Risk & ReturnRisk & Return

Return is what makes you eat Return is what makes you eat wellwell

Risk is what makes you sleep Risk is what makes you sleep wellwell

Our goal is to Our goal is to understand...understand...

r

Recall that ’r’ has many Recall that ’r’ has many names....names....

• Interest rateInterest rate• Rate of return / Required rate of Rate of return / Required rate of

returnreturn– stocksstocks

• Yield / Yield to maturity Yield / Yield to maturity – BondsBonds

• (Opportunity) cost of capital (Opportunity) cost of capital – Capital budgetingCapital budgeting

‘‘r’ compensates investors r’ compensates investors for..for..

• Impatience! Time Value of MoneyImpatience! Time Value of Money– people rather have things now than people rather have things now than

laterlater

• Risk!Risk!– investors dislike uncertaintyinvestors dislike uncertainty

Expected vs. Realized Expected vs. Realized returnsreturns

• 60 - 40 chance that return on 60 - 40 chance that return on Microsoft next year will be +25% Microsoft next year will be +25% OROR -10%.-10%.

• Expected return is _____%Expected return is _____%

• The actual return will be The actual return will be eithereither ____ ____ oror ____ ____

Surprises....Surprises....

• Surprise, Surprise, after the factafter the fact, in return is , in return is eithereither

+14% = (25% - 11%)+14% = (25% - 11%) oror-21% = (-10% - 11%)-21% = (-10% - 11%)

Actual return = E(r) + unexpected returnActual return = E(r) + unexpected return e.g. -10% = 11% - 21% e.g. -10% = 11% - 21%

• Investors don’t like surprisesInvestors don’t like surprises

What is the expected What is the expected surprise?surprise?

• multiply probability times each multiply probability times each surprisesurprise

• Answer = _____Answer = _____

• Expected surprise is always ____ !!!Expected surprise is always ____ !!!

• TrickTrick: take expected : take expected squaredsquared surprisessurprises

VarianceVariance• Expectation of squared surprises Expectation of squared surprises

is called is called VarianceVariance

• Square root of variance is called Square root of variance is called Standard deviationStandard deviation– Easier to understandEasier to understand

• Calculate variance and std. dev. of Calculate variance and std. dev. of Microsoft returnsMicrosoft returns

Microsoft...Microsoft...• Variance:Variance:

.6 (.25 - .11).6 (.25 - .11)22 + .4 (-.1 - .11) + .4 (-.1 - .11)22 = _____ = _____

• Standard Deviation:Standard Deviation:SQRT(____) = ____SQRT(____) = ____

• Standard deviation is easier to Standard deviation is easier to interpretinterpret– Has the same units as return (%)Has the same units as return (%)

Variance / Std. Dev. Variance / Std. Dev. FormulaeFormulae

2

1

2

p r rj jj

N

Std dev. . 2

Another example..Another example..

pi ri

Probability Return inState of Economy of state state

+1% change in GNP .25 -5%

+2% change in GNP .50 15%

+3% change in GNP .25 35%

i (pi x ri)

i = 1 -0.0125

i = 2 0.075

i = 3 0.0875

Expected return = (-0.0125 + 0.0750 + 0.0875)

= 0.15 or 15%

Expected Return

i pi (ri - r)2 pi x (ri - r )2

i=1 .25 .04 .01

i=2 .5 0 .00

i=3 .25 .04 .01

Var(R) = Std. Dev. = ___

Variance

Std. Dev. of selected Std. Dev. of selected stocksstocks

Company AT & T 24.2%Digital Equip. 38.4Ford Motor 28.7Genentech 51.8McGraw Hill 29.3Tandem Comp. 50.7MARKET PORTFOLIO 20.8

Total RiskTotal Risk

• Standard Deviation (or variance) Standard Deviation (or variance) is a measure of is a measure of total risktotal risk

• It gives us an idea of how likely It gives us an idea of how likely one is likely to get ‘burned’ if one is likely to get ‘burned’ if he/she invests in any single stock he/she invests in any single stock or portfolio of stocksor portfolio of stocks

The Normal The Normal DistributionDistribution

Historical Returns and Standard Historical Returns and Standard DeviationsDeviations

Value of a $1 investmentValue of a $1 investment

What do you notice?What do you notice?• From the previous two graphs, we From the previous two graphs, we

notice that...notice that...

• The _________ the standard deviation, The _________ the standard deviation, the ___________ is the value of a $1 the ___________ is the value of a $1 investment in the long runinvestment in the long run

• In general, the _______ the risk, the In general, the _______ the risk, the _______ the return!_______ the return!

