Risk And Valuation Lectures 13 & 14 PDF

Topic 6 Risk and valuation Lectures 13 & 14 1 Risk and valuation: course aims and objectives 1. To describe how to calculate ‘ex ante’ expected returns and variance of returns 2. To understand a key result on the effect of portfolio diversification on risk 3. To consider the important distinction be...

Topic 6 Risk and valuation Lectures 13 & 14 1 Risk and valuation: course aims and objectives 1. To describe how to calculate ‘ex ante’ expected returns and variance of returns 2. To understand a key result on the effect of portfolio diversification on risk 3. To consider the important distinction between systematic and unsystematic risk and how this affects expected returns 2 Introduction and motivation 1. In Topic 3 we considered how to value shares using dividend discount models – discount future dividends by investors’ required/expected rate of return Q : what determines the required rate of return for different shares? 3 2. LSE shares have historically delivered higher and more volatile returns than gilts or T-Bills Q : what do investors expect to receive in the future when they invest now? Q : what about returns on shares of individual companies? 4 Expected returns and variances How do we model uncertainty about future returns?  future return depends on the state of the world 5 Example State of the world s1 s2 s3  (si ) ~ R1 ~ R2 1/5 3/5 1/5 −13% 32% 42% 17% 23% 9% States are exhaustive and mutually exclusive  only one of s1, s2, and s3 can occur  (s ) = 1 i i 6 Expected return—is the return you expect to receive on average State of the world s1 s2 s3 investors’ required return  (si ) ~ R1 ~ R2 1/5 3/5 1/5 −13% 32% 42% 17% 23% 9% ~ 1  3  1  E[ R1 ] =   −13 +   32  +   42  = 25% 5  5  5  ~ 1  3  1  E[ R2 ] =   17  +   23 +   9  = 19% 5  5  5  ~ E[ R ] =   ( si ) R( si ) si 7 Variance and standard deviation Actual return = expected return + unexpected return ~ ~ ~ R = E[ R ] + U 0.7 0.6 0.5 0.4 ~ U2 0.3 0.2 ~ U1 0.1 0.0 –13% ~ U3 25% 32% 42% ~ E [ R1 ] 8 ~ ~ ~ R1 = E [ R1 ] + U1 State s1 s2 s3 ~ E[R ]  (si ) ~ R1 ~ U1 ~ R2 ~ U2 1/5 −13% −38% 17% −2% 3/5 32% 7% 23% 4% 1/5 42% 17% 9% −10% = 25% 19% ~  Ui  0 ~ E[U i ] = 0 ~ Summary measures of U i are the variance of 2 ~ return,  [ Ri ] , and the standard deviation of ~ return,  [ Ri ] ……. 9   ~ ~ 2  [ R ] =   ( si ) R( si ) − E[ R ] 2 si ~  [ R1 ] : ( −13 − 25)22 = 1444  1 / 5 = 288.8 (32 − 25) = 49  3 / 5 = 29.4 2 ( 42 − 25) = 289  1 / 5 = 57.8 2 = 376.0 ~  [ R1 ] = 376 = 19.39% ~  [ R2 ] : (17 − 19 )22 = 4  1 / 5 = 0.8 ( 23 − 19 ) = 16  3 / 5 = 9.6 2 (9 − 19 ) = 100  1 / 5 = 20.0 2 = 30.4 ~ [R 2 ] = 30.4 = 5.51 % 10 Portfolio diversification and risk Examples of portfolios 1. Divide investment equally between 5 shares  portfolio weights 1 1 1 1 1   , , , ,  5 5 5 5 5 2. Invest £200K in total, £25K each in 4 shares, £50K in share 5, £40K in share 6 and £10K in share 7  portfolio weights 1 1 1 1 1 1 1   , , , , , ,   8 8 8 8 4 5 20  11 SAQ The amounts invested in four securities are, £25,000, £5,000, £10,000, £20,000 What is the total amount invested and what are the individual portfolio weights? Total investment = 25+5+10+20=£60,000 Portfolio weights = {0.42, 0.08, 0.17, 0.33} 12 Portfolio return State of the world s1 s2 s3  (si ) ~ R1 ~ R2 1/5 3/5 1/5 −13% 32% 42% 17% 23% 9% portfolio: 1 = 3 / 5, portfolio return: 3~ 2 ~ ~ R p = R1 + R2 5 5 2 = 2 / 5 … a weighted average of security returns 13 Portfolio expected return State  (si ) s1 s2 s3 ~ E[R ] 1/5 3/5 1/5 = ~ R1 ~ R2 ~ 3~ 2~ Rp = R1 + R2 5 5 −13% 17% 32% 23% 42% 9% 25% 