Random Walks vs Random Variables. The Random Walk Model. Simple rate of return to an asset is: Simple rate of return

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1 The Random Walk Model Assume the logarithm of 'with dividend' price, ln P(t), changes by random amounts through time: ln P(t) = ln P(t-1) + µ + ε(it) (1) where: P(t) is the sum of the price plus dividend payments made in period t, µ = E {ln[p(t)/p(t-1)]} is the expected continuously compounded return, and Random Walks vs Random Variables If changes in (log) stock prices are random, then (log) price levels follow a random walk Distinction between random variables and random walks is confusing for many (most) students ε(it) is the random change in the stock price from period t-1 to period t Simple rate of return to an asset is: R(t) = [P(t) - P(t-1) + D(t)] / P(t-1) (2) which is close to the continuously compounded return r(t) for small values of the returns Simple rate of return Equation (1) can be rewritten in terms of returns: r(t) = µ + ε(it) (3) where the 'unexpected' return in period t is the error term ε(it) As with the errors from regression models (e.g., APS 402), the errors should be random Some Properties of the Random Walk Model The mean and variance of returns are proportional to the length of the measurement interval k E ln[p(t) / P(t-k)] = k µ Var ln[p(t) / P(t-k)] = k σ² (4a) (4b) More Properties of the Random Walk Model Autocorrelation tests on returns are equivalent to testing whether the errors are indeed random corr[r(t), R(t-k)] is the autocorrelation coefficient at lag k Most evidence finds that common stock returns have very small autocorrelations for daily or monthly data for many lags k thus, these tests are consistent with the random walk model for stock prices Professor Schwert 1-6 Spring 1997

2 Prediction of Dow Jones Industrial Average (Random Walk Model, 1997 = 7000) [Mean = 3.0%, Std Dev = 18.0%] Prediction of Dow Jones Industrial Average (Random Walk Model, 1997=7000) [Mean = 7.0%, Std Dev = 15.1%] Based on historical estimates, Low (-2 sd) High (+2 sd) Based on historical estimates, Low (-2 sd) High (+2 sd) U.S. Monthly Stock Returns, Volatility of Monthly US Stock Returns, Percent Return per Month 50% 40% 30% 20% 10% 0% -10% -20% -30% Percent Return per Month 30% 25% 20% 15% 10% 5% (Based on Daily Returns Within the Month) -40% 0% Percentage of Returns % 30% 25% 20% 15% 10% 5% 0% Histograms of Monthly US Stock Returns, >5 Standard Deviations from the Mean Distribution of Stock Returns A statistic which measures the 'fat-tailedness' of a sample is the studentized range, SR: SR = (Max - Min) / Standard Deviation Jarque-Bera statistic (in Eviews) measures whether the skewness & kurtosis statistics are consistent with a Normal distribution skewness should equal 0 and kurtosis should equal 3 Professor Schwert 7-12 Spring 1997

3 Short-term Interest Rates, Avg 0.79% 0.41% 0.37% Std 4.89% 1.30% 0.24% Max 38.28% 8.89% 1.67% Min % -7.35% -0.02% SR T-stat Short-term Interest Rates, Avg 0.67% 0.46% 0.42% Std 4.36% 1.04% 0.20% Max 17.59% 6.92% 1.67% Min % -7.35% 0.13% SR T-stat Short-term Interest Rates, Avg 0.85% 0.29% 0.09% Std 8.06% 1.32% 0.13% Max 38.28% 5.06% 0.43% Min % -7.83% -0.02% SR T-stat Short-term Interest Rates, Avg 1.01% 0.48% 0.38% Std 4.08% 1.72% 0.26% Max 16.53% 9.84% 1.35% Min % -7.20% 0.03% SR T-stat Returns Over More than One Period When analyzing returns over more the one period it is convenient to use continuously compounded returns, r(t) = ln [1+R(t)], since these returns add up over time Returns Over More than One Period Thus, the k-period return [P(t+k)/P(t)]-1, is just [1+R(t+1)][1+R(t+2)]...[1+R(t+k)] - 1 = R(t+1) + R(t+2) R(t+k) + cross-product terms These cross-product terms can be important, however Professor Schwert Spring 1997

