.14 LMO^Lasmo Canada (66%)^0.06 MHI^Morgan Hydrocarbons 0.07 MKC^Mark Resources (36%)^0.04 MRP^Morrison Petroleums (77%)^0.08 NCN^Norcen Energy (63%) 0.24 NCN.A Norcen Energy CL A (70%)^0.29 NCO^North Canadian Oils (48%) 0.11 NMC^Numac Oil & Gas (53%)^0.05 NWS^Nowsco Well Service 0.10 PCP^PanCanadian Petroleum (13%)^0.23 PNN^Pinnacle Resources^0.05 POC^Poco Petroleums (65%) 0.13 RES^Renaissance Energy 0.49 RGO^Ranger Oil 0.49 RRL^Ranchmen's Resources (44%)^0.02 SKO^Saskatchewan Oil and Gas (76%)^0.18 SRL^Sceptre Resources (55%)^0.05 ULP^Ulster Petroleums 0.05 86 4.0 PAPER & FOREST PRODUCTS (18)^2.23 4.1 Paper & Forest Products (18) 2.23 A^Abitibi-Price (18%)^ 0.10 CAS^Cascades (41%) 0.08 CFI^Crestbrook Forest (46%) 0.04 CFP^Canfor (55%)^ 0.21 DHC.B Donohue CL B (46%)^0.10 DOM.B Doman Industries CL B (74%) 0.03 DTC^Domtar (57%) 0.25 FCC.A Fletcher Challenge Canada CL A (28%)^0.16 IFP.A^International Forest Products CL A (76%) 0.07 MB^MacMillan Bloedel (51%)^0.61 NF^Noranda Forest (18%) 0.12 PFP^Canadian Pacific Forest (20%)^0.12 RPP^Repap Enterprises (62%) 0.04 SPL^Scott Paper (50%)^ 0.08 TBC.A Tembec CL A (59%) 0.05 WFT^West Fraser Timber (58%)^0.08 WGL^Westar Group (70%) 0.04 WLW^Weldwood of Canada (15%) 0.03 5.0 CONSUMER & PRODUCTS (25)^ 9.47 5.1 Food Processing (5)^ 0.69 BCS.A B.C. Sugar Refinery CL A,B 0.12 CFL^Corporate Foods (33%) 0.07 CMG^Canada Malting (60%)^0.10 FPL^FPI Ltd^ 0.06 MFL^Maple Leaf Foods (44%) 0.35 5.2 Tabacco (2) 1.56 IMS^Imasco (60%)^ 1.49 ROC^Rothmans (29%) 0.07 5.3 Distilleries (2) 4.56 CDL.A Corby Distilleries CL A (49%)^0.08 VO^Seagram (62%)^ 4.48 5.4 Breweries St Beverages (4) 1.65 KOC^Coca-Cola Beverages (51%)^0.09 LBT^Labatt, John (62%)^ 0.71 MOL.A Molson CL A 0.71 MOL.B Molson CL B (50%) 0.13 87 5.5 Household Goods (4)^0.28 COC^Camco Inc (29%) 0.03 CRW^Cinram Ltd (62%) 0.05 DTX^Dominion Textile^0.15 NMA.A Noma industries CL A (71%)^0.05 5.6 Autos & Parts (4) 0.50 FMC^Ford Motor of Canada (6%)^0.04 HAY^Hayes-Dana (43%)^0.05 MG.A^Magna International CL A^0.36 UAP.A UAP Inc CL A (68%) 0.05 5.7 Packaging Products (4)^0.23 CCQ.B CCL Industries CL B (73%)^0.12 CGC^Consumers Packaging (25%) 0.01 INO^International Innnopac (63%)^0.01 LMP.A Lawson Mardon CL A^0.09 6.0 INDUSTRIAL & PRODUCTS (38) 11.05 6.1 Steel (5)^ 1.42 CE!^Co-Steel (84%)^0.17 DFS^Dofasco 0.75 ISP^Ipsco^ 0.19 IVA.A^Ivaco CL A 0.03 STE.A Stelco CL A,B^ 0.29 6.2 Metal Fabricators (5) 0.32 DRE.A Dreco Energy Services CI A^0.04 DRL^Derlan Industries^0.05 HLY^Haley Industries (20%)^0.01 SHL.A Shaw Industries (73%) 0.09 SNU.A SNC Group CL A 0.13 6.3 Machinery (1)^ 0.07 UDI^United Dominion Industries (44%) 0.07 6.4 Transportation Equipment (3)^1.00 BBD.A Bombardier CL A (20%) 0.07 BBD.B Bombardier CL B^0.88 HSC^Hawker Siddeley Canada (41%)^0.05 6.5 Electrical/Electronic (10)^4.29 AAZ^Archer Communications (63%)^0.01 CAE^CAE Industries 0.41 CMW^Canadian Marconi (48%)^0.10 GAN^Gandalf Technologies 0.02 KGL.B Kaufel Group CL B 0.03 88 MLT^Mite! (49%)^ 0.02 NNC^Newbridge Networks (58%)^0.10 