risk and retrun theory
TRANSCRIPT
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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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Beta and Unique Risk
2
m
im
iB
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Beta and Unique Risk
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