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Risk, Return, and

Portfolio Management

• Measures of Risk and Return

• Portfolio Construction and

Management

• Capital Markets and Pricing of Risk

• Optimal Portfolio Choice

• Capital Asset Pricing Model

Value = + + + FCF1 FCF2 FCF∞

(1 + WACC)1 (1 + WACC)∞ (1 + WACC)2

Free cash flow (FCF)

Market interest rates

Firm’s business risk Market risk aversion

Firm’s debt/equity mix Cost of debt

Cost of equity

Weighted average cost of capital

(WACC)

Net operating profit after taxes

Required investments in operating capital

−

=

Determinants of Intrinsic Value: The Cost of Equity

...

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What is investment risk?

• Two types of investment risk

– Stand-alone risk

– Portfolio risk

• Investment risk is related to the probability of

earning a low or negative actual return.

• The greater the chance of lower than

expected, or negative returns, the riskier the

investment.

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Probability Distributions

• A listing of all possible outcomes, and the

probability of each occurrence.

• Can be shown graphically.

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Expected Rate of Return

Rate of

Return (%) 100 15 0 -70

Firm X

Firm Y

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Selected Realized Returns, 1926-2010

Source: Based on Ibbotson Stocks, Bonds, Bills, and Inflation: 2011

Classic Yearbook (Chicago: Morningstar, Inc., 2011), p. 32.

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Value of $100 Invested at the End of 1925 in U.S.

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What are investment returns?

• Investment returns measure the financial results

of an investment.

• Returns may be historical or prospective

(anticipated).

• Returns can be expressed in:

– Dollar terms.

– Percentage terms.

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An investment costs $1,000 and is sold after

1 year for $1,100.

Dollar return:

Percentage return:

$ Received - $ Invested

$1,100 - $1,000 = $100.

$ Return/$ Invested

$100/$1,000 = 0.10 = 10%.

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How Does One Measure Returns?

• Realized (Historical) Returns

– The return that actually occurs over a particular time

period.

Rate Gain Capital YieldDividend

1

t

tt

t

t

t

ttt

P

PP

P

Div

P

PDivR

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Hypothetical Investment Alternatives

Economy Prob. T-Bills HT Coll USR MP

Recession 0.1 5.5% -27.0% 27.0% 6.0% -17.0%

Below avg 0.2 5.5% -7.0% 13.0% -14.0% -3.0%

Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%

Above avg 0.2 5.5% 30.0% -11.0% 41.0% 25.0%

Boom 0.1 5.5% 45.0% -21.0% 26.0% 38.0%

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Why is the T-bill return independent of the economy?

Do T-bills promise a completely risk-free return?

• T-bills will return the promised 5.5%, regardless of

the economy.

• No, T-bills do not provide a completely risk-free

return, as they are still exposed to inflation.

Although, very little unexpected inflation is likely to

occur over such a short period of time.

• T-bills are also risky in terms of reinvestment risk.

• T-bills are risk-free in the default sense of the word.

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How do the returns of High Tech and

Collections behave in relation to the market?

• High Tech: Moves with the economy, and

has a positive correlation. This is typical.

• Collections: Is countercyclical with the

economy, and has a negative correlation.

This is unusual.

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Calculating the Expected Return

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12.4%

(0.1)(45%)(0.2)(30%)

(0.4)(15%)(0.2)(-7%)-27%))(1.0(ˆ

r̂

return of rate Expectedˆ

N

1i

r

rP

r

ii

Summary of Expected Returns

Expected Return

High Tech 12.4%

Market 10.5%

US Rubber 9.8%

T-bills 5.5%

Collections 1.0%

High Tech has the highest expected return, and appears

to be the best investment alternative, but is it really?

Have we failed to account for risk?

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Calculating Standard Deviation

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N

1ii

2

2

P)r̂r(

Variance

deviation Standard

Standard Deviation for Each Investment

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%0.0

)1.0()5.55.5(

)2.0()5.55.5()4.0()5.55.5(

)2.0()5.55.5()1.0()5.55.5(

)ˆ(

bills-T

2/1

2

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22

bills-T

1

2

N

i

iPrr

σM = 15.2% σUSR = 18.8%

σColl = 13.2% σHT = 20%

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Comparing Standard Deviations

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USR

Prob. T-bills

HT

0 5.5 9.8 12.4 Rate of Return (%)

Comments on Standard Deviation as a

Measure of Risk

• Standard deviation (σi) measures total, or

stand-alone, risk.