Five Largest One-Day Percentage Declines Five Largest One-Day Percentage Declines in the Dow-Jones Industrial Averagein the Dow-Jones Industrial Average

The Five Worst DaysThe Five Worst Days

DateDate Point LossPoint Loss % Loss% Loss

October 19, 1987October 19, 1987 508.00 508.00 22.6%22.6%

October 28, 1929October 28, 1929 38.88 38.88 12.8 12.8

October 29, 1929October 29, 1929 30.57 30.57 11.7 11.7

November 6, 1929November 6, 1929 25.55 25.55 9.9 9.9

December 18, 1899December 18, 1899 5.57 5.57 8.7 8.7

Why do returns fluctuate?Why do returns fluctuate?

New Information!New Information!

• Market-wide infoMarket-wide info..– WarWar– Oil shockOil shock– Rise in interest Rise in interest

raterate– Exchange rate Exchange rate

changeschanges

• Asset-specific infoAsset-specific info– StrikesStrikes– LawsuitLawsuit– Death of CEODeath of CEO– FDA approval of a FDA approval of a

drugdrug

Hence...Hence...Total Risk

Market Risk Asset-specific Risk

a.k.aSystematic Risk

Non-diversifiable Risk

a.k.aUnsystematic riskDiversifiable Risk

Unique Risk

+

Portfolio Expected Returns and Portfolio Expected Returns and VariancesVariances

•Portfolio Portfolio weightsweights: put 50% in Asset A : put 50% in Asset A and 50% in Asset B:and 50% in Asset B:

State of the State of the ProbabilityProbabilityReturnReturnReturnReturnReturn Return ononeconomyeconomy of stateof state on Aon A on Bon B portfolioportfolio

BoomBoom 0.400.40 30%30% -5%-5% 12.5%12.5%

BustBust 0.600.60 -10%-10% 25%25% 7.5%7.5%

1.001.00

Portfolio Expected Returns and Portfolio Expected Returns and Variances (continued)Variances (continued)

• E(RE(RPP) ) = = 0.40 x (.125) + 0.60 x (.075) = .095 0.40 x (.125) + 0.60 x (.075) = .095 = 9.5%= 9.5%

• Var(RVar(RPP) ) = = 0.40 x (.125 - .095)0.40 x (.125 - .095)22 + 0.60 x (.075 + 0.60 x (.075 - .095)- .095)22

= .0006 = .0006

• SD(RSD(RPP) ) = .0006 = .0245 = 2.45%= .0006 = .0245 = 2.45%

• Note:Note: E(RE(RPP) ) = = .50 x E(R.50 x E(RAA) + .50 x E(R) + .50 x E(RBB) = 9.5%) = 9.5%

• BUTBUT:: Var (RVar (RPP)) .50 x Var(R.50 x Var(RAA) + .50 x ) + .50 x Var(RVar(RBB))

Portfolios....Portfolios....

Stock A returns

0.05

0.04

0.03

0.02

0.01

0

-

0.01

-

0.02

-

0.03

Stock B returns

0.04

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

Portfolio returns:50% A and 50% B

0.05

0.04

0.03

0.02

0.01

0

-

0.01

-

0.02

-

0.03

Portfolio Expected Returns and Variances Portfolio Expected Returns and Variances (continued)(continued)

• New portfolio weights: put 3/7 in A and 4/7 in B:New portfolio weights: put 3/7 in A and 4/7 in B:

State of the State of the ProbabilityProbabilityReturnReturnReturnReturnReturn onReturn oneconomyeconomy of stateof stateon Aon A on Bon Bportfolioportfolio

BoomBoom 0.400.4030%30%-5%-5% 10%10%

BustBust 0.600.60-10%-10%25%25% 10%10%

1.001.00

Portfolio Expected Returns and Portfolio Expected Returns and Variances (concluded)Variances (concluded)

•A.A. E(RE(RPP) ) = = 10%10%

•B.B. SD(RSD(RPP) ) = = 0%0%

Amazing, eh?Amazing, eh?

E(R) of PortfolioE(R) of Portfolio

• Weighted average of E(R) of stocks Weighted average of E(R) of stocks in portfolioin portfolio

For 2-stock portfolio, For 2-stock portfolio,

• E(Rp) = WE(Rp) = W11 x E(R x E(R11) + W) + W22 x E(R x E(R22))

In general....In general....• E(RE(Rpp) = weighted average of ) = weighted average of

expected return on individual stocksexpected return on individual stocks

BUT!!BUT!!

