33 Course Review1. The regressions.(a) Forecasting regression for predicting returns o ver tim e.+1= + + +1; =1 2(b) Time series regression Explaining variation in returns over time; characterizing correla-tion between returns, finding betas for factor models,= + + ; =12(c) Cross-sectional regression Explaining variation in average returns across stocks by vari-ation in their betas (b,h,s,..) or characteristics()=+ ; =1 2(d) With factor models, you can find the cross-sectional implications of the time-series re-gression³´= + ()Week by week high points1. Market return forecast(a) D/P can predict market returns. “Low” P gives high returns.+1= + ()++1≈ 4+1= + (− )++1 ≈ 01This means expected returns ¡+1¢vary over time(b) Economic significance:i. [(+1)] is large relative to (+1). (2measures 2[(+1)] / 2(+1).ii. 2rise with horizon.1950 1960 1970 1980 1990 2000 20100510152025 4 x D/PReturn671iii. Stronger forecasts at long horizons result from a persistent forecasting variable(D/P) (we did this algebraically)D/PReturnAdd these up to get large long-horizon return forecastHigh D/P today forecasts high returns for many future daysHigh D/P today is persistent, so return forecast will be high in the futureForecastsWhy D/P forecasts long horizon returnsiv. (Coming) return forecastability is “enough” to explain price volatility, that’s eco-nomic significance!(c) DP does not forecast ∆ as it “should.”∆+1= +0× (− )++1(d) Linearized present value formulas, useful tools.=1 −∆− = ∞X=1−1(∆+− +)Much better than= ∞X=1Ã1+11+21+!+i. Interpretation of the regressions: Prices today reflect expected dividend growth andexpected returns for many periods in the future. If +rises, then − willdecline, and low − will be followed by high returns on a verage, generating theregression.ii. Source: a useful return identity. Return must come from price rise or dividends!+1=+1+ +1...(algebra)...+1≈ (+1− +1) − (− )+(+1− )= −+1+ + ∆+1(46)(e) Volatility and bubbles.i. If − or − vary, they must foreast long-run returns, long-run dividend growth,or their own long-run movements. The regression coefficients must add up. Thismeans that we can account for price volatility with return forecasts, dividend growthforecasts or future prices.672ii. Run both sides of(− )=∞X=1−1+−∞X=1−1∆+on − and1=− Long run return forecast and long run dividend growth forecast must add up.iii. 1 and 0. “should be” = −1, =0. “Is”=1=0iv. Regression coefficients are covariance over variance, so m u ltiply through by ()()=(− )=⎡⎣X=1−1∆+⎤⎦− ⎡⎣X=1−1+⎤⎦Measured variation in expected returns is just enough to account for all price-dividend volatility. Another measure of “economically large”v. If we only look out steps(− )=X=1−1+− X=1−1∆++ (+− +)1=()− ()+ ()High prices could mean prices that rise at forever, a “bubble.” At 1 y ear, the lastterm is h uge. A t 15 y ears, it’s gone. This sense of “bubble” is not there.(f) VAR and impulse-response.i. VAR+1= + +1∆+1= + +1+1= + +1ii. We can find implied long run foreasts and other statistics by iterating forward, forexample,+2= +³+1+ +2´iii. “Impulse response.” Price mo vements with no dividend change melt away and cor-respond to higher expected returns. Here stocks are like bonds. Price movementswith a dividend change are permanent and represent “cashflow risk”6730 5 10 15 2000.20.40.60.81Response to Δd shock returndiv growth0 5 10 15 2000.20.40.60.81Cumulative response to Δd shock returndividendprice0 5 10 15 20−0.0500.050.1Response to dp shockΣρj−1rt+j=1.08Σρj−1Δdt+j=0.08rt+1 = −0.960 5 10 15 20−1−0.8−0.6−0.4−0.200.2Cumulative response to dp shock returndividendpriceResponse to and shocks from∆+1= + + +1+1= + + +1+1= + + +1with = − You can do this by hand: From+1= × + +1=01 × + +1∆+1= × + +1=0× + +1+1= × + +1=094 × + +1and the identity (46)+1= −+1+ +1plot the responses to =1= 0 hence = −096 = , and the response to=1=0andhence=1.(g) Good expected return news lowers actual returns (return decomposition).(h) Preview: The “random walk” is overturned in many markets. D/P forecasts stock re-turns, yield spreads forecast bond returns, interest rate spreads forecast fx returns.(i) Surve y: Many other variables help to predict both stock returns and dividend growth.For example, cay (consumption/wealth)+1= + × + × + +1a high value of another variable raises both expected dividends and expected returns, orhigher short run returns but lower long run returns, to leave unchanged.(− )=∞X=1−1+− ∞X=1−1∆+674(j) Interpretation: the premium for holding risk varies over time, higher in economic badtimes
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