Risk and Return. Calculating Return - Single period. Calculating Return - Multi periods. Uncertainty of Investment.

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1 Chater 10, 11 Risk and Return Chater 13 Cost of Caital Konan Chan, 018 Risk and Return Return measures Exected return and risk? Portfolio risk and diversification CPM (Caital sset Pricing Model) eta Calculating Return - Single eriod Holding eriod return (HPR) Ending rice - eginning rice Dividend HPR eginning rice This assumes we only have one investment eriod. What about multile eriods? Calculating Return - Multi eriods rithmetic average rithmetic mean of returns Good measure for future erformance Geometric average Geometric mean of returns The return measure that gives the same cumulative erformance as actual returns (buyand-hold) Required for mutual fund erformance Konan Chan 3 Konan Chan 4 Returns -Examle Year End Year-end Holding Period Return Price Div (P t P t-1 + D t ) / P t ( ) / 100 = ( ) / 110 = ( ) / 90 = rithmetic verage: ( ) / 3 = 3.15% Geometric verage: (1 + R G ) 3 = ( )*( )*( ) R G = [( )*( )*( )] 1/3 1=.33% R G R, R G is a better measure for ast erformance Uncertainty of Investment Return and risk tradeoff (every investment has its uncertainty) t the time when we measure the exected level of returns, we need to quantify the uncertainty (risk) How to estimate the exected return and risk? based on robability distribution based on historical data Konan Chan 5 Konan Chan 6

The Normal Distribution Exected Return & Risk Exected Return (Mean) Find out ossible future states Estimate robability and outcome for each state Sum of all ossible outcomes by multilying

2 The Normal Distribution Exected Return & Risk Exected Return (Mean) Find out ossible future states Estimate robability and outcome for each state Sum of all ossible outcomes by multilying robabilities Er () s () rs () Risk (Variance or Standard deviation) The degree of various outcomes, or deviation from mean Standard deviation is the square root of variance s Var() r () s r() s E() r s Konan Chan 7 Konan Chan 8 Exected Return & Risk - Examle Initial investment : $100 State of Economy Probability End Price HPR oom % Normal % Recession % Exected return = 0.3(0.5)+0.5(0.)+0.(-0.4)=17% Variance = 0.3( ) +0.5( ) +0.( ) = Standard deviation = (0.0981) 0.5 = Return & Risk - Historical data Treat each historical outcome equally and assign a robability of 1/n ( n is number of observations) Return 1 Use samle average r rt n Risk Use samle variance 1 ˆ ( rt r ) n 1 Konan Chan 9 Konan Chan 10 Return & Risk -Historical data (examle) Using Excel functions C D E F G H I J K K M 1 Month Return 6.0% 3.0% -.0%.0%.5% -1.5% 0.5% 0.%.3% -5.5% 1.0% 1.3% Return and Risk Two ssets State of Probability Stock ond economy oom 5% 80% 5% Mean Variance Std.Dev. =VERGE(:M) =VR(:M) =STDEV(:M) 0.8% 0.08%.89% Normal 60% 30% 10% Recession 15% -30% 15% Konan Chan 11 Konan Chan 1

3 Return and Risk - Examle r S = 0.5* * *(-0.3) = r = 0.5* * *0.15 = Standard deviation S.5*(.8.335).6*(.3.335).15*(.3.335) 34%.5*( ).6*(.1.095).15*( ) 3.1% Portfolio Risk and Return What is the exected return of a ortfolio consisting of 60% stock and 40% bond? Given r S =33.5% and r =9.5% r P = 0.6*33.5%+0.4*9.5%=3.9% How about ortfolio risk? It s not a weighted average of standard deviations Konan Chan 13 Konan Chan 14 Portfolio Risk - Examle State of Prob. Portfolio (60% S+40% ) economy oom 5% 50% Normal 60% % Recession 15% -1% Exected return=3.9% Standard deviation=19.1% Portfolio Risk We need to account for covariance Variance for a two-asset ortfolio : w w w w Cov( r ),r w w ww P.5*(.5.39).6*(..39).15*(.1.39) 19.1% Konan Chan 15 Konan Chan 16 Covariance and Correlation What is covariance? Measures how closely two variables move together Cov( r, r ) [ r ( s) E( r )] [ r ( s) E( r )] Correlation coefficient s standardize covariance by dividing standard deviations of individual returns Cov( r, r ), is between +1 and means erfect ositive correlation and -1 means erfect negative correlation Covariance Covariance and Coefficient (Examle).5*(.8.335).6*(.3.335).15*(.3.335) 34%.5*( ).6*(.1.095).15*( ) 3.1% Cov (r S, r ) = 0.5( ) ( )+0.6 ( ) ( )+0.15 ( ) ( ) = % Coefficient (r S, r )=-1.58%/[(34%)* (3.1%)] = Konan Chan 17 Konan Chan 18 S

