Lecture: 4 - Measuring Risk (Return Volatility) I.Uncertain Cash Flows - Risk Adjustment II.We Want a Measure of Risk With the Following Features a. Easy

Lecture: 4 - Measuring Risk (Return Lecture: 4 - Measuring Risk (Return Volatility)Volatility)

I. Uncertain Cash Flows - Risk Adjustment

II. We Want a Measure of Risk With the Following

Features

a. Easy to Calculate

b. Ranks Assets According to Compensated Risk

c. Can be Translated into a Discount Rate, k

III. Economy-Wide or Systemic Risk -> Beta

Works best for a portfolio of assets.

IV. Non-Systematic or Company Specific Risk ->

Variance

Works best for a single asset.


TO MEASURE RISK WE NEED A GOAL VARIABLE.

INVESTORS’ GOAL VARIABLE IS RETURNS

% Return =

= +

= % capital gain (loss) + % Dividend Yield

% R = = .40 or 40%

[ ]P P D

P1 0 1

0

( )P P

P1 0

0

D

P1

0

( )12 10 2

10

QUESTION: What is risk , or, what does risk-free mean?

ANSWER: If exante expected returns always equal expost returns for an investment then we say it is risk- free. If actual returns are sometimes larger and sometimes smaller than expected, the investment carries risk. (we are happy with large ones but unhappy with small ones).

A measure of risk should tell us the likelihood that we will not get what we expect and the magnitude of how different our returns will be from the expected.

HOW TO MEASURE WHAT TO EXPECT

• Enumerate outcomes i.e., the different riskscenarios.

• Generate a probability distribution - attachprobabilities to each scenario that sum to 1

(remember statistics course)

EXAMPLE

Economic Scenarios Prob IBM Return

High growth ( 5%) .30 .25Low growth (3%) .40 .15Recession (-3%) .30 .05

Sum = 1

Get mean return - expected return - best guess - (Note: Book uses k instead of R)

E(R) = Pi Ri =

E(R) = (.3 * .25) + (.4 * .15) + (.3 * .05) = .15


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USE VARIANCE TO MEASURE TOTAL RISK

2 = (Ri - )2 Pi

or standard deviation;

= [ 2].5

For IBM

2IBM = (.25 - .15)2(.3) + (.15 - .15)2(.4) + (.05 - .15)2(.3)

= .006


R_

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QUESTION: What is the variance of a stock which has a mean of .15 and returns of .15 in all states of the economy?

- Zero!

VARIANCE FOR A SINGLE ASSET CONTAINS

a. Diversifiable Risk (Firm Specific) -

easily by diversification at little or no cost.

b. Undiversified (System) Risk -

cannot be eliminated through diversification.

Variance Measures the Dispersion of a Variance Measures the Dispersion of a Distribution Around Its MeanDistribution Around Its Mean


1

2

These two distributions have the same mean but 1’s variance is smaller than 2’s.

If these represent stock returns, a risk averse investor should choose stock 1.

A Standarized Risk Measure A Standarized Risk Measure

Coefficient of Variation = Standard Coefficient of Variation = Standard Deviation/MeanDeviation/Mean


1

2

When two stock return distributions have different means and variances, a risk averse investor choosing between them needs a method that compares mean return relative to risk, such as coefficient of variation or the capital asset pricing model.

Portfolio Mean Return and VariancePortfolio Mean Return and Variance


TO GET THE VARIANCE OF A PORTFOLIO WE NEED TO CALCULATE THE PORTFOLIO MEAN RETURN.

Portfolio mean return is a linear, weighted average of individual mean returns of the assets in the portfolio.

GETTING THE WEIGHTS INVESTMENT $ INVESTED Wi

1 100 100/500 = .2 .102 200 200/500 = .4 .053 200 200/500 = .4 .15

E(Rp) = W1 + W2 + W3

= .2(.1) + .4(.05) + .4(.15) = .10

GENERAL => E(Rp) = Wi = R i_

R_

3R_

2R_

1

R i_

R p

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n

1

For a two asset portfolio:

p2 = W1

2 1

2 + W22

22 + 2W1W2Cov12

where: Cov12 = covariance = Corr12 1 2 and Corr12 = correlation

QUESTION: Diversification reduces variance of portfolio even when corr=0. WHY?- Some asset-specific risk offset one another.

Portfolio Variance is More Complex -Portfolio Variance is More Complex -A Nonlinear FunctionA Nonlinear Function


PortfolioRisk

Number of securities in the portfolio

Diversifiable risk drops as more securitiesare added to a portfolio.

Diversifiable Risk

Nondiversifiable Risk

It’s usually best to diversify, except in this case.

Correlation

• Statistical Measure of the Degree of Linear

Relationship Between Two Random Variables

• Range: + 1.0 to -1.0

• + 1.0 - Move Up and Down Together - Exactly

the Same Rate

• 0.0 - No Relationship Between the Returns

• - 1.0 - Move Exactly Opposite Each Other


Stock 1Return

NegativeCorrelation

PositiveCorrelation

ZeroCorrelation

Stock 2 Return

Stock 1Return

Stock 1Return

Stock 2 Return Stock 2 Return

Covariance Measures How Closely Returns For Two AssetsTrack Each Other Other (Closeness to the Regression Line)

All else equal, covariance is large when the data points fall along the regression line instead of away from it because, on the line, the deviations from the means of each variable are equal – the products are squares - larger than otherwise.

