Chapter 5 Risk and Return: Past and Prologue Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

5.1 Rates of Return 5-2

Holding-period return (HPR) The holding-period return (HPR) is the total return received from holding an asset or portfolio of assets over a period of time, generally expressed as a percentage.

Holding-period return (HPR) Holding period return (HPR): HPR= where P S = Sale price/Ending price (or P 1 ) P B = Buy price/Beginning price (or P 0 ) CF= Cash flow during holding period CF consideration!! The rate of return: is annualizing the HPR: convert a rate of n into a yearly rate. 5-4 [P S - P B + CF] / P B

Example 1 Suppose you buy one share of a stock today for $45 and you hold it for one year and sell it for $52.You also received $8 in dividends at the end year. HPR=( )/45=33.33% 5-5

Measuring Investment Returns 1-Arithmetic Average Or simple average: the sum of returns in each period divided by the number of periods. AAR or Arithmetic Average Return: n = number of time periods 5-6

Example 2 AAR = 17.51% 5-7

Measuring Investment Returns 2-Geometric average or GAR(time-weighted average return): 15.61% GAR = 15.61% 5-8

Measuring Investment Returns Q: When should you use the GAR and when should you use the AAR? A1:When you are evaluating PAST RESULTS (ex-post): Use the AAR (average without compounding) if you ARE NOT reinvesting any cash flows received before the end of the period. Use the GAR (average with compounding) if you ARE reinvesting any cash flows received before the end of the period. A2: When you are trying to estimate an expected return (ex-ante return):use the AAR A3: if you calculate for long period, you must used GAR but if the investment period is between 1 year to 2 years, the simple average is enough

Measuring Investment Returns 3-Dollar-weighted return procedure (DWR)=IRR of a project Find the internal rate of return IRR for the cash flows (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.) Break even point= cash outflow+ cash inflow 5-10

Tips on Calculating Dollar Weighted Returns This measure of return considers both changes in investment and security performance Initial Investment is an _______ Ending value is considered as an ______ Additional investment is an _______ Security sales are an ______ outflow inflow outflow inflow 5-11

Measuring Ex-Post (Past) Returns i. Dollar-weighted return procedure (DWR): Find the internal rate of return for the cash flows (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.) NPV = Solve for IRR: IRR = $0 = -$50/(1+IRR) 0 - $51/(1+IRR) 1 + $112/(1+IRR) % average annual dollar weighted return The DWR gives you an average return based on the stock’s performance and the dollar amount invested (number of shares bought and sold) each period. 5-12

The Interest Rates: APR, EAR And APY Some can over report the return, to avoid this technique, apply the three interest measurements: 1-APR(annual percentage rates: simple) 2-EAR (effective annual rate: compound) 3-APY=(annual percentage yield)=EAR

5.2 Risk and Risk Premiums 5-14

Scenario or Subjective Returns Scenario, Scenario analysis, Probability distribution 1-Expected return(mean return) E(r) = Expected Return p(s) = is the probability for the return r in scenario r(s) = return in scenario E(r) = p(s) r(s)  s

Subjective or Scenario Variance surprise return: it’s the different between the possible outcome and the expected return 2-variance: The expected value of the squared deviation from the mean 3-standard deviation: defined as the square root of the variance.

Numerical Example: Subjective or Scenario Distributions StateProb. of State Return E(r) = (.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%  2 = [(.2)( ) 2 + (.5)( ) 2 + (.3)( ) 2 ]  2 = % 2  = [ ] 1/2 =.07 or 7% 5-17

Risk Premium & Risk Aversion The risk free rate is the rate of return that can be earned with certainty. Is the assumption of (zero risk) true??? The risk premium is the difference between the expected return of a risky asset and the risk-free rate. 4- Excess Return or Risk Premium asset = 5-Risk aversion is an investor’s unwillingness to accept risk. How is the aversion to accept risk overcome? By offering investors a higher risk premium E[r asset ] – rf

5.4 Inflation and Real Rates of Return 5-19

Inflation, Taxes and Returns The average inflation rate from 1966 to 2005 was 4.29% Taxes are paid on nominal investment income. This reduces real investment income even further. You earn a 6% nominal, pre-tax rate of return and you are in a 15% tax bracket and face a 4.29% inflation rate. What is your real after tax rate of return? r real  [6% x ( )] – 4.29%  0.81%

Real vs. Nominal Rates Fisher effect: Approximation real rate  nominal rate - inflation rate r real  r nom - i Exampler nom = 9%, i = 6% r real  3% Fisher effect: Exact r real = or r real = The exact real rate is less than the approximate real rate. [(1 + r nom ) / (1 + i)] – 1 (r nom - i) / (1 + i) (9% - 6%) / (1.06) = 2.83% r real = real interest rate r nom = nominal interest rate i = expected inflation rate 5-21

5.5Asset Allocation Across Risky and Risk Free Portfolios 5-22

Allocating Capital Between Risky & Risk-Free Assets Possible to split investment funds between safe and risky assets Risk free asset rf : proxy; ________________________ Risky asset or portfolio r p : _______________________ Example. Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows: –Stock A you put ______ –Stock B you put ______ –Stock C you put ______ $2,500 $3,000 $2,000 T-bills or money market fund risky portfolio $7,

