The Relationship Between Risk and Return

Goal of Financial Management:

Maximize the value of the firm as determined by: the present value of its expected cash flows, discounted back at a rate that reflects both the riskiness of the firms projects and the financing mix used to fund the projects.

Goals of Risk Analysis

A good risk and return modelShould apply to all assetsExplain the types of risk that are rewardedDevelop standardized risk measuresTranslate risk into a rate of return demanded by the investorShould do well explaining past returns and forecasting future returns.

Issues Relating to Risk

Riskiness of the expected future cash flowsStand Alone vs. Portfolio RiskDiversifiable Risk vs. Non-Diversifiable (Market) RiskHigher Market Risk Implies Higher ReturnThe same principles apply to physical assets

Stand Alone Risk

The risk faced from owning the asset by itself. (there are no other assets which help to spread risk)

The return from owning the asset varies based on outcomes in the market

Need to look at the expected return and standard deviation

Probability Distributions

Probability Distribution provides the combinations of outcomes and the probability that the outcome will occur

Example: Weather Forecast Outcome Probability Rain .6 = 60% No Rain .4 = 40%

Probability

The probability tells the likelihood that it will rain. The probability is based upon the current conditions. Given 100 days with the current conditions (the history), it will rain on 60 of the following days.We want to use the same logic when discussing the possible return from owning the stock - what is the history?

An Example

Intel has decided to introduce its new computer chip. There are three possible outcomes and three possible returns

Outcome Return Prob 1) High Demand 90% 40% 2) Average Demand 30% 20% 3) Low Demand -80% 40%

Example Continued

Assume MidAmerican Energy is also facing three outcomes

Outcome Return Prob 1) High Demand 15% 25% 2) Average Demand 10% 50% 3) Low Demand 5% 25% How would you compare the two stocks?

Expected Rate of Return

To compare the two stocks you would need to find the expected rate of return

n

1ttt ))(Return(Prob

Intel

Demand Ret Prob Ret x Prob High 90% 40% (.9)(.4) =

.36 Average 30% 20% (.3)(.2) = .06 Low -80% 40% (-.8)(.4) = -.32 expected return 10%

Mid American Energy

Demand Ret Prob Ret x Prob High 15% 25% (.15)(.25) = .0375 Average 10% 50% (.1)(.5) = .0500

Low 5% 25% (.25)(.05) = .0125 expected return 10%

The expected return for each stock is 10% Which would you prefer to own?

Measuring Stand Alone Risk

To compare the stand alone risk you need to look at the standard deviation:

To calculate Standard Deviation:1) Find Expected return2) Subtract expected return from each

outcomes return3) Square the number in 2) 4) Multiply the squares by the probabilities and

sum them together5) Take the square root of the number in 4

Intel

Demand (Ret-ExpectRet)2 x Prob High (90% - 10%)2 x (.40) = .2560 Average (30% - 10%)2 x (.20) = .0080 Low (-80% - 10%)2 x (.40) = .3240 .5880 take the square root (.5880)1/2

standard deviation = .7668 =76.68%

Mid American Energy

Demand (Ret-ExpectRet)2 x Prob High (15%-10%)2 x (.25) = .000625 Average (10%- 10%)2 x (.50) = 0.00 Low (5% - 10%)2 x (.25) = .000625 .00125 take the square root (.00125)1/2

standard deviation = .035355 = 3.54%

Interpreting Standard Deviation

What does the standard deviation tell us?Assuming that the returns are normally distributed:

The actual return will be within one standard deviation 68.26% of the time.This means that we can expect the return to fall in a range between the expected return plus and minus the standard deviation 68% of the time

Prob Ranges for Normal Dist.

68.26%95.46%99.74%

Our Example

Intel had an expected return of 10% and standard deviation of 76.64%. Therefore we expect the return to be between 10-76.64 = - 66.64% and 10+76.64 = 86.64% 68% of the time

Mid American Energy had an expected return of 10% and standard deviation of 3.536 implying an interval form 6.464% to 13.536%

Which would you rather own?