4 Cov. and Coef. - historical data Investment Oortunity Set Emirically, we estimate covariance & correlation by using historical time series data 1 r 1 Cov 1 1 N ( 1 1t t1 Cov1 ˆ ˆ r r )( r 1 t r ) /( N 1) Return S.D Weight Corr.coef Weight Weight Risk 40% 35% 31% 7% 3% 1% 0% 1% 3% 6% 30% Return 0% 19% 18% 18% 17% 16% 15% 14% 14% 13% 1% 0% 10% 15% 0% 5% 30% 35% 40% 45% Risk (Standard Deviation) Konan Chan 19 Konan Chan 0 Return 15% Minimum variance ortfolio, Z Mean-Variance nalysis Diversification Effect What will haen if 0.3 0% = -1 w w ww = 1.0 : = -1.0 : w w w w Return 15% = -0.3 = 1 = 0 10% 0% 5% 10% 15% 0% 5% 30% 35% 40% 45% Risk (standard deviation) s long as < 1, the standard deviation of a ortfolio of two asset is less than the weighted average of the standard deviations of the individual assets Konan Chan 1 Konan Chan Return Efficient Frontier Minimum variance ortfolio Efficient Frontier Z x x x x x x x Efficient ortfolio is the ortfolio with the highest return for a given amount of risk. the lowest risk for a given amount of return Efficient Frontier with Three Stocks Versus Ten Stocks Risk, return combination of a ortfolio or a single stock Risk Konan Chan 3 Cororate Finance Konan Chan 4

5 Exected Return Otimal Risky Portfolio Risk-free Rate Caital llocation Line M Efficient Frontier Otimal Risky Portfolio M Caital llocation Line Standard Deviation Otimal Portfolio Selection Otimal Portfolio Selection requires 3 stes: Construct efficient frontier Pick otimal risky ortfolio by Caital llocation Line with risk-free asset Choose aroriate weights for otimal risky ortfolio and risk-free asset (deend on risk aversion of investors) Searation roerty : ste and 3 are indeendent ll rational risk-averse investors will assively index holdings to an equity fund (ortfolio M ) and a money market fund Konan Chan 5 Konan Chan 6 Terminology of Return and Risk Risk-free rate The rate of return that can be earned with certainty Risk remium Difference between return and risk-free asset return Risk aversion The degree to which an investor is unwilling to accet risk Risk-Free sset Only the government can issue default-free bonds. T-bills viewed as the risk-free asset Money market funds also considered risk-free in ractice Konan Chan 7 Konan Chan 8 sset llocation (continued) Caital llocation Line (CL) varying the weights between a risk-free asset and a risky ortfolio gives us all ortfolio combinations, which fall on a single line The sloe of the CL is the Reward-to- Variability Ratio, or the Share ratio E( r ) r S f Risk version and llocation ssume investors are risk averse, they invest a risky security if it rovides risk remium. Greater (lower) levels of risk aversion lead investors to choose larger (smaller) roortions of the riskfree rate If the reward-to-variability ratio increase, then investors might well decide to take on riskier ositions. Konan Chan 9 Konan Chan 30

Caital Market Line Caital allocation line formed from 1-month T-bills and a broad index of common

Historical Evidence on CML From 196 to 009, the assive risky ortfolio offered an average risk

Konan Chan 31 Konan Chan 3 verage Market Risk Premiums Historical Returns, 196-014 Risk-return

1% Small Comany Stocks 16.7 3.1 Long-Term Cororate onds 6.4 8.4 Long-Term Government onds 6.1 10.

1 90% 0% + 90% Konan Chan 33 Konan Chan 34 CPM Caital sset Pricing Model (CPM) Theory of

6 Caital Market Line Caital allocation line formed from 1-month T-bills and a broad index of common stocks (e.g. the S&P 500). Historical Evidence on CML From 196 to 009, the assive risky ortfolio offered an average risk remium of 7.9% with a standard deviation of 0.8%, resulting in a rewardto-volatility ratio of.38. Konan Chan 31 Konan Chan 3 verage Market Risk Premiums Historical Returns, Risk-return trade-off verage Standard Series nnual Return Deviation Distribution Large Comany Stocks 1.1% 0.1% Small Comany Stocks Long-Term Cororate onds Long-Term Government onds U.S. Treasury ills Inflation % 0% + 90% Konan Chan 33 Konan Chan 34 CPM Caital sset Pricing Model (CPM) Theory of relationshi between risk and return Exected (required) return = risk-free rate + beta * market risk remium Market risk remium = r m r f Risk free rate = r f eta = (measure of market risk) r i = r f + (r m r f ) Konan Chan 35 Konan Chan 36

Risk and Diversification Risk and Diversification Portfolio standard deviation 0 Unique risk Market comensates investors for taking risk Only market risks are comensated Unique risk should be

factors affecting only that firm Market Risk (systematic risk) Economy-wide sources of risk that affect the overall stock market Measured by eta Konan Chan 37 Konan Chan 38 eta Stock etas Sensitivity

