Risk and CapitalBudgeting

Chapter 13

Chapter 13 - Outline

What is Risk?Risk Related MeasurementsCoefficient of CorrelationThe Efficient FrontierCAPM and Beta

What is Risk?Risk means uncertainty about a future

outcome

Risk varies greatly depending on the investment:–A T-Bill has zero or no risk– A gold-mining expedition in Africa has high risk

Most investors and financial managers are risk averse, meaning they don’t like risk - Preference: relative certainty as opposed to uncertainty- Expectation: higher value or return for risky investments

Stocks, Bonds, Bills, and Inflation

Hypothetical value of $1 invested at year-end 1925. Assumes reinvestment of income and no transaction costs or taxesThis is for illustrative purposes only and not indicative of any investment.Past performance is no guarantee of future results. 3/1/2008. Copyright © 2008 Ibbotson Associates, Inc.

1925 - 2009

AverageReturn

EndingWealth

1925

1935

1945

1955

1965

1975

1985

1995

2009

$12,130$2,526$99.17

$20.51$12.14

Rates of Return 1926-2009

Source: Ibbotson Associates Year

Per

cent

age

Ret

urn

Stocks, Bonds, Bills, and Inflation

CompoundAnnualReturn

ArithmeticAnnualReturn

Risk(StandardDeviation)

*The 1933 Small Company Stock total return was 142.9%.This is for illustrative purposes only and not indicative of any investment.Past performance is no guarantee of future results. 3/1/2008. Copyright © 2008 Ibbotson Associates, Inc.

Summary Statistics 1926 - 2009

Distribution of Annual Returns

LargeCompanyStocks

9.66% 11.8% 20.5%

Small CompanyStocks

*11.7% 16.5% 33.0%

GovernmentBonds 5.7% 6.1% 9.4%

Inflation 2.9% 3.1% 4.2%

Treasury Bills 3.7% 3.8% 3.1%

Some Risk Related Statistical Measurements

Expected Value:– equal to weighted average of outcomes x probabilities

Standard Deviation:– measure of dispersion or variability around the

expected value– the larger the standard deviation the greater the risk

Coefficient of Variation:– equal to standard deviation / expected value– the larger the coefficient of variation the greater the

risk

Variabilityand risk

Probabilitydistribution withdiffering degreesof risk

Calculating mean (or expected) return

Probability Probability Return x return

.20 -10% -2%

.50 +10 5

.30 +30 9

Total 12%

Mean or expected return

Calculating variance and standard deviationof Merck returns from past monthly data Deviation from mean Squared Month Return return deviation

1 5.4% 2.6% 6.76 2 1.7 - 1.1 1.21 3 - 3.6 - 6.4 40.96 4 13.6 10.8 116.64 5 - 3.5 - 6.3 39.69 6 3.2 0.4 0.16

Total 16.8 205.41

Mean: 16.8/6 = 2.8% Variance: 205.41/6 = 34.235 Std dev: Sq root of 34.235 = 5.85% per month

Calculating variance and standard deviation

Deviation Probability from mean x squared Probability Return return deviation

.20 -10% -22% 96.8

.50 +10 - 2 2.0

.30 +30 +18 97.2

Total 196.0

Variance

Standard deviation = square root of variance = 14%

Mean and standard deviation mean measures average (or expected return)

standard deviation (or variance) measures the spread or variability of returns

risk averse investors prefer high mean & low standard deviation

20

standard deviation

expected

return

15

10

5

05 10 15 20

better

HOWEVER, INVESTOR FOCUS IS ON PORTFOLIO RISK & RETURN

Expected portfolio return

Portfolio Expected Proportion x proportion (x) return (r) return (xr)

Merck .40 10% 4%

McDonald .60 15 9

Total 1.00 13%

Expectedportfolioreturn

Risk and Diversification

Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments.

Unique Risk - Risk factors affecting only that firm. Also called “diversifiable risk.”

Market Risk - Economy-wide sources of risk that affect the overall stock market. Also called “systematic risk.”

Diversification eliminates unique risk

deviationstandardPortfolio

Unique risk

Market risk

Number ofsecurities

5 10

Stock Diversification

1 2 4 6 8 16 30 50 100 1000Number of Stocks in Portfolio

Ris

k

Market Risk

Diversifiable Risk

This is for illustrative purposes only and not indicative of any investment.

Past performance is no guarantee of future results. 3/1/2000. Copyright © 2000 Ibbotson Associates, Inc.

Where do Diversification Benefits Come from?

Concepts of Correlation and Covariance

Expected Return on portfolio

Standard Deviation of Portfolio Returns

Coefficient of Correlation

Shows the extent of correlation among projects

Has a numerical value of between -1 and +1Its value shows the risk reduction between

projects:Negative correlation (-1) Large risk reductionNo correlation (0) Some risk reductionPositive correlation (+1) No risk reduction

Coefficient of Correlation Coefficient of Variation

The Efficient Frontier

Combinations of projects with the best trade-off between risk and return

2 objectives:– Achieve the highest possible return at a

given risk level– Provide the lowest possible risk at a given

return level

The Efficient Frontier is the best risk-return line or combination of

possibilities

Mean & standard deviation: Portfolio of Merck & McDonald

8

9

10

11

12

13

14

15

16

4 4.5 5 5.5 6 6.5 7 7.5 8

Std dev %

Expected return %

100% McD

100% Merck

40% Merck

Calculating covariance and correlation betweenMerck and McDonald from past monthly data

