Learning about return and risk from the historical record and beta estimation Reference: Investments, Bodie, Kane, and Marcus, and Investment Analysis and Behavior, Nofsinger and Hirschey Nattawut Jenwittayaroje, Ph.D., CFA NIDA Business School 1 Rationale Risk is as important to investors as expected return. Though we have CAPM, the level of risk faced by investors need to be estimated from historical experience. Neither expected returns nor risk are directly observable. Only realized rates of return and risk can be observed after the fact. Essential tools for estimating expected returns and risk from the historical record is needed. 2 T Bills and Inflation History of T bill Rates, Inflation and Real Rates for Generations, 1926 2005 T Bills and Inflation Interest Rates and Inflation, 1926 2005 3 4
T Bills and Inflation Rates of Return: Single Period Nominal and Real Wealth Indexes, 1966 2005 Approx. 10 Approx. 1.6 HPR = Holding Period Return P 0 = Beginning price P 1 = Ending price D 1 = Dividend during period one Example: Ending Price = 48 Beginning Price = 40 Dividend = 2 HPR = (48 40 + 2)/40 = 25% HPR = capital gain yield + dividend yield = 8/40 + 2/40 = 20% + 5% 5 6 Expected Return and Standard Deviation Expected Returns: Example Expected Return = p(s) = probability of a state r(s) = return if a state occurs 1 to s states State Prob. of State r in State 1.1 -.05 (or 5%) 2.2.05 3.4.15 4.2.25 5.1.35 E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) =.15 = 15% 7 8
Variance or Dispersion of Returns Standard deviation = [variance] 1/2 Using Our Example: Var =[(.1)(-.05-.15) 2 +(.2)(.05-.15) 2...+.1(.35-.15) 2 ] Var=.01199 S.D.= [.01199] 1/2 =.1095 = 10.95% Mean and Variance of Historical Returns In forward looking analysis so far, we determine a set of relevant scenarios and associated investment outcomes (i.e., rates of return) and probability. In contrast, asset and portfolio return histories come in the form of time series of past realized returns that do not explicitly provide the probabilities of those observed returns; we observe only dates and associated holding period returns. Therefore, when we use historical data, we treat each observation as an equally likely scenario. 9 10 Mean and Variance of Historical Returns The Normal Distribution Expected return is arithmetic average or arithmetic average of rates of return 11 A graph of the normal curve with mean of 10% and the standard deviation of 20%. 12
The Normal Distribution Normal and Skewed Distribution (mean = 6% SD = 17%) Investment management is far more tractable when asset rates of return can be well approximated by the normal distribution. First, it s symmetric. Therefore, measuring risk as the SD of returns is adequate. Second, when assets with normally distributed returns are mixed, the resulting portfolio return is also normally distributed. Third, only two parameters (mean and SD) have to be estimated to obtain the probabilities of future scenarios. How closely actual return distributions fit the normal curve. 13 Skewness measures the degree of asymmetry Positive (negative) skewness SD overestimates (underestimate) risk. 14 Normal and Fat Tails Distributions (mean =.1 SD =.2) Kurtosis is a measure of the degree of fat tails. History of Rates of Returns of Asset Classes for Generations, 1926 2005 15 The asset classes with higher volatility (i.e., SD) provided higher average returns investors demand a risk premium to bear risk. 16
Histograms of Rates of Return for 1926 2005 Excess Returns and Risk Premiums How much should you invest in a risky asset (e.g., stocks).. How much of an expected reward is offered for the risk involved in investing money in a stock. We measure the reward as the difference between the expected holding period return on the stock and the risk free rate risk premium. The difference in any particular period between the actual rate of return on a risky asset and the riskfree rate excess return. 17 18 History of Excess Returns of Asset Classes for Generations, 1926 2005 History of Excess Returns of Asset Classes for Generations, 1926 2005 The average excess return was positive for every subperiods. Average excess returns of large stocks in the last 40 years suggest a risk premium of 6% 8% The skews of the two large stock portfolios are significantly negative, 0.62 and 0.70. Negative skews imply SD underestimates the actual level of risk. Fat tails are observed for five assets during 1926 2005. The serial correlation is practically zero for four of the five portfolios, supporting market efficiency. 19 20
Estimating Beta Capital Asset Pricing Model Beta Estimation Need Risk free rate data Treasury bill. Market portfolio data S&P 500, DJIA, NASDAQ, SET Index, etc. Stock return data Interval Daily, monthly, annual, etc. Length One year, five years, ten years, etc. 21 Market Index variations Interval variations Constant 0.005 Std Err of Y Est 0.006 R Squared 24.71% No. of Observations 52 Degrees of Freedom 50 Constant 0.004 Std Err of Y Est 0.001 R Squared 20.11% No. of Observations 52 Degrees of Freedom 50 Constant 0.0001 Std Err of Y Est 0.0002 R Squared 31.37% No. of Observations 9591 Degrees of Freedom 9589 Constant 0.019 Std Err of Y Est 0.054 R Squared 31.41% No. of Observations 38 Degrees of Freedom 36 Beta estimate 0.995 Std Err of Coef. 0.246 t-statistic 4.05 Beta estimate 0.737 Std Err of Coef. 0.208 t-statistic 3.550 Beta estimate 1.047 Std Err of Coef. 0.016 t-statistic 66.26 Beta estimate 1.301 Std Err of Coef. 0.320 t-statistic 4.06