Slide 1

Return, Risk, and the Security Market Line

Chapter 13Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin13-#1Chapter OutlineExpected Returns and VariancesPortfoliosAnnouncements, Surprises, and Expected ReturnsRisk: Systematic and UnsystematicDiversification and Portfolio RiskSystematic Risk and BetaThe Security Market LineThe SML and the Cost of Capital: A Preview13-#Expected ReturnsExpected returns are based on the probabilities of possible outcomes

In this context, expected means average if the process is repeated many times

The expected return does not even have to be a possible return

13-#3Example: Expected ReturnsSuppose you have predicted the following returns for stocks C and T in three possible states of the economy.

1. What is the probability of Recession?

StateProbabilityC TBoom0.315%25%Normal0.510% 20%Recession??? 2% 1%

Probabilities add up to 100% (or 1.0) thus 1.0 0.3 0.5 = 0.2 or 20%13-#Example: Expected ReturnsSuppose you have predicted the following returns for stocks C and T in three possible states of the economy.

2. What are the expected returns?

StateProbabilityC TBoom0.315%25%Normal0.510% 20%Recession0.2 2% 1%

RC = .3(15%) + .5(10%) + .2(2%) = 9.9%RT = .3(25%) + .5(20%) + .2(1%) = 17.7%13-#Example: Expected Returns The three states of the economy still apply to stocks C and T.

3. If the risk-free rate is 4.15%, what is the risk premium for C & T?RC = .3(15%) + .5(10%) + .2(2%) = 9.9%RT = .3(25%) + .5(20%) + .2(1%) = 17.7%

Stock Cs risk premium: 9.9 - 4.15 = 5.75%Stock Ts risk premium: 17.7 - 4.15 = 13.55%13-#Variance and Standard DeviationVariance and standard deviation measure the volatility of returns

Using unequal probabilities for the entire range of possible outcomes

Weighted average of squared deviations

13-#7Example: Variance and Standard DeviationStateProbabilityC TBoom0.315%25%Normal0.510% 20%Recession0.2 2% 1%Expected return 9.9%17.7%Considering the previous example of stocks C and T:What are the variance and St. Dev of Stock C?2 = .3(15%-9.9%)2 + .5(10%-9.9%)2 + .2(2%-9.9%)2 = 0.2029% = 4.50%13-#Example: Variance and Standard DeviationWhat is the variance and standard deviation for T?Stock T2 = .3(25%-17.7%)2 + .5(20%-17.7%)2 + .2(1%-17.7%)2 = 0.7441% = 8.63%StateProbabilityC TBoom0.315%25%Normal0.510% 20%Recession0.2 2% 1%Expected return 9.9%17.7%13-#9Another ExampleConsider the following information:StateProbabilityABC, Inc. (%)Boom.2515%Normal.50 8%Slowdown .15 4%Recession.10- 3%

What is the expected return?E(R) = .25(15%) + .5(8%) + .15(4%) + .1(-3%) = 8.05%13-#Another ExampleConsider the following information:StateProbabilityABC, Inc. (%)Boom.2515%Normal.50 8%Slowdown .15 4%Recession.10- 3%

What is the variance?Variance = 2= .25(15%-8.05%)2 + .5(8%-8.05%)2 + .15(4%-8.05%)2 + .1(-3%-8.05%)2 = 0.267475Standard Deviation = = 26.7475= 5.17%13-#PortfoliosA portfolio is a collection of assets

An assets risk and return are important in how they affect the risk and return of the portfolio

13-#12PortfoliosThe risk/return trade-off for a portfolio is measured by the portfolios expected return and standard deviation, just as with individual assets

RiskReturn13-#13Example: Portfolio WeightsSuppose you have $15,000 to invest and you have purchased securities in the following amounts:$2000 of DCLK$3000 of KO$4000 of INTC$6000 of KEI13-#14Example: Portfolio WeightsWhat are your portfolio weights in each security?$2,000 of DCLK$3,000 of KO$4,000 of INTC$6,000 of KEI$15,000

DCLK: 2/15 = .133KO:3/15 = .200INTC:4/15 = .267KEI:6/15 = .400 15/15 = 1.00013-#15Portfolio Expected ReturnsThe expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio

You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

13-#Example: Expected Portfolio ReturnsConsider the portfolio weights computed previously. The individual stocks have the following expected returns:

DCLK:19.69%KO: 5.25%INTC:16.65%KEI: 18.24%13-#Example: Expected Portfolio Returns1. What is the expected return on this portfolio?

