Diversification and Diversification and Portfolio Management (Ch. Portfolio Management (Ch.

8)8)

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How investors view risk and returnHow investors view risk and return

• Investors Investors like returnlike return. They seek to . They seek to maximize return.maximize return.

• But investors But investors dislike riskdislike risk. They seek to . They seek to avoid or minimize risk. Why?avoid or minimize risk. Why?– Because human beings possess the Because human beings possess the

psychological trait of “psychological trait of “risk aversionrisk aversion” which is a ” which is a dislike for taking risks.dislike for taking risks.

Implications of risk aversionImplications of risk aversion

• The The “risk-return tradeoff”“risk-return tradeoff” - Risk averse investors - Risk averse investors require higher rates of return to induce them to invest require higher rates of return to induce them to invest in higher risk securities.in higher risk securities.

• The higher a security’s risk, the higher the return The higher a security’s risk, the higher the return investors demand. Thus, the less they are willing to pay investors demand. Thus, the less they are willing to pay for the investment, i.e. for the investment, i.e. as risk increase, Pas risk increase, P00 decreases decreases..

• Risk averse investors will Risk averse investors will diversifydiversify their investments in their investments in order to reduce risk.order to reduce risk.

DiversificationDiversification

• Definition - An investment strategy designed to Definition - An investment strategy designed to reduce riskreduce risk by spreading the funds invested across by spreading the funds invested across many securities.many securities.

• It is holding a broad It is holding a broad portfolioportfolio of securities so as “not of securities so as “not to have all your eggs in one basket.” to have all your eggs in one basket.”

• Since people hold Since people hold diversified portfoliosdiversified portfolios of securities, of securities, they are not very concerned about the risk and they are not very concerned about the risk and return of a return of a single securitysingle security. They are more concerned . They are more concerned about the risk and return of their about the risk and return of their entire portfolioentire portfolio..

Two components of an asset’s risk Two components of an asset’s risk (standard deviation)(standard deviation)

• Unique RiskUnique Risk - Also called “ - Also called “diversifiable riskdiversifiable risk” and ” and ““unsystematic risk.”unsystematic risk.” The part of a security’s risk associated The part of a security’s risk associated with random outcomes generated by events specific to the with random outcomes generated by events specific to the firm. This risk can be eliminated by proper diversification.firm. This risk can be eliminated by proper diversification.

• Market RiskMarket Risk – Also called “ – Also called “systematic risksystematic risk.” The part of a .” The part of a security’s risk that cannot be eliminated by diversification security’s risk that cannot be eliminated by diversification because it is associated with economic or market factors that because it is associated with economic or market factors that systematically affect most firms.systematically affect most firms.

Market risk reflects economy-wide sources of risk that Market risk reflects economy-wide sources of risk that affect most firms and, hence, the affect most firms and, hence, the overall stock overall stock marketmarket. .

The expected return on a The expected return on a portfolio of stocksportfolio of stocks

• AssumeAssume N N stocks are held in the portfolio. stocks are held in the portfolio.

• Stock Stock ii is held in the proportion, is held in the proportion, wwi i

• Then the expected return on the portfolio of Then the expected return on the portfolio of stocks is the weighted average of the individual stocks is the weighted average of the individual stock expected returns:stock expected returns:

)()(1

jj rEwrEN

jp

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The standard deviation of The standard deviation of returns for a portfolio of returns for a portfolio of stocksstocks• The standard deviation of returns for the portfolio of stocks is given by:The standard deviation of returns for the portfolio of stocks is given by:

where where returnreturni i is the return of the portfolio in state is the return of the portfolio in state i i and and nn represents the number of states of the economy. represents the number of states of the economy.

n

iiip

1

2 yprobabilit*))E(r - (return p

The standard deviation of The standard deviation of returns for a portfolio of returns for a portfolio of stocksstocks

jiijj

N

i

N

jip ww

1 1

• We can also calculate the standard deviation We can also calculate the standard deviation of returns for the portfolio of stocks as:of returns for the portfolio of stocks as:

where where ρρijij= = the correlation coefficient for the correlation coefficient for stocks stocks ii and and jj

Correlation coefficientCorrelation coefficient

• The “Correlation Coefficient” is a measure of The “Correlation Coefficient” is a measure of the extent that the extent that twotwo variables move or vary variables move or vary together.together.

• It ranges between –1.0 and +1.0It ranges between –1.0 and +1.0– Positive correlation: a Positive correlation: a highhigh value on one variable is value on one variable is

likely to be associated with a likely to be associated with a highhigh value on the value on the other.other.