Trade off Between Risk and Return

Avg Ret

Stnd DevExcess Ret

Small Co Stocks

17.6% 33.6% 12.1%

Large Co Stocks

13.3% 20.1% 7.8%

L-T Corp Bonds 5.9% 8.7% 0.4%

L-T Gov Bonds 5.5% 9.3%

US Treas Bills 3.8% 3.2%

Inflation 3.2% 4.5%

Risk Aversion

Generally, people are risk averse. (They avoid risk)In our example the expected return is the same for both stocks, but Intel is much riskier (as measured by the standard deviation)What if the expected returns were not the same?

Which do you prefer?

Project A expected return of 50% with standard deviation of 30%

Project B expected return of 8% with standard deviation of 15%

Coefficient of Variation

The amount of risk per unit of return which is equal to:

Calculating the Coefficient of Variation:Project A 30/50 = .6 Project B 15/8 =

1.875

Return Expected

Deviation Standard

Semi Variance

If stocks are normally distributed they are symmetric about the mean.This teats upside and downside risk equally. An investor is often more concerned about the chance that a return falls below what is expected – or in other words the downside risk.

Semi Variance

n

tn

1

2t Return) Average(R

Where: n = number of periods where actual return<average return

Sources of Risk

Project Risk – Factors influencing the realized cash flows of the project – error in estimation Competitive Risk – Cash flows impacted by the actions of a competitorIndustry-Specific Risk – Technology, Legal, and Commodity RiskInternational risk – Political risk and exchange rate riskMarket Risk – Impacts all firms, marcoeconomic changes such as inflation and interest rates

Risk Intuition

Diversification – It is possible to decrease the impact of some of the risks through diversification.

Example: Project risk can be offset by other projects undertaken by the firm.

Which of the risks on the previous slide can be diversified? Which Can’t?As an investor, which risks should you be more concerned with (which can be diversified)?

Risk and Diversification

ProjectRisk

Competitive Risk

Industry WideRisk

International

Risk

Market Risk

FirmSpecificAffects

One Firm

MarketRisk

Affects All

Firms

Multiple

Projects

Acquiring

Competitors

DiversifyingAcross Sectors

Diversifying Across

Countries

Cannot

Affect

Diversifying AcrossDomestic Firms &

Markets

Diversifying Globally

Diversifying Across

Asset Classes

Investors Can Mitigate Risk By:

Firm Can Reduce Risk By:

Quick Stats Review

Covariance:

Combines the relationship between the stocks with the volatility.

(+) the stocks move together (-) The stocks move opposite of each other

iBBiAAi PrrrrABCov ))(()(

Stats Review 2

Correlation coefficient: The covariance is difficult to compare when looking at different series. Therefore the correlation coefficient is used.

The correlation coefficient will range from -1 to +1

)/()( BAAB ABCov

Risk in a Portfolio Context

The expected return of a portfolio of assets is equal to the weighted average of the expected returns of the individual assets.Example four stocks 25% of your $ in each

Intel 25% Disney 10% BP 15% Citicorp 16%

Portfolio Expect Ret (.25)(.25)+(.1)(.25)+(.15)(.25)+(.16)(.25)=.165

Standard Deviation

The standard deviation of the portfolio will not equal the weighted average of the standard deviations of the stocks in the portfolio.The standard deviation can be calculated from each years portfolio expected return just like for an individual asset.

Example 1

Two stocks with correlation coefficient = -1

Year Stock A Stock B Portfolio2004 26% -6% 10%2005 6% 14% 10%2006 -4% 24% 10%2007 12% 8% 10%Avg Ret 10% 10% 10%

Stand dev 10.86 10.86 0

Example 2

Two stocks with correlation coefficient =+1

Year Stock A Stock B Portfolio2004 16% 19% 17.5%2005 8% 7% 7.5%2006 12% 13% 12.5%2007 4% 1% 2.5%

Avg Ret 10% 10% 10%Stand dev 4.47 6.71 5.59

Example 3

Two stocks with correlation coefficient =+.571

Year Stock A Stock B Portfolio 2004 18% 22% 20% 2005 -4% 12% 4% 2006 24% 18% 21%

2007 2% -12% -5% Avg Ret 10% 10% 10% Stand dev 11.4 13.19 10.9

Real World

Most stocks have a correlation between +0.5 and +0.7

Why is it usually positive?

