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= [P 1 – P 0 + CF] / P 0 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): HPR ann = –With compounding: EAR –HPR ann = where n = number of years held HPR/n [(1+HPR) 1/n ]-1 5-3

Measuring Ex-Post (Past) Returns Must be Geometric Means for multi- periods 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 ___ share of the stock at the beginning of each interim period and sell ___ share at the end of each interim period. TWRs are thus ___________ of the amount invested in a given period. To calculate TWRs: Calculate the return for each time period, typically a year. Then calculate either an arithmetic (AAR) or a geometric average (GAR) of the returns. one independent 5-6

Measuring Ex-Post (Past) Returns HPR for year 1: HPR for year 2: [$54 - $53 +$2] / $53 = 5.66% [$53 + $2 - $50] / $50 = 10% AAR = [ ] / 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): A2: When you are trying to estimate an expected return (ex- ante return): 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. Use the AAR 5-8

Characteristics of Probability Distributions 1.Mean: __________________________________ _ 2.Median: _________________ 3.Variance or standard deviation: 4.Skewness:_______________________________ 5.Leptokurtosis: ______________________________ If a distribution is approximately normal, the distribution is fully described by _____________________ Arithmetic average & usually most likely Dispersion of returns about the mean Long tailed distribution, either side Too many observations in the tails Characteristics 1 and 3 Middle observation 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 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 r real  r nom - i Exampler nom = 9%, i = 6% r real  3% Fisher effect: Exact r real = or r real = The exact real rate is less than the approximate real rate. [(1 + r nom ) / (1 + i)] – 1 (r nom - i) / (1 + i) (9% - 6%) / (1.06) = 2.83% r real = real interest rate r nom = nominal interest rate i = expected inflation rate 5-12

Historical Real Returns & Sharpe Ratios (p121) Real Returns%Sharpe Ratio Series World Stk US Lg. Stk Sm. Stk World Bnd LT Bond 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

5.5Asset Allocation Across Risky and Risk Free Portfolios 5-14

r f = 5%  rf = 0% E(r p ) = 14%  rp = 22% y = % in r p (1-y) = % in rf Example 5-15

E(r C ) = Expected Returns for Combinations E(r C ) = For example, let y = ____ E(r C ) = E(r C ) =.1175 or 11.75%  C = y  rp + (1-y)  rf  C = (0.75 x 0.22) + (0.25 x 0) = or 16.5%  c = yE(r p ) + (1 - y)r f y  rp + (1-y)  rf Return for complete or combined portfolio 0.75 (.75 x.14) + (.25 x.05) 5-16

Quantifying Risk Aversion p121 E(r p ) = Expected return on portfolio p r f = the risk free rate 0.5 = Scale factor A x  p 2 = Proportional risk premium The larger A is, the larger will be the _________________________________________ Many studies have concluded that investors’ average risk aversion is between 2 and 4 (or 1 and 2 w/o scaling) investor’s added return required to bear risk 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 PeriodAverage  Sharpe Ratio 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