– Negative correlation: a Negative correlation: a highhigh value on one variable is value on one variable is likely to be associated with a likely to be associated with a lowlow value on the other. value on the other.

– No correlation: values of each are independent of No correlation: values of each are independent of the other the other

Correlation coefficientCorrelation coefficient

• It is denoted by the Greek letter, “rho”: It is denoted by the Greek letter, “rho”: ρρ– If If ρρ = +1.0, = +1.0, perfect positiveperfect positive correlation correlation– If If ρρ = -1.0, = -1.0, perfect negativeperfect negative correlation correlation– If If ρρ = 0, uncorrelated or independent = 0, uncorrelated or independent

How diversification reduces riskHow diversification reduces risk

• Combining stocks into a Combining stocks into a portfolioportfolio reduces reduces the variability of possible returns as long the variability of possible returns as long as the returns on the individual stocks as the returns on the individual stocks are are notnot perfectly correlated, i.e. as long perfectly correlated, i.e. as long as their correlation coefficients are as their correlation coefficients are lessless than +1.0.than +1.0.

Portfolio risk falls as you add Portfolio risk falls as you add securitiessecurities

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You can’t eliminate “market risk”You can’t eliminate “market risk”

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Market risk

Uniquerisk

This pattern occurs because of the This pattern occurs because of the two components of a stock’s risktwo components of a stock’s risk

• Total Risk = Market risk + unique riskTotal Risk = Market risk + unique risk

• The unique risk is “diversified away” when The unique risk is “diversified away” when individual stocks are combined in a portfolio.individual stocks are combined in a portfolio.

• Only market risk remains.Only market risk remains.

• The amount of the market risk is determined by The amount of the market risk is determined by the market risk of the individual stocks in the the market risk of the individual stocks in the portfolio.portfolio.

How should we measure portfolio How should we measure portfolio risk now?risk now?

• Since diversification eliminates Since diversification eliminates unique riskunique risk and leaves and leaves market or non-diversifiable riskmarket or non-diversifiable risk, the latter is the only , the latter is the only relevant risk for a diversified investor.relevant risk for a diversified investor.

• Therefore, the relevant measure of risk for a Therefore, the relevant measure of risk for a portfolioportfolio is a is a measure of the sensitivity of the portfolio’s returns to measure of the sensitivity of the portfolio’s returns to changes in the return on the “market portfolio”.changes in the return on the “market portfolio”.

• This is known as the portfolio’s This is known as the portfolio’s beta (beta (ββ))

• By definition, the market portfolio has a beta of 1 and the By definition, the market portfolio has a beta of 1 and the risk-free asset has a beta of 0.risk-free asset has a beta of 0.

How to interpret a betaHow to interpret a beta

• If If ββii > > 1, returns to stock 1, returns to stock ii are amplified are amplified relative to the market.relative to the market.

• If If ββii is between 0 and 1.0, returns to stock is between 0 and 1.0, returns to stock ii tend to move in the same direction as tend to move in the same direction as the market but not as far.the market but not as far.

• If If ββii < 1(very rare), returns to stock < 1(very rare), returns to stock ii tend tend to move in the opposite direction as the to move in the opposite direction as the market.market.

How to interpret a beta-cont’dHow to interpret a beta-cont’d

• A stock with A stock with β = 1 has average market risk.β = 1 has average market risk.– A well-diversified portfolio of such stocks A well-diversified portfolio of such stocks

tends to move by the same percentage as tends to move by the same percentage as the overall market moves.the overall market moves.

• A stock with β = +.5 has below average A stock with β = +.5 has below average market risk.market risk.– A well-diversified portfolio of these stocks A well-diversified portfolio of these stocks

tends to move half as far as the overall tends to move half as far as the overall market moves.market moves.

Measuring individual security Measuring individual security betasbetas

• Security betas are estimated by running a Security betas are estimated by running a regression between the historical returns on regression between the historical returns on the security and the historical returns on the the security and the historical returns on the market portfolio over the same period of time.market portfolio over the same period of time.

• Typically, betas are estimated using 5 years of Typically, betas are estimated using 5 years of historical monthly returns.historical monthly returns.

• The slope of the regression represents the The slope of the regression represents the betabeta

General comments about General comments about riskrisk• Most stocks are positively correlated Most stocks are positively correlated

with the market (with the market (ρρi,mi,m 0.65). 0.65).

• σσ 0.35 for an average stock. 0.35 for an average stock.

• Combining stocks in a portfolio Combining stocks in a portfolio generally lowers risk. generally lowers risk.

