FIN 40153: Advanced Corporate FinanceFall 2012

RISK, RETURN, & CAPM

BASED ON RWJ CHAPTERS (10 &11)

There is general agreement that average returns should increase with risk.

AverageReturn

Risk

But, how should risk be measured? At what rate does the line slope up? Is the relation linear?

Lets look at some key historical evidence.

How are risk and expected return related?

The Accumulated Value of $1 Invested in 1926

0.1

10

1000

1930 1940 1950 1960 1970 1980 1990 2000

Common StocksLong T-BondsT-Bills

$59.70

$17.48

34.1775$)1()1()1(1$ 200219271926 rrr

Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

$59.70

Rates of Return 1926-2002

-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 95 '00

Common Stocks

Long T-Bonds

T-Bills

Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

How Can We Assess Risk?

Many people think intuitively about risk as the possibility of an outcome which is worse than anticipated.

For those who hold more than one asset, is it the risk of each asset they care about, or the risk of the portfolio?

Historical Volatility and Return for 500 Individual Stocks by Size

Appears to be a relation between portfolios volatilities and returns but no precise relationship between volatility and average return for individual stocks. Individual stocks have higher volatility and lower average returns compared to large portfolios. WHY?

Risk and Probability Distributions

The Expected Return Is A Weighted Sum Of The Possible Returns, where Each Return Is Weighted By Its Probability Of Occurrence, p.

The Variance of Returns Measures the dispersion of returns around the expectation.

The Standard Deviation (STD or ) Is The Square Root Of The Variance. It Is In The Same Units As The Returns

n

iii pRREturnExpected

1

][Re

n

iii RERpVariance

1

22 ])[(

Calculating Sample Statistics What if we don’t know the probabilities. We can estimate the

variance and expected return using the arithmetic mean of past returns and the sample variance.

We can use historical data

Mean = R = (R1 + R2 + R3 + ... + RT)/T

Sample Variance = 2 = "Average" of [RT - R]2

T

1t

2_

t2 )R(R

1T1

Var

Risk and Return in Portfolios

When we have more that one asset (i.e. a portfolio) risk and return are more complex. We will show that there are two kinds of risk: Diversifiable/idiosyncratic/nonsystematic risk Nondiversifiable/systematic risk

We can eliminate diversifiable risk by carefully combining assets into portfolios.

It is usually possible to improve the risk return tradeoff by

diversifying.

Amazon vs. S&P 500

Covariances and Correlations: The keys to Understanding Diversification.

• When thinking in terms of probability distributions, the covariance between two assets’ (A and B) returns is

Covariance = Cov(AB) = AB=

• When estimating covariances from historical data, the estimate is given by:

p])E[R-R])(E[R-R( iBBiAAi

n

1=i

)R-R)(R-R(1T

1BBiAAi

T

1=i

Correlation Coefficients

• Covariances are difficult to interpret. Is a covariance of 20 big or small? The correlation coefficient is a normalized version of the covariance given by:

• Correlation = CORR(A,B) =

• The correlation will always lie between 1 and -1.

– A correlation of 1.0 implies ...

– A correlation of -1.0 implies ...

– A correlation of 0.0 implies ...

BA

AB

BA

=

Cov(AB)

Return (%)Probability A B P

0.2 18 25 21.50.2 30 10 200.2 -10 10 00.2 25 20 22.50.2 17 -5 6

E(Return) 16 12 14

Risk and Return Example:Two Assets, A and BA portfolio, P, comprised of 50% of funds in A and 50% of funds in B.Only five equally probable future outcomes, summarized below.

In this case:VAR(A) = 191.6 and STD(A) = 13.84.VAR(B) = 106.0 and STD(B) = 10.29.COV(A,B) = 21CORR(A,B) = 21/(13.84*10.29) = .1475.VAR(P) = 84.9 and STD(P) = 9.21 -- less than either component!

In general for portfolios

The expected return on a portfolio is the weighted average of the expected returns on each asset. If xi is the proportion of the portfolio in asset j, then

]E[Rw = ]RE[ ii

n

1=ip

Note that this is a ‘linear’ relationship.