What type of risk does this represent?

Portfolio Effects

Each stock has two types of risk Market Related (Non diversifiable) Firm Specific (Diversifiable)

Increasing the number of stocks in your portfolio should increase the diversification, lowering the portfolio risk.However there is a limit to the decrease in risk, since most stocks are positively correlated you can not eliminate all of the market risk

Calculations of Standard Deviation

Variance and Standard Deviation can be calculated if you know the correlation coefficient and standard deviation of each asset.For two assets:

BAABAABBAAportfolio wwww )1(2)1( 22222

Marginal Investor

The investor “trading at the margin” who has the most influence on the price.The type of marginal investor plays a key role in determining how a firm may respond to different circumstancesUsually it is assumed that the marginal investor is well diversified.

Measuring Market Risk and The Market Portfolio

A market portfolio of all stocks available still has a positive standard deviation. The market portfolio would represent the return on the “average” stock.

Capital Asset Pricing Model

CAPM relates an assets market risk to the expected return from owning the asset.Major components:

Risk Free Rate - the return earned on an asset that is risk free (US Treasuries)Beta - A measure of the firms market risk compared to the “average” firmMarket Return - the expected return on a portfolio of all similar assets

Beta - Intuition

Beta measures the sensitivity of the individual asset to movements in the market for similar assets.Stock example:

Assume the S&P500 increases by 10% If a stock also increase by 10% over the same

period it would have a beta equal to 1. If a stock increases by more than 10% its beta

will be greater than 1.

Beta - Intuition

A higher beta implies that the stock is more sensitive to an economy wide fluctuation than the market portfolio.In other words the stock has a higher amount of Non-diversifiable risk.Since the Market risk for the stock is higher it should also have a higher return...

Risk and Return

The CAPM compares the return on the market portfolio to a risk free rate, the difference is the market risk premium.The Market Risk Premium represents the extra return for accepting the market risk related to the riskier asset (the extra return on the “average” stock).

CAPM

ri=rRF+Bi(rM-rRF)Where:ri = The return on asset i

rRF = The return on the Risk Free Asset

rM = The return on the Market Portfolio

Bi = the beta on asset i

ri=rRF+Bi(rM-rRF)

Example:Bank of America has a beta of 1.55

Let If rRF = 7% and rM = 9.2%

The return on Bank of America stock is:ri= rRF + Bi ( rM - rRF )r = .07 +1.55 (.092-.07) = .104

Market Risk Premium

The Market Risk Premium is the extra return from investing in the “average” stock. In the CAPM this is equal to rM-rRF

The market risk premium represents the market risk. If a stock had a beta of 1 it would earn

ri= rRF + Bi ( rM - rRF )r = .07 + 1.0 (.092-.07) = .092

which is the market return

Risk and Return

Given the inputs to the CAPM you can develop the relationship between the risk of an asset (as measured by beta) and its return.An easy way to demonstrate this is to graph the possible risk and return combinations.

Graphing the Security Market Line

ri= rRF + Bi ( rM - rRF )

Let risk (Bi) be on the horizontal axis and return (ri) be on the vertical axis.

The slope of the line is then equal to the market risk premium (rm-rRF)

Then you can graph all the possible combinations of risk and return.

ri= rRF + Bi ( rM - rRF )

Lets put in some numbers for beta and ki

beta = 0 ri = .07+0(.092-.07)=.07= rRF

beta = 1 ri = .07+1(.092-.07)=.092= rM

beta =1.55 ri = .07 +1.55(.092-.07) = .104

B=0,r=rRF B=1,r=0.092 B=1.55,r=.104

Beta

Return

rRF

0 1.0 1.55

0.092.104

Security MarketLine

Note:

The market risk premium measures the risk aversion of the investors. If investors become more risk averse the risk premium widens (investors require a higher return to accept risk)In this case the slope of the security market line will become steeper.