4 Returns to Different Assets Returns Over More than One Period Period Asset A Asset B Asset C Average 2-period Return Assets with the highest standard deviation of simple returns have the lowest terminal value, for a given level of average simple returns Value of $1 Investment in Period 0 at the end of Period 2 $ 1.21 $ 1.20 $ 1.17 Simple returns are easiest to use when measuring the returns to many different securities at the same point in time Define a portfolio return as the weighted average of the returns to the N securities in the portfolio: R(pt) = Σ w(it) R(it) where Σ w(it) = 1. The portfolio weights w(it) represent the proportion of wealth invested in asset i at the beginning of period t. If an investor put equal dollar amounts in each of N securities, this would be an equal-weighted portfolio w(it) = w(i) = 1 / N If one invested in proportion to the outstanding market value (i.e., price times shares outstanding) of each of N securities, this would be a value-weighted portfolio w(it) = value of asset i / total value of all N assets The return to an equal-weighted portfolio is the average return to the assets in the portfolio in period t, so it is easy to compute but it is hard to maintain an equal-weighted portfolio through time you must rebalance every period as the value of the holdings change Professor Schwert Spring 1997

5 The return to a value-weighted portfolio is difficult to compute because the value-weights change every period but it is easy to maintain a value-weighted portfolio it requires no rebalancing (except to account for new issues or retirements of securities) The S&P 500 and the CRSP value-weighted portfolios are examples of value-weighted portfolios By construction, a value-weighted portfolio places larger weight on large firms therefore smaller weight on small firms relative to an equal-weighted portfolio A comparison of the returns to the CRSP valueand equal-weighted portfolios of all NYSE stocks from shows this effect for , the mean monthly returns are.96% and 1.31% for the value and equal-weighted portfolios the monthly standard deviations of returns are 5.53% and 7.61% Thus, the risks and returns to small stocks are probably higher than for large stocks. Distribution of CRSP Value & Equal-weighted Stock Returns, Cont Comp Simple Simple CRSP VW CRSP VW CRSP EW Returns Returns Returns Avg 0.80% 0.96% 1.31% Std 5.52% 5.53% 7.61% Max 32.41% 38.28% 65.51% Min % % % SR t-test Distribution of CRSP Value & Equal-weighted Stock Returns, Cont Comp Simple Simple CRSP VW CRSP VW CRSP EW Returns Returns Returns Avg 0.53% 0.85% 1.64% Std 7.99% 8.04% 11.66% Max 32.41% 38.28% 65.51% Min % % % SR t-test Distribution of CRSP Value & Equal-weighted Stock Returns, Cont Comp Simple Simple CRSP VW CRSP VW CRSP EW Returns Returns Returns Avg 0.91% 1.00% 1.18% Std 4.13% 4.12% 5.16% Max 15.33% 16.56% 29.92% Min % % % SR t-test Professor Schwert Spring 1997

6 Market model regression: R(it) = α(i) + β(i) R(mt) + ε(it), t = 1,...,T (5) where R(it) is the return to asset i and R(mt) is the return to the "market" portfolio of assets in period t is a time series regression model the slope coefficient β(i) ("beta") is a measure of the relative nondiversifiable risk of the asset which is the dependent variable in the regression [R(it)] as part of the portfolio which is the independent variable [R(mt)] Weighted average beta Σ β(i) β equals 1 by construction for the set of assets i which make up the regressor portfolio [R(mt)] Weighted average intercept Σ α(i) α in the market model regression (5) must equal 0 by construction If the Capital Asset Pricing Model (CAPM) is true, then the intercept α(i) = [1 β(i)] R(f) where R(f) is the risk-free rate of return so betas β(i) > 1 would typically be associated with α(i) < 0 Table below shows the market model regression using the CRSP value-weighted portfolio as the regressor, R(et) = α + β R(vt) + ε(it) where R(et) is the simple return to the equal-weighted portfolio and R(vt) is the simple return to the value-weighted portfolio in period t Market Model: Equal-weighted Returns on Value-weighted Returns α 0.10% 0.49% 0.08% β t(β=1) R² Risk Premium Market Model: CAPM Test for the Equal-weighted CRSP Portfolio α 0.18% 0.52% 0.11% t(α=0) β t(β=1) Professor Schwert Spring 1997