Variance of a Two-Asset Portfolio

For a portfolio of two assets, A and B, the portfolio variance is:

ABBA2B

2B

2A

2A

2p ww2+w+w = VariancePortfolio

For the two-asset example considered above:Portfolio Variance = .52(191.6) + .52(106.0) + 2(.5)(.5)21

= 84.9 (check)

or,

BABA2B

2B

2A

2A

2p BAcorrww2+w+w = VariancePortfolio ),(

The Variance of an n-asset Portfolio

Consider the two asset case again ( N = 2) and wi , j = 1/2

2p = (1/2)2 cov(R1, R1) + (1/2)(1/2)cov(R1, R2) + (1/2)(1/2)cov(R2, R1) + (1/2)2 cov(R2, R2)

When i = j the cov(Ri, Rj) = var(Ri)

= (1/2)2 var(R1) + (1/2)2 var(R2) + 2(1/2)(1/2)cov(R1, R2)

Same as previous slide

= w1w1cov(R1R1) + w1w2cov(R1R2) + w1w3cov(R1R3) +…+ w2w1cov(R2R1) + w2w2cov(R2R2) + w2w3cov(R2R3) + … + w3w1cov(R3R1) +

w3w2cov(R3R2) + w3w3cov(R3R3) + ….

So there will be n variance terms and n2 – n covariance terms when calculating the variance of a portfolio

Risk in n-asset Portfolios

n variances - one per asset

(n2 – n) covariances - one between each pair of assets.

+

What happens when the number of risky assets in the portfolio, n, becomes large?

This is easiest to demonstrate if we assume: that all the assets have the same variance of return, VAR. The covariance between each pair of assets is the same, COV. The portfolio is equal-weighted (each asset has a weight 1/n)

Then, Portfolio Return Variance is: (1/n)VAR + (1 - 1/n)COV What happens if n approaches infinity? Graphical Illustration:

020406080

100120

0 20 40 60

# of Assets

Po

rtfo

lio V

ari

an

ce

CORR=1.0

CORR=0.5

CORR=0.25

CORR=0.0

What happens when the number of assets in the portfolio, n, becomes large?

DIVERSIFICATION ELIMINATES UNIQUE RISK

deviationstandardPortfolio

Nonsystematic risk

Systematic/Market risk

Number ofsecurities

5 10

Diversification

Diversification reduces risk. If asset returns were uncorrelated on average diversification could eliminate risk. But, they are actually positively correlated on average. Diversification will reduce risk but will not remove all of

the risk. So, There are two kinds of risk

Diversifiable/nonsystematic risk Disappears in well diversified portfolios

Nondiversifiable/systematic risk Does not disappear in well diversified portfolios

Summary statements regarding risk and return for a portfolio. We will show:

(i) The expected return on a portfolio is always a weighed average of the expected returns on the portfolio's components.

(ii) The risk of a portfolio's return, as measured by standard deviation, is generally less that the weighted average of the risks of the portfolio’s components .

(iii) Risk generally declines when new assets are added to a portfolio. Risk declines more: The lower the new asset's standard deviation, The lower the correlations between the new asset's

payoffs and payoffs to the various existing assets.

Nonsystematic/diversifiable risks

Examples Firm discovers a gold mine beneath its property Lawsuits Technological innovations Strikes

The key is that these events are random and unrelated across firms. Some surprises are positive, some are negative. On average, the surprises offset each other

BIOGEN and MS Drug Failure

Eli Lilly Lawsuit with Ariad

Share Price Falls about $0.50 on Announcement (Close to Open). $0.50 x 1B Shares Outstanding = $500M

Ariad Pharm

Stock Went Up By $2. $2 x 62 M Shares = $124 M

Systematic/Nondiversifiable risk We know that the returns on different assets are positively

correlated with each other on average. This suggests that economy-wide influences affect all assets.

Examples: Business Cycle Inflation Shocks Productivity Interest rates

These are economic events that affect all assets. The risk associated with these events is systematic, and does not disappear in well diversified portfolios.

GE, DJI, S&P 500, & NASDAQ

Measuring Systematic Risk

How can we estimate the amount or proportion of an asset's risk that is diversifiable or non-diversifiable?

Incremental risk a new asset adds to the risk of a portfolio is through its Beta Coefficient:

)Var(R)R,Cov(R

M

Mii

As such, Beta is a measures of sensitivity: it describes how strongly the stock return moves with the market.

Example: A Stock with =2 will on average go up 20% when the market goes up 10%, and vice versa.

Some Additional Insight: Restate Beta in terms of correlations.