Increased Risk Aversion

rRF

Beta

Return

Bi

Estimating the Components of the CAPM

Risk Free Return Usually long Term treasury bonds are

used to approximate the risk free returnMarket return

The market return uses historical data on a market index, the S&P 500 is a commonly used

Estimating Beta

Two main approaches to estimating beta Historical Data (Top Down Beta) Utilizes the price history for the stock to

estimate beta. Problems? Bottom Up Beta Comparing the firm to others in the

same industry.

Estimating a Top Down Beta

The most common approach is to use linear regression analysis.Regression -- Attempts to explain the relationship between two variables by estimating the line that best describes the relationship.

Regression Review

Equation of a line: Y = a + bXGraphing combinations of X and Y form a line.X is the independent variable and placed on the horizontal axis. Y the dependent variable and placed on the vertical axis (The value of Y depends upon X)a is the Y intercept and b the slope of the line.

Observations of X and Y variables

X

Y

Regression Estimates the line that best explains the relationship between the

variables

The Line is the one that minimizes the sum of the

squared residuals

Estimating the Regression

The slope of the line is then equal to

The Intercept is:

XVariance

y)Cov(x,

)( XY AverageslopeAverage

Confidence in the ResultsR-Squared (R2)

R2 will range up to one. It is the portion of the relationship explained by the regressionR-Squared (R2) = correlationYX

2=b2x2/Y

2

Examples: An R2 of one implies all the points are on

the line An R2 of 0.5 would mean that half of the

relationship is explained by the line.

Confidence in the ResultsT-statistic

The t-statistic tells us whether or not we can reject the hypothesis that the variable is equal to zero.The higher the t-statistic the higher the confidence that we can reject the hypothesis that the slope is zero.If you cannot reject the hypothesis -- It implies that the dependent variable has no impact on the independent variable.

T-Statistic

A Rule of Thumb:The confidence levels are based upon the number of observations, but in general:If you have a t-statistic above 2.0 you can reject the null hypothesis at the 95% level.(With 120 observations a t-statistic of 2.36 allows rejection at the 99% level)

Standard Error

Provides a measure of “spread” around each variable. Provides a confidence band “similar to standard deviation)We can use standard error to estimate the T- Statistic (Assuming a normal distribution)T-Statintercept=A/SEA T-Statslope = B/SEB

Quick Review

Linear Regression - Provides line the best describes the relationship between two variablesR2 - Portion of relationship explained by the estimated lineT-Statistic - Confidence in the estimate of the variable (Is is statistically significant?)Standard Error - Confidence Interval

Estimating Beta

The basic CAPM can be rearranged to allow the use of regression analysis to

estimate Beta. ri=rRF+Bi(rM-rRF)

ri=rRF+BirM -BirRF

ri=rRF-BirRF +BirM

ri=rRF(1-Bi)+BirM

Estimating Beta

ri=rRF(1-Bi)+BirM

We know that rRF(1-Bi) is a constant let it = a

ri=a+BirM

Dependent IndependentVariable Variable

Estimating Beta

Given Historical data on the return of the market portfolio and the individual asset we can estimate Beta.

)Variance(r

)r,Cov(rBeta

M

iM

Estimating Jensen’s Alpha

We can also gain insight by looking at the intercept term.The goal is to compare the intercept term to the value we should have gotten for it given the historical data.From the rearranged CAPM the intercept should equal

rRF(1-B)

Jensen’s Alpha

rRF(1-B)Given the historical data to estimate kRF and the B we found from the regression we can find an estimate of the interceptThe difference between the estimate in the regression and the one from the historical data is called Jensen’s Alpha.

Jensen’s Alpha

The estimate from the regression comes from the historical data on the returns on the market and stock -- It is an estimates of the actual return received.The theoretical estimate of Jensen’s Alpha comes form the risk free rate and the assets beta - It measures what you would have expected to receive.

Interpreting Jensen’s Alpha

Ifa > rRF(1-B) The intercept from the regression

is higher than what we would have expected. This implies that the stock did better than expected.

a < rRF(1-B) The intercept from the regression is less than what we would have expected. This implies that the stock worse than expected.