ReturnWeightDCLK:19.69%.133KO: 5.25%.200INTC:16.65%.267KEI: 18.24%.400E(RP) = .133(19.69%) + .2(5.25%) + .267(16.65%) + .4(18.24%) = 15.41%13-#Portfolio VarianceCompute the expected portfolio return, the variance, and the standard deviation using the same formula as for an individual asset

Compute the portfolio return for each state:RP = w1R1 + w2R2 + + wmRm

13-#19Example: Portfolio VarianceConsider the following information:

StateProbabilityA B Boom.430%-5% Bust.6-10%25%13-#Example: Portfolio VarianceConsider the following information:StateProb.A B Boom.430%-5% Bust.6-10%25%What is the expected return for asset A? VarA, St. DevA?E(RA) = .4(30%) + .6(-10%) = 6%Variance(A) = .4(30% - 6%)2 + .6(-10% - 6%)2 = 3.84%Std. Dev.(A) = 19.6%13-#21Example: Portfolio VarianceConsider the following information:StateProb.A B Boom.430%-5% Bust.6-10%25%What is E(RB) ? VarB? St. DevB? E(RB) = .4(-5%) + .6(25%) = 13%Variance(B) = .4(-5%-13%)2 + .6(25%-13%)2 = 2.16%Std. Dev.(B) = 14.7%13-#Example: Portfolio VarianceConsider the following information:StateProbabilityA B Boom.430%-5% Bust.6-10%25%If you invest 50% of your money in Asset A, what is the expected return for the portfolio?If 50% of the investment is in Asset A, then 50% (100% - 50%) must be invested in Asset B as the total asset allocation must be 100%Wi = 100%13-#Example: Portfolio VarianceConsider the following information:StateProb.A B.. Boom.430%-5%Bust.6-10%25%E(R)6%13%Sigma19.6%14.7%If you invest 50% of your money in Asset A, what is the expected return for the portfolio If a boom? If a bust?What is expected return for the whole portfolio (that is, considering both states of the economy)?a. E(RP_Boom) = .5(30%) + .5(-5%) = 12.5% E(RP_Bust) = .5(-10%) + .5(25%) = 7.5%b. E(RP) = .4 (12.5% ) + .6(7.5%) = 9.5%13-#Example: Portfolio VarianceAnother way of computing Portfolio returns :

Exp. portfolio return = .5(6%) + .5(13%) = 9.5%E(RP) = WA E(RA) + WB E(RB) + WC E(RC) + . + WN E(RN)N: number of assets in the portfolioWi : weight of asseti in the portfolioConsider the following information:StateProb.A B.. Boom.430%-5%Bust.6-10%25%E(R)6%13%Sigma19.6%14.7%13-#25Example: Portfolio VarianceWhat is the variance of the portfolio?Variance of portfolio = .4(12.5%-9.5%)2 + .6(7.5%-9.5%)2 = 0.06%

Standard deviation = 2.45%13-#26Another ExampleWhat are the expected return and standard deviation for a portfolio that is equally distributed between X, Y, & Z?

Consider the following information:13-#27Expected vs. Unexpected ReturnsRealized returns are generally NOT equal to expected returns

There is the expected component and the unexpected component

At any point in time, the unexpected return can be either positive or negative

Over time, the average of the unexpected component is zero13-#Announcements and NewsAnnouncements and news contain both an expected component and a surprise component

It is the surprise component that affects a stocks price and therefore its return

13-#29Announcements and NewsThis surprise is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated

13-#30Systematic RiskRisk factors that affect a large number of assets

Also known as non-diversifiable risk or market risk

Includes such things as changes in GDP, inflation, interest rates, etc.