Deviation Product Return: from mean: of Month Merck McD Merck McD deviations

1 5.4% 10.7% 2.6% 8.9% 23.2 2 1.7 - 8.4 - 1.1 -10.2 11.2 3 - 3.6 1.6 - 6.4 - 0.2 1.2 4 13.6 10.2 10.8 8.4 90.9 5 - 3.5 4.4 - 6.3 2.6 -16.5 6 3.2 - 7.8 0.4 - 9.6 - 3.8 Total 16.8 10.7 106.1 Mean 2.8 1.78

Covariance: 106.1/6 = 17.7 Std dev Merck: 5.9% Std dev McD: 7.7% Corr. co-effic: Cov/(sdMe . sdMcD )

= 17.7/(5.9 x 7.7) = .39

Calculating covariance and correlation

Deviation from Probability Return on: mean return: x product of Prob. A B A B deviations

.15 -10% -10% -22% -22% + 72.6 .05 -10 +10 -22 - 2 + 2.2 .45 +10 +10 - 2 - 2 + 1.8 .05 +10 +30 - 2 +18 - 1.8 .25 +30 +30 +18 +18 + 81.0 .05 +30 -10 +18 -22 - 19.8

Mean +12 +12 Total 136 Std dev +14 +14 Covariance

Correlation = = = .6944 coefficient (sd A) x (sd B) 14 x 14

covariance 136

Effect of changing correlations: Portfolio of Merck & McDonald

8

9

10

11

12

13

14

15

16

0 1 2 3 4 5 6 7 8 9

Std dev %

Expected return %100% McD

100% Merck

corr = .4

corr = -1

corr = 1

With a correlation of one, same as a weighted average.

Portfolio RiskExample

Suppose you invest 60% of your portfolio in Wal-Mart and 40% in IBM. The expected dollar return on your Wal-Mart stock is 10% and on IBM is 15%. The standard deviation of their annualized daily returns are 19.8% for Wal-Mart and 29.7% for IBM. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

% 23.76 54.564 Deviation Standard

54.5641)19.8x29.7x2(.40x.60x

]x(29.7)[(.40)

]x(19.8)[(.60) Variance Portfolio22

22

Portfolio Risk with lower correlation:

Example

Suppose you invest 60% of your portfolio in Wal-Mart and 40% in IBM. The expected dollar return on your Wal-Mart stock is 10% and on IBM is 15%. The standard deviation of their annualized daily returns are 19.8% for Wal-Mart and 29.7% for IBM. Assume a correlation coefficient of 0.70 and calculate the portfolio variance.

% 21.91 86.479 Deviation Standard

86.479.7)19.8x29.7x2(.40x.60x

]x(29.7)[(.40)

]x(19.8)[(.60) Variance Portfolio22

22

With a correlation less than one, less than a weighted average.

The set of portfoliosExpected return

Risk

x

x

x

xxxxxxxx

xxxxx

xxx

xx

x

A

B

The set of portfolios between A and B

are efficient portfolios

Adding a riskless asset to the efficient frontier

8

9

10

11

12

13

14

15

16

4 4.5 5 5.5 6 6.5 7 7.5 8

Std dev %

Expected return %

riskless rate

tangency portfolio

Capital asset pricing model

Expectedreturn

Expectedmarketreturn

Riskfreerate

0 .5 1.0 Beta

r = rf +

(rm - rf )

Beta

Beta is a statistical measure of volatilityIt measures how responsive or sensitive a stock is to market

movements in general

An individual stock’s beta shows how it compares to the market as a whole:beta = 1 means equal risk with the marketbeta > 1 means more risky than the marketbeta < 1 means less risky than the market

Beta Computation

2m

imiB

Covariance with the market

Variance of the market

Beta

(1) (2) (3) (4) (5) (6) (7)Product of

Deviation Squared deviationsDeviation from average deviation from average

Market Anchovy Q from average Anchovy Q from average returnsMonth return return market return return market return (cols 4 x 5)

1 -8% -11% -10% -13% 100 1302 4 8 2 6 4 123 12 19 10 17 100 1704 -6 -13 -8 -15 64 1205 2 3 0 1 0 06 8 6 6 4 36 24

Average 2 2 Total 304 456

Variance = σm2 = 304/6 = 50.67

Covariance = σim = 736/6 = 76

Beta (β) = σim/σm2 = 76/50.67 = 1.5

Calculating the variance of the market returns and the covariance between the returns on the market and those of Anchovy Queen. Beta is the ratio of

the variance to the covariance (i.e., β = σim/σm2)

Market risk (beta) for common stocks1994

Stock Beta Stock Beta

AT&T .92 Exxon .51Biogen 2.20 Ford Motor Co. 1.12Bristol Myers Squibb .97 General Electric 1.22Coca Cola 1.12 McDonald’s 1.07Compaq 1.18 Microsoft 1.23

Market risk (beta) for common stocks2010*

Stock Beta Stock Beta

AT&T .63 Exxon .39Biogen .68 Ford Motor Co. 2.79Bristol Myers Squibb .59 General Electric 1.72Coca Cola .51 McDonald’s .52Hewlett-Packard 1.03 Microsoft 1.06

* Source: Finance.yahoo.com, which is based on 36 months of data and S&P500 as market index.