• The larger σi is, the lower the probability that

actual returns will be close to expected

returns.

• Larger σi is associated with a wider

probability distribution of returns.

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Comparing Risk and Return

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Security Expected Return, Risk,

T - bills 5 .5% 0.0% High Tech 12.4 20.0 Collections* 1.0 13.2 US Rubber* 9.8 18.8 Market 10.5 1 5.2

*Seems out of place.

r̂

Coefficient of Variation (CV)

• A standardized measure of dispersion about

the expected value, that shows the risk per

unit of return.

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r̂return Expected

deviation StandardCV

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Illustrating the CV as a Measure of Relative Risk

σA = σB , but A is riskier because of a larger probability

of losses. In other words, the same amount of risk (as

measured by σ) for smaller returns.

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0

A B

Rate of Return (%)

Prob.

Risk Rankings by Coefficient of Variation

CV

T-bills 0.0

High Tech 1.6

Collections 13.2

US Rubber 1.9

Market 1.4

• Collections has the highest degree of risk per unit of

return.

• High Tech, despite having the highest standard

deviation of returns, has a relatively average CV.

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Investor Attitude Towards Risk

• Risk aversion: assumes investors dislike

risk and require higher rates of return to

encourage them to hold riskier securities.

• Risk premium: the difference between the

return on a risky asset and a riskless asset,

which serves as compensation for investors

to hold riskier securities.

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Portfolio Construction: Risk and Return

• Assume a two-stock portfolio is created with

$50,000 invested in both High Tech and

Collections.

• A portfolio’s expected return is a weighted

average of the returns of the portfolio’s

component assets.

• Standard deviation is a little more tricky and

requires that a new probability distribution for

the portfolio returns be constructed.

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Calculating Portfolio Expected Return

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%7.6%)0.1(5.0%)4.12(5.0ˆ

ˆˆ

:average weighteda is ˆ

1

p

N

i

iip

p

r

rwr

r

An Alternative Method for Determining

Portfolio Expected Return

Economy Prob HT Coll Port

Recession 0.1 -27.0% 27.0% 0.0%

Below avg 0.2 -7.0% 13.0% 3.0%

Average 0.4 15.0% 0.0% 7.5%

Above avg 0.2 30.0% -11.0% 9.5%

Boom 0.1 45.0% -21.0% 12.0%

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6.7% (12.0%) 0.10 (9.5%) 0.20

(7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 r̂p

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Calculating Portfolio Standard Deviation and CV

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51.0%7.6

%4.3

%4.3

6.7) - (12.0 0.10

6.7) - (9.5 0.20

6.7) - (7.5 0.40

6.7) - (3.0 0.20

6.7) - (0.0 0.10 21

2

2

2

2

2

p

p

CV

Comments on Portfolio Risk Measures

• σp = 3.4% is much lower than the σi of either

stock (σHT = 20.0%; σColl = 13.2%).

• σp = 3.4% is lower than the weighted

average of High Tech and Collections’ σ

(16.6%).

• Therefore, the portfolio provides the average

return of component stocks, but lower than

the average risk.

• Why? Negative correlation between stocks.

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General Comments about Risk

• σ 35% for an average stock.

• Most stocks are positively (though not

perfectly) correlated with the market (i.e., ρ

between 0 and 1).

• Combining stocks in a portfolio generally

lowers risk.

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Correlation

• The degree to which the stocks face common

risks and their prices move together.

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Returns Distribution for Two Perfectly

Negatively Correlated Stocks (ρ = -1.0)

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Returns Distribution for Two Perfectly

Positively Correlated Stocks (ρ = 1.0)

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Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

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Partial Correlation, ρ = +0.35

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Creating a Portfolio: Beginning with One Stock and

Adding Randomly Selected Stocks to Portfolio

• σp decreases as stocks are added, because

they would not be perfectly correlated with the

existing portfolio.

• Expected return of the portfolio would remain

relatively constant.

• Eventually the diversification benefits of

adding more stocks dissipates (after about 40

stocks), and for large stock portfolios, σp

tends to converge to 20%.

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Diversification in Stock Portfolios

# Stocks in Portfolio 10 20 30 40 2,000+

Diversifiable Risk

Market Risk

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Firm Specific Risk, p

p (%)

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Breaking Down Sources of Risk

Stand-alone or total risk

= Market risk + Diversifiable risk

• Market risk: portion of a security’s stand-

alone risk that cannot be eliminated through

diversification. Measured by beta.

• Diversifiable risk: portion of a security’s

stand-alone risk that can be eliminated

through proper diversification.