• SD(RSD(Rpp) < weighted average of ) < weighted average of SD(RSD(Rii) on individual stocks) on individual stocks

• This is the essence of diversification This is the essence of diversification benefitsbenefits

How Diversification How Diversification Works...Works...

• Asset-specific risk is reducedAsset-specific risk is reduced

• Adding more assets reduces this risk Adding more assets reduces this risk furtherfurther

• good news and bad news ‘cancel’ out..good news and bad news ‘cancel’ out..– with less than perfect correlation among with less than perfect correlation among

stock returnsstock returns

Portfolio DiversificationPortfolio DiversificationAverage annualstandard deviation (%)

Number of stocksin portfolio

Diversifiable risk

49.2

23.9

19.2

1 10 20 30 40 1000

Non-diversifiableRisk

Diversification Diversification continued..continued..

• In the limit, ALL In the limit, ALL asset-specificasset-specific risk risk can be eliminated by holding a slice can be eliminated by holding a slice of the MARKET portfolioof the MARKET portfolio

• Market portfolioMarket portfolio is a portfolio of is a portfolio of allall assets in an economyassets in an economy– In practice, S&P 500 is a reasonable In practice, S&P 500 is a reasonable

approximation of market portfolioapproximation of market portfolio

Believe me no, I thank my fortune for itMy ventures are not in one bottom trustedNor to one place; nor is my whole estateUpon the fortune of this present year.Therefore, my merchandise makes me not sad.

- Antonio, in The Merchant of Venice

CCaptial aptial AAsset sset PPricing ricing MModelodel

• Is…one of the most important ideas in Is…one of the most important ideas in finance in this centuryfinance in this century

• Is…based on the idea that all investors will Is…based on the idea that all investors will diversify because it makes sense to do sodiversify because it makes sense to do so

• Says…hence investors only care about Says…hence investors only care about systematic risksystematic risk

• Says…systematic risk is measured by Says…systematic risk is measured by BETABETA

CAPM says..CAPM says..• If investors can get rid of asset-specific risk If investors can get rid of asset-specific risk

(without sacrificing returns) they will do so(without sacrificing returns) they will do so

• Corollary 1Corollary 1: Market : Market rewardsrewards investors (in investors (in terms of higher return) only for bearing risk terms of higher return) only for bearing risk they cannot avoid - i.e. they cannot avoid - i.e. systematic risksystematic risk (or (or beta)beta)

• Corollary 2Corollary 2: All investors (should??) hold : All investors (should??) hold market portfoliomarket portfolio

CAPM EquationCAPM Equation

• E(RE(Rii) ) = = Expected return on asset iExpected return on asset i

• E(RE(Rmm) ) = = Expected return on Expected return on market portfoliomarket portfolio

• RRff = = Risk-free rateRisk-free rate

• [E(R[E(Rmm) - R) - Rff] = ] = Market risk premiumMarket risk premium

In general...In general...• Beta can be any number, but Beta can be any number, but

typically it is between 0.4 to 3.0typically it is between 0.4 to 3.0

By Definition...By Definition...

• Beta of risk-free asset is always Beta of risk-free asset is always ZEROZERO– Risk-free assets have no risk!Risk-free assets have no risk!– not even systematic risknot even systematic risk

• Beta of the market portfolio is always Beta of the market portfolio is always ONEONE– Market risk is ‘average’ riskMarket risk is ‘average’ risk– S&P 500 is often taken as an approximation of S&P 500 is often taken as an approximation of

market portfoliomarket portfolio

The Security Market Line (SML)The Security Market Line (SML)Asset expectedreturn E (Ri)

Assetbeta

E (RM)

Rf

M = 1.0

= E (RM ) – Rf

0

Using CAPMUsing CAPM

• The formula can be used to estimate The formula can be used to estimate – ‘‘r’ for projects in DCF methodsr’ for projects in DCF methods– ‘‘r’ for expected / required return in r’ for expected / required return in

valuing stocksvaluing stocks

• CAPM equation is one of the most CAPM equation is one of the most widely used equations in financewidely used equations in finance

ExampleExample• Beta of Intel = 1.73Beta of Intel = 1.73

• T-bills rate = RT-bills rate = Rff = 5% = 5%

• E(RE(Rmm) = 12%) = 12%– (estimated from past average of market (estimated from past average of market

return)return)

Hence,Hence,

• Market risk premium = [E(Rm) - Rf] = 7%Market risk premium = [E(Rm) - Rf] = 7%

• E(R) = .05 + [.12 - .05] x 1.73 = _____ E(R) = .05 + [.12 - .05] x 1.73 = _____