NTL^Northern Telecom (47%) 3.43 SHK^SHL Systemhouse (33%) 0.06 SPZ Spar Aerospace^ 0.11 6.6 Cement/Concrete Products (3)^0.26 GYP^CGC Inc (24%) 0.03 LCI.PR.E Lafarge Canada E^ 0.12 ST.A^St. Lawrence Cement CL A (63%)^0.11 6.7 Chemicals (6)^ 2.03 ANG^Alberta Natural Gas (50%)^0.08 CCL^Celanese Canada (44%) 0.14 DUP.A^Du Pont Canada CL A (23%)^0.17 NVA^Nova Corp^ 1.36 OIL Ocelot Industries 0.12 SE^Sherritt Gordon 0.15 6.9 Business Services (5)^ 1.65 COS^Corel Systems (53%) 0.07 CSN^Cognos Inc. (61%) 0.06 IIT.A^Intera Information CL A (72%)^0.03 MCL^Moore Corp^ 1.40 XXC.B^Xerox Canada CL B^0.11 7.0 REAL ESTATE & CONSTRUCTION (11)^0.91 7.1 Developers & Contractors (4) 0.18 COT^Coscan (43%)^ 0.04 CXA.A^Consolidated HCI Holdings CL A (42%) 0.01 ITW^Intrawest Development^0.10 RLG^Royal Lepage (42%) 0.03 7.2 Property Managers (7) 0.74 BCD^Bramalea (30%)^ 0.08 CBG^Cambridge Shopping Centres (75%)^0.25 CDN^Carena Development (33%)^0.10 MKP^Markborough Properties (19%) 0.06 RPC^Revenue Properties (29%) 0.02 TZC.A^Trizec CL A (28%)^ 0.15 TZC.B^Trizec CL B (21%) 0.08 89 8.0 TRANSPORTATION & ENVIRONMENTAL SERVICES (8) 2.12 8.1 Transportation & Environmental Services (8) ^ 2.12 AC^Air Canada^ 0.34 ALC^Algoma Central (59%) ^ 0.01 GEL^Greyhound Lines of Canada (31%) ^ 0.05 LDM.A Laidlaw Inc Cl A (53%) 0.15 LDM.B Laidlaw Ind Cl B ^ 1.20 PEN^Philip Environmental (64%) ^ 0.09 PWA^PWA Corporation 0.16 TMA^Trimac (70%) ^ 0.12 9.0 PIPELINES (4) 2.06 9.1 Oil Pipelines (2) 0.29 IPL^Interprovincial Pipe Line (36%) ^ 0.27 TMP^Transmountain Pipelines (33%) 0.03 9.2 Gas Pipelines (2) ^ 1.77 TRP^TransCanada Pipelines (78%) ^ 1.34 W^Westcoast Energy (63%) 0.43 10.0 UTILITIES (17) ^ 13.72 10.1 Gas Utilities (3) 0.54 BCG^BC Gas Inc 0.38 GWT^GW Utilities (11%) ^ 0.04 UEI^Union Energy (39%) 0.12 10.2 Electrical Utilities (5) 1.77 ACO.X Atco CL I (80%) ^ 0.14 CU^Canadian Utilities CL A (60%) ^ 0.26 CU.X^Canadian Utilities CL B (35%) 0.10 FTS^Fortis Inc^ 0.14 TAU^TransAlta Utilities^ 1.12 10.3 Telephone Utilities (9) 11.41 AGT^Telus Corp 1.26 B^BCE Inc^ 8.51 BCT^British Columbia Telephone (50%) ^ 0.70 BCX^BCE Mobile Communications (26%) 0.30 BRR^Bruncor (67%) 0.14 MTT^Maritime Telephone & Telegraph (66%) ^ 0.22 NEL^Newtel Enterprises (44%) ^ 0.08 QT^Quebec Telephone (49%) 0.09 TGO^Teleglobe Inc (49%) 0.12 90 11.0 COMMUNICATIONS & MEDIA (19)^4.74 11.1 Broadcasting (5)^ 0.31 BNB^Baton Broadcasting (47%)^0.05 CF^CFCF (71%) 0.03 CHM.B CHUM CL B^ 0.12 TM.B^Tele-Metropole CL B (73%)^0.03 WIC.B WIC Western CL B (67%) 0.08 11.2 Cable Se Entertainment (6)^0.79 CPX Cineplex Odeon (43%) 0.04 RCI.A Rogers Communications CL A (9%) 0.04 RCI.B Rogers Communications CL B (35%) 0.52 SAT Canadian Satellite (26%) 0.02 SCL.B Shaw Cablesystems CL B (78%)^0.10 VDO^Le Groupe Videotron (29%)^0.06 11.3 Publishing Se Printing (8) 3.64 HLG^Hollinger (32%)^ 0.12 MHP^Maclean Hunter (80%)^1.01 QBR.A Quebeccor CL A (44%) 0.09 QBR.B Quebeccor CL B 0.14 STM^Southam (76%)^ 0.45 TOC^Thompson Corporation (30%)^1.57 TSP^Toronto Sun Publishing (36%) 0.07 