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Understanding Investment Risk and Return Relationships

Learn about calculating rates of return, dollar-weighted returns, measuring mean, variance, and skewness, and the Fisher effect. Understand capital allocation between risky and risk-free assets.

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Understanding Investment Risk and Return Relationships

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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

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