Issues in Estimation

What estimation period should be used?

What interval should be used to calculate the returns (monthly, weekly, daily)?

Calculating Dividends in the return

Estimating Beta: An example

Disney 5 years of monthly returns Example:

March 37.87 April 36.42 Dividend in April 0.05

Return=((36.42+.05)-37.87)/37.87 =-3.69%Monthly return over the same period on the S&P 500 served as the market return

Regression Results

rDisney= -0.0001+1.40(rM)+

Beta = 1.40rM(1-B) = -.0001=-.01%

R2=.32

Standard error of Beta = .27

Interpreting the results

Beta, The stock is more responsive to market risk than the market average.R2=.32 The line explains 32% of the relationship between the variables (32% of the Disney’s return is explained by market risk factors the rest is firm or industry risk).SE = .27 Beta ranges from 1.4+.27 = 1.67 to 1.4-.27 = 1.13 with 68% confidence

Interpreting Jensen’s Alpha

During the 5 years, the average monthly return on long term treasuries was .4%

rRF(1-B) = .004(1-1.4) = -.0016 a = -.01%

Jensen’s Alpha a- rRF(1-B) =-.0001 - (-.0016) =.0015

On average Disney performed .15% better than expected each month.

That translated into (1.0015)12-1 =.0181=1.81% better than expected each year.

Adjusted Beta

Many analysts adjust the regression estimate of beta. Beta has been shown to move toward one over time as the firm matures. The data would not represent this well.A common adjustment is the following is to find a weighted average beta as follows: .67(regression estimate)+.33(1)Disney .67(1.4)+.33(1) = 1.27

Regression Example (2)

SUMMARY OUTPUT Cisco

Regression Statistics

R Square 0.24397973Observations 59

Coefficients Standard Error t StatIntercept 0.03358372 0.01311694 2.56033182S&P500 1.28470379 0.299540417 4.28891635

Regression Results

The coefficient on S&P 500 is the beta, Beta = 1.2847, Intercept = .0335Standard Error on Beta = 0.2995

T-Statistic on Beta = 4.2889R2=.2439

Can you explain each of these?Can you Calculate Jensen’s Alpha?

Financial Leverage and Beta

The amount of borrowing that the firm uses to finance its capital projects plays a key role in determining beta.A higher use of debt should increase the riskiness of the firm and increase its beta.The use of debt concentrates risk on the shareholder (the residual claimant).

Financial Leverage and Risk

Given the same level of earnings, increasing the use of debt creates a fixed payment that must be paid prior to the shareholder claimsBecause of this the return required by the shareholders increases to compensate them for extra risk.The firm is more responsive to market changes (implying a higher beta..)

Fundamental Beta

The fundamental beta is the beta the firm would have if it used no debt to finance its operations.When we ran the regression, the firm most likely was using debt. Therefore the data does not provide us with a measure of risk that is independent of the use of debt.

UnLevered Beta

Assume that the impact of financial leverage is fairly straight forward.

BL = BU(1+(1-t)Debt/Equity)

BL = Levered Beta BU = Unlevered Beta t = corporate tax rate

Disney’s Unlevered Beta

L = U(1+(1-t)(D/E))

we estimated the leveraged beta to be 1.4historically its Debt to equity ratio is 14% and its marginal corporate tax rate is 36%We can find the unlevered beta1.4 = U(1+(1-.36)(.14)) the solve for U = 1.2849

Then we could find the Beta based upon different levels of debt/equity.

Disney’s Unlevered Beta

BL = BU(1+(1-t)Debt/Equity) we estimated the leveraged beta to be 1.4 Historically Disney’s Debt to Equity ratio is 14%

and its marginal corporate tax rate is 36%. 1.4 = U(1+(1-.36)(.14))

then solve for U = 1.2849 As the Debt/Equity ratio changes we can

estimate the levered beta.

Bottom Up Beta

The bottom up beta is a weighted average of the average beta in the firms core industries.

The bottom up beta will usually provide a better estimate of market risk when:

There is a high standard error in the regressionThere have been structural changes in the firm

(reorganization or merger for example)When the firm’s equity is not traded or traded

infrequently.

