Lesson 4

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

3

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?


4

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


( ) ∑=

=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

7


( ) ∑=

=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


8


( )[ ] ii

n

i

PRER 2

1

2 −=∑=

σ

154902 .=σ

Variance and standard deviation of our example

4959.=σ


9

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.

10

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

11

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.

15

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

16

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

01/31/13 17

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

18

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.


19

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 ×+×=


20

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


21

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


22

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.


23

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


24

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

=

−=×

−==


25

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.


26

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

27

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


28

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


29

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


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

31

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

32

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.

33

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.

36

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

39

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

41

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

45

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

Lesson 4

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