Chapter 5 investments risk and return past and prologue

Chapter 5 Risk and Return: Past and Prologue 5.1 Rates of Return 5-2 5.1 Rates of Return  Holding-Period Return (HPR) • Rate of return over given investment period  HPR= [PS PB + CF] / PB− • PS = Sale price, PB = Buy price, CF = Cash flow during holding period 5.1 Rates of Return  Measuring Investment Returns over Multiple Periods • Arithmetic average - Sum of returns in each period divided by number of periods • Geometric average - Single per-period return that would gives the same cumulative performance as the sequence of actual returns - Compound period-by-period returns; find per-period rate that compounds to same final value - Called a time-weighted average return Table 5.1 Quarterly Cash Flows/Rates of Return of a Mutual Fund 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Assets under management at start of quarter ($ million) 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 ($ million) 1.2 2 0.8 1.56 5.1 Rates of Return From Table 5.1 • Arithmetic average of quarterly return (10+25-20+20)/4 = 8.75% • Geometric average of quarterly return (1+0.10) X (1+0.25) X (1-0.20) X (1+0.20) = (1+rg)4 rg = 0.0719 or 7.19% 5.1 Rates of Return Conventions for Annualizing Rates of Return  Returns on assets with regular cash flows usually are quoted as an nual percentage rates, or APRs. - Mortgage: Monthly payments - Bonds: Semiannual coupons Without compounding: APR = Per-period rate × Periods per year With compounding: 1 + EAR = (1 + Rate per period) n = (1 + ) n  APR = [(1 + EAR)1/ n – 1] X n APR n Conventions for Annualizing Rates of Return Example: Suppose you have a 5% HPR on a 3 month investment. What is the annual rate of return with and without compounding? Without: With: n = 12/3 = 4, so HPRann(APR) = HPR*n = 0.05*4 = 20% HPRann(EAR) = (1.054) - 1 = 21.55% 5-8 Example: Suppose you buy one share of a stock today for $45 and you hold it for two years and s ell it for $52. You also received $8 in dividends at the end of the two years. (PB = , PS = , CF = ): HPR = HPRann (APR)= The annualized HPR assuming annual compounding is (n= ): HPRann (EAR)= $45 $52 $8 (52 - 45 + 8) / 45 = 33.33% 0.3333/2 = 16.66% 1/2 (1+0.3333)1/2 - 1 = 15.47% Annualized w/out compounding 5-9 Conventions for Annualizing Rates of Return 5.2 Risk and Risk Premiums 5-10 [...]... between risky and risk free assets 5- 31 rf = 5% σrf = 0% E(rp) = 14% σrp = 22% y = % in rp (1-y) = % in rf 5- 32 Expected Returns for Combinations E(rC) = yE(rp) + (1 - y)rf σc = yσrp + (1-y)σrf E(rC) = Return for complete or combined portfolio For example, let y = 0. 75 E(rC) = (. 75 x 14) + (. 25 x 05) E(rC) = 11 75 or 11. 75% σC = yσrp + (1-y)σrf σC = (0. 75 x 0.22) + (0. 25 x 0) = 0.1 65 or 16 .5% 5- 33 Complete... 40.00% $2,000 / $7 ,50 0 = The complete portfolio includes the riskless investment and rp 26.67% 100.00% Wrf = 25% ; Wrp = 75% In the complete portfolio WA = 0. 75 x 33.33% = 25% ; WB = 0. 75 x 40.00% = 30% WC = 0. 75 x 26.67% = 20%; Wrf = 25% 5- 30 Allocating Capital Between Risky & Risk- Free Assets Issues in setting weights - Examine risk & return tradeoff - Demonstrate how different degrees of risk aversion... s E(r) = Expected Return p(s) = probability of a state r(s) = return if a state occurs 1 to s states 5- 12 : The risk to the investment Var(r) = σ 2 = ∑ p(s) × [rs − E(r)]2 s SD(r)=σ = [σ2]1/2 E(r) = Expected Return p(s) = probability of a state rs = return in state “s” 5- 13 Numerical Example State Prob of State Return 1 2 - 05 2 5 05 3 3 15 E(r) = (.2)(-0. 