Estimating BetaEstimating Beta• Slope of regression line of RSlope of regression line of Rii on R on Rmm

Upper 95.000%

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Ri

Rm

Beta on calculatorBeta on calculator

• Clear Statistical Memory: Clear Statistical Memory: CLCL

• Enter (x,y) pair: xEnter (x,y) pair: xii INPUTINPUT y yii

• Get slope: 0 Get slope: 0 y-hat, m y-hat, m SWAPSWAP

+

Intercept Slope

Beta of PortfoliosBeta of Portfolios

• Weighted average of beta of Weighted average of beta of individual stocksindividual stocks

• For 2-stock portfolio:For 2-stock portfolio:

ExampleExample1. Compaq Beta:1. Compaq Beta: 1.601.60 Invest: $ 7,000Invest: $ 7,000

2. Nordstrom Beta:2. Nordstrom Beta: 1.201.20 Invest: $21,000Invest: $21,000

• WW11 = 7000/28000 = 7000/28000 = 0.25= 0.25

• WW22 = 21000/28000 = 21000/28000 = 0.75 = 0.75

• Portfolio Beta =Portfolio Beta =.25(1.6) + .75(1.2) = ______.25(1.6) + .75(1.2) = ______

Another use of CAPMAnother use of CAPM

• Used to identify undervalued or Used to identify undervalued or overvalued securitiesovervalued securities

• If If youryour estimate of expected return is estimate of expected return is greater than CAPM-based required greater than CAPM-based required return --> asset is return --> asset is undervalued undervalued

• If If youryour estimate of expected return is estimate of expected return is less than CAPM-based required less than CAPM-based required return --> asset is return --> asset is overvaluedovervalued

SML: Undervalued vs. OvervaluedSML: Undervalued vs. OvervaluedAsset expectedreturn E (Ri)

Assetbeta

E (RM)

Rf

M = 1.00

Undervalued

Overvalued

SML Continued...SML Continued...

• If an asset plots above SML, it is If an asset plots above SML, it is UNDERVALUEDUNDERVALUED– E(R) is E(R) is higherhigher than justified by risk than justified by risk

• If an asset plots below SML, it is If an asset plots below SML, it is OVERVALUEDOVERVALUED– E(R) is E(R) is lowerlower than justified by risk than justified by risk

Market Equilibrium & Market Equilibrium & SMLSML

• In equilibrium, buying and selling In equilibrium, buying and selling pressure forces all assets to plot pressure forces all assets to plot exactly on SMLexactly on SML– mispricing does not last for very longmispricing does not last for very long

• Hence, in equilibrium, the Hence, in equilibrium, the Reward-to-RiskReward-to-Risk ratio for all assets ratio for all assets is the same:is the same:(Re )

( )ward to Risk

E R R

Betaii f

i

In Equilibrium...In Equilibrium...

Which gives usWhich gives us

E R R

Beta

E R R

Beta

E R R

Betai f

i

j f

j

m f

m

( ) ( ) ( )

( )

1

RecapRecap

• Measure of total risk is Measure of total risk is variancevariance or or standard deviationstandard deviation

• The essence of The essence of diversificationdiversification is is reducing total risk of a portfolioreducing total risk of a portfolio

• Total risk of a portfolio is always Total risk of a portfolio is always reduced when security returns are reduced when security returns are less than perfectly correlatedless than perfectly correlated

RecapRecap

• Market portfolio is the portfolio of ALL Market portfolio is the portfolio of ALL risky securitiesrisky securities

• Market portfolio has no unsystematic riskMarket portfolio has no unsystematic risk

• Market portfolio only has systematic riskMarket portfolio only has systematic risk

• CAPM says all investors hold market CAPM says all investors hold market portfolioportfolio

RecapRecap

• The contribution an individual security The contribution an individual security makes to the risk of the market makes to the risk of the market portfolio is measured by portfolio is measured by betabeta

• Hence for Hence for individual securitiesindividual securities risk is risk is measured by measured by betabeta

• For For diversified portfoliosdiversified portfolios, risk is , risk is measured by measured by standard deviationstandard deviation (or (or variance)variance)

One final point…One final point…

• Corporations finance their activities by a Corporations finance their activities by a mix of debt, preferred, and common stockmix of debt, preferred, and common stock

• Hence for a corporation’s activities as a Hence for a corporation’s activities as a whole, the required rate of return is whole, the required rate of return is measured by measured by WWeighted eighted AAverage verage CCost of ost of CCapitalapital

• WACCWACC