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## Chapter 6

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**Chapter6**Risk and Return: Past and Prologue**Rates of Return: Single Period**HPR = Holding Period Return P1 = Ending price P0 = Beginning price D1 = Dividend during period one**Rates of Return: Single Period Example**Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25%**Data from Text Example p. 154**1 2 3 4 Assets(Beg.) 1.0 1.2 2.0 .8 HPR .10 .25 (.20) .25 TA (Before Net Flows 1.1 1.5 1.6 1.0 Net Flows 0.1 0.5 (0.8) 0.0 End Assets 1.2 2.0 .8 1.0**Returns Using Arithmetic and Geometric Averaging**Arithmetic ra = (r1 + r2 + r3 + ... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1 rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29%**Dollar Weighted Returns**Internal Rate of Return (IRR) - the discount rate that results present value of the future cash flows being equal to the investment amount • Considers changes in investment • Initial Investment is an outflow • Ending value is considered as an inflow • Additional investment is a negative flow • Reduced investment is a positive flow**Dollar Weighted Average Using Text Example**Net CFs 1 2 3 4 $ (mil) - .1 - .5 .8 1.0 Solving for IRR 1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 + 1.0/(1+r)4 r = .0417 or 4.17%**Quoting Conventions**APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period)Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68%**Characteristics of Probability Distributions**1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2**Normal Distribution**s.d. s.d. r Symmetric distribution**Skewed Distribution: Large Negative Returns Possible**Median Negative Positive r**Skewed Distribution: Large Positive Returns Possible**Median Negative r Positive**S**E ( r ) = p ( s ) r ( s ) s Measuring Mean: Scenario or Subjective Returns Subjective returns p(s) = probability of a state r(s) = return if a state occurs 1 to s states**Numerical Example: Subjective or Scenario Distributions**StateProb. of State rin State 1 .1 -.05 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35 E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) = .15**S**2 Variance = p ( s ) [ r - E ( r )] s s Measuring Variance or Dispersion of Returns Subjective or Scenario Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095**Real vs. Nominal Rates**Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)**Annual Holding Period ReturnsFrom Figure 6.1 of Text**Geom Arith Stan. Series Mean% Mean% Dev.% Lg Stk 11.01 13.00 20.33 Sm Stk 12.46 18.77 39.95 LT Gov 5.26 5.54 7.99 T-Bills 3.75 3.80 3.31 Inflation 3.08 3.18 4.49**Annual Holding Period Risk Premiums and Real Returns**Risk Real Series Premiums% Returns% Lg Stk 9.2 9.82 Sm Stk 14.97 15.59 LT Gov 1.74 2.36 T-Bills --- 0.62 Inflation --- ---**Allocating Capital Between Risky & Risk-Free Assets**• Possible to split investment funds between safe and risky assets • Risk free asset: proxy; T-bills • Risky asset: stock (or a portfolio)**Allocating Capital Between Risky & Risk-Free Assets (cont.)**• Issues • Examine risk/ return tradeoff • Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets**rf = 7%**srf = 0% E(rp) = 15% sp = 22% y = % in p (1-y) = % in rf Example Using the Numbers in Chapter 6 (pp 171-173)**E(rc) = yE(rp) + (1 - y)rf**rc = complete or combined portfolio For example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13% Expected Returns for Combinations**Possible Combinations**E(r) E(rp) = 15% P rf = 7% F 0 s 22%**s**Since = 0, then rf = y c p Variance on the Possible Combined Portfolios s s**If y = .75, then**= .75(.22) = .165 or 16.5% c If y = 1 = 1(.22) = .22 or 22% c If y = 0 = 0(.22) = .00 or 0% c Combinations Without Leverage s s s**Using Leverage with Capital Allocation Line**Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage rc = (-.5) (.07) + (1.5) (.15) = .19 sc = (1.5) (.22) = .33**CAL**(Capital Allocation Line) E(r) P E(rp) = 15% E(rp) - rf = 8% ) S = 8/22 rf = 7% F s 0 P = 22%**Risk Aversion and Allocation**• Greater levels of risk aversion lead to larger proportions of the risk free rate • Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets • Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations**Quantifying Risk Aversion**E(rp) = Expected return on portfolio p rf = the risk free rate .005 = Scale factor A x sp = Proportional risk premium The larger A is, the larger will be the addedreturn required for risk**Quantifying Risk Aversion**Rearranging the equation and solving for A Many studies have concluded that investors’ average risk aversion is between 2 and 4**Chapter7**Efficient Diversification**n**S W = 1 i i =1 Two-Security Portfolio: Return rp = W1r1 +W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected return on Security 2**s12 = Variance of Security 1**s22 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk sp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2)**Covariance**Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of returns s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2**Correlation Coefficients: Possible Values**Range of values for r1,2 -1.0 <r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated**Three-Security Portfolio**rp = W1r1 +W2r2 + W3r3 s2p = W12s12 + W22s22 + W32s32 + 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3)**In General, For an n-Security Portfolio:**rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures)**Two-Security Portfolio**E(rp) = W1r1 +W2r2 sp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2) sp= [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2**E(r)**13% r = -1 r = .3 r = -1 8% r = 1 St. Dev 12% 20% TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS r = 0**Portfolio Risk/Return Two Securities: Correlation Effects**• Relationship depends on correlation coefficient • -1.0 <r< +1.0 • The smaller the correlation, the greater the risk reduction potential • If r = +1.0, no risk reduction is possible**s**Sec 1 E(r1) = .10 = .15 r = .2 12 s Sec 2 E(r2) = .14 = .20 2 Minimum Variance Combination 1 s 2 - Cov(r1r2) 2 = W1 s 2 s 2 - 2Cov(r1r2) + 2 1 = (1 - W1) W2**Minimum Variance Combination: r = .2**(.2)2 - (.2)(.15)(.2) = W1 (.15)2 + (.2)2 - 2(.2)(.15)(.2) W1 = .6733 W2 = (1 - .6733) = .3267**Minimum Variance: Return and Risk with r = .2**rp = .6733(.10) + .3267(.14) = .1131 s = [(.6733)2(.15)2 + (.3267)2(.2)2 + p 1/2 2(.6733)(.3267)(.2)(.15)(.2)] 1/2 = [.0171] = .1308 s p**Minimum Variance Combination: r = -.3**(.2)2 - (.2)(.15)(.2) = W1 (.15)2 + (.2)2 - 2(.2)(.15)(-.3) W1 = .6087 W2 = (1 - .6087) = .3913**Minimum Variance: Return and Risk with r = -.3**rp = .6087(.10) + .3913(.14) = .1157 s = [(.6087)2(.15)2 + (.3913)2(.2)2 + p 1/2 2(.6087)(.3913)(.2)(.15)(-.3)] 1/2 = [.0102] = .1009 s p**Extending Concepts to All Securities**• The optimal combinations result in lowest level of risk for a given return • The optimal trade-off is described as the efficient frontier • These portfolios are dominant**The minimum-variance frontier of risky assets**E(r) Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.**Extending to Include Riskless Asset**• The optimal combination becomes linear • A single combination of risky and riskless assets will dominate**ALTERNATIVE CALS**CAL (P) CAL (A) E(r) M M P P CAL (Global minimum variance) A A G F s P P&F A&F M**Dominant CAL with a Risk-Free Investment (F)**CAL(P) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of P & F dominate