Risk, Return and the Historical Record PDF

Chapter Risk, Return and the 5 Historical Record Bodie, Kane, and Marcus Essentials of Investments 2024 Release © McGraw Hill LLC. All rights reserved. Authorize...

Chapter Risk, Return and the 5 Historical Record Bodie, Kane, and Marcus Essentials of Investments 2024 Release © McGraw Hill LLC. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC.  5.1 Rates of Return Holding-Period Return (HPR) Discrete rate of return over given investment period: PEnding − PBeginning + DivCash HPR = PBeginning © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 2  5.1 Rates of Return: Example What is the HPR for a share of stock that was purchase for $25, sold for $27 and distributed $1.25 in dividends? $27.00 – $25.00 + $1.25 HPR = = 0.13 = 13.00% $25.00 The HPR is the sum of the dividend yield plus the capital gains yield © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 3  5.1 Rates of Return: Measuring over Multiple Periods Arithmetic average Sum of returns in each period divided by number of periods Geometric average Single per-period return Gives same cumulative performance as sequence of actual returns Dollar/SAR-weighted average return Internal rate of return on investment © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 4  Table 5.1 Annual Cash Flows & Rates of Return of a Mutual Fund © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 5  5.1 Rates of Return: Measuring over Multiple Periods Arithmetic average: The sum of the returns divided by the number of years. r1 + r2 +... + rn rArithmetic = n 10 + 25 − 20 + 20 = =.0875 = 8.75% 4 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 6  5.1 Rates of Return: Measuring over Multiple Periods Geometric average: Single period return that gives the same cumulative performance as the sequence of actual returns rGeometric = [(1 + r1 )  (1 + r2 ) ...  (1 + rn )]1/ n − 1 = 1.10  1.25 .80  1.20  1/ 4 − 1 = 0.0719 = 7.19% © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 7  5.1 Rates of Return Dollar-weighted average return The internal rate of return on an investment Annualizing Rates of Return APR = Annual Percentage Rate Per-period rate × Periods per year Ignores Compounding EAR = Effective Annual Rate Actual rate an investment grows Does not ignore compounding © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 8  5.1 Rates of Return: EAR vs. APR n-Periods of Compounding: Continuous Compounding:  APR  n EAR = e APR − 1 EAR =  1 +  −1  n  APR = [( EAR + 1)1/ n − 1]  n APR = ln( EAR + 1) where n = compounding per period © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 9 5.2 Inflation and The Real Rates of Interest Nominal Interest and Real Interest 1 + rNom 1 + rReal = 1+ i where rReal = Real Interest Rate rNom = Nominal Interest Rate i = Inflation Rate Example: What is the real return on an investment that earns a nominal 10% return during a period of 5% inflation? 1 +.10 1 + rReal = = 1.048 1 +.05 r =.048 or 4.8% © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 10  5.2 Inflation and The Real Rate of Interest U.S. History of Interest Rates, Inflation, and Real Interest Rates Since the 1950s, nominal rates have increased roughly in tandem with inflation 1930s/1940s: Volatile inflation affects real rates of return Figure 5.1 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 11 Figure 5.1 Inflation and Interest rates (1927-2022) © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 12  5.3 Risk and Risk Premiums Scenario Analysis and Probability Distributions Scenario analysis: Possible economic scenarios Probability distribution: Possible outcomes with probabilities Expected return: Mean value Variance: Expected value of squared deviation from mean Standard deviation: Square root of variance © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 13  Spreadsheet 5.1 Scenario Analysis for a Stock Index Fund © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 14  Figure 5.3 Normal Distribution r = 10% and σ = 20% © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 15  5.3 Risk and Risk Premiums 2 The Normal Distribution: Transform normally distributed return into standard deviation score (normally distributed with mean = 0 and SD=1): ri − E ( ri ) sri = i Original return, given standard normal return: ri = E ( ri ) + sri   i 16 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC.  5.3 Risk and Risk Premiums Normality over Time When returns over very short time periods are normally distributed, HPRs up to 1 month can be treated as normal © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 17  5.3 Risk and Risk Premiums: Value at Risk Value at risk (VaR): Measure of downside risk Worst loss with given