7 Risk and Diversification Risk and Diversification Portfolio standard deviation 0 Unique risk Market comensates investors for taking risk Only market risks are comensated Unique risk should be diversified away Market risk Number of Securities Diversification Strategy designed to reduce risk by sreading the ortfolio across many investments Unique Risk (diversifiable risk) Risk factors affecting only that firm Market Risk (systematic risk) Economy-wide sources of risk that affect the overall stock market Measured by eta Konan Chan 37 Konan Chan 38 eta Stock etas Sensitivity of stock s return to the market return How stock s return changes with market return changes Proxy for market risk β = 1.0: same risk as the market (average stock) β < 1.0: less risky than market (defensive stock) β > 1.0: riskier than market (aggressive stock) Konan Chan 39 Konan Chan 40 Market Equilibrium In equilibrium, all assets and ortfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market E( R ) R E( R ) R i f M f E( RM ) Rf i M Security Characteristic Line eta is the sloe of 0.60 the regression line R i = +β R M, 0.50 regressing a stock s 0.40 return (R i ) on the market return (R M ) 0.30 Security's (Excess) Return = Intercet = Sloe Market (Excess) Return R i = i + i R M + e i Konan Chan 41 Konan Chan 4

CPM and Valuation Return = [dividend + caital gain]/rice = dividend yield + % caital gain In equilibrium, the exected return defined above should equal CPM return.

8 CPM and Valuation Return = [dividend + caital gain]/rice = dividend yield + % caital gain In equilibrium, the exected return defined above should equal CPM return. Exected return = exected dividend yield + caital gain yield = CPM exected return CPM & Valuation - Examle Your stockbroker calls you to buy Fearfree Inc. The stock is currently selling for $15 a share The risk free rate is 5%, and you demand a 17% return on the market. Fearfree's current dividend is $4 a share Some analyst has estimated that Fearless's beta is.0 and that the stock's dividend will grow at a constant 8% Is recommendation to buy Fearfree a good one? What do you think the stock is worth? Konan Chan 43 Konan Chan 44 CPM & Valuation - Examle Factor Models D 0 =4, g=8%, r m =17%, r f =5%,β= From CPM, r = 5%+*(17%-5%) = 9% P 0 = $4*(1 + 8%) / (9% - 8%)= $0.57 Intrinsic value $0.57 > market value $15 Price may areciate by 5.57 later! Single factor model R i = i + i R M + e i Usually, use market index as the single factor i is factor loading (sensitivity) Multifactor model R i = i + 1i R 1f + i R f + + e i Use different factors, such as GNP, inflation, Fama-French three-factor model R it = a i + b i R Mt + s i SM t + h i HML t + e i Market, size factor, book-to-market factor Four factor model (add momentum) Konan Chan 45 Konan Chan 46 Exected Return by Factor Model Cost of Caital WCC Required rate of return Konan Chan 47

9 Cost of Caital Cost of getting caital Think about a comany as a ortfolio comosed by all its debt and equity securities Return on comany = return on ortfolio Investors required return on comany = investors required return on ortfolio = comany cost of caital WCC Weighted verage Cost of Caital (WCC) The exected return on a ortfolio of all securities within a firm Comany cost of caital Konan Chan 49 Konan Chan 50 WCC Three stes to calculate cost of caital Calculate the value of each security as a roortion of the firm s market value Determine the required rate of return on each security Calculate a weighted average of these required returns D P E WCC= *(1 TC ) rdebt + * rreferred + * requity V V V Konan Chan 51 fter-tax Cost of Debt Consider two firms with 35% tax rate: has no debt, has $1000 debt with 10% interest rate Comany EIT Interest Exense Pre-tax income Taxes (35%) Net Income saves 35 of taxes due to debt, making the after-tax interest rate 65/1000 or 6.5% = 10%(1-0.35) Konan Chan 5 Measure Caital Structure In estimating WCC, Use market values of the securities, not book values, to comute weights Cost of caital must be based on what investors are actually willing to ay for the comany s securities ook values are often not equal to true market value of securities Measure Market Value Market value of onds Price (or PV) er bond times number of bonds Market value of Equity Market rice er share times number of shares Konan Chan 53 Konan Chan 54

10 Required Return of Debt Required return on debt (cost of debt) (1-T) * r d = (1-T) * YTM Recall yield to maturity (YTM) is total annual exected return if you buy the bond today and hold u to maturity Required Return of Equity Required return on equity (cost of equity), CPM, r e = r f + (r m r f ) Constant growth DDM, Required return on referred stock D1 D1 P0 rreferred r P referred D1 D1 P0 re g r g P e 0 0 Konan Chan 55 Konan Chan 56 How to Use WCC? comany s WCC is for average risk rojects, i.e., for rojects that are in the firms existing business The return that investors could exect to earn if they invested in securities with comarable degrees of risk. WCC djustment? Most financial managers adjust WCC uward for riskier than average rojects and downward for safer than average rojects Comanies with diverse divisions might use industry average WCCs, which is obtained from other comanies in the same line of business, as their individual cororate divisions Konan Chan 57 Konan Chan 58 Why Care about WCC? What is the goal (objective) for a cororate manager? Shareholders desire wealth maximization Max market rice of stock n equivalent way of firm value maximization is to minimize WCC Konan Chan 59

Chapter 13 Return, Risk, and Security Market Line

1 Chapter 13 Return, Risk, and Security Market Line Konan Chan Financial Management, Spring 2018 Topics Covered Expected Return and Variance Portfolio Risk and Return Risk & Diversification Systematic