TS.B^Torstar CL B (43%)^0.20 12.0 MERCHANDISING (31) 5.13 12.1 Wholesale Distributors (5)^0.37 ACK^Acklands (76%)^ 0.03 EML^Emco Ltd (51%) 0.02 FTT^Finning Ltd 0.26 UWB^United Westburne (28%)^0.04 WJX.A Wajax CL A,B (46%) 0.02 12.2 Food Stores (6)^ 1.46 EMP.A Empire CL A 0.12 L^Loblaw Companies (24%)^0.19 OSH.A Oshawa Group CL A 0.47 PGI^Provigo (48%)^ 0.23 UGO.B Unigesco CL B 0.03 WN^Weston, George (42%)^0.42 12.3 Department Stores (3) 0.49 HBC^Hudson's Bay Co (27%)^0.26 SCC^Sears Canada (37%) 0.20 WDS^Woodward's Ltd (49%) 0.03 91 12.4 Clothing Stores (3)^ 0.17 DLX.A^Dylex CL A (76%) 0.10 GFG.A^Grafton Group CL A 0.00 RET.A^Reitman's CL A^ 0.07 12.5 Specialty Stores (5) 1.38 CTR.A^Canadian Tire CL A 1.12 GDS.A^Gendis CL A,B (43%)^0.09 PCJ.A^Peoples Jewellers CL A 0.01 PJC.A^Jean Coutu Group CL A 0.14 TOG^TOG International (41%) 12.6 Health Hospitality (9)^1.27 BCH^Biochem (IAF) (61%) 0.30 CAO.A^Cara Operations CL A (63%)^0.12 CAO^Cara Operations (38%)^0.07 DEP^Deprenyl Research (68%) 0.15 FSH^Four Seasons Hotel 0.15 LWN^Loewen Group (77%)^0.23 MHG.A MDS Health Group CL A (42%)^0.03 MHG.B MDS Health Group CL B^0.18 QLT^Quadra Logic (67%) 0.04 13.0 FINANCIAL SERVICES (35) 20.98 13.1 Banks (7)^ 18.12 BMO^Bank of Montreal^3.01 BNS^Bank of Nova Scotia 2.51 CM Cdn Imperial Bank of Commerce^3.62 LB^Laurentian Bank of Canada (38%)^0.07 NA National Bank of Canada^0.86 RY^Royal Bank of Canada 4.88 TD Toronto-Dominion Bank 3.17 13.2 Trust, Savings & Loan (6)^0.60 CEH^Central Capital (32%) 0.00 CEH.A^Central Capital CL A (78%)^0.00 CGA^Central Guaranty Trustco (12%)^0.00 NT National Trustco (53%) 0.24 RYL^Royal Trustco (47%)^0.33 TTG^General Trustco (35%) 0.02 13.3 Investment Co's & Funds (6)^0.61 CGI^Canadian General Investments (52%)^0.05 CNN.UN Canada Trust Income Investments^0.02 FMS.A^First Marathon CL A^0.12 IGI^Investors Group (25%) 0.15 MKF^Mackenzie Financial Corp^0.21 UNC^United Corporations (50%) 0.06 92 13.5 Insurance (6)^ 0.49 CRX^Crownx (22%) 0.00 CRX.A Crownx CL A 0.01 ELF^E-L Financial (64%)^0.08 FFH^Fairfax Financial 0.05 GWO^Great West Lifeco (14%)^0.09 LON^London Insurance Group (43%) 13.6 Financial Management Co's (10)^1.15 CXE^Consol. Canadian Express (57%)^0.01 CXS^Counsel Corp (53%)^0.03 DBC.A Dundee Bancorp CL A 0.04 FC^FCA International (74%)^0.02 GLZ^Great Lakes Group (8%) 0.05 HIL^Hees International Bancorp (37%)^0.25 LGC.B Laurentian Group CL B (31%)^0.03 PGC.A Pagurian CL A (65%)^0.18 PWF^Power Financial Corp (31%)^0.30 TFC.A Trilon Financial CL A (42%) 14.0 CONGLOMERATES (12)^5.76 14.1 Conglomerates (12) 5.76 AGR.B Agra Industries CL B (74%)^0.05 BL.A^Brascan CL A (49%)^0.45 CAM.A Canam Manac CL A (31%)^0.01 CP^Canadian Pacific 3.31 FIL.A^Federal Industries CL A,B^0.14 HSM^Horsham^ 0.49 ISE^International Semi-Tech (65%)^0.07 JN^Jannock (73%) 0.19 OCX^Onex Corporation (75%)^0.07 POW^Power Corp of Canada (61%)^0.60 SRC^Scott's Hospitality 0.33 SRC.0 Scott's Hospitality CL C^0.06 TSE Composite^100.00 93

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