19% −1.0% 28.4% 28.8% ~ 1  3  1  E[ R p ] =   −1 +   28.4  +   28.8  = 22.6% 5  5  5  3 2 ~ ~ = 1E[ R1 ] +  2 E[ R2 ] = 25 + 19 = 22.6% 5 5 14 …the expected return of a weighted average of securities (a portfolio) equals a weighted average of the constituent security expected returns ~ E[ R p ] 25% 22.6% 19% 0 3/5 1 15 1 Portfolio variance and standard deviation of return ~ 1 5 3 5 1 5  2 [ R p ] = ( −1 − 22.6)2 + (28.4 − 22.6)2 + (28.8 − 22.6)2 = 139.26 ~  [ R p ] = 139.26 = 11.8% 3 2 ~ Note,  [ R p ] = 11.8%  19.39 + 5.51( = 13.84) 5 5  p  1 1 +  2 2 ~ ~ … if R1 and R2 are less than perfectly positively correlated In general, 16 …the standard deviation of a weighted average of securities (a portfolio) is less than a weighted average of the constituent security standard deviations  portfolio diversification reduces risk 17 p 19.39% 11.8% 5.51% 0 3/5 (0.08, 5.28%) 18 1 1 Formula summary: ~ ~ ~ R p = 1R1 +  2 R2 ,  2 = 1 − 1 ~ ~ ~ E[ R p ] = 1E[ R1 ] +  2 E[ R2 ] ~ ~ 2 ~ 2 2 ~ 2 2 ~  [ R p ] = 1  [ R1 ] +  2  [ R2 ] + 21 2 12 [ R1 ] [ R2 ] ~ ~ ~ ~ ~ ~ cov( R1 , R2 ) E[ R1  R2 ] − E[ R1 ]E[ R2 ] 12 = ~ ~ = ~ ~  [ R1 ] [ R2 ]  [ R1 ] [ R2 ] ~ ~ E[ R1  R2 ] = (1 / 5)( −13)(17 ) + (3 / 5)(32 )( 23) + (1 / 5)( 42 )(9) = 473 ( ) 12 = [473 − (25  19 )] / 376 30.4 = −0.0187 ~  [ R p ] = 0.36(376 ) + 0.16(30.4) + 0.48( −0.0187 ) 374 30.4 = 11.8% 19 How portfolio diversification affects risk Average annual standard deviation 55% diversifiable risk 28.4% non-diversifiable risk 1 10 50 Number of randomly selected UK stocks in portfolio 20 Risk unsystematic systematic non-diversifiable v. diversifiable market non-market ~ ~ ~ R = E[ R ] + U Information = expected + unexpected announcement (surprise) part part ~ What are the sources of risk (U ) ? Which risks are systematic and which unsystematic? Which types of information surprises give rise to systematic and unsystematic risk? 21 Systematic risk: sources of risk potentially affecting many securities to various extents – eg. information surprises about inflation, interest rates, industrial productivity, entry to the Euro ….. Unsystematic risk: sources of risk affecting individual, or a small group of, securities – eg. information surprises about drug discoveries, corporate boardroom problems, takeover bids ….. ~ ~ ~ ~ ~ ~ R = E[ R ] + U = E[ R ] + m +  ~ ~ Total risk (U )=systematic m +unsystematic (~ ) risk 22 Pricing systematic risk 1. Riskier securities give expected returns that include a risk premium over the riskfree interest rate 2. Portfolio diversification eliminates unsystematic risk leaving only systematic risk  systematic risk principle: investors require higher expected return to compensate them only for systematic risk Eg. Biotechnology/internet stocks—high total risk, much lower systematic risk 23 Summary We model uncertain returns by their return in different states (of the world) Summary measures of uncertain returns are expected return and standard deviation A portfolio is described by its portfolio weights (summing to 1) Portfolio return is a weighted average of returns on the constituent securities Portfolio expected returns is a weighted average of expected returns on the constituent securities …... 24 portfolio standard deviation is less than a weighted average of standard deviation of returns on the constituent securities (   1 ) portfolio diversification eliminates diversifiable risk, leaving only systematic risk the systematic risk principle says that security prices are determined so that securities with higher systematic risk promise higher expected returns 25