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.