Calculating Bottom up Beta

Determine the key industries in which the firm operatesFind the average unlevered beta of other firms in the key industriesCalculate a weighted average of the unlevered betas (weighted by the % of the firm in each industry)Use the firm’s debt equity ratio to find the current beta

Calculating Bottom Up Beta

1. Look at the firm’s financial statements to breakdown the firm into business units.

2. Estimate the average unlevelered beta of other publicly traded firms

3. Calculate the weighted average of the unlevered betas

4. Calculate the debt/equity ratio of the firm5. Combine 3 and 4 to find the levered beta.

Financial Statements

Look at the annual report and or 10-K (firms website or Edgar, or Mergent)From Disney 10-K

“The Walt Disney Company, together with its subsidiaries, is a diversified worldwide entertainment company with operations in four business segments: Media Networks, Parks and Resorts, Studio Entertainment, and Consumer Products.”

Calculating unlevered beta

To find the unlevered beta for each business unit you would need to find the unleverd beta of firms who are concentrated in the same business as the business unit.As an example we will use the parks and resorts business line.Disney’s parks are destination resorts, family friendly, focus on amusement rides etc. They also have a small portion of their business in cruise lines.

All data from Yahoo - D/E are book All data from Yahoo - D/E are book valuesvalues

Disney –Parks and Resorts Comparable firms

Firm Beta D/EUnlevered

Beta6 Flags 2.87 3.33 0.91Cedar Fair 1.28 1.52 0.64Royal Caribbean 1.88 0.82 1.22Carnival 1.71 0.42 1.34Great Wolf 0.59 0.53 0.44Average 0.91

Other business units

Media- Time Warner (enterprise competitor), Univision, ACME communications, Gray TelevisionConsumer goods (toys) – Matel, Hasbro, Action Products, Action GamesStudios – Marvel (X-Men movie…),Lions Gate, Graymark, Image (DVD production intermediary), Time Warner (enterprise competitor)

Calculating the weight in each business unit

Simple approaches - % revenue, % assets, % capital expenditureMultiple approach – Use industry averages for revenue multiple.enterprise value (EV)=MVequity+BVdebt-Cash EV/sales multiple used to aggregate revenuesRev*EV/Sales = est. value per business unitthen find % of total est. value

% of BusinessSimple approaches

Media Networks

13,207 41.3% 26,926 54.3% 2,228 74.2%

Parks and Resorts

9,023 28.2% 15,807 31.9% 726 24.2%

Studio 7,587 23.8% 5,965 12.0% 37 1.2%

Consumer 2,127 6.7% 877 1.8% 10 0.3%

Revenues (2005) Capital ExpendituresIdentifiable Assets

D/E Book or Market Value?

Book Value is based on the balance sheetMarket Value would be based upon the current value. For equity this is easy – it is the market capitalization of the firm. For Debt it is much harder due to a lack of pricing data for debt. It is possible to estimate a market value for debt, based on a portion of debt- if you can find a price.Book value often over emphasizes the impact of debt, since market value of equity will be more undervalued by book value .

Disney Bottom Up Beta

Unlevered Beta RevenueIdentifaible

AssetsCapital

Expenditures Damodaran

Media Networks1.120 46.31% 60.83% 83.15% 49.25%

Parks and Resorts 0.911 25.72% 29.04% 22.03% 20.09%

Studio1.081 25.67% 13.00% 1.33% 25.62%

Consumer1.182 7.87% 2.09% 0.39% 5.04%

Unlevered Beta 1.123 1.111 1.151 1.071

D/E Yahoo 0.389 0.389 0.389 0.389Disney 1.399 1.383 1.433 1.334

D/E Damodaran 0.250 0.250 0.250 0.250Disney 1.306 1.292 1.338 1.245

Source of Weights

Other methods

Beta can also be estimated in other ways for example:Accounting Betas -- found by analyzing the financial statement of the firm and similar firmsAlternate regression -- You can replace equity returns with a proxy (% change in earnings or cash flows for example)

Measuring Beta - Summary

Two main methods Top Down (regression) and Bottom Up. Bottom up is better when we do not have good data.Beta is an estimate of the firms sensitivity to market risk.The use of financial leverage plays a key role in determining the beta

What’s Next?