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Failure to Diversify

• If an investor chooses to hold a one-stock

portfolio (doesn’t diversify), would the investor

be compensated for the extra risk they bear? – NO!

– Stand-alone risk is not important to a well-

diversified investor.

– Rational, risk-averse investors are concerned with

σp, which is based upon market risk.

– There can be only one price (the market return) for

a given security.

– No compensation should be earned for holding

unnecessary, diversifiable, risk.

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The Capital Asset Pricing Model (CAPM)

William F. Sharpe

b. 1934

Nobel Laureate, 1990

The risk premium for

any security is

proportional to its beta

with the market

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Capital Asset Pricing Model (CAPM)

• Model linking risk and required returns. CAPM

suggests that there is a Security Market Line

(SML) that states that a stock’s required return

equals the risk-free return plus a risk premium

that reflects the stock’s risk after diversification.

ri = rRF + (rM – rRF)βi

ri = rRF + (RPM)βi

• Primary conclusion: The relevant riskiness of a

stock is its contribution to the riskiness of a well-

diversified portfolio.

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Beta

• Measures a stock’s market risk, and shows a

stock’s volatility relative to the market.

• Indicates how risky a stock is if the stock is

held in a well-diversified portfolio.

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Comments on Beta

• If beta = 1.0, the security is just as risky as

the average stock.

• If beta > 1.0, the security is riskier than

average.

• If beta < 1.0, the security is less risky than

average.

• Most stocks have betas in the range of 0.5 to

1.5.

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How is Beta Interpreted?

EXCESS RETURN

ON STOCK

EXCESS RETURN

ON MARKET PORTFOLIO

Beta < 1

(defensive)

Beta = 1

Beta > 1

(aggressive)

Each characteristic

line has a

different slope.

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Capital Asset Pricing Model (CAPM)

• CAPM allows us to find the required returns for a given level of risk.

• SML shows relationship between risk (measured by asset’s covariance with market) and its required return.

• Risk measure for market is market’s covariance with itself, i.e., its variance, Var (RM).

• For any risky asset, required return = RF + risk premium which is based on the covariance of the asset with the market.

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Estimating the Required Rate of Return

• Estimating the required return takes two steps:

1. Measure the investment’s systematic risk, β

2. Determine the risk premium required to compensate

for that amount of systematic risk

• Systematic risk depends on the beta of the

security.

– Beta: a measure of the sensitivity of a stock’s returns

relative to the market returns.

R Rit i i Mt

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Estimating the Required Return (cont’d)

rj = rF + βj(rM - rF)

M = 1.0

Systematic Risk (Beta)

rF

rM

Req

uir

ed

Retu

rn

Risk

Premium

SML • Market Risk Premium

– Historical average

excess return of the

market over rF

– Researchers

generally report

estimates of 4–8%

for the future equity

risk premium.

– SML: Security Mkt Line

)(

),(

)(

),( )(

market h thecommon wit is that of Volatility

i

Mkt

Mkti

Mkt

i

MktiiMkt

iRVar

RRCov

RSD

RRCorrRSD

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Example on Required Return

• Estimate the required rate of return for a stock

with a beta of 1.14 when the risk-free rate is 8%

and an expected market return of 14%.

rF=8%

rM=14%

M = 1.0

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Can the beta of a security be negative?

• Yes, if the correlation between Stock i and

the market is negative (i.e., ρi,m < 0).

• If the correlation is negative, the regression

line would slope downward, and the beta

would be negative.

• However, a negative beta is highly unlikely.

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Calculating Betas

• Well-diversified investors are primarily concerned

with how a stock is expected to move relative to the

market in the future.

• Without a crystal ball to predict the future, analysts

are forced to rely on historical data. A typical

approach to estimate beta is to run a regression of

the security’s past returns against the past returns of

the market.

• The slope of the regression line is defined as the

beta coefficient for the security.

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Illustrating the Calculation of Beta

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.

.

.

ri

_

rM -5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ri = -2.59 + 1.44 rM ̂ ̂

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

Beta Coefficients for High Tech, Collections,

and T-Bills

50

rM

ri

-20 0 20 40

40 20

-20

HT: b = 1.32

T-bills: b = 0

Coll: b = -0.87

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Web Sites for Beta

• http://finance.yahoo.com

– Enter the ticker symbol for a “Stock Quote”, such as

IBM or Dell, then click GO.

– When the quote comes up, select Key Statistics from

panel on left.

– Can also download historical prices to compute returns.

