Bài giảng Essentials of Investments - Chapter 5 Risk and Return: Past and Prologue

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus1 Chapter 5 Risk and Return: Past and Prologue Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus2 Rates of Return: Single Period HPR P P D P    1 0 1 0 HPR = Holding Period Return P1 = Ending price P0 = Beginning price D1 = Dividend during period one Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus3 Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus4 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus5 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% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus6 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus7 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% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus8 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% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus9 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus10 r Symmetric distribution Normal Distribution s.d. s.d. Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus11 rNegative Positive Skewed Distribution: Large Negative Returns Possible Median Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus12 rNegative Positive Skewed Distribution: Large Positive Returns Possible Median Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus13 Subjective returns p(s) = probability of a state r(s) = return if a state occurs 1 to s states Measuring Mean: Scenario or Subjective Returns E(r) = p(s) r(s)S s Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus14 Numerical Example: Subjective or Scenario Distributions State Prob. 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus15 Standard deviation = [variance]1/2 Measuring Variance or Dispersion of Returns Subjective or Scenario Variance = S s p(s) [rs - E(r)] 2 Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095 Using Our Example: Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus16 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) Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus17 Annual Holding Period Returns From 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus18 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 --- --- Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus19 • 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus20 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus21 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) Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus22 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus23 E(r) E(rp) = 15% rf = 7% 22%0 P F Possible Combinations s Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus24 pc = Since rf y Variance on the Possible Combined Portfolios = 0, thens ss Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus25 c = .75(.22) = .165 or 16.5% If y = .75, then c = 1(.22) = .22 or 22% If y = 1 c = 0(.22) = .00 or 0% If y = 0 Combinations Without Leverage s s s Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus26 Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock (while not really possible, lets assume we can do it) Using 50% Leverage rc = (-.5) (.07) + (1.5) (.15) = .19 sc = (1.5) (.22) = .33 Note that we assume the T-bill is totally risk free (bear with me again) Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus27 E(r) E(rp) = 15% rf = 7% = 22%0 P F P ) S = 8/22 E(rp) - rf = 8% CAL: (Capital Allocation Line) s Capital Allocation Line Slope: Reward to variability ratio: ratio of risk premium to std. dev. Risk premium This graph is the risk return combination available by choosing different values of y. Note we have E(r) and variance on the axis. Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus28 Problem 9: Portfolio Return Stock price and dividend history Year Beginning stock price Dividend Yield 2001 $100 $4 2002 110 $4 2003 90 $4 2004 95 $4 An investor buys three shares at the beginning of 2001, buys another 2 at the beginning of 2002, sells 1 share at the beginning of 2003, and sells all 4 remaining at the beginning of 2004. A. What are the arithmetic and geometric average time-weighted rates of return? B. What is the dollar weighted rate of return? Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus29 Answer • Time weighted return – 2001 (110-100+4)/100 = 14% – 2002 (90-110+4)/110 = - 14.6% – 2003 (95-90+4)/90 = 10% • Arithmetic mean return (14-14.6+10)/3 = 3.13% • Geometric mean return (1+.14)*(1-.146)*(1+.1)]1/3 = 1.078.33 –1 = 2.3% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus30 Problem 11: Risk Premiums • Using the historical risk premiums as your guide from the chart earlier, what is your estimate of the expected annual HPR on the S&P500 stock portfolio if the current risk-free interest rate is 5.0%. What does the risk premium represent? Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus31 Answer For the period of 1926- 2004 the large cap stocks returned 10.0%, less t- bills of 3.7% gives a risk premium of 6.3%. – If the current risk free rate is 5.0%, then – E(r) = Risk free rate + risk premium – E(r) = 5.0% + 6.3% = 11.3% – The risk premium represents the additional return that is required to compensate you for the additional risk you are taking on to invest in this asset class. Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus32 Problem 12: Client Portfolios • You manage a risky portfolio with an expected return of 12% and a standard deviation of 25%. The T-bill rate is 4%. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected return and standard deviation of your client’s portfolio? – Clients Fund E(r) (expected return) =.7 x 12% + .3 x 4% = 9.6% σ (standard deviation) = .7 x .25 = 17.5% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus33 Problem 13: Portfolio Allocations • Suppose your risky portfolio includes investments in the following proportions. What are the investment proportions in your clients portfolio Stock A 27% Stock B 33% Stock C 40% • Investment proportions: T-bills = 30% Stock A = .7 x 27% = 18.9% Stock B = .7 x 33% = 23.1% Stock A = .7 x 40% = 28.0% Check: 30 + 18.9 + 23.1 + 28 = 100% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus34 Problem 14: Reward to Variability C. What is the reward-to-variability ratio (s) of your risky portfolio and your clients portfolio? – Reward to Variability (risk premium / standard deviation) – Fund = (12.0% – 4%) / 25 = .32 – Client = (9.6% – 4%) / 17.5 = .32 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus35 Problem 15: The CAL Line D. Draw the CAL of your portfolio. What is the slope of the CAL? Slope of the CAL line % Slope = .3704 17 P 14 Client Standard Deviation 18.9 27 7 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus36 Problem 16: Maximizing Standard Deviation Suppose the client in Problem 12 prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%. What is the investment proportion? What is the expected return on the portfolio? Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus37 Answer Portfolio standard deviation 20% = (y) x 25% Y = 20/25 = 80.0% Mean return = (.80 x 12%) + (.20 x 4%) = 10.4% Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus38 Problem 17: Increasing Stock Volatility • What do you think would happen to the expected return on stocks if investors perceived an increased volatility of stocks due to some recent event, i.e. Hurricane Katrina? Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus39 Answer • Assuming no change in risk aversion, investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before. If we assume the risk-free rate is unchanged, the increase in the risk premium would require a higher expected rate of return in the equity market. Review of Objectives • A. Do you understand rates of return? • B. Do you know how to calculate return using scenario, probabilities, and other key statistics used to describe your portfolio? • C. Do you understand the implications of using a risky and a risk- free asset in a portfolio?