Download - Markowitz

AMITY GLOBALBUSINESS SCHOOL Chandigarh

Basic principles• There are many asset classes out there and

many of them are useful to investors. • Some asset classes are noted for their long

term stability (low risk), others for their high returns.

• Generally speaking, the higher the reward you are after, the more risk you’ll need to take.

• Portfolios can be constructed out of multiple asset classes that exhibit superior risk and return relationships to any single asset, because diversification can significantly reduce risk.


The point of portfolio construction• A portfolio is often more than the sum of its parts.

Because not all asset classes perform the same way over the short term, a portfolio of many asset classes usually offers a superior overall relationship between risk and return to any single asset.

• A portfolio consisting only of shares would have done badly in the last few years since the US market crashed, but property and bonds have performed very well. This is quite typical, more “defensive” asset classes often do well when equities are falling.

• A diversified portfolio has a reasonable long term growth rate because over time all asset classes offer a positive return, but being invested across different asset classes smooths out returns and offers a more predictable growth rate.


Naïve Diversification• This type of diversification focus on

selecting the securities that best fit the needs and desires of investors.


Markowitz diversification• Markowitz diversification strategy primarily

focuses on degree of correlation between the assets returns in a portfolio rather than simply investing in a large number of securities.


Risk and return on portfolio• Expected return on portfolio is computed

as the weighted average of expected returns on stocks.

N

• E(Rp)= ∑ WiE(Ri) i=1Where,E(Rp)=the expected return on portfolioN= The number of stocks in a portfolioWi=The proportion of portfolio invested in stock IE(Ri)=the expected return on stock i.


• For a portfolio consisting of two assets the above equation can be expressed as :-

• E(Rp)=W1E(R1)+(1-W1)E(R2)

• Calculate expected return on a portfolio of stocks 50%a and 50%b and 75%a and 25%b.

• E(Ra)=12.5% and E(Rb) = 20%


• Solution:- Return on portfolio consisting 50% stock a and 50% stock b.

• E(Rp)=.50(12.5%)+(1-.50)20%


Portfolio variance and standard deviation, portfolio risk

Portfolio return volatility (standard deviation):


For a two asset portfolio: Portfolio risk:


For a three asset portfolio: Portfolio risk


Ques: Calculate return on the portfolio from the following data.

Security Expected Return,R1% Proportion X1%

1 10 25

2 20 75


• Solution:-

• Rp = R1X1 + R2X2

• Rp= .10(.25)+.20(.75)

• =17.5%


Ques: Calculate degree of risk in a portfolio from

the following:• σa =4• σb =7• Wa =.5• Wb =.5

When coeficient correlation rx= -1

rx = -0.5

rx = 0

rx =1


• Ans: 1.5

• Ans:3.04

• Ans:4.03

• Ans:5.5


Formula to find out ideal combination of securities

• Wa= σb/ σa + σb

• =7/4+7

• Wa= .64

• When Wa = .64

• Wb = .36

• Therefore when rx= -1

• Ans =0


Q- Compute risk and return of portfolio from following information:

Security Expected return% Proportion %

1 10 20

2 15 20

3 20 60


• Other information:-• Standard deviation• σ1 =.2• σ2 =.3• σ 3 =.5• r12 =.5• r13 =.1• r23 =-0.3


Calculate risk of the portfolio• E(Ra)=12.5%,E(Rb)=20%, Sda=5.12%,

SDb=20.49% and rab=-1.

• Risk of a portfolio