probability, usually 1% or 5% © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 18  5.3 Risk and Risk Premiums Risk Premiums and Risk Aversion Risk-free rate: Rate of return that can be earned with certainty Risk premium: Expected return in excess of that on risk-free securities Excess return: Rate of return in excess of risk- free rate Risk aversion: Reluctance to accept risk Price of risk: Ratio of risk premium to variance © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 19  5.3 Risk and Risk Premiums Mean-Variance Analysis Ranking portfolios by Sharpe ratios (reward to volatility) A higher Sharpe ratio is better when comparing different portfolios Portfolio Risk Premium E ( rp ) − rf SP = Standard Deviation of Excess Returns P where E ( rp ) = Expected Return of the portfolio rf = Risk Free rate of return  P = Standard Deviation of portfolio excess return © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 20  5.4 The Historical Record Using Time Series of Return Scenario analysis derived from sample history of returns Variance and standard deviation estimates from time series of returns: 1   ( rt − rt ) 2 Var ( rt ) = n −1 SD ( rt ) = Var ( rt ) 1 rt =  rt n © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 21  5.4 The Historical Record: World Portfolios World Large stocks: 24 developed countries, ~6000 stocks U.S. large stocks: Standard & Poor's 500 largest cap U.S. small stocks: Smallest 20% on NYSE, Nasdaq, and Amex World bonds: Same countries as World Large stocks U.S. Treasury bonds: Barclay's Long-Term Treasury Bond Index © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 22  Table 5.3: Historical Return and Risk © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 23  Figure 5.4: Frequency Distribution of Annual Returns on United States Treasury Bills , 1927-2022 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 24  Figure 5.4: Frequency Distribution of Annual Returns on 30- Year Treasury Bonds , 1927-2022 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 25  Figure 5.4: Frequency Distribution Of Annual Returns on Common Stocks, 1927-2022 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 26  5.5 Asset Allocation across Portfolios Asset Allocation Portfolio choice among broad investment classes (stocks, bonds, Treasury bills, etc.) Complete Portfolio Entire portfolio, including risky and risk-free assets Capital Allocation Choice between risky and risk-free assets © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 27  5.5 Asset Allocation across Portfolios The Risk-Free Asset Treasury bonds (still affected by inflation) Price-indexed government bonds Money market instruments effectively risk-free Risk of CDs and commercial paper is miniscule compared to most assets © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 28  5.5 Portfolio Asset Allocation: Expected Return and Risk Expected Return of the Complete Portfolio E (rC ) = y  E (rp ) + (1 − y )  r f where E ( rC ) = Expected Return of the complete portfolio E ( rp ) = Expected Return of the risky portfolio rf = Return of the risk free asset y = Percentage assets in the risky portfolio Standard Deviation of the Complete Portfolio  C = y  p where  C = Standard deviation of the complete portfolio  P = Standard deviation of the risky portfolio © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 29 Figure 5.7 Investment Opportunity Set © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 30  5.5 Asset Allocation across Portfolios Capital Allocation Line (CAL) Plot of risk-return combinations available by varying allocation between risky and risk-free Risk Aversion and Capital Allocation y: Preferred capital allocation Available risk premium to variance ratio y= Required risk premium to variance ratio [ E ( rP ) − rf ] /  P2 [ E ( rP ) − rf ] = = A A P2 © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 31  5.6 Passive Strategies and the Capital Market Line Passive Strategy Investment policy that avoids security analysis Capital Market Line (CML) Capital allocation line using market-index portfolio as risky asset © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 32 Table 5.5: Excess Returns Statistics for the Market Index © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 33 5.6 Passive Strategies and the Capital Market Line Cost and Benefits of Passive Investing Passive investing is inexpensive and simple Expense ratio of active mutual fund averages 1% Expense ratio of hedge fund averages 1%-2%, plus 10% of returns above risk-free rate Active management offers potential for higher returns © McGraw Hill LLC. All rights reserved. No reproduction or further distribution permitted without the prior written consent of McGraw Hill LLC. 34