lesson 4
TRANSCRIPT
Risk and Return Relationship
FIN 3321Investment and Portfolio Management I
01/31/13 1Prepared by P D Nimal
2
ObjectivesOn satisfactory completion of this topic student wil l be
able to:
Understand the relationship between risk and return of assets
Portfol io Risk and Return
Importance of covariance and correlation between returns of assets
Diversif ication advantage
Mean Variance Eff ic ient Frontier
Capital Market Line
Prepared by P D Nimal01/31/13
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Expected Return and Risk
Do not use Historical Data
Use Forecasted Data
Suppose you are considering investing in
shares of HNB. Market price is Rs. 200.
You want to hold the share for one year.
What is your expected rate of return?
Prepared by P D Nimal01/31/13
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Expected Return and Risk cont…
This wil l depend on the
Actual dividend you would receive and
The market price at which you could sell the share
These two wil l decide the rate of return that you could
earn
Both dividend and the price at which you can
sell wil l depend on the possible state of
economic condit ions.Prepared by P D Nimal01/31/13
01/31/13 5
Expected Return and Risk cont…
( ) ∑=
=n
iiiPRRE
1
The average dispersion of the return is measured by the
variance or standard deviat ion. The equation is as follows.
( )[ ] ii
n
i
PRER 2
1
2 −= ∑=
σ
Calculate the E(R) and the Standard Deviat ion of assets
given in the table.
6
Expected Return and Risk cont…Suppose the state of economic conditions and the possible rates of return with probabil i t ies of the occurrence of each state of economic condition are as follows
Return and Probabilities
Economic Rate of ProbabilityRate of
ReturnConditions Return *Probability
Growth 17.5 0.2 3.5Expansion 11.2 0.3 3.36Stagnation 5.4 0.25 1.35Decline -8.9 0.25 -2.225
1 ER=5.985Prepared by P D Nimal01/31/13
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Expected Return and Risk cont…
( ) ∑=
=n
iiiPRRE
1
( )9855
2598254532112517
.
........RE
=×−×+×+×=
The average dispersion of the return is measured by the variance or standard deviation. The equation is as follows.
( )[ ] ii
n
i
PRER 2
1
2 −=∑=
σ
Prepared by P D Nimal01/31/13
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Expected Return and Risk cont…
( )[ ] ii
n
i
PRER 2
1
2 −=∑=
σ
154902 .=σ
Variance and standard deviation of our example
4959.=σ
Prepared by P D Nimal01/31/13
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Risk and Return Investment Alternatives
Econ. Prob. T-Bill Alta Repo Am F. MPBust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%Below avg.
0.20 8.0 -2.0 14.7 -10.0 1.0
Avg. 0.40 8.0 20.0 0.0 7.0 15.0Above avg.
0.20 8.0 35.0 -10.0 45.0 29.0
Boom 0.10 8.0 50.0 -20.0 30.0 43.0 1.00
Calculate the Risk and Return of assets given in the table.
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Expected Return versus Risk
SecurityExpectedreturn% Risk, σ%
Alta Inds. 17.4 20.0Market 15.0 15.3Am. Foam 13.8 18.8T-bills 8.0 0.0Repo Men
1.7 13.4
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What is unique about the T-bill return?
The T-bill will return 8% regardless of the state of the economy.
Is the T-bill riskless? Explain.
12
Alta Inds. and Repo Men vs. the Economy
Alta Inds. moves with the economy, so it is positively correlated with the economy. This is the typical situation.
Repo Men moves counter to the economy. Such negative correlation is unusual.
13
Stand-Alone Risk
Standard deviation measures the stand-alone risk of an investment.
The larger the standard deviation, the higher the probability that returns will be far below/above the expected return.
14
Coefficient of Variation (CV)
CV = STD/E(R) CVT-BILLS = 0.0 / 8.0 = 0.0. CVAlta Inds = 20.0 / 17.4 = 1.1. CVRepo Men = 13.4 / 1.7 = 7.9. CVAm. Foam = 18.8 / 13.8 = 1.4. CVM = 15.3 / 15.0 = 1.0.
