Defining and Measuring Risk Risk is the chance that an unexpected outcome will occur A probability...

Defining and Measuring Risk

Risk is the chance that an unexpected outcome will occur

A probability distribution is a listing of all possible outcomes with a probability assigned to each;

must sum to 1.0 (100%).

Probability Distributions

It either will rain, or it will not.Only two possible outcomes.

Outcome (1) Probability (2)

Rain 0.40 = 40%

No Rain 0.60 = 60%

1.00 100%

Probability Distributions

Martin Products and U. S. Electric

Martin Products U.S. Electric

Boom 0.2 110% 20%Normal 0.5 22% 16%Recession 0.3 -60% 10%

1.0

Probability of This State Occurring

State of the Economy

Rate of Return on Stock if This State Occurs

Expected Rate of Return

Rate of return expected to be realized from an investment during its life

Mean value of the probability distribution of possible returns

Weighted average of the outcomes, where the weights are the probabilities

(1) (2) (3) = (4) (5) = (6)Boom 0.2 110% 22% 20% 4%Normal 0.5 22% 11% 16% 8%Recession 0.3 -60% -18% 10% 3%

1.0 km = 15% km = 15%

State of the Economy

Martin Products U. S. Electric

Return if This State Occurs (ki)

Product: (2) x (5)

Probability of This State

Occurring (Pr i)

Return if This State

Occurs (ki)Product: (2) x (3)

^ ^


n

1ii

n21

k

kkkk

i

n21

Pr

PrPrPr ˆ


Discrete Probability Distributions

Discrete Probability Distribution:number of possible outcomes is limited, or finite.

Continuous Probability Distributions

Continuous Probability Distribution:number of possible outcomes is unlimited, or infinite.

Probability Density

-60 0 15 110Rate of Return

(%)Expected Rate of Return

Martin ProductsMartin Products

U. S. ElectricU. S. Electric

Continuous Probability Distributions

Measuring Risk: The Standard Deviation

Calculating Martin Products’ Standard Deviation

(1) (2) (1) - (2) = (3) (4) (5) (4) x (5) = (6)110% 15% 95 9,025 0.2 1,805.0 22% 15% 7 49 0.5 24.5 -60% 15% -75 5,625 0.3 1,687.5

Payoff

ki(ki - k)

2Pr iProbability

Expected Return

kki - k (ki - k)

2^^

^ ^

%3.59 517,3 Deviation Standard

0.517,3 Variance

2mm

2

^^^^

n

1iiikk̂ return of rate Expected Pr

Variance 2 k i - ˆ k 2Prii 1

n

Standard deviation 2 k i - ˆ k 2Prii 1

n

Measuring Risk: The Standard Deviation

Standardized measure of risk per unit of return

Calculated as the standard deviation divided by the expected return

Useful where investments differ in risk and expected returns

k̂Return

Risk CV Coefficient of variation

Measuring Risk: Coefficient of Variation

Risk Aversion

Risk-averse investors require higher rates of return to invest in higher-risk securities

Risk Aversion and Required Returns

Risk Premium (RP):

The portion of the expected return that can be attributed to an investment’s risk beyond a riskless investment

The difference between the expected rate of return on a given risky asset and that on a less risky asset

Portfolio Risk and theCapital Asset Pricing Model

CAPM:A model based on the proposition that any

stock’s required rate of return is equal to the risk-free rate of return plus a risk premium, where risk is based on diversification.

PortfolioA collection of investment securities

ˆ k p w1ˆ k 1 w 2

ˆ k 2 w Nˆ k N

w jˆ k j

j1

N

Portfolio ReturnsExpected return on a portfolio,pk̂

The weighted average expected return on the stocks held in the portfolio

Portfolio Returns

Realized rate of return, kThe return that is actually earnedActual return usually different from

expected return

Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM:

25

15

0

-10

Stock W

-10

0

15

25

Portfolio WM

-10

0

15

25

Stock M

Stock M

0

15

25

-10

0

15

25

-10

Stock M’

0

15

25

-10

Stock MM’

Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’:

Portfolio RiskCorrelation Coefficient, r

Measures the degree of relationship between two variables.

Perfectly correlated stocks have rates of return that move in the same direction.

