Risk and Rates of Risk and Rates of ReturnReturn

Chapter 8Chapter 8

Historical Risk and Return Historical Risk and Return Info.Info.

Based on annual returns from 1926-Based on annual returns from 1926-20042004

Avg. ReturnAvg. Return Std Dev.Std Dev.

Small StocksSmall Stocks 17.5%17.5% 33.1%33.1%

Large Co. StocksLarge Co. Stocks 12.4%12.4% 20.3%20.3%

L-T Corp BondsL-T Corp Bonds 6.2%6.2% 8.6%8.6%

L-T Govt. BondsL-T Govt. Bonds 5.8%5.8% 9.3%9.3%

U.S. T-BillsU.S. T-Bills 3.8%3.8% 3.1%3.1%

Chapter 8 Risk & Rates of Chapter 8 Risk & Rates of ReturnReturn

Risk: The Big Risk: The Big PicturePicture

Expected ReturnExpected Return Stand Alone RiskStand Alone Risk Portfolio Return Portfolio Return

and Riskand Risk

Risk DiversificationRisk Diversification Market RiskMarket Risk

– BetaBeta CAPM/Security CAPM/Security

Market Line Market Line Equation (SML)Equation (SML)

What is Risk? The big What is Risk? The big picture.picture.

Risk Risk is an uncertain outcome or is an uncertain outcome or chance of an adverse outcome.chance of an adverse outcome.

Concerned with the riskiness of cash Concerned with the riskiness of cash flows from financial assets.flows from financial assets.

Stand Alone Risk: Single AssetStand Alone Risk: Single Asset– relevant risk measure is the relevant risk measure is the total risktotal risk

of expected cash flows measured by of expected cash flows measured by standard deviationstandard deviation . .

Risk: The Big Picture (cont.)Risk: The Big Picture (cont.)

Portfolio Context: A group of assets. Portfolio Context: A group of assets. Total risk consists of:Total risk consists of:– Diversifiable Risk (company-specific, Diversifiable Risk (company-specific,

unsystematic)unsystematic)– Market Risk (non-diversifiable, systematic)Market Risk (non-diversifiable, systematic)

Small group of assets with Diversifiable Small group of assets with Diversifiable Risk remaining: interested in portfolio Risk remaining: interested in portfolio standard deviation. standard deviation. – correlation (correlation ( or r) between asset returns or r) between asset returns

which affects portfolio standard deviationwhich affects portfolio standard deviation

finishing the Big Picture on finishing the Big Picture on RiskRisk

Well-diversified PortfolioWell-diversified Portfolio Large Portfolio (10-15 assets) eliminates Large Portfolio (10-15 assets) eliminates

diversifiable risk for the most part.diversifiable risk for the most part. Interested in Interested in Market RiskMarket Risk which is the which is the

risk that cannot be diversified away.risk that cannot be diversified away. The relevant risk measure is The relevant risk measure is BetaBeta which which

measures the riskiness of an individual measures the riskiness of an individual asset in relation to the market portfolio.asset in relation to the market portfolio.

Holding Period (Realized) Holding Period (Realized) ReturnReturn

HPR = (End of Period Price - Beginning HPR = (End of Period Price - Beginning Price Price + Dividends)/Beginning + Dividends)/Beginning PricePrice

HPR = Capital Gains Yield + Dividend HPR = Capital Gains Yield + Dividend YieldYield

HPR = (P1-P0)/P0 + D/P0HPR = (P1-P0)/P0 + D/P0Example: Bought at $50, Receive $3 in Example: Bought at $50, Receive $3 in

dividends, current price is $54dividends, current price is $54HPR = (54-50+3)/50 = .14 or 14%HPR = (54-50+3)/50 = .14 or 14%CGY = 4/50 = 8%, DY = 3/50 = 6%CGY = 4/50 = 8%, DY = 3/50 = 6%

Expected Return: Single Expected Return: Single AssetAsset

Expected Rate of Return given a probability Expected Rate of Return given a probability distribution of possible returns(rdistribution of possible returns(rii): E(r)): E(r)

nn

E(r) = E(r) = PPi i rrii

i=1i=1

Realized or Average Return on Historical Realized or Average Return on Historical Data: Data:

-- n n

r = 1/n r = 1/n rrii

i=1i=1

Standard DeviationStandard Deviation

Relevant Risk Measure for single assetRelevant Risk Measure for single asset

Variance = Variance = 22 = = ( ( rri i - E(r))- E(r))2 2 PPii

Standard Deviation = Square Root of Standard Deviation = Square Root of VarianceVariance

Historical Variance = Historical Variance = 22 = 1/n = 1/n(r(ri i – r– rAVGAVG ) )22

Sample Variance = sSample Variance = s22 = 1/(n-1) = 1/(n-1) (r(ri i – r– rAVGAVG ) )22

Example: Exp. Return and Example: Exp. Return and

State of ContraryEconomy Probability MAD Inc. Co. (CON)Boom 0.25 80% -6%Normal 0.60 30% 10%Recession 0.15 -30% 20%

Example: Standard Example: Standard DeviationDeviation

Decision Time: Coefficient of Decision Time: Coefficient of VariationVariation

Most investors are Most investors are Risk AverseRisk Averse, , meaning they don’t like risk and demand meaning they don’t like risk and demand a higher return for bearing more risk.a higher return for bearing more risk.

