Transcript Slide 1
Chapter 5 Risk and Return: Past and Prologue Measuring Ex-Post (Past) Returns One period investment: regardless of the length of the period. Must be in %. Holding period return (HPR): HPR = [P1 – P0 + CF] / P0 5-2 Annualizing HPRs Q: Why would you want to annualize returns? 1. Annualizing HPRs for holding periods of greater than one year: – Without compounding (Simple or APR): HPRann = HPR/n – – With compounding: EAR HPRann = [(1+HPR)1/n]-1 where n = number of years held 5-3 Measuring Ex-Post (Past) Returns • Must be Geometric Means for multiperiods • Must be weighted means for portfolios Dollar-Weighted Return i. Dollar-weighted return procedure (DWR): = IRR we learned e.g. a stock pays $2 dividend was purchased for $50. The prices at the end of the year 1 and year 2 are $53 and $54, respectively. What’s DWR (or IRR)? 5-5 Time-Weighted Returns AAR or GAR ii. Time-weighted returns (TWR): TWRs assume you buy one ___ share of the stock at the beginning of each one share at interim period and sell ___ the end of each interim period. TWRs are thus ___________ independent of the amount invested in a given period. To calculate TWRs: Calculate the return for each time period, typically a year. calculate either an arithmetic (AAR) or a geometric Then average (GAR) of the returns. 5-6 Measuring Ex-Post (Past) Returns HPR for year 1: [$53 + $2 - $50] / $50 = 10% HPR for year 2: [$54 - $53 +$2] / $53 = 5.66% AAR = [0.10 + 0.0566] / 2 = 7.83% or 7.81% for GAR 5-7 Measuring Ex-Post (Past) Returns Q: When should you use the GAR and when should you use the AAR? A1: When you are evaluating PAST RESULTS (ex-post): Use the AAR (average without compounding) if you ARE NOT reinvesting any cash flows received before the end of the period. Use the GAR (average with compounding) if you ARE reinvesting any cash flows received before the end of the period. A2: When you are trying to estimate an expected return (exante return): Use the AAR 5-8 Characteristics of Probability Distributions Arithmetic average & usually most likely _ 1. Mean: __________________________________ 2. Median: Middle observation _________________ 3. Variance or standard deviation: Dispersion of returns about the mean 4. Skewness:_______________________________ Long tailed distribution, either side 5. Leptokurtosis: ______________________________ Too many observations in the tails If a distribution is approximately normal, the distribution 1 and 3 is fully described by Characteristics _____________________ 5-9 Value at Risk (VaR) VaR versus standard deviation: • For normally distributed returns VaR is equivalent to standard deviation (although VaR is typically reported in dollars rather than in % returns) • VaR adds value as a risk measure when return distributions are not normally distributed. – Actual 5% probability level will differ from 1.68445 standard deviations from the mean due to kurtosis and skewness. 5-10 5.3 The Historical Record You read 5-11 Inflation, Taxes and Returns Real vs. Nominal Rates Fisher effect: Approximation real rate nominal rate - inflation rate rreal rnom - i rreal = real interest rate Example rnom = 9%, i = 6% rnom = nominal interest rate rreal 3% i = expected inflation rate Fisher effect: Exact rreal = [(1 + rnom) / (1 + i)] – 1 or rreal = (rnom - i) / (1 + i) rreal = (9% - 6%) / (1.06) = 2.83% The exact real rate is less than the approximate real rate. 5-12 Historical Real Returns & Sharpe Ratios (p121) Series World Stk US Lg. Stk Sm. Stk World Bnd LT Bond Real Returns% 6.00 6.13 8.17 Sharpe Ratio 0.37 0.37 0.36 2.46 2.22 0.24 0.24 • Real returns have been much higher for stocks than for bonds • Sharpe ratios measure the excess return to standard deviation. • The higher the Sharpe ratio the better. • Stocks have had much higher Sharpe ratios than bonds. 5-13 5.5 Asset Allocation Across Risky and Risk Free Portfolios 5-14 Example rf = 5% srf = 0% E(rp) = 14% srp = 22% y = % in rp (1-y) = % in rf 5-15 Expected Returns for Combinations E(rC) = yE(rp) + (1 - y)rf sc = ysrp + (1-y)srf E(rC) = Return for complete or combined portfolio For example, let y = 0.75 ____ E(rC) = (.75 x .14) + (.25 x .05) E(rC) = .1175 or 11.75% sC = ysrp + (1-y)srf sC = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5% 5-16 Quantifying Risk Aversion E rp rf 0.5 A s p p121 2 E(rp) = Expected return on portfolio p rf = the risk free rate 0.5 = Scale factor A x sp2 = Proportional risk premium The larger A is, the larger will be the _________________________________________ investor’s added return required to bear risk Many studies have concluded that investors’ average risk aversion is between 2 and 4 (or 1 and 2 w/o scaling) 5-17 A Passive Strategy • Investing in a broad stock index and a risk free investment is an example of a passive strategy. – The investor makes no attempt to actively find undervalued strategies nor actively switch their asset allocations. – The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML. 5-18 Excess Returns and Sharpe Ratios implied by the CML Excess Return or Risk Premium Time Period 1926-2008 1926-1955 1956-1984 1985-2008 Average 7.86 11.67 5.01 5.95 s 20.88 25.40 17.58 18.23 Sharpe Ratio 0.37 0.46 0.28 0.33 The average risk premium implied by the CML for large common stocks over the entire time period is 7.86%. • How much confidence do we have that this historical data can be used to predict the risk premium now? 5-19 Active versus Passive Strategies • Active strategies entail more trading costs than passive strategies. • Passive investor “free-rides” in a competitive investment environment. • Passive involves investment in two passive portfolios – Short-term T-bills – Fund of common stocks that mimics a broad market index – Vary combinations according to investor’s risk aversion. 5-20