CAPM measures the impact of market risk on the return of an individual security.

So far we have concentrated on Stand Alone Risk, but we know that combining assets into a portfolio can reduce stand alone risk.

Portfolios

We showed earlier that it was possible to reduce risk by combining assets into a portfolio.There is a limit to the amount of risk a portfolio can eliminateGiven a set of assets, different weighting of the assets will produce different returns for the portfolio (and different risk)

Efficient Frontier

By changing the weights in a portfolio you get different return and risk combinations. It is often possible to rearrange a portfolio and produce a higher return without changing the risk.The efficient frontier provides the set of portfolios that produces the highest return at each level of risk.

Efficient Frontier

Given four assets, the next slide shows a graph of 76 different portfolios created by changing only the weights in the portfolio.The vertical axis is the return on the portfolio , the horizontal axis represents the standard deviation of the portfolio.The efficient frontier is the set of points that provides the highest return for each level of risk.

0

1

2

3

4

5

6

7

0 2 4 6 8 10

Arbitrage Pricing Model

The CAPM and APM both make a distinction between stand alone and market riskThe CAPM assumes that the market risk is captured by the market portfolio.The APT assumes that there are many risk factors that help to determine the market risk.

Arbitrage Pricing Model

APM assumes that several factors contribute to market risk (interest rate, inflation, exchange rates …). Just like the CAPM it assumes we can measure the sensitivity of an asset to each factor (Beta did this in the case of the CAPM)In the APM let Bi represent the sensitivity of the asset to factor i

Arbitrage Pricing Model

The expected return of the asset is then:E(R)=RRF+B1(E(R)1-rRF) +B2(E(R)2-rRF)+ +Bn(E(R)n-rRF)

+

The CAPM is actually a one factor version of the APM

The APM is difficult to implement due to need to identify the relevant factors and returns.

Arbitrage Pricing Model

AssumptionsEqual portfolios of risk should provide equal expected returnsInvestors will drive the return of those that do not compensate for their risk up and those that provide too much return down.

Sources of Market Wide RiskThere are different sources of market risk relating to the different factors investigated.

Arbitrage Pricing Model

Arbitrage illustrationAssume one factor and 3 portfolios

A=2.0 B=1.0 C=1.5

Portfolio with 50% in A and 50% in B has same beta as CWhat is portfolio of A and B paid 16% but Portfolio C paid 15%?

APM in practice

Use of factor analysis to determine the factors that impact a broad group of stocksBenefits

Specifies number of factorsMeasures beta relative to the common factors# of factors, factor betas, factor risk premium

WeaknessesThe factors are “unspecified”

*Chen, Roll, & Ross 1986 J of *Chen, Roll, & Ross 1986 J of BusinessBusiness

Multifactor Models

# of factors of identified by the APM – a multifactor model attempts to identify the factorsPossible factors*:

Industrial productionUnanticipated inflationShifts in term structure of interest ratesReal rate of return

Proxy Models

Attempting to identify financial or other multiples that are linked to returnsExample: Fama and French – low price to book ratios and low market capitalization result in higher returns.

Rt=1.77% - .11ln(MV)+.35ln(MV/MV)

(-1.99) (4.44)

The Risk in Borrowing

The risk of default is a primary concern for the debt market.Again with added risk there should be added return.Default risk includes firm specific risk, unlike the equity risk model we have been discussing.Bonds have a much larger downside potential than upside potential.

Default Risk and Bond Ratings

Moody’s investors services and Standard and Poor’s Corporation provide ratings for corporate bonds based upon the quality of the bond.

The ratings allow investors to compare the safety of bonds to each other. A large part of the rating is based upon default risk.

The highest rating, AAA or Aaa, represents a very low probability of default.

Bond Ratings

As the probability of default increases, the rating drops from AAA to AA (or Aaa to Aaa).After A the ratings go to BBB then BB etc. Bonds rated below BB are considered high risk or “Junk Bonds”.