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Expected Return versus Coefficient of Variation
SecurityExpectedreturn%
Risk:σ%
Risk:CV
Alta Inds 17.4 20.0 1.1Market 15.0 15.3 1.0Am. Foam 13.8 18.8 1.4T-bills 8.0 0.0 0.0Repo Men 1.7 13.4 7.9
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Return vs. Risk (Std. Dev.): Which investment is best?
T-bills
Repo
MktAm. Foam
Alta
0.0%2.0%4.0%6.0%8.0%
10.0%12.0%14.0%16.0%18.0%20.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Risk (Std. Dev.)
Ret
urn
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Portfolio Risk and Return
The return of a portfol io is equal to the weighted average of
the returns of individual assets in the portfol io.
Two-Asset Case
State of Economy
Probability ReturnsX Y
1234
0.100.200.500.20
-8.57.26.54.2
8.5-5.44.37.5
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Risk and Return Portfolio Investment
( )684
224556227158
.
........RE X
=×+×+×+×−=
( )423
257534245158
.
........RE Y
=×+×+×−×=
Suppose we invest in an equally weighted portfolio of these two assets. i .e., 50% of the investment in X and 50% in Y.
Prepared by P D Nimal01/31/13
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Expected Return of the Portfolio
State of Economy
Probability Portfolio Return
1234
0.100.200.500.20
(-8.5*.5+8.5*.5)=0(7.2*.5-5.4*.5)=.9
(6.5*.5+4.3*.5)=5.4(4.2*.5+7.5*.5)=5.85
( ) 0545042350684 .....RE p =×+×=
( ) 05485524559201 ........RE p =×+×+×+×=
( ) ( ) ( ) 5050 .RE.RERE YXp ×+×=
Prepared by P D Nimal01/31/13
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Risk of the Portfolio
514.X =σ 674.Y =σ
Lets Compute the Standard deviation of X and Y separately
State of Economy
Probability Portfolio Return
1234
0.100.200.500.20
(-8.5*.5+8.5*.5)=0(7.2*.5-5.4*.5)=0.9(6.5*.5+4.3*.5)=5.4(4.2*.5+7.5*.5)=5.85
We wil l consider the same example
Prepared by P D Nimal01/31/13
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Risk of the Portfolio cont…
( ) ( ) ( ) ( )
282
1845
0548552054455054902054012
22222
.
.
...........
p
P
=
=−+−+−+−=
σσσ
( ) 054.RE p =
59.4
5.67.45.51.4
≠×+×≠Pσ
Standard deviation of the portfolio
Important:This is not the Weighted average of the standard
deviations
( )( )∑=
−=n
iipipp PRER
1
2σ
Prepared by P D Nimal01/31/13
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Risk of the Portfolio cont…
212122
22
21
21
2 2 ,P CovWWWW ++= σσσ
∑∑= =
=n
kkj
n
jjkP WW
1 1
2 σσ
Portfol io standard deviation can be calculated as
follows
When there are two stocks in the portfolio
( )
28.2
184.5
73.105.286.215.34.205. 2222
==
−×+×+×=
p
P
σ
σAccording to our ex.
Prepared by P D Nimal01/31/13
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Covariance between two assets
[ ][ ] ijjkk
n
ijk PRERRER )()(
1, −−= ∑
=
σ
6 7
X Y P XP YP X-ERx Y-Ery P*6*7
-8.5 8.5 0.1 -0.85 0.85 -13.18 5.08 -6.69544
7.2 -5.4 0.2 1.44 -1.08 2.52 -8.82 -4.44528
6.5 4.3 0.5 3.25 2.15 1.82 0.88 0.8008
4.2 7.5 0.2 0.84 1.5 -0.48 4.08 -0.39168
1 ER 4.68 3.42 Cov -10.7316
Prepared by P D Nimal01/31/13
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Risk Of the Portfolio cont…
21212122
22
21
21
2 2 σσσσσ ,P CorrWWWW ++=
( )
28.2
184.5
67.451.451.05.286.215.34.205. 2222
==
×−×+×+×=
p
P
σ
σ
This can be written in a different way
According to our ex.