• www.valueline.com

– Enter a ticker symbol at the top of the page.

• Most stocks have betas in the range of 0.5 to 1.5.

Comparing Expected Returns and Beta Coefficients

Security Expected Return Beta

High Tech 12.4% 1.32

Market 10.5 1.00

US Rubber 9.8 0.88

T-Bills 5.5 0.00

Collections 1.0 -0.87

Riskier securities have higher returns, so the rank

order is OK.

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The Security Market Line (SML): Calculating

Required Rates of Return

SML: ri = rRF + (rM – rRF)βi

ri = rRF + (RPM)βi

• If the expected return on the market is

10.5% and the risk free rate is 5.5%, the

market risk premium is

RPM = rM rRF = 10.5% 5.5% = 5.0%.

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What is the market risk premium?

• Additional return over the risk-free rate

needed to compensate investors for

assuming an average amount of risk.

• Its size depends on the perceived risk of the

stock market and investors’ degree of risk

aversion.

• Varies from year to year, but most estimates

suggest that it ranges between 4% and 8%

per year.

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Calculating Required Rates of Return

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rHT = 5.5% + (5.0%)(1.32) = 5.5% + 6.6% = 12.10%

rM = 5.5% + (5.0%)(1.00) = 10.50%

rUSR = 5.5% +(5.0%)(0.88) = 9.90%

rT-bill = 5.5% + (5.0)(0.00) = 5.50%

rColl = 5.5% + (5.0%)(-0.87) = 1.15%

r

High Tech 12.4% 12.1% Undervalued

Market 10.5 10.5 Fairly valued

US Rubber 9.8 9.9 Overvalued

T-bills 5.5 5.5 Fairly valued

Collections 1.0 1.15 Overvalued

Expected vs. Required Returns

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r̂

)rr̂(

)rr̂(

)rr̂(

)rr̂(

)rr̂(

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Illustrating the Security Market Line

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. .

Coll

. HT

T-bills

. USR

SML

rM = 10.5

rRF = 5.5

-1 0 1 2

.

SML: ri = 5.5% + (5.0%)βi

ri (%)

Risk, bi

.

An Example:

Equally-Weighted Two-Stock Portfolio

• Create a portfolio with 50% invested in High

Tech and 50% invested in Collections.

• The beta of a portfolio is the weighted

average of each of the stock’s betas.

βP = wHTβHT + wCollβColl

βP = 0.5(1.32) + 0.5(-0.87)

βP = 0.225

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Calculating Portfolio Required Returns

• The required return of a portfolio is the weighted average of each of the stock’s required returns.

rP = wHTrHT + wCollrColl

rP = 0.5(12.10%) + 0.5(1.15%)

rP = 6.625% • Or, using the portfolio’s beta, CAPM can be

used to solve for expected return.

rP = rRF + (RPM)βP

rP = 5.5% + (5.0%)(0.225)

rP = 6.625%

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Computing a Portfolio’s Variance and Volatility

• Variance of a Two-Stock Portfolio

),(2 )( )( )( 21212

2

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2

1 rrCovwwrVarwrVarwrVar P

Exercise: Portfolio application problem

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Portfolio Application Problem

• F&N (F) has an expected return of 10% and a

standard deviation of 20%; CapitaLand (C) has an

expected return of 16% and a standard deviation

of 40%. If the correlation between F&N and

CapitaLand is 0.4, what are the expected return

and standard deviation for a portfolio comprised of

30% in F&N and 70% in CapitaLand?

Factors That Change the SML

• What if investors raise inflation expectations

by 3%, what would happen to the SML?

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SML1

ri (%) SML2

0 0.5 1.0 1.5

13.5

10.5 8.5

5.5

ΔI = 3%

Risk, bi

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Factors That Change the SML

• What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML?

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SML1

ri (%) SML2

0 0.5 1.0 1.5

ΔRPM = 3%

Risk, bi

13.5

10.5

5.5

Verifying the CAPM Empirically

• The CAPM has not been verified completely.

• Statistical tests have problems that make

verification almost impossible.

• Some argue that there are additional risk

factors, other than the market risk premium,

that must be considered.

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More Thoughts on the CAPM

• Investors seem to be concerned with both

market risk and total risk. Therefore, the SML

may not produce a correct estimate of ri.

ri = rRF + (rM – rRF)bi + ???

• CAPM/SML concepts are based upon

expectations, but betas are calculated using

historical data. A company’s historical data

may not reflect investors’ expectations about

future riskiness.

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Questions and Answers