# 09_Oheads - Lecture Notes 9 Introduction to Risk Returns on...

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1 Lecture Notes 9 Introduction to Risk Returns on stock • There are two components to the return on most stocks: i. most, but not all, stocks pay a cash dividend ; and ii. a capital gain or loss which, if not realized , is re-invested. • If D t +1 is the dividend paid at the end of the current period, and P t is the price of the stock, the total ex-post percentage return is (1) Example . Suppose you bought 100 shares of Wal-Mart (WMT) one year ago today at \$25. Over the last year, you received \$20 in dividends (= 20 cents per share × 100 shares). At the end of the year, the stock sells for \$30. • What was the return? You invested \$2,500. After 1 year you have \$3,020, which is a percentage return of 520/2500 = 20.8% R t 1 + D t 1 + P t ------------ P t 1 + P t () P t --------------------------- + =

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2 • If a stock is held for n years, and the return in year t is R t , the holding period return is • For a sample of annual percentage returns R 1 , R 2 , R 3 , …, R T over T years, the average percentage return R is defined by (2) • The sample variance of these returns, or the expected value of ( R t R ) 2 , measures the variation in percentage returns: (3) [This formula has T –1 as a divisor and not T to ensure that s 2 is an unbiased estimate of the underlying population variance σ 2 .] • The square root of (3) is the sample standard deviation s , and it has the same dimensions as R and R t . R 1 R 1 + () 1 R 2 + 1 R n + 1 = R R 1 R 2 R T +++ T -------------------------------------------- 1 T --- R t t 1 = T = s 2 1 T 1 ------------ R t R 2 t 1 = T =
3 Example . For the two samples of returns on assets A and B in Table 1 the sample means, variances and standard deviations are (4) (5) (6) so that s A = 0.008333 or about 9.13% and (7) so that s B = 0.086667 or about 29.44%. Table 1: Sample percentage returns on two assets Year A B 1991 15% 30% 1992 0% –20% 1993 5% 20% 1994 20% 50% R A 0.15 0.0 0.05 0.20 ++ + 4 --------------------------------------------------------- 0 . 1 == R B 0.30 0.20 0.20 0.50 4 ------------------------------------------------------------ 0 . 2 s A 2 0.15 0.1 () 2 0.0 0.1 2 0.05 0.1 2 0.2 0.1 2 + 3 ------------------------------------------------------------------------------------------------------------------------------------------ 0.00833333 = = s B 2 0.3 0.2 2 0.2 0.2 2 0.2 0.2 2 0.5 0.2 2 + 3 ---------------------------------------------------------------------------------------------------------------------------------------- 0.08666667 = =

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4 Empirical observation based on these statistics: Over longer ho- rizons, assets or portfolios with higher variation in returns from year to year tend to have higher average return. Figure 1: \$1 investment in different types of assets, 1926–2001 (Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago) 0.01 0.1 1 10 100 1000 10000 1926 1929 1932 1935 1938 1941 1944 1947 1950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 Cash \$0.10 Treasury bills \$17.20 Long-term government bonds \$50.72 Large-company stocks \$2279.18 Small-company stocks \$7861.05
5 Table 2: Annual return statistics 1926-2001 Series Mean Risk prem. Std. dev. Histogram Inflation ( negative return on cur- rency) 3.21 4.39 Large company stocks 12.65 8.79 20.24 Small company stocks 17.34 13.48 33.23 Long-term corporate debt 6.12 2.26 8.63

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6 Long-term government bonds 5.60 1.74 9.23 Intermediate-term gov-
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09_Oheads - Lecture Notes 9 Introduction to Risk Returns on...

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