Calculating portfolio betasCalculating portfolio betas

• AssumeAssume N N stocks are held in the portfolio. stocks are held in the portfolio.

• Stock Stock ii is held in the proportion, is held in the proportion, wwi i

• Then the portfolio beta is the weighted Then the portfolio beta is the weighted average of the individual stock betas:average of the individual stock betas:

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Risk-return trade-off Risk-return trade-off revisitedrevisited• For a diversified investor whose only For a diversified investor whose only

concern is non-diversifiable risk, concern is non-diversifiable risk, measured by beta, this investor will measured by beta, this investor will now want higher return for a security now want higher return for a security with a higher beta.with a higher beta.

• This linear relationship between a This linear relationship between a security’s expected return and beta is security’s expected return and beta is formalized by the formalized by the Security Market Security Market Line (SML).Line (SML).

The Security Market LineThe Security Market Line

E(r)

BETA

rf

SML

E(rm)

1

where rf is the return on the risk-free security and E(rm) is the expected return on the market portfolio

The Security Market LineThe Security Market Line

E(r)

BETA

rf

SML

E(rm)

1 βA

E(rA)

Reward-to-risk ratioReward-to-risk ratio

• The reward-to-risk ratio is calculated as the ratio of The reward-to-risk ratio is calculated as the ratio of the excess return (beyond that of the risk-free the excess return (beyond that of the risk-free return) that is required or expected for a particular return) that is required or expected for a particular security given its level of risk:security given its level of risk:

Reward to risk ratio Reward to risk ratio

• This excess return (E(rThis excess return (E(rAA) – r) – rff) is referred to as the ) is referred to as the asset’s asset’s risk premium, risk premium, which is the return investors which is the return investors require beyond that of the risk free rate for security Arequire beyond that of the risk free rate for security A

A

fA rrE

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Capital Asset Pricing Model Capital Asset Pricing Model (CAPM)(CAPM)• The CAPM assumes that the reward-to-risk ratio of all The CAPM assumes that the reward-to-risk ratio of all

securities are equal….securities are equal….giving us the following giving us the following model to estimate the expected or required model to estimate the expected or required return on a stockreturn on a stock::

where where • rrf f is the return on the risk-free security and is often proxied is the return on the risk-free security and is often proxied

by the 3-month U.S. Treasury Bill or Treasury Bond rate; by the 3-month U.S. Treasury Bill or Treasury Bond rate;

• E(rE(rmm) ) is the expected return on the market portfolio and is is the expected return on the market portfolio and is often proxied by the return on the S&P 500 index and;often proxied by the return on the S&P 500 index and;

• E(rE(rmm) - r) - rf f represents the represents the market risk premiummarket risk premium

fmAfA rrErrE )()(

Jensen’s alpha (Jensen’s alpha (αα))

• Using the CAPM, and assuming that securities are priced based on Using the CAPM, and assuming that securities are priced based on this model, one can measure whether a particular security this model, one can measure whether a particular security performed better or worse than expected by this model, i.e., did the performed better or worse than expected by this model, i.e., did the security provide a return greater than that required and expected security provide a return greater than that required and expected by investors?by investors?

• This performance measure is called This performance measure is called Jensen’s alpha Jensen’s alpha and is and is calculated as follows:calculated as follows:

where where kkAA represents the represents the actual return actual return achieved by security A and achieved by security A and E(kE(kAA) ) represents the expected return based on CAPM.represents the expected return based on CAPM.

AA rEr

The Security Market Line and The Security Market Line and Jensen’s alphaJensen’s alpha

E(r)

BETA

rf

SML

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1 βA

E(rA)

rA

Jensen’s alpha

CAPM assumptionsCAPM assumptions

• The CAPM relies on historical data to The CAPM relies on historical data to calculate its inputs, thus it implicitly calculate its inputs, thus it implicitly assumes that the past is a good measure assumes that the past is a good measure of the futureof the future

• The model assumes no transaction costs, The model assumes no transaction costs, identical and complete information, identical and complete information, rational investors and that securities that rational investors and that securities that are mispriced will self-adjust. These are are mispriced will self-adjust. These are all all efficient market assumptions.efficient market assumptions.

CAPM limitationsCAPM limitations

• Theoretically, the market portfolio Theoretically, the market portfolio should consist of all assets.should consist of all assets.

• Studies have shown that additional Studies have shown that additional explanatory (risk) factors must be explanatory (risk) factors must be considered in explaining security considered in explaining security returns.returns.