Risk and Return: Past and Prologue

Chapter 5 Risk and Return: Past and Prologue Bodie, Kane, and Marcus Essentials of Investments Tenth Edition

What is in Chapter 5 5.1 Rates of Return HPR, arithmetic, geometric, dollar-weighted, APR, EAR 5.2 Inflation and Real Rate of Interest 5.3 Risk and Risk Premium Risk calc, risk premium = extra return per risk unit 5.4 Historical Records 5.5 Asset Allocation 5.6 Passive Strategies and CML 2

5.1 Rates of Return Holding-Period Return (HPR) Rate of return over given investment period HPR = where P S P B P P + Div S B P Div = Cash Dividend B = Sale Price = Buy Price 3

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? HPR PS PB + Div $27 25 + 1.25 = = = 13% P 25 B Capital Gains Yield? $27 25 = 8% 25 Dividend Yield? $1.25 25 = 5% 4

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 Compound period-by-period returns; find per-period rate that compounds to same final value Dollar-weighted average return Internal rate of return on investment 5

Table 5.1 Rates of Return of a Mutual Fund: Example 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Assets under management at start of quarter 1 1.2 2 0.8 Holding-period return (%) 10 25 20 20 Total assets before net inflows 1.1 1.5 1.6 0.96 Net inflow ($ million) 0.1 0.5 0.8 0.6 Assets under management at end of quarter 1.2 2 0.8 1.56 Arithmetic Average 10% + 25% + ( 20%) + 20% = 8.75% 4 Geometric Average [(1.1) (1.25) (.8) (1.2)] = 7.19% 1 4 Dollar-Weighted 0 1.0 1.0.5.8.96 = + + + + 1 + IRR (1 + IRR ) (1 + IRR ) (1 + IRR ) IRR = 3.38% 2 3 4 6

5.1 Rates of Return Conventions for 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 8

5.1 Rates of Return: EAR vs. APR For n periods of compounding: APR n EAR = (1 + ) 1 n APR = [( EAR + 1) 1] n where n = compounding per period For Continuous Compounding: APR EAR = e 1 APR = ln( EAR + 1) 1 n 9

5.2 Inflation and The Real Rates of Interest Nominal Interest and Real Interest 1+ R 1+ r = 1 + i where r = Real Interest Rate R = 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+ r = = 1.048 1 +.05 r =.048 or 4.8% 11

5.2 Inflation and The Real Rates of Interest Equilibrium Nominal Rate of Interest Fisher Equation R = r + E(i) E(i): Current expected inflation R: Nominal interest rate r: Real interest rate 12

Interest Rates, Inflation, and Real Interest Rates 1926-2015 13

5.2 Inflation and The Real Rates 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 14

5.3 Risk and Risk Premiums Scenario Analysis and Probability Distributions Scenario analysis: Possible economic scenarios; specify likelihood and HPR Probability distribution: Possible outcomes with probabilities Expected return: Mean value of distribution of HPR Variance: Expected value of squared deviation from mean Standard deviation: Square root of variance 16

Spreadsheet 5.1 Scenario Analysis for the Stock Market 17

5.3 Risk and Risk Premiums 18

Figure 5.1 Normal Distribution r = 10% and σ = 20% 19

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 Use continuously compounded rates where normality plays crucial role 20

5.3 Risk and Risk Premiums Deviation from Normality and Value at Risk Kurtosis: Measure of fatness of tails of probability distribution; indicates likelihood of extreme outcomes Skew: Measure of asymmetry of probability distribution Using Time Series of Return Scenario analysis derived from sample history of returns Variance and standard deviation estimates from time series of returns: 21

Figure 5.2 Comparing Scenario Analysis to Normal Distributions with Same Mean and Standard Deviation 22

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 riskfree rate Risk aversion: Reluctance to accept risk Price of risk: Ratio of risk premium to variance 23

5.3 Risk and Risk Premiums: Sharpe Ratios The Sharpe (Reward-to-Volatility) Ratio Ratio of portfolio risk premium to standard deviation Mean-Variance Analysis Ranking portfolios by Sharpe ratios S P where Portfolio Risk Premium Standard Deviation of Excess Returns E( r ) = Expected Return of the portfolio r f σ P = p = Risk Free rate of return E( r ) r = Standard Deviation of portfolio excess return p σ P f 25

5.3 Risk and Risk Premiums: Historical Sharpe Ratios World Markets U.S. Markets Sharpe Large Government Small Large U.S. Long-Term ratios Stocks Bonds Stocks Stocks Treasuries 1926-2013 0.33 0.26 0.37 0.41 0.23 1926-1955 0.43 0.22 0.40 0.46 0.59 1956-1985 0.19 0.05 0.38 0.28-0.11 1986-2013 0.35 0.52 0.37 0.48 0.41 26

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 28

5.4 The Historical Record: World Portfolios Returns World Markets U.S. Markets Total Returns Large Stocks Government Bonds Small Stocks Large Stocks U.S. Long-Term Treasuries Geometric average (%) 8.24 5.37 11.82 9.88 5.07 Lowest return -39.94 (1931) -13.50 (1946) -54.27 (1937) -45.56 (1931) -13.82 (2009) Highest return 70.81 (1933) 34.12 (1985) 159.05 (1933) 54.56 (1933) 32.68 (1985) 29

Figure 5.4 Rates of Return on Stocks, Bonds, and Bills 30

5.5 Asset Allocation across Portfolios Asset Allocation Portfolio choice among broad investment classes Complete Portfolio Entire portfolio, including risky and risk-free assets Capital Allocation Choice between risky and risk-free assets 32

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 33

5.5 Portfolio Asset Allocation: Expected Return and Risk Expected Return of the Complete Portfolio E( r ) = y E( r ) + (1 y) r C where E( r ) = Expected Return of the complete portfolio C E( r ) = Expected Return of the risky portfolio p r f p = Return of the risk free asset f y = Percentage assets in the risky portfolio Standard Deviation of the Complete Portfolio σ C where = y σ p σ C = Standard deviation of the complete portfolio σ = Standard deviation of the risky portfolio P 34