Introduction to Risk and Return Common stocks 13.0% 9.2% 20.3% Small-company stocks Long-term corporate bonds Long-term government bonds Intermediate-term government bonds U.S. Treasury bills Inflation Risk premium Arithmetic (relative to U.S. Standard Series mean Treasury bills) deviation -90%90%0%

The Value of an Investment of $1 in 1926 Source: Ibbotson Associates Index Year End

The Value of an Investment of $1 in 1926 Source: Ibbotson Associates Index Year End Real returns

Rates of Return Source: Ibbotson Associates Year Percentage Return

Measuring Risk Return % # of Years Histogram of Annual Stock Market Returns

Variance and Standard Deviation  VAR (r i ~ ) = E [ r i ~ - E(r i ~ ) ] 2 =  ri 2,  COV( r 1 ~, r 2 ~ ) = E [(r 1 ~ - E(r 1 ~ )) (r 2 ~ - E(r 2 ~ ))] where r i ~ is actual return governed by probability distribution  EX:The return of asset i next period is ether.2 with prob. 60% or -.1 with prob. 40%  E(r i ~ ) =.6*.2 +.4*(-.1) =.08  Var(r i ~ ) =.6*(.2-.08) 2 +.4*( ) 2 =.0216

Return and Variance of Two Assets  Calculating Portfolio risks of two stocks E(r ~ )Weight SD(r ~ ) Stock A Stock B  E(r p ~ ) = x 1 *E(r 1 ~ ) + x 2 *E(r 2 ~ ), where x 1 + x 2 = 1 E(r p ~ ) =.6* *.21 =.174  What about variance? x 1 *  x 2  2 2 ? No!!!

Covariance of a portfolio of two assets   p 2 = E [ r p ~ - E(r p ~ ) ] 2 = E [x 1 r 1 ~ + x 2 r 2 ~ - x 1 *E(r 1 ~ ) - x 2 *E(r 2 ~ )] 2 = E[x 1 *(r 1 ~ -E(r 1 ~ )) + x 2 *(r 2 ~ -E(r 2 ~ )] 2 = E[ x 1 2 (r 1 ~ -E(r 1 ~ ) 2 + x 2 2 (r 2 ~ -E(r 2 ~ ) 2 + x 1 x 2 (r 1 ~ -E(r 1 ~ )(r 2 ~ -E(r 2 ~ ) + x 1 x 2 (r 1 ~ -E(r 1 ~ )(r 2 ~ -E(r 2 ~ )] = x 1 2  x 2 2  x 1 x 2 COV(r 1 ~, r 2 ~ ) Define COV(r 1 ~, r 2 ~ ) = E[(r 1 ~ -E(r 1 ~ ) (r 2 ~ -E(r 2 ~ )] =  12

Correlation Coefficient To get rid of the unit, we define Correlation coefficient  12 = COV(r 1 ~, r 2 ~ ) /  1  2, where -1<=  <= 1 Thus,  p 2 = x 1 2  x 2 2  x 1 x 2  1  2  12  If  12 = 1, then  p = X 1  1 + X 2  2  If  12 < 1, then  p < X 1  1 + X 2  2 Stock 1Stock 2 Stock 1x 1 2  1 2 x 1 x 2 COV(r 1 ~, r 2 ~ ) Stock 2x 1 x 2 COV(r 1 ~, r 2 ~ )x 2 2  2 2

The composition of portfolio variance  Two risky assets  Three assets  Four assets  N risky assets

Variance of a Diversified Portfolio  What is the variance of portfolio if the number of stock increases?  General Formula: a portfolio with equally weighted N stocks  Portfolio variance: = N (1/N) 2 * average var. + (N 2 -N)(1/N) 2 * average cov. = 1/N * average var. + (1-1/N) * average cov.  As N increases, the variance of each individual stock becomes less important, and the average covariance becomes dominant.

How does diversification reduce risks?  The central message: total risk can be decomposed into two parts: systematic and unsystematic risks.  Therefore diversification can only eliminate unique risks (or unsystematic risks, diversifyable risks), can not eliminate market risk (systematic risks, undiversificable risk)  What is unsystematic risks?  RD program, new product introduction, labor relations, personal changes, lawsuits.  The risk of a well-diversified portfolio depends on the market risk of the securities included in the portfolio.

Measuring Risk

How individual securities affect portfolio risk? ABAB  Row 1A..6 2 * *.4*.2*18.6*28  Row 2B.6*.4*.2*18.6* *28 2 Row 1 =.6 * [.6* *.2*18.6*28] =.6 * 249 Row 2 =.4 * [.6*.2*18.6*28 +.4*28 2 ] =.4 * 376 Total = 300  The contribution of stock A to portfolio risk is WEIGHT * COVARIANCE WITH ALL THE SECURITIES IN THE PORTFOLIO (INCLUDING ITSELF)  The risk of a stock not only depend on its own risks, but also its contribution to the risk of whole portfolio.

Stock’s Beta  If the portfolio is the market portfolio, then we have the formal definition of Beta  Beta - Sensitivity of a stock’s return to the return on the market portfolio.  = Cov (r i ~, r m ~ ) / Var(r m ~ ) =  i,m  i  m /  m 2 =  i,m [  i /  m ]

Conclusions  Markets risk accounts for most of the risk of a well-diversified portfolio.  The beta of an individual security measure its sensitivity to market movement.  A nice property of Beta:  p =  X i  I, where Xi is the weight of market value of asset I  Does corporate diversification add value for shareholders?