Risk and Retrun Theory

7/29/2019 Risk and Retrun Theory

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Financial Management

Risk and Return& CAPM


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Dictionary meaning A hazard, a peril; exposure to loss

Skydiving, betting , investment.

Type of analyzing risk : On a stand alone basis and on portfolio basis.

No investment should be undertaken unless the expected rate of

return in high enough to compensate the investor for perceived risk

of the investment

Risk and return go hand in hand.

What is risk ?


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Total Return = Income + Capital gain

The stock price for Stock A was $9.50 per share 1 year ago. Thestock is currently trading at $10 per share, and shareholders just

received a $1 dividend. What return was earned over the past year?

Return on single asset

1 1 01 011

0 0 0

Rate of return Dividend yield Capital gain yield

DIVDIV

P PP PR

P P P


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An events probability is defined as chance that the event will

occur

If all possible events, or outcomes, are listed, and if a probability is

assigned to each event, the listing is called probability

distribution.

Lets add some statistics


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Probability distribution


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Expected payoff


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Expected return


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Expected Return


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Are two companies same


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The tighter or more peaked the probability distribution is it is more

likely that actual outcome would closed to expected value and vise a

versa. So more spread in the distribution more risky is the

asset.

Probability distribution


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The average rate of return is the sum of the various one-period ratesof return divided by the number of period.

Formula for the average rate of return is as follows:

Average Rate o f Return

1 2

=1

1 1= [ ]n

n t

t

R R R R Rn n


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Risk refers to uncertainty of return. (Actual Return is differentthen expected return)

Uncertainty could be in terms of time and difference in return.

Difference in return can be found out with the help of Standard

deviation and variance.

Risk of Return

Standard deviation = Variance

22

1

1

1

n

t

tR Rn


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Measuring Risk

Coin Toss Game-calculating variance and standard deviation

(1) (2) (3)

Percent Rate of Return Deviation from Mean Squared Deviation

+ 40 + 30 900

+ 10 0 0

+ 10 0 0

- 20 - 30 900

Variance = average of squared deviations = 1800 / 4 = 450

Standard deviation = square of root variance = 450 = 21.2%


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Portfolio Risk


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Measuring Risk

Portfolio rate

of return=

fraction of portfolio

in first assetx

rate of return

on first asset

+fraction of portfolio

in second assetx

rate of return

on second asset

((

(())

))


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Portfolio Risk

2

2

2

2

211221

1221

211221

122121

21

xxx

xx2Asset

xxxxx1Asset

2Asset1Asset

The variance of a two assets portfolio is the sum of these fourboxes


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jk=

j

krjk

jis the standard deviation of thejth asset in the portfolio,

kis the standard deviation of the kth asset in the portfolio,

rjkis the correlation coefficient between thejth and kth assets in theportfolio.

Correlation coefficient

A standardized statistical measure of the linear relationship between

two variables. Correlation is a scaled version of covariance

Its range is from -1.0 (perfect negative correlation), through 0 (no

correlation), to +1.0 (perfect positive correlation).

Covariance and correlation coef ficient

Covariance :


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The ratio of the standard deviation of a distribution to the mean of

that distribution.

It is a measure of RELATIVE risk.

CV = / R

The Coefficient of variation shows the risks per unit of return,

and it provides a more meaningful basis for comparison when

the expected returns on two alternatives are not the same.

Higher the ratio riskier is the asset !!

Coefficient of variation

Di ifi i d l i


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Diversification an d correlation

coefficient

Combining securities that are not perfectly,positively correlated reduces risk.

INVESTMENTRETURN

TIME TIMETIME

SECURITY E SECURITY F

As both the securities will not move in same direction.

CombinationE and F


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%12)1540(.)1060(.ReturnExpected

Portfolio Risk

ExampleSuppose you invest 60% of your portfolio in Wal-Mart and 40% inIBM. The expected dollar return on your Wal-Mart stock is 10%and on IBM is 15%. The expected return on your portfolio is:


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Portfolio Risk

222

2

2

2

211221

211221222

1

2

1

)7.29()40(.x7.298.191

60.40.xxIBM

7.298.191

60.40.xx)8.19()60(.xMart-Wal

IBMMart-Wal

ExampleSuppose you invest 60% of your portfolio in Wal-Mart and 40% inIBM. The expected dollar return on your Wal-Mart stock is 10% andon IBM is 15%. The standard deviation of their annualized dailyreturns are 19.8% and 29.7%, respectively. Assume acorrelation coefficient of 1.0 and calculate the portfoliovariance.


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Portfolio Risk

ExampleSuppose you invest 60% of your portfolio in Wal-Mart and 40% inIBM. The expected dollar return on your Wal-Mart stock is 10% andon IBM is 15%. The standard deviation of their annualized dailyreturns are 19.8% and 29.7%, respectively. Assume a correlationcoefficient of 1.0 and calculate the portfolio variance.