Weights in rp –W A = –W B = –W C = $2,500 / $7,500 =33.33% $3,000 / $7,500 =40.00% $2,000 / $7,500 =26.67% % W rf = ; W rp = In the complete portfolio W A = 0.75 x 33.33% = 25%; W B = 0.75 x 40.00% = 30% W C = 0.75 x 26.67% = 20%; 25% 75% W rf = 25% Allocating Capital Between Risky & Risk-Free Assets 5-24

the probability distribution of the rate of return on the risky portfolio VS the probability distribution of the rate of return on the complete portfolio of both risky and risk-free assets. Issues in setting weights –Examine risk &return tradeoff –Demonstrate how different degrees of risk aversion will affect allocation between risky and risk free assets Allocating Capital Between Risky & Risk-Free Assets 5-25

r f = 5%  rf = 0% E(r p ) = 14%  rp = 22% y = % in r p (1-y) = % in rf Example 5-26

E(r C ) = Expected Returns for Combinations E(r C ) = For example, let y = ____ E(r C ) = E(r C ) =.1175 or 11.75%  C = y  rp + (1-y)  rf  C = (0.75 x 0.22) + (0.25 x 0) = or 16.5%  c = yE(r p ) + (1 - y)r f y  rp + (1-y)  rf Return for complete or combined portfolio 0.75 (.75 x.14) + (.25 x.05) 5-27

Complete portfolio Varying y results in E[r C ] and  C that are ______ ___________ of E[rp] and rf and  rp and  rf respectively. E(r c ) = yE(r p ) + (1 - y)rf  c = y  rp + (1-y)  rf linear combinations This is NOT generally the case for the  of combinations of two or more risky assets. 5-28

E(r) E(r p ) = 14% r f = 5% 22% 0 P F Possible Combinations  E(r p ) = 11.75% 16.5% y =.75 y = 1 y =

Combinations Without Leverage Since σ rf = 0 σ c = y σ p If y =.75, then σ c = If y = 1 σ c = If y = 0 σ c = 75(.22) = 16.5% 1(.22) = 22% 0(.22) = 0% E(r c ) = yE(r p ) + (1 - y)rf y =.75 E(r c ) = y = 1 E(r c ) = y = 0 E(r c ) = (.75)(.14) + (.25)(.05) = 11.75% (1)(.14) + (0)(.05) = 14.00% (0)(.14) + (1)(.05) = 5.00% 5-30

Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage E(r c ) =  c = (1.5) (.14) + (-.5) (.05) = = 18.5% (1.5) (.22) = 0.33 or 33% E(r C ) =18.5% 33% y = 1.5 y =

Risk Aversion and Allocation Greater levels of risk aversion lead investors to choose larger proportions of the risk free rate Lower levels of risk aversion lead investors to choose 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 y = 0 y =

E(r)E(r) E(r p ) = 14% r f = 5% = 22% = 22% 0 P FF  rp ) Slope = 9/22 ) Slope = 9/22 E(r p ) - r f = 9% CAL(CapitalAllocationLine)  5-33

CML Plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky portfolio. The slope, S, of the CAL equals the increase in expected return that an investor can obtain per unit of additional standard deviation. the reward-to-volatility ratio is the same for all complete portfolios that plot on the capital allocation line. While the risk-return combinations differ, the ratio of reward to risk is constant. S=risk premium/standard deviation

5.6 Passive Strategies and the Capital Market Line 5-35

A Passive Strategy The investor makes no attempt to actively find undervalued strategies nor paying the costs involved in undertaking security analysis. To avoid this, they used a natural strategy and select a diversified portfolio of common stocks that mirrors the corporate sector of the broad economy. Such strategies are called indexing Investing in a broad stock index and a risk free investment is very popular strategy for passive investor. the capital allocation line provided by one-month T-bills and a broad index of common stocks that employs the market (or an index that mimics overall market performance) called the capital market line (CML) 5-36

Active versus Passive Strategies Active strategies : stock picking Costly trading Management fees. Passive involves investment in two passive portfolios –Short-term T-bills –Fund of common stocks that mimics a broad market index –Vary combinations according to investor’s risk aversion. 5-37

Selected Problems 5-38

Problem 2 a. The holding period returns for the three scenarios are: Boom: Normal: Recession: E(HPR) =  2 (HPR) (50 – )/40 = 0.30 = 30.00% (43 – )/40 = 0.10 = 10.00% (34 – )/40 = – = –13.75% [(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (–13.75%)] = 8.75%

Problem 2 Cont. b. E(r) =  = (0.5 x 8.75%) + (0.5 x 4%) = 6.375% 0.5 x 17.88% = 8.94% Risky E[r p ] = 8.75% Risky  p = 17.88% 5-40

Problem 5 (95 – )/90 = 10.00% (90 – )/110 = –14.55% (110 – )/100 = 14.00% Return = [(capital gains + dividend) / price] a. TWR Year a. TWR 2.33% 3.15% % 5-41