The The Coefficient of Variation (CV)Coefficient of Variation (CV) scales risk per unit of expected return. scales risk per unit of expected return.

CV = CV = /E(r)/E(r) CV is a measure of relative risk, where CV is a measure of relative risk, where

standard deviation measures absolute standard deviation measures absolute risk.risk.

Back to our Example: CVBack to our Example: CV

MAD Inc.MAD Inc. E(r) = 33.5%E(r) = 33.5% = 34.0%= 34.0% CV = 34%/33.5%CV = 34%/33.5% CV = 1.015CV = 1.015

Contrary Co.Contrary Co. E(r) = 7.5%E(r) = 7.5% = 8.9%= 8.9% CV = 8.9%/7.5%CV = 8.9%/7.5% CV = 1.187CV = 1.187

Portfolio Risk and ReturnPortfolio Risk and Return

E(rE(rpp) = ) = wwiiE(rE(rii) = weighted average of ) = weighted average of the expected return of each asset in the the expected return of each asset in the portfolioportfolio

In our example, MAD E(r) = 33.5% and In our example, MAD E(r) = 33.5% and CON E(r) = 7.5%CON E(r) = 7.5%

What is the expected return of a portfolio What is the expected return of a portfolio consisting of 60% MAD and 40% CON?consisting of 60% MAD and 40% CON?

E(rE(rpp) = ) = wwiiE(rE(rii) = .6(33.5%) + .4(7.5%) ) = .6(33.5%) + .4(7.5%) = 23.1%= 23.1%

Portfolio RiskPortfolio Risk

Looking at a 2-asset portfolio for Looking at a 2-asset portfolio for simplicity, the riskiness of a portfolio is simplicity, the riskiness of a portfolio is determined by the relationship determined by the relationship between the returns of each asset over between the returns of each asset over different states of nature or over time.different states of nature or over time.

This relationship is measured by the This relationship is measured by the correlation coefficient( correlation coefficient( rr ): -1<= ): -1<= rr < < =+1=+1

All else constant: Lower All else constant: Lower rr = less = less portfolio riskportfolio risk

Example Portfolio Example Portfolio

Each MAD-CON rEach MAD-CON rii = .6(MAD)+.4(CON); = .6(MAD)+.4(CON);

E(RE(Rpp)) = 23.1%= 23.1%

State of Contrary MAD-CONEconomy Probability MAD Inc. Co. (CON) PortfolioBoom 0.25 80% -6% 45.6%Normal 0.60 30% 10% 22.0%Recession 0.15 -30% 20% -10.0%

Market RiskMarket Risk

As more and more assets are added As more and more assets are added to a portfolio, risk measured by to a portfolio, risk measured by decreases.decreases.

HoweverHowever, we could put every , we could put every conceivable asset in the world into conceivable asset in the world into our portfolio and still have risk our portfolio and still have risk remaining. (See Fig. 8-8, pg. 265)remaining. (See Fig. 8-8, pg. 265)

This remaining risk is called This remaining risk is called Market Market RiskRisk and is measured by and is measured by Beta.Beta.

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Diversifiablerisk

As you add stocks to your As you add stocks to your portfolio, diversifiable risk portfolio, diversifiable risk

is reduced.is reduced.

The Concept of BetaThe Concept of Beta Beta(b)Beta(b) measures how the return of an individual measures how the return of an individual

asset (or even a portfolio) varies with the market.asset (or even a portfolio) varies with the market. b b = 1.0 : same risk as the market= 1.0 : same risk as the market b b < 1.0 : less risky than the market< 1.0 : less risky than the market b b > 1.0 : more risky than the market> 1.0 : more risky than the market Beta is the slope of the regression line (y = a + Beta is the slope of the regression line (y = a +

bx) between a stock’s return(y) and the market bx) between a stock’s return(y) and the market return(x) over time, return(x) over time, bb from simple linear from simple linear regression. regression.

Sources for stock betas: ValueLine Investment Sources for stock betas: ValueLine Investment Survey (at BEL), Yahoo Finance, MSN Money, Survey (at BEL), Yahoo Finance, MSN Money, Standard & PoorsStandard & Poors

Relating Market Risk and Relating Market Risk and Return: the CAPM and SML Return: the CAPM and SML

equationequation The story is the same as Chapter 6: a The story is the same as Chapter 6: a

stock’s required rate of return = stock’s required rate of return = nominal risk-free rate + the stock’s nominal risk-free rate + the stock’s risk premium.risk premium.

