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Risk and Rates of Return

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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

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ExampleExample

1.0000

0.06254

0.25003

0.37502

0.25001

0.06250

P(X)X The mean of this distribution is 2,

since it is a symmetrical distribution.E

xam

ple

Q:If you toss a coin four times what is the chance of getting x heads?

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Calculate the Mean

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Discrete Probability Discrete Probability DistributionDistribution

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Variance and Standard DeviationVariance and Standard Deviation

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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%

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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

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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

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Expected Rate of Return

(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 r m = 15% r m = 15%

State of the Economy

Martin Products U. S. ElectricReturn if This

State Occurs (ri)Product: (2) x (5)

Probability of This State

Occurring (Pr i)Return if This State

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

^ ^

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ˆ r Pr1r1 Pr2r2 Prnrn

Pririi1

n

Expected Rate of Return

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Continuous versus Discrete Probability Distributions

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

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Discrete Probability DistributionsDiscrete Probability Distributionsa. Martin Products

Probability of Occurrence

b. U. S. ElectricProbability of Occurrence

-60 -45 -30 -15 0 15 22 30 45 60 75 90 110Rate of

Return (%)Expected Rate of Return (15%)

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

-10 -5 0 5 10 16 20 25Rate of

Return (%)Expected Rate

of Return (15%)

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

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Continuous Versus Discrete Probability Distributions

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

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Probability Density

-60 0 15 110Rate of Return

(%)Expected Rate of Return

Martin ProductsMartin Products

U. S. ElectricU. S. Electric

Continuous Probability Distributions

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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

r i(r i - r)

2Pr iProbability

Expected Return

rri - r (r i - r)

2^ ^ ^

%3.59 517,3 Deviation Standard

0.517,3 Variance

2mm

2

^^^^

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Expected rate of return ˆ r ri ˆ r 2 Prii1

n

Variance 2 ri - ˆ r 2Pri

i1

n

Standard deviation ri - ˆ r 2Pri

i1

n

Measuring Risk: The Standard Deviation

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Measuring Risk: Coefficient of Variation

Calculated as the standard deviation divided by the expected return

Useful where investments differ in risk and expected returns

Coefficient of variation = CV = Risk

Return

ˆ r

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Risk Aversion and Required Returns

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

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

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Expected Rate of Return

Expected ending value - Beginning value

Beginning valueExpected rate

of return =

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Expected return on a portfolio,

ˆ r p w1ˆ r 1 w2

ˆ r 2 wNˆ r N

w jˆ r j

j1

N

ˆ r pThe weighted average expected return on the stocks held in the portfolio

Portfolio Returns

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Portfolio Returns

Realized rate of return, rThe return that is actually earned

Actual return usually different from expected return

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

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15

0

-10

Stock W

-10

0

15

25

Portfolio WM

-10

0

15

25

Stock M

25

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 ( = +1.0) and for Portfolio MM’:

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Portfolio RiskCorrelation Coefficient, 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.

∑xy) – (∑x)(∑y) / √ {n(∑x2) – (∑x)2}{n(∑y2) – (∑y)2}

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Portfolio Risk

Risk ReductionCombining 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.

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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.

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Firm-Specific Risk versus Market Risk

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.

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Firm-Specific Risk versus Market Risk

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.

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The Concept of Beta

Beta 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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N

1jjj

nn2211

w

www

p

Portfolio Beta Coefficients

The beta of any set of securities is the weighted average of the individual securities’ betas

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ˆ r j expected rate of return on the jth stock

rj required rate of return on the jth stock

rRF risk free rate of return

RPM rM - rRF market risk premium

RPj rM - rRF j risk premium on the jth stock

The Relationship Between Risk and Rates of Return

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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 = rM - rRF = 14% - 6% = 8%.

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Risk Premium for a Stock

Risk Premium for Stock j

= RPM x j

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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.

The Required Rate of Return for a Stock

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Security Market Line

SML :rj rRF rM rRF j

rhigh = 22

rM = rA = 14

rLOW = 10

rRF = 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%

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The Impact of Inflation

rRF is the price of money to a riskless borrower.

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

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

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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.

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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.

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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

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Changes in Equilibrium Stock Prices

Stock prices are not constant due to changes in:

Risk-free rate, rRF,

Market risk premium, rM – rRF,

Stock X’s beta coefficient, x,

Stock X’s expected growth rate, gX, and

Changes in expected dividends, D0.

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Physical Assets Versus Securities

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

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Word of Caution

CAPM (Capital Asset Price Model)Based on expected conditions

Only have historical data

As conditions change, future volatility may differ from past volatility

Estimates are subject to error