%23.85.564DeviationStandard

5.56419.8x29.7)2(.40x.60x

]x(29.7)[(.40)

]x(19.8)[(.60)VariancePortfolio

22

22


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Portfolio Risk

)rx()r(xReturnPortfolioExpected 2211

)xx(2xxVariancePortfolio211221

2

2

2

2

2

1

2

1


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Example Correlation Coefficient = .4

Stocks % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Standard Deviation = weighted avg = 33.6

Standard Deviation = Portfolio = 28.1

Real Standard Deviation:

= (282)(.62) + (422)(.42) + 2(.4)(.6)(28)(42)(.4)

= 28.1 CORRECT

Return : r = (15%)(.60) + (21%)(.4) = 17.4%

Portfolio Risk


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Example Correlation Coefficient = 0.4

Stocks s.d % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Standard Deviation = weighted avg = 33.6

Standard Deviation = Portfolio = 28.1

Return = weighted avg = Portfolio = 17.4%

Lets Add stock New Corp to the portfolio

Portfolio Risk


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Example Correlation Coefficient = .3

Stocks % of Portfolio Avg Return

Portfolio 28.1 50% 17.4%

New Corp 30 50% 19%

NEW Standard Deviation = weighted avg = 31.80NEW Standard Deviation = Portfolio = 23.43

NEW Return = weighted avg = Portfolio = 18.20%

NOTE: Higher return & Lower risk

How did we do that? DIVERSIFICATION

Portfolio Risk


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Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk is the variability of return on stocks or portfolios

associated with changes in return on the market as a whole.

It is also called Market riskor Non diversifiable risk

Unsystematic Risk is the variability of return on stocks or portfolios

not explained by general market movements. It is stock/ company

specific. It is avoidable through diversification.

It it also called Uniqueriskor diversifiablerisk

Systematic risk and unsystematic risk


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To t a l R i s k = S y s t e m a t i c R i s k + U n s y s t e m a t i c

R i s k

Total

Risk

Unsystematic risk

Systematic risk

STDDEVOF

PORTFOLIORETU

RN

NUMBER OF SECURITIES IN THE PORTFOLIO

Factors such as changes in nations

economy, Inflation, tax policy,

or a change in the world situation.

T t l R i k S t t i R i k U t t i


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To t a l R i s k = S y s t e m a t i c R i s k + U n s y s t e m a t i c

R i s k

Total

Risk

Unsystematic risk

Systematic risk

STDDEVOF

PORTFOLIORETU

RN

NUMBER OF SECURITIES IN THE PORTFOLIO

Factors unique to a particular company

or industry. For example, the death of a

key executive or loss of a governmental

defense contract.


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risk-averse : He doesnt like riskInvestor will choose among investments with the equal rates ofreturn, the investment with lowest standard deviation. Similarly, ifinvestments have equal risk (standard deviations), the investorwould prefer the one with higher return

risk-neutral: He is indifferent

Investor does not consider risk, and would always prefer investmentswith higher returns

risk-seeking : He loves risk

Investor likes investments with higher risk irrespective of the rates ofreturn.

In reality, most (if not all) investors are risk-averse

Expected Risk and Preference

30


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CAPM is a model that describes the relationship between risk and

expected (required) return; in this model, a securitys expected(required) return is the risk-free rate plus a premium based on the

systematic riskof the security.

CAPM Assumptions :

Capital markets are efficient.

Homogeneous investor expectations over a given period.

Risk-free asset return is certain(use short to intermediate-term

Treasuries as a proxy).

Market portfolio contains only systematic risk (use S&P 500 Index orsimilar as a proxy).

Capital asset pricing model


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Characteristic l ine

EXCESS RETURN

ON STOCK

EXCESS RETURN

ON MARKET PORTFOLIO

Beta =Rise

Run

Narrower spreadis higher correlation

Characteristic Line


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An index of systematic risk.

It measures the sensitivity of a stocks returns to changes in returns on the

market portfolio.

The beta for a portfolio is simply a weighted average of the individual stock

betas in the portfolio.

What is Beta?

EXCESS RETURN

ON STOCK

EXCESS RETURN

ON MARKET PORTFOLIO

Beta < 1

(defensive)

Beta = 1

Beta > 1

(aggressive)

Each characteristicline has a

different slope.


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Rj

= Rf

+ bj

(RM

- Rf

)

Rj is the required rate of return for stock j,

Rf is the risk-free rate of return,bjis the beta of stock j (measures systematic

risk of stock j),

RM is the expected return for the marketportfolio

Security Market l ine


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Security market l ine

Rj = Rf+ bj(RM - Rf)

bM = 1.0

Systematic Risk (Beta)

Rf

RM

RequiredRetur

n

Risk

Premium

Risk-free

Return


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Market Portfolio - Portfolio of all assets in the economy. In practice abroad stock market index, such as the S&P Composite, is used to

represent the market.

Beta - Sensitivity of a stocks return to the return on the market

portfolio.

Beta and Unique Risk


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2

m

im

iB


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2

m

im

iB

Covariance with themarket

Variance of the market


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

Product of

Deviation Squared deviations

Deviation from average deviation from averageMarket Anchovy Q from average Anchovy Q from average returns

Month return return market return return market return (cols 4 x 5)

1 -8% -11% -10% -13% 100 130

2 4 8 2 6 4 12

3 12 19 10 17 100 170

4 -6 -13 -8 -15 64 120

5 2 3 0 1 0 0

6 8 6 6 4 36 24

Average 2 2 Total 304 456

Variance = m2 = 304/6 = 50.67

Covariance = im = 456/6 = 76

Beta () = im/m2

= 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 ofthe variance to the covariance (i.e., = im/m

2)

Beta


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Thank You

Risk and Retrun Theory

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