212,12,1
21
2,12,1 51.0
67.451.4
73.10
σσσσ
CorrCov
CovCorr
=
−=×
−==
Prepared by P D Nimal01/31/13
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Risk-Return Relationship of Portfolios on Correlation
21212122
22
21
21
2 2 σσσσσ ,P CorrWWWW ++=
When the correlation is 1, what is the standard
deviation of the portfolio?
( )
594
0821
6745141528621534205 2222
.
.
.......
p
P
==
××++×=
σ
σ
According to our ex.
Prepared by P D Nimal01/31/13
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Portfolio ER & STD when Correlation coefficient is 1
Wx Wy ER STD
1 0 5.6 5.2
0 1 2.6 3.5
0.5 0.5 4.1 4.35
When Correlation (1)
Y
X
0
1
2
3
4
5
6
0 2 4 6
Std
ER
( ) ∑=
=n
iiip RWRE
1
∑∑= =
=n
iij
n
jjiP WW
1 1
2 σσ
21122122
22
21
21
2 2 σσρσσσ WWWWP ++=Prepared by P D Nimal01/31/13
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Portfolio ER & STD when Correlation coefficient is -1
Wx Wy ER STD1 0 5.6 5.2
0.5 0.5 4.1 0.850.4 0.6 3.8 0
0.25 0.75 3.35 1.3250 1 2.6 3.5
When Correlation -1
Y
X
0
1
2
3
4
5
6
0 2 4 6
STD
ER
Prepared by P D Nimal01/31/13
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Portfolio ER &STD when Correlation coefficient is 0
Wx Wy ER STD1 0 5.6 5.2
0.5 0.5 4.1 3.130.25 0.75 3.35 2.93
0 1 2.6 3.5
When Correlation 0
Y
X
0
1
2
3
4
5
6
0 2 4 6
STDE
R
Prepared by P D Nimal01/31/13
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Portfolio ER & STD on Correlation Coefficient- Summary
ER Vs. STD on Corr
Corr+0Corr=-1
Y
X
Corr=1
0
1
2
3
4
5
6
0 2 4 6
STD
ER
Prepared by P D Nimal01/31/13
01/31/13 30
Portfolio Risk cont…
Therefore, the standard deviat ion of portfol io return is
dependent on the correlat ion or covariance structure of stocks
in the portfol io
When the correlation of two stocks is 1, the standard deviation is the weighted
average of standard deviations of the stocks.
When the correlation of two stocks is less than 1, the standard deviation of the
portfolio is less than the weighted average of standard deviations of the stocks.
Since the correlations of stocks are in general less than 1, the standard deviation of
the portfolio is less than the weighted average of standard deviations of the stocks
This effect is called diversification advantage
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Calculate the Expected return and std of the portfolio of 60% Alta and 40% Repo
Econ. Prob. T-Bill Alta Repo Am F. MPBust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%Below avg.
0.20 8.0 -2.0 14.7 -10.0 1.0
Avg. 0.40 8.0 20.0 0.0 7.0 15.0Above avg.
0.20 8.0 35.0 -10.0 45.0 29.0
Boom 0.10 8.0 50.0 -20.0 30.0 43.0 1.00
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Portfolio ER & STD - Mean-Variance Efficient Frontier (Markowitz-1959)
Me an - Va rian c e Ef f ic ie n t Fro n t ie r
C
A
B
STD
Mean
= E
R
EF is From B to CBecause, it gives • The Highest ER at a
given level of STD and
• The lowest STD at a given level of ER
•When we draw the efficient frontier of all the stocks in the market, it looks like bellow.
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Portfolio ER & STD- Mean-Variance Efficient Frontier (Markowitz-1959)
Me an - Varian c e Ef f ic ie n t Fro n t ie r
C
A
B
STD
Mea
n =
ER
•The line from B to C is Called the Capital Market Line (CML) (without risk-free lending & borrowing).
34
Feasible and Efficient Portfolios
The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks.
An efficient portfolio is one that offers: the most return for a given amount of risk, or the least risk for a give amount of return.
The collection of efficient portfolios is called the efficient set or efficient frontier.
