Portfolio Theory

Himanshu PuriFaculty

DIAS

HARRY MARKOWITZ MODEL

• The portfolio theory developed deals with the selection of portfolio that maximizes expected returns consistent with the individual investor acceptable level of risk.

• Model provides a conceptual framework and analytical tool for selection of optimal portfolio.

• As the model is based on the expected returns (means) and the standard deviation (variance) of different portfolios it is also called MEAN-VARIANCE MODEL

ASSUMPTIONS

• An investor is basically risk averse.• The risk of a portfolio is estimated on the basis of

variability of expected returns of the portfolio.• The decision of the investor regarding selection of

the portfolio is made on the basis of expected returns and risk of the portfolio.

• An investor attempts to get maximum return from the investment with minimum risk. That is for a given level of risk he attempts to earn a higher return.

HM Model can be presented in 3 steps :

• Setting the Risk-Return Opportunity Set

• Determining the Efficient Set

• Selecting the Optimal portfolio

Setting the Risk-Return Opportunity Set

• Starts with the identification of the opportunity set of various portfolios in terms of risk and return of each portfolio.

• Say ‘x’ number of securities available in which an investor can invest his funds.

• An infinite number of combinations of all or a few of these securities are possible. Each such combination has an expected average rate of return and a level of risk.

RISK – RETURN OF NUMBER OF POSSIBLE PORTFOLIOS

• The shaded area AEHA includes all possible combinations of risk and return of portfolios.

• Combination R represents risk level of r1 and the return level of r2.

Determining the Efficient Set• Efficient portfolio is one which provides the

maximum expected return for any particular degree of risk or the lowest possible degree of risk for any given rate of return.

• The portfolios which lie along the boundary AGEH are efficient portfolios.

• For given level of risk r3 there are three portfolios L, M and N. but the portfolio L is called an efficient portfolio.

• Also L is called the dominating portfolio.

• The boundary AGEH is called the Efficient Frontier.

Selecting the optimal portfolio• The HM model does not specify one optimal

portfolio.

• It rather generates the efficient set of portfolios, which by definition are all optimal.

• To select the expected risk-return combination that will satisfy investor’s preferences, indifference curve are used.

• All the points lying on a particular indifference curve represent different combinations of risk and return which provide same level of utility or satisfaction to the investor.

• Now, the efficient frontier can be combined with the indifference curve to determine the investor’s optimal portfolio.

• The investor’s optimal portfolio is found at the tangency point of efficient frontier with indifference curve.

• This tangency point marks the highest level of satisfaction, the investor can attain.

Finding the Efficient Frontier.

• The efficient frontier contains a very large number of portfolios.

• Not all portfolios contain all securities.• The upper right edge of the efficient frontier

corresponds to the single security which has highest expected return H.

• Any other portfolio would have a lower expected return because at least a part of the investor fund would be placed in other securities that have expected return lower than H.

• As he moves along the efficient frontier to the left, new securities enter the portfolio mix and some securities may leave the portfolio and this is known as corner portfolio.

Risk free Lending and borrowing

• It is possible in the HM model to build portfolios with higher utilities by combining risk free investments

• On Risk free lending or investment return is certain

• Standard deviation of the risk free asset is zero

• Examples: treasury bills, government securities

Risk-Free Asset

Covariance between two sets of returns is

n

1ijjiiij )]/nE(R-)][RE(R-[RCov

Because the returns for the risk free asset are certain,

0RF Thus Ri = E(Ri), and Ri - E(Ri) = 0

Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero.

Combining a Risk-Free Asset with a Portfolio

Expected return is the weighted average of the two returns

))E(RW-(1(RFR)W)E(R iRFRFport

This is a linear relationship

Combining a Risk-Free Asset with a Portfolio

Standard deviation: The expected variance for a two-asset portfolio is

211,22122

22

21

21

2port rww2ww)E(

Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become

iRFiRF iRF,RFRF22

RF22

RF2port )rw-(1w2)w1(w)E(

Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

22RF

2port )w1()E( i

Combining a Risk-Free Asset with a Portfolio

Given the variance formula22

RF2port )w1()E( i

22RFport )w1()E( i the standard deviation is

i)w1( RF

Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

Combining a Risk-Free Asset with a Portfolio

Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.

Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port

RFR

M

C

AB

D

Risk-Return Possibilities with Leverage

• To attain a higher expected return than is available at point M (in exchange for accepting higher risk)

• Either invest along the efficient frontier beyond point M, such as point D

• Or, add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point M

Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port

RFR

M

CML

Borrowing

Lending

The Market Portfolio• Portfolio M lies at the point of tangency, so it has the

highest portfolio possibility line

• This line of tangency is called the Capital Market Line (CML)

• Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML (the CML is a new efficient frontier)– Therefore this portfolio must include all risky assets

The CML

• Individual investors should differ in position on the CML depending on risk preferences (which leads to the Financing Decision)

– Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio (points left of M)

– Aggressive investors would borrow funds at the RFR and invest everything in the market portfolio (points to the right of M)

Sharpe’s

Optimization

Model

Uses a single number to decide whether a security should be a part of portfolio or not.

i

Fi RR

A security is preferred to another if it excess return to beta ratio is more than the other security

Sharpe computes a number which is compared to the above ratio for all the securities. Only those securities are selected which have excess return to beta ratio above this number.

Steps in arriving at the optimal portfolio:

1. Calculate the excess return to beta ratio for each stock and rank it in descending order.

2.Find out all the stocks for which the excess return to beta ratio is more than a cut-off rate.

3. Determine the weightages in which the investments have to be made in the stocks in the optimal portfolio

SecurityMean return

1 7

2 15

3 17

4 12

5 11

6 5.6

7 17

8 11

9 7

10 11

Risk-free Rate=5%

Excess return

2

10

12

7

6

0.6

12

6

2

6

Beta

0.8

1

1.5

1

1.5

0.6

2

1

1

2

Ratio

2.50

10.00

8.00

7.00

4.00

1.00

6.00

6.00

2.00

3.00

Now arrange the securities in the descending order of the excess return to beta ratio

SecurityMean return

Excess return beta ratio

2 15 10 1 10.00

3 17 12 1.5 8.00

4 12 7 1 7.00

7 17 12 2 6.00

8 11 6 1 6.00

5 11 6 1.5 4.00

10 11 6 2 3.00

1 7 2 0.8 2.50

9 7 2 1 2.00

6 5.6 0.6 0.6 1.00

The securities are then selected using a cut-off rate.

Starting from the top, portfolios are constructed with the first portfolio including only the first security, the second portfolio including the first and second security and so on.

For each of these portfolios a number C(i) is computed where C(i) is given by the following equation:

i

i e

im

i

i e

iFim

i

i

i

RR

C

12

22

12

2

1

*)(

Where:

iancemarketm var:2

SecurityMean return

Excess return beta ratio

2 15 10 1 10.00

3 17 12 1.5 8.00

4 12 7 1 7.00

7 17 12 2 6.00

8 11 6 1 6.00

5 11 6 1.5 4.00

10 11 6 2 3.00

1 7 2 0.8 2.50

9 7 2 1 2.00

6 5.6 0.6 0.6 1.00

50

40

20

10

40

30

40

16

20

6

2

ie

Security

2 0.200

3 0.450

4 0.350

7 2.400

8 0.150

5 0.300

10 0.300

1 0.100

9 0.100

6 0.060

2

*)(

ie

iFi RR

0.020

0.056

0.050

0.400

0.025

0.075

0.100

0.040

0.050

0.060

2

2

ie

i

0.200

0.650

1.000

3.400

3.550

3.850

4.150

4.250

4.350

4.410

i

i e

iFi

i

RR

12

*)(

0.020

0.076

0.126

0.526

0.551

0.626

0.726

0.766

0.816

0.876

i

i e

i

i12

2

1.667

3.688

4.420

5.429

5.451

5.301

5.023

4.906

4.748

4.517

iC

Computing iC

Now only those securities are selected for which the excess return to beta is more than the corresponding C(i) value. So the first 5 securities are selected

i

Fi RR

)(

10.00

8.00

7.00

6.00

6.00

4.00

3.00

2.50

2.00

1.00

102 mfor

The cut-off ratio C* has to be such that all the securities above the lowest selected security are selected. In this case it turns out to be 5.45.

Determining the Weightages

The percentage X(i) to be invested in security say (i) is given:

N

ii

ii

Z

ZX

1

Where:

*)(2

CRR

Zi

Fi

e

ii

i

Security beta ratio   C

2 1 10.00 50 1.667

3 1.5 8.00 40 3.688

4 1 7.00 20 4.420

7 2 6.00 10 5.429

8 1 6.00 40 5.451

Z

0.090979

0.095585

0.077447

0.109789

0.013724

X

0.23477

0.246657

0.199851

0.283309

0.035414

2

ie