Since CORR(Ri,Rm) = COV(Ri,Rm)/STD(Rm)STD(Ri), the market beta can be restated as:

i = STD(Ri)CORR(Ri,Rm)/STD(Rm).

Interpret the components this way: STD(Ri) is a measure to the total risk of asset i.

CORR(Ri,Rm) is a measure of the proportion of asset i's risk that is systematic.

STD(Ri)CORR(Ri,Rm) measures the systematic risk of asset i.

STD(Rm) measures the total risk of the market, all of which is systematic.

So, i is the systematic risk of asset i, relative to the systematic risk of the market.

This also implies that the average of all betas is 1.0

What is Home Depots Beta?

Negative BetaCitigroup Inc

Betas and Portfolios Also, the Beta of a portfolio is the weighted average of the

component assets’ Betas. Example: You have 30% of your money in Asset X, which

has X = 1.4 and 70% of your money in Asset Y, which has Y = 0.8.

Your portfolio Beta is: P = .30(1.4) + .70(0.8) = 0.98.

Why do we care about this feature of betas?

It implies that beta measures the contribution of each asset to the risk of a well-diversified portfolio!

Betas for selected common stocks (based on monthly data for 2000 – 2005)

Industry Beta Ticker Firm Beta

Gold & Silver -0.04 NEM Newmont Mining Corp 0.02

Personal & Household Products 0.25 PG Proctor Gamble 0.19

Utilities 0.48 EIX Edison International 0.50

Restaurants 0.69 SBUX Starbucks Corp 0.60

Retail Grocery 0.74 SWY Safeway 0.67

Forestry & Wood Products 0.95 WY Weyerhaeuser Company 0.96

Recreational Products 1.00 HDI Harley-Davidson Inc 1.14

Apparel 1.12 LIZ Liz Claiborne Inc 0.90

Auto & Truck Manufacturers 1.44 GM General Motors 1.20

Computer Hardware AAPL Apple Computer Inc 1.35

Software and Programming 1.74 ADBE Adobe Systems 1.84

Computer Services 1.77 YHOO Yahoo! Inc 2.80

Communications Equipment 2.20 CSCO Cisco Systems Inc 2.28

Semiconductors 2.59 AMDINTC

Advanced Micro Devices IncIntel Corporation

3.232.17

The Capital Asset Pricing Model (CAPM)

Given that some risk can be diversified, diversification is easy, most investors do diversify,

What is the equilibrium relation between risk and expected return in the capital markets?

The CAPM is the best-known and most-widely used equilibrium model of the risk/return relation.

CAPM Intuition

E[Ri] = RF (risk free rate) + Risk Premium

= Appropriate Discount Rate Risk free assets earn the risk-free rate (think of this

as a rental rate on capital). If the asset is risky, we need to add a risk premium. The size of the risk premium depends on the

amount of systematic risk for the asset (stock, bond, or investment project).

Could a risk premium ever be negative?

The CAPM Intuition Quantified

]R][E[R)Var(R

)R,Cov(RR]E[R FM

M

MiFi

]R][E[RR]E[R FMiFi

The expression above is referred to as the “Security Market Line” (SML).

Amount of Risk ()Market Risk Premium

or,

The Security Market Line

Expected Return (%)

Risk: i1.0

SML: Ri = RF + i( E[RM] - RF )

RF

RM

Expected Rates of ReturnTicker Firm Beta Expected Return

NEM Newmont Mining Corp 0.02 4.2% + 0.20×5% = 5.2%

PG Proctor Gamble 0.19 4.2% + 0.19×5% = 5.15%

EIX Edison International 0.50 6.70%

SBUX Starbucks Corp 0.60 7.20%

SWY Safeway 0.67 7.55%

WY Weyerhaeuser Company 0.96 9.00%

HDI Harley-Davidson Inc 1.14 9.90%

LIZ Liz Claiborne Inc 0.90 8.70%

GM General Motors 1.20 10.20%

AAPL Apple Computer Inc 1.35 10.95%

ADBE Adobe Systems 1.84 13.40%

4.20%

YHOO Yahoo! Inc 2.80 18.20%

CSCO Cisco Systems Inc 2.28 15.60%

AMDINTC

Advanced Micro Devices IncIntel Corporation

3.232.17

20.35%15.05%

Assumption Rf = 4.2% and MRP = 5%