Negatively correlated stocks have rates of return that move in opposite directions.

Portfolio RiskRisk Reduction

Combining stocks that are not perfectly correlated will reduce the portfolio risk through diversification.

The riskiness of a portfolio is reduced as the number of stocks in the portfolio increases.

The smaller the positive correlation, the lower the risk.

Firm-Specific Risk versus Market Risk

Firm-Specific Risk:That part of a security’s risk

associated with random outcomes generated by events, or behaviors, specific to the firm.

Firm-specific risk can be eliminated through proper diversification.


Market Risk:That part of a security’s risk that

cannot be eliminated through diversification because it is associated with economic, or market factors that systematically affect all firms.


Relevant Risk:The risk of a security that cannot be

diversified away, or its market risk.This reflects a security’s

contribution to a portfolio’s total risk.

The Concept of BetaBeta Coefficient,

A measure of the extent to which the returns on a given stock move with the stock market.

= 0.5: Stock is only half as volatile, or risky, as the average stock.

= 1.0: Stock has the same risk as the average risk.

= 2.0: Stock is twice as risky as the average stock.

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p

Portfolio Beta CoefficientsThe beta of any set of securities is the

weighted average of the individual securities’ betas

stock j on thereturn of rate k̂ thj expected

The Relationship Between Risk and Rates of Return

stock j on thereturn of rate k

stock j on thereturn of rate k̂th

j

thj

required

expected


return of rate k


stock j on thereturn of rate k̂

RF

thj

thj

freerisk

required

expected


premiumrisk market k-k RP

return of rate freerisk k

stock j on thereturn of rate required k

stock j on thereturn of rate expected k̂

RFMM

RF

thj

thj


stock j on the premiumrisk k-k RP

premiumrisk market k-k RP

return of rate k


stock j on thereturn of rate k̂

thjRFMj

RFMM

RF

thj

thj

freerisk

required

expected


Market Risk Premium

RPM is the additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.

Assuming: Treasury bonds yield = 6%, Average stock required return = 14%, Then the market risk premium is 8 percent:

RPM = kM - kRF = 14% - 6% = 8%.

Risk Premium for a Stock

Risk Premium for Stock j = RPj = RPM xj

jstock for return of rate k j required

jRFMRF

jMRFj

kk k

RP k k

The Required Rate of Return for a Stock

Security Market Line (SML):The line that shows the relationship

between risk as measured by beta and the required rate of return for individual securities.

Security Market Line jRFMRFj kk k k:SML

khigh = 22

kM = kA = 14

kLOW = 10

kRF = 6

Risk, j0 0.5 1.0 1.5 2.0

Required Rate of Return (%)

Risk-Free Rate: 6%

Safe Stock Risk Premium: 4%

Market (Average Stock) Risk Premium: 8%

Relatively Risky Stock’s Risk Premium: 16%

The Impact of Inflation

kRF is the price of money to a riskless borrower.

The nominal rate consists of: a real (inflation-free) rate of return, and an inflation premium (IP).

An increase in expected inflation would increase the risk-free rate.

Changes in Risk Aversion

The slope of the SML reflects the extent to which investors are averse to risk.

An increase in risk aversion increases the risk premium and increases the slope.

Changes in a Stock’s Beta Coefficient

The Beta risk of a stock is affected by:composition of its assets,use of debt financing, increased competition, andexpiration of patents.

Any change in the required return (from change in beta or in expected inflation) affects the stock price.

Stock Market Equilibrium

The condition under which the expected return on a security is just equal to its required return

Actual market price equals its intrinsic value as estimated by the marginal investor, leading to price stability

Changes in Equilibrium Stock Prices

Stock prices are not constant due to changes in:

Risk-free rate, kRF,

Market risk premium, kM – kRF,

Stock X’s beta coefficient, bx,

Stock X’s expected growth rate, gX, and

Changes in expected dividends, D0.

Physical AssetsVersus Securities

Riskiness of corporate assets is only relevant in terms of its effect on the stock’s risk.

Word of Caution

CAPMBased on expected conditionsOnly have historical dataAs conditions change, future volatility

may differ from past volatilityEstimates are subject to error

Defining and Measuring Risk Risk is the chance that an unexpected outcome will occur A probability...

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