Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 1 Chapter 6 Risk and Return: Past and Prologue

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 2 Rates of Return: Single Period HPR = Holding Period Return P 1 = Ending price P 0 = Beginning price D 1 = Dividend during period one

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 3 Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( )/ ( 20) = 25%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 4 Data from Text Example p Assets(Beg.) HPR (.20).25 TA (Before Net Flows Net Flows (0.8) 0.0 End Assets

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 5 Returns Using Arithmetic and Geometric Averaging Arithmetic r a = (r 1 + r 2 + r r n ) / n r a = ( ) / 4 =.10 or 10% Geometric r g = {[(1+r 1 ) (1+r 2 ).... (1+r n )]} 1/n - 1 r g = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 =.0829 = 8.29%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 6 Dollar Weighted Returns Internal Rate of Return (IRR) - the discount rate that results present value of the future cash flows being equal to the investment amount Considers changes in investment Initial Investment is an outflow Ending value is considered as an inflow Additional investment is a negative flow Reduced investment is a positive flow

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 7 Dollar Weighted Average Using Text Example Net CFs $ (mil) Solving for IRR 1.0 = -.1/(1+r) /(1+r) 2 +.8/(1+r) /(1+r) 4 r =.0417 or 4.17%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 8 Quoting Conventions APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period) Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01) = 12.68%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 9 Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 10 r r Symmetric distribution Normal Distribution s.d.

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 11 r r NegativePositive Skewed Distribution: Large Negative Returns Possible Median

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 12 r rNegativePositive Skewed Distribution: Large Positive Returns Possible Median

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 13 Subjective returns p(s) = probability of a state r(s) = return if a state occurs 1 to s states p(s) = probability of a state r(s) = return if a state occurs 1 to s states Measuring Mean: Scenario or Subjective Returns E(r) = p(s) r(s)  s

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 14 Numerical Example: Subjective or Scenario Distributions StateProb. of Stater in State E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) =.15

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 15 Standard deviation = [variance] 1/2 Measuring Variance or Dispersion of Returns Subjective or Scenario Variance=  s p(s) [r s - E(r)] 2 Var =[(.1)( ) 2 +(.2)( ) ( ) 2 ] Var= S.D.= [.01199] 1/2 =.1095 Using Our Example:

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 16 Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 17 Annual Holding Period Returns From Figure 6.1 of Text GeomArithStan. SeriesMean%Mean%Dev.% Lg Stk Sm Stk LT Gov T-Bills Inflation

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 18 Annual Holding Period Risk Premiums and Real Returns Risk Real SeriesPremiums%Returns% Lg Stk Sm Stk LT Gov T-Bills Inflation

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 19 Possible to split investment funds between safe and risky assets Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) Allocating Capital Between Risky & Risk-Free Assets

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 20 Allocating Capital Between Risky & Risk-Free Assets (cont.) Issues –Examine risk/ return tradeoff –Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 21 r f = 7%  rf = 0% E(r p ) = 15%  p = 22% y = % in p (1-y) = % in r f Example Using the Numbers in Chapter 6 (pp )

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 22 E(r c ) = yE(r p ) + (1 - y)r f r c = complete or combined portfolio For example, y =.75 E(r c ) =.75(.15) +.25(.07) =.13 or 13% Expected Returns for Combinations

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 23 E(r) E(r p ) = 15% r f = 7% 22% 0 P F Possible Combinations 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 24 p p c c = = Since rfrf rfrf y y Variance on the Possible Combined Portfolios = 0, then      

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 25 c c =.75(.22) =.165 or 16.5% If y =.75, then c c = 1(.22) =.22 or 22% If y = 1 c c = 0(.22) =.00 or 0% If y = 0 Combinations Without Leverage      

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 26 Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage r c = (-.5) (.07) + (1.5) (.15) =.19  c = (1.5) (.22) =.33

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 27 E(r) E(r p ) = 15% r f = 7% = 22% = 22% 0 P F F P P ) S = 8/22 ) S = 8/22 E(r p ) - r f = 8% CAL(CapitalAllocationLine) 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 28 Risk Aversion and Allocation Greater levels of risk aversion lead to larger proportions of the risk free rate Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 29 Quantifying Risk Aversion E(r p ) = Expected return on portfolio p r f = the risk free rate.005 = Scale factor A x  p = Proportional risk premium The larger A is, the larger will be the added return required for risk

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 30 Quantifying Risk Aversion Rearranging the equation and solving for A Many studies have concluded that investors’ average risk aversion is between 2 and 4