Covariance is a Measure of Risk and Beta Covariance is a Measure of Risk and Beta is a Standardized Covarianceis a Standardized Covariance


Beta Is a Standardized Covariance Beta Is a Standardized Covariance MeasureMeasure


We Need Beta (Standardized Covariance Measure) in Order to Make Comparisons of Risk Between Assets or Portfolios.

• Measured Relative to the Market Portfolio (the most

diversified portfolio is the standard)

• Slope of the Regression Line

• Slopes Measured Relative to Market Return

General Formula

Betai =

=

Beta and the Market (Illustration)

• Beta = 1 - Same as Market Risk

• Beta > 1 - Riskier than Market

• Beta < 1 - Less Risky than Market

• Beta = 2 - Twice as Risky as Market

Covim

m 2

Corrim i m

m

2

Positive Beta Positive Beta


1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

Homestake's Return

0.40.30.20.10-0.1

S&P 500 Return

The correlation between Homestake and the S&P 500 is 0.18 and its beta is 0.54

Annual return pairs for the S&P 500 and Homestake Mining's stock

Slope is 0.54

Year S&P Homestake1983 0.23 0.011984 0.06 -0.261985 0.32 0.11986 0.18 0.091987 0.05 0.391988 0.17 -0.271989 0.31 0.551990 -0.03 -0.091991 0.3 -0.161992 0.08 -0.251993 0.1 0.831994 0.01 -0.19

Negative Beta Negative Beta


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5

Gasoline Return

0.40.30.20.10-0.1

S&P 500 Return

Gasoline's correlation with the S&P 500 is -0.47 and its beta is -2.11.

Annual return pairs for the S&P 500 and gasoline

Slope is -2.11

Year S&P Gas1983 0.23 0.081984 0.06 -0.11985 0.32 0.091986 0.18 -0.451987 0.05 0.191988 0.17 -0.041989 0.31 -0.081990 -0.03 0.731991 0.3 -0.331992 0.08 -0.071993 0.1 -0.291994 0.01 0.2

Zero Beta Zero Beta


Year S&P Gold1983 0.23 -0.11984 0.06 -0.161985 0.32 01986 0.18 0.251987 0.05 0.21988 0.17 -0.171989 0.31 01990 -0.03 -0.051991 0.3 -0.051992 0.08 -0.061993 0.1 0.121994 0.01 0.02

0.4

0.2

0

-0.2

0.40.30.20.10-0.1

S&P 500 Return

The correlation between gold and the S&P 500 and its beta is approximately zero.

Annual return pairs for the S&P 500 and Gold

Slope is zero

Go ld

Return

Positive Beta

Negative Beta

StockReturn

Positive and negative beta stock returns move opposite one another.

High Beta

Low Beta

Market

StockReturn

During this time period the market rises, falls, and then rises again. A high (low) beta stock varies more (less) than the market.

Port folio Beta Port folio Beta


GENERAL FORMULA

Bp = WiBi

Example: Beta for a portfolio containing three stocks.

INVESTMENT $ INVESTED Wi Bi

1 100 100/1000 = .1 2.02 400 400/1000 = .4 1.53 500 500/1000 = .5 0.5

Bp = W1B1 + W2B2 + W3B3

= .1(2) + .4(1.5) + .5(.5) = 1.05

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CAPM CAPM

““Beta is Useful Because it Can Be Precisely Translated into Beta is Useful Because it Can Be Precisely Translated into a Required Return, k, Using the Capital Asset Pricing Model”a Required Return, k, Using the Capital Asset Pricing Model”

Lecture 4 - Measuring Risk (Return Volatility) Lecture 4 - Measuring Risk (Return Volatility)

CAPM (Capital Asset Pricing Model)

General Formula

Ri = ki = Rf + Bi(Rm - Rf)

= time value + (units of risk) x (price per unit)

= time value + risk premium

where, Rf = Risk-Free Rate -> T-Bill

Rm = Expected Market Return -> S&P 500

Bi = Beta of Stock i

Example:

Suppose that a firm has only equity, is twice as

risky as the market and the risk free rate is 10%

and expected market return is 15%. What is the

firm’s required rate?


= 10% + 2(15% - 10%)

= 10% + 10%

= 20%

QUESTION: If an asset has a B = 0, what is its return?-> Rf

QUESTION: If an asset has a B = 1, what is its return? -> Rm

QUESTION: Suppose E(R1) > E(R2) AND B1 < B2, which asset do you choose? -> 1

QUESTION: How about if E(R1) > E(R2) and B1 > B2 ?

Now we need to know B1 and B2 and use the CAPM


CONSIDER STOCK PRICE AND CAPM

P =


QUESTIONS:What happens to price as growth increases? P increases!What happens to price if k increases? P Decreases!What happens to price if Beta decreases? P increases!What happens to price if Rf increases? B >1->P increase

B<1 -> P decreaseWhat happens to price if Rm decreases? P increases!

QUESTION: As financial managers, what variables should we try to change and in what directions?

1. Increase cash flows - or growth in CF’s - make superior investment decisions, use the lowest cost financing or manipulate debt/equity ratio

2. Bring cash flows in closer to the present

3. Decrease Beta - Manipulate assets (Labor- Capital ratio).


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Documents

Lecture: 4 - Measuring Risk (Return Volatility) I.Uncertain Cash Flows - Risk Adjustment II.We Want a Measure of Risk With the Following Features a. Easy