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Chapter 6

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  1. Chapter6 Risk and Return: Past and Prologue

  2. Rates of Return: Single Period HPR = Holding Period Return P1 = Ending price P0 = Beginning price D1 = Dividend during period one

  3. Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

  4. 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

  5. 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%

  6. 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

  7. 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%

  8. 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%

  9. 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

  10. Normal Distribution s.d. s.d. r Symmetric distribution

  11. Skewed Distribution: Large Negative Returns Possible Median Negative Positive r

  12. Skewed Distribution: Large Positive Returns Possible Median Negative r Positive

  13. 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

  14. 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

  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

  16. 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)

  17. 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

  18. 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 --- ---

  19. 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)

  20. 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

  21. 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)

  22. 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

  23. Possible Combinations E(r) E(rp) = 15% P rf = 7% F 0 s 22%

  24. s Since = 0, then rf = y c p Variance on the Possible Combined Portfolios s s

  25. 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

  26. 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

  27. CAL (Capital Allocation Line) E(r) P E(rp) = 15% E(rp) - rf = 8% ) S = 8/22 rf = 7% F s 0 P = 22%

  28. 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

  29. 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

  30. Quantifying Risk Aversion Rearranging the equation and solving for A Many studies have concluded that investors’ average risk aversion is between 2 and 4

  31. Chapter7 Efficient Diversification

  32. 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

  33. 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)

  34. 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

  35. 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

  36. Three-Security Portfolio rp = W1r1 +W2r2 + W3r3 s2p = W12s12 + W22s22 + W32s32 + 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3)

  37. In General, For an n-Security Portfolio: rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures)

  38. Two-Security Portfolio E(rp) = W1r1 +W2r2 sp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2) sp= [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2

  39. E(r) 13% r = -1 r = .3 r = -1 8% r = 1 St. Dev 12% 20% TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS r = 0

  40. 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

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. The minimum-variance frontier of risky assets E(r) Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.

  48. Extending to Include Riskless Asset • The optimal combination becomes linear • A single combination of risky and riskless assets will dominate

  49. 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

  50. 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