13-#Unsystematic RiskRisk factors that affect a limited number of assets

Also known as unique risk and asset-specific risk

Includes such things as labor strikes, part shortages, etc.

13-#32Computing ReturnsTotal Return = expected return + unexpected return

Unexpected return = systematic portion + unsystematic portion

Therefore, total return can be expressed as follows:

Total Return = expected return + [systematic portion + unsystematic portion]13-#DiversificationPortfolio diversification is the investment in several different asset classes or sectors

Diversification is not just holding a lot of assets

13-#34DiversificationFor example, if you own 5 airline stocks, you are not diversifiedHowever, if you own 50 stocks that span 20 different industries, then you are diversified

13-#35Total RiskTotal risk = systematic risk + unsystematic risk

The standard deviation of returns is a measure of total risk

For well-diversified portfolios, unsystematic risk is very small

Consequently, the total risk for a diversified portfolio is essentially equivalent to just the systematic risk13-#

Diversification13-#

Diversification13-#The Principle of DiversificationDiversification can substantially reduce the variability of returns without an equivalent reduction in expected returns

This reduction in risk arises because worse-than-expected returns from one asset are offset by better-than-expected returns from another

However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion13-#39Measuring Systematic RiskHow do we measure systematic risk?

We use the beta coefficient

What does beta tell us?

A beta of 1 implies the asset has the same systematic risk as the overall market

A beta < 1 implies the asset has less systematic risk than the overall market

A beta > 1 implies the asset has more systematic risk than the overall market13-#40

Actual Company Betas

13-#41Total vs. Systematic RiskConsider the following information: St.DevBetaSecurity C20%1.25Security K30%0.95

1. Which security has more total risk?K because the standard deviation is greater than C2. Which security has more systematic risk?C because the beta is larger than K13-#42Total vs. Systematic RiskConsider the following information: St.DevBetaSecurity C20%1.25Security K30%0.95

3. Which security should have the higher expected return?

C because a well diversified investor cars about systematic risk. These investors would require higher returns for higher risk, and beta is higher for C13-#43Example: Portfolio BetasConsider the previous example with the following four securities:

SecurityWeightBetaDCLK.1332.685KO.20.195INTC.2672.161KEI.42.434

What is the portfolio beta?

.133(2.685) + .2(.195) + .267(2.161) + .4(2.434) = 1.94713-#44Beta and the Risk PremiumRemember that the risk premium = expected return risk-free rate

The higher the beta, the greater the risk premium should be

Can we define the relationship between the risk premium and beta so that we can estimate the expected return?YES!13-#

Example: Portfolio Expected Returns and Betas: The SML

RfAE(RA)13-#46Security Market LineThe security market line (SML) is the representation of market equilibrium

The slope of the SML is the reward-to-risk ratio: (E(RM) Rf) / M

But since the beta for the market is ALWAYS equal to one, the slope can be rewritten:

Slope = E(RM) Rf = market risk premium13-#47Reward-to-Risk Ratio: Definition and ExampleThe reward-to-risk ratio is the slope of the line illustrated in the previous example

Slope = (E(RA) Rf) / (A 0)Reward-to-risk ratio for previous example = (20 8) / (1.6 0) = 7.5

What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)?

What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)?13-#48Market EquilibriumIn equilibrium, all assets and portfolios must have the same reward-to-risk ratio, and they all must equal the reward-to-risk ratio for the market

13-#The Capital Asset Pricing Model (CAPM)

The capital asset pricing model defines the relationship between risk and return:

E(RA) = Rf + A(E(RM) Rf)

If we know an assets systematic risk, we can use the CAPM to determine its expected return

This is true whether we are talking about financial assets or physical assets13-#Example - CAPMSecurity BetaExpected Return DCLK2.685 4.15 + 2.685(8.5) = 26.97% KO0.1954.15 + 0.195(8.5) = 5.81% INTC2.161 4.15 + 2.161(8.5) = 22.52% KEI2.434 4.15 + 2.434(8.5) = 24.84%

Consider the betas for each of the assets given earlier. If the risk-free rate is 4.15% and the market risk premium is 8.5%,

What is the expected return for each?13-#51The CAPM

13-#Quick QuizHow do you compute the expected return and standard deviation for an individual asset? For a portfolio?

What is the difference between systematic and unsystematic risk?

What type of risk is relevant for determining the expected return?

Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%.What is the expected return on the asset?Expected Return = 5% + 1.2 (8%) = 14.6%

13-#53Comprehensive ProblemThe risk free rate is 4%, and the required return on the market is 12%. What is the required return on an asset with a beta of 1.5?

What is the reward/risk ratio?

13-#54TerminologyPortfolioExpected ReturnUnsystematic RiskSystematic RiskSecurity Market Line (SML)BetaCapital Asset Pricing Model (CAPM)

13-#Formulas

Expected return on an investment

Variance of an entire population, not a sample

Expected return on a portfolio

Slope = E(RM) Rf = market risk premiumCAPM = E(RA) = Rf + A(E(RM) Rf)13-#

Key Concepts and SkillsCalculate expected returnsDescribe the impact of diversificationDefine the systematic risk principleConstruct the security market lineEvaluate the risk-return trade-offCompute the cost of equity using the Capital Asset Pricing Model13-#Measuring portfolio returns2. Using Std. Dev. and Variance to measure portfolio risk3. Diversification can significantly reduce unsystematic riskBeta measures systematic risk5. The slope of the Security Market Line = the market risk premium6. The Capital Asset Pricing Model (CAPM) provides us a measurement of a stocks required rate of return.What are the most important topics of this chapter?

13-#58Sheet1StateProbabilityC TBoom0.315%25%0.260%0.533%Normal0.510%20%0.000%0.053%Recession0.22%1%0.624%2.789%9.90%17.70%0.2029%0.7441%4.50%8.63%StateProbabilityABC, Inc. (%)Boom0.2515%0.483%Normal0.58%0.000%Slowdown0.154%0.164%Recession0.1-3%1.221%8.05%0.267475%5.17%StateProb.A B PortfolioBoom0.430%-5%12.50%5.760%3.240%0.090%Bust0.6-10%25%7.50%2.560%1.440%0.040%E(R)=6.0%13.0%9.5%3.8400%2.1600%0.0600%Var3.8%2.2%0.060%St.Dev19.60%14.70%2.45%

Sheet2

Sheet3

Sheet1StateProbabilityC TBoom0.3015%25%0.260%0.533%Normal0.5010%20%0.000%0.053%Recession0.202%1%0.624%2.789%9.90%17.70%0.2029%0.7441%4.50%8.63%StateProbabilityABC, Inc. (%)Boom0.2515%0.483%Normal0.58%0.000%Slowdown0.154%0.164%Recession0.1-3%1.221%8.05%0.267475%5.17%StateProb.A B PortfolioBoom0.430%-5%12.50%5.760%3.240%0.090%Bust0.6-10%25%7.50%2.560%1.440%0.040%E(R)=6.0%13.0%9.5%3.8400%2.1600%0.0600%Var3.8%2.2%0.06%St.Dev19.60%14.70%2.45%50%30%20%State Prob.XYZPortfolioRportf*ProbBoom0.2525%22%-10%17.10%0.001705690Normal0.5010%9%5%8.70%0.000000980Slowdown0.151%4%15%4.70%0.000257094Recession0.10-8%-15%18%-4.90%0.0018878761E(RP)=8.840%Var=0.3852%St.Dev=6.2062%

Sheet2

Sheet3

Chart10.080000.110.40.40.40.140.80.80.80.171.21.21.20.21.61.61.60.232220.262.42.42.4

Expected ReturnBetaExpected Return

Sheet1BetaExpected Return08%0.411%0.814%1.217%1.620%223%2.426%