01/31/13 35
Capital Asset Pricing Model (CAPM) Sharpe (64), Lintner (65)
Sharpe and Lintner introduced two basic assumptions to the Markowitz’s EF.
1. Unlimited lending and borrowing at Risk-Free rate.
2. Homogeneous expectations or complete agreement about the ER and STD of securities. This leads to have a similar EF for all rational investors.
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M
Z
.ArRF
σM Risk, σp
The Capital MarketLine (CML):
New Efficient Set
. .B
rM^
ExpectedReturn, rp
Efficient Set with a Risk-Free AssetWith risk-free lending and borrowing, the CML is as follows (Rf-M-Z). The tangency portfolio would be the market portfolio.
01/31/13 37
Capital Market Line (CML)
ERe
Rf
0
M
σmSTD
CML
38
p = RF +
SlopeIntercept
ER σp.ERM - RF
σM
Risk measure
The CML Equation
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What does the CML tell us?
The expected rate of return on any efficient portfolio is equal to the risk-free rate plus a risk premium.
The optimal portfolio for any investor is the point of tangency between the CML and the investor’s indifference curves.
What doesn’t the CML tell us?
01/31/13 Prepared by P D Nimal 40
CML gives the ER and STD of efficient portfolios
The problem is that it only gives ER and Risk (STD) of efficient portfolios.
ER & STD of Inefficient portfolios and individual stocks are not given
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rRF
σMRisk, σp
I1
I2
CML
R = Optimal Portfol io
.R .MrR
rM
σR
^^
ExpectedReturn, rp
Capital Market Line & Investor portfolio selection
I1-Risk Averse I2-Risk Taker
01/31/13 42
Capital Market Line (CML)cont…
• According to this analysis, the optimal portfolio of risky assets would be the market portfolio (M).
• The portfolios from Rf to M are lending portfolios because they lend a portion of their investment at Rf and
• The portfolios from M to upwards are borrowing portfolios because they borrow some money at Rf and invest both their capital and borrowed money in the market portfolio.
• Depending on the risk preference investor can choose a lending or borrowing portfolio.
01/31/13 43
Lending & Borrowing Portfolios
mmfrf ERWRWER +=
•ER & Risk of lending and borrowing portfolios.
mmW σσ =Weight on the market portfolio is •Less than 1 for lending portfolios and•Greater than 1 for borrowing portfolios.
01/31/13 44
Calculate the ER & STD of (0.5 Rf and 0.5 M) (a lending portfolio) and (-0.5 Rf and 1.5 M) (a borrowing portfolio).
Rf Rm Prob. P(50:50)3 1 0.1 23 0.9 0.2 1.953 5.4 0.5 4.23 5.8 0.2 4.4
•ER & Risk of lending and borrowing portfolios.
Lending & Borrowing Portfolios
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Adding Stocks to a Portfolio
What would happen to the risk of a portfolio as more randomly selected stocks were added?
σp would decrease because the added stocks would not be perfectly correlated.
46
σ1 stock ≈ 35%σMany stocks ≈ 20%
-75 -60 -45 -30 -15 0 15 30 45 60 75 90 105
Returns (% )
1 st ock2 st ocksMany st ocks
4710 20 30 40 2,000 stocks
Company Specif ic (Diversif iable) Risk
Market Risk20%
0
Stand-Alone Risk, σ p
σp
35%
Risk vs. Number of Stock in Portfolio
48
Market risk & Diversifiable risk
Market risk is that part of a security’s risk that cannot be eliminated by diversification.
Firm-specific, or diversifiable, risk is that part of a security’s risk that can be eliminated by diversification.
49
Market risk & Diversifiable riskConclusions
As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio.
σp falls very slowly after about 40 stocks are included. The lower limit for σp is σM
By forming well-diversified portfolios, investors can eliminate about half the risk of owning a single stock.
The Problem of CMLInvestment and Portfolio Management II
01/31/13 Prepared by P D Nimal 50
CML gives the ER and STD of efficient portfolios
The problem is that it only gives ER and Risk (STD) of efficient portfolios.
ER & STD of Inefficient portfolios and individual stocks are not given
The SML of CAPM will solve this problem which will be discussed in the Investment and Portfolio Management II