05) + ( .5) (0. 05) + (.3)(0. 15) = 6% σ 2 = ∑ p(s)...  Risky asset or portfolio rp: _ Risky portfolio  Example Your total wealth is $10,000 You put $2 ,50 0 in ris k free T-Bills and $7 ,50 0 in a stock portfolio invested as follo ws: Stock A you put $2 ,50 0 Stock B you put $3,000 Stock C you put $2,000 $7 ,50 0 5- 29 Allocating Capital Between Risky & Risk- Free Assets Weights in rp WA = $2 ,50 0 / $7 ,50 0 = WB = 33.33% WC = $3,000 / $7 ,50 0 = 40.00%... (E(rp) – rf) / σp (A risk- free asset: a risk premium=0, a standard deviation=0) Quantify the incremental reward for each increase of 1% in the standard deviation of that portfolio A higher sharp ratio indicates a better reward per unit of volatility (a more efficient po rtfolio) 5- 19 5- 20 5- 21 Figure 5. 4 Rates of Return on Stocks, Bonds, and Bills 5. 4 Inflation and Real Rates of Return Inflation rate... ratios than bonds 5- 26 5- 27 • Asset Allocation - Portfolio choice among broad investment classes • Capital Allocation - Choice between risky and risk- free assets • Complete Portfolio - Entire portfolio, including risky and risk- free assets Allocating Capital Between Risky & Risk- Free Assets  Possible to split investment funds between safe and risky as sets T-bills or money market fund  Risk free asset... 3 The standard deviation is the appropriate measure of risk for a portfolio of assets with normally distributed returns 5- 17 Risk Premium & Risk Aversion 5- 18 The Sharpe(Reward-to-Volatility) Measure A statistic commonly used to rank portfolios in terms of risk- return trade-off is Sharp e (or reward-to-volatility) measure S = Portfolio risk premium / Standard deviation of portfolio excess return =... 5- 33 Complete portfolio E(rc) = yE(rp) + (1 - y)rf σc = yσrp + (1-y)σrf = yσrp 5- 34 E(r) Possible Combinations E(rp) = 14% P E(rp) = 11. 75% y=1 y =. 75 rf = 5% F y=0 0 16 .5% 22% σ 5- 35 E(r) CAL (Capital Allocation Line) P E(rp) = 14% Risk Premium E(rp) - rf = 9% ) Slope = 9/22 rf = 5% F 0 Slope = [E(rp) – rf] / σrp σrp = 22% σ 5- 36 ... rate is less than the approximate real rate 5- 24 Figure 5. 5 Interest Rates, Inflation, and Real Interest Rates 1926-2010 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... (.3)(0. 15) = 6% σ 2 = ∑ p(s) × [rs − E(r)] 2 s σ2 = [(.2)(-0. 05- 0.06)2 + ( .5) (0. 05- 0.06)2 + (.3)(0. 15- 0.06)2] σ2 = 0.0049%2 σ = [ 0.0049]1/2 = 07 or 7% 5- 14 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 Long tailed distribution, either . Chapter 5 Risk and Return: Past and Prologue 5. 1 Rates of Return 5- 2 5. 1 Rates of Return  Holding-Period Return (HPR) • Rate of return over given investment period  HPR=. $ 45 $52 $8 (52 - 45 + 8) / 45 = 33.33% 0.3333/2 = 16.66% 1/2 (1+0.3333)1/2 - 1 = 15. 47% Annualized w/out compounding 5- 9 Conventions for Annualizing Rates of Return 5. 2 Risk and Risk Premiums 5- 10 Scenario. Example State Prob. of State Return 1 .2 - . 05 2 .5 . 05 3 .3 . 15 E(r) = (.2)(-0. 05) + ( .5) (0. 05) + (.3)(0. 15) = 6% σ2 = [(.2)(-0. 05- 0.06)2 + ( .5) (0. 05- 0.06)2 + (.3)(0. 15- 0.06)2] σ2 = 0.0049%2 σ