Fabozzi Bond Markets Analysis and Straegies 2004Fabozzi Bond Markets Analysis and Straegies 2004

Summary of Bond Ratings

Moody’s

S&P Fitch

Aaa AAA AA Maximum safety

Investment Grade

Aa AA AA Very High Grade

A A A Upper Med Grade

Baa BBB BBB Lower Med Grade

Ba BB BB Low Grade Speculative Low Credit

WorthinessB B B Highly Speculative

Caa CCC CCC In poor standing Substantial Risk close to default

or in defaultCa CC CC May be in default

C C C More risky than CC

D D D In Default In default

Yield Spread Monthly Data Jan 1919 – June 2004 (Moodys)

0

2

4

6

8

10

12

14

16

18

20

9/8/1913 1/24/1941 6/11/1968 10/28/1995Date

%

Aaa Baa

Long Term Average Yearly Yields Over Time (Moody’s)

4

6

8

10

12

14

16

18

1980 1985 1990 1995 2000

(%)

Aaa Aa A Baa

Yield Spreads 1994 - 2003

0

1

2

3

4

5

6

7

8

9

10

Jan-94 Nov-94 Sep-95 Jul-96 May-97 Mar-98 Jan-99 Nov-99 Sep-00 Jul-01 May-02 Mar-03

Date

Yie

ld

0

1

2

3

4

5

6

Sp

read

AAA

BBB

Treas

AAA-Treas

BBB-Treas

Determination of Default Risk

Generally Higher cash flow generation relative to financial obligation – lowers default riskMore stability in cash flows – lowers default riskHigher liquidity of assets lowers default risk

Yield Spreads

Yield SpreadsThe difference in required return between two assets, the difference in required return represents the difference in risk.Often bonds that are the same except for the possibility of default are compared, implying that the yield spread is a measure of the default risk

Bond Rating Criteria

Financial RatiosMortgage ProvisionsGuarantee Provisions

Sinking fundsMaturityStability

RegulationOthers

Yield Spreads and Risk Premiums

The difference in yield between any two assets should represent differences in risk. The extra return earned on a riskier security is termed the risk premium.Generally the risk premium is quoted in basis points.

Yield Spread = Yield on Bond A – Yield on Bond BWhere yield on bond B is being used as a

benchmark

Bond Ratings and Average Yield Spreads vs. US Treasuries (long term

bonds Jan 2004)

Rating Spread Rating Spread AAA .30% B+ 3.25% AA .50% B 4.00% A+ .70% B- 6.00% A .85% CCC 8.00% A- 1.00% CC 10.00% BBB 1.5% C 12.0% BB 2.5% D 20.0%

Relative Yield Spreads

Spreads are also measured relative to a base rate

b bondon yield

B bondon yield -A bondon yield

spread

yield

relative

B bondon yield

A bondon yield

ratio

yield

General Factors Impacting Yield Spreads

1. Type of issuer2. Issuers creditworthiness3. Maturity4. Embedded options5. Taxability6. Liquidity7. Other risks associated with

previously discussed premiums

Linking Yield Spreads to Financial Performance

One of the key things impacting the rating is the financial condition of the firm.Changes in the financial condition obviously impact the ability of the firm to pay its debt obligations.Often the most commonly used measure is an interest coverage ratio. However use of interest coverage by itself may mislead. Therefore composite scores of credit risk may be used.

Bond Rating Criteria and Financial Ratios 1998-2000

AAA AA A BBB BB B EBIT int cov 17.5 10.8 6.8 3.9 2.3 1.0EBITDA Int Cov 21.8 14.6 9.6 6.1 3.8 2.0NetCF/TotDebt 90% 67% 50% 32% 20% 11%FCF/TotDebt 41% 22% 17% 6% 1% -4%ROC 28.2% 22.9% 19.9% 14% 11.7% 7.2%LTDebt/TotCap 15% 26.4% 32.5% 41% 56% 71%TotDebt/TotCap 27% 36% 40% 47.4% 61% 75%