The main assumption is investors hold The main assumption is investors hold well diversified portfolios = only well diversified portfolios = only concerned with market risk. concerned with market risk.

A stock’s risk premium = measure of A stock’s risk premium = measure of market risk X market risk premium.market risk X market risk premium.

SML EquationSML Equation

RPRPMM = market risk premium = r = market risk premium = rMM - r - rRFRF

RPRPii = stock risk premium = (RP = stock risk premium = (RPMM))bbii

rrii = r = rRFRF + (r + (rMM - r - rRF RF ))bbii

= r= rRFRF + (RP + (RPMM))bbii

CAPM ExampleCAPM Example

What is Intel’s required return if its B What is Intel’s required return if its B = 1.2 (from ValueLine Investment = 1.2 (from ValueLine Investment Survey), the current 3-mo. T-bill rate Survey), the current 3-mo. T-bill rate is 5%, and the historical US market is 5%, and the historical US market risk premium of 8.6% is expected?risk premium of 8.6% is expected?

Portfolio BetaPortfolio Beta The beta of a portfolio of stocks is The beta of a portfolio of stocks is

equal to the weighted average of equal to the weighted average of their individual betas: their individual betas: bbpp = = wwiibbii

ExampleExample: What is the portfolio beta for : What is the portfolio beta for a portfolio consisting of 25% Home a portfolio consisting of 25% Home Depot with b = 1.1, 40% Hewlett-Packard Depot with b = 1.1, 40% Hewlett-Packard with b = 1.4, and 35% Disney with b = with b = 1.4, and 35% Disney with b = 1.35. What is this portfolio’s required 1.35. What is this portfolio’s required (expected) return if the risk-free rate is (expected) return if the risk-free rate is 5% and the market expected return is 5% and the market expected return is 13%?13%?

Continuing our ExampleContinuing our Example

Coca-Cola currently sells for $44. Should Coca-Cola currently sells for $44. Should we add Coca-Cola with an expected we add Coca-Cola with an expected price and dividend in a year of $48.27 & price and dividend in a year of $48.27 & $1.24 and a b = 0.6 to our portfolio?$1.24 and a b = 0.6 to our portfolio?

To make our decision find Coke’s To make our decision find Coke’s expected return using the holding expected return using the holding period return formula and compare to period return formula and compare to Coke’s SML return.Coke’s SML return.

Recall that rRecall that rRFRF = 5% and r = 5% and rMM = 13% = 13%

Drink Coke?Drink Coke?

The Security Market Line The Security Market Line (SML)(SML)

A graphical representation of the A graphical representation of the CAPM/SML equation.CAPM/SML equation.

Gives required (expected) returns for Gives required (expected) returns for investments with different betas.investments with different betas.

Y axis = expected return, X axis = betaY axis = expected return, X axis = beta Intercept = risk-free rate = 3-month T-bill Intercept = risk-free rate = 3-month T-bill

rate (B = 0)rate (B = 0) Slope of SML = market risk premiumSlope of SML = market risk premium For the following SML graph, let’s use a 3-For the following SML graph, let’s use a 3-

month T-bill rate of 5% and assume month T-bill rate of 5% and assume investors require a market return of 13%.investors require a market return of 13%.

Graph r = 5% + B(13%-5%)Graph r = 5% + B(13%-5%) Market risk premium = 13% - 5% = 8%Market risk premium = 13% - 5% = 8%

Our SML and Coke: rOur SML and Coke: rRFRF = 5%, r = 5%, rMM = 13%= 13%

12.50%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0 0.5 1 1.5 2 2.5 3Beta

Return

Changes to SML EquationChanges to SML Equation

What happens if inflation increases?What happens if inflation increases?

What happens if investors become What happens if investors become more risk averse about the stock more risk averse about the stock market?market?

Check out the following graphs with Check out the following graphs with our base SML = 5% + (13%-5%)bour base SML = 5% + (13%-5%)b

SML change, increase in SML change, increase in inflation and rinflation and rRFRF: r: rRFRF = 7%, r = 7%, rMM = =

15%15%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0 0.5 1 1.5 2 2.5 3Beta

Return

SML increase in risk aversion SML increase in risk aversion (market risk premium: r(market risk premium: rRFRF = 5%, r = 5%, rMM

= 16%= 16%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

0 0.5 1 1.5 2 2.5 3Beta

Return

Finding Beta using Excel.Finding Beta using Excel.

There are two functions in Excel that There are two functions in Excel that will find the X coefficient (beta).will find the X coefficient (beta).

The functions are LINEST and SLOPE.The functions are LINEST and SLOPE. The format is The format is =LINEST(y range,x =LINEST(y range,x

range)range) The above format is the same for The above format is the same for

SLOPE.SLOPE. Remember the stock’s returns is the y Remember the stock’s returns is the y

range, and the market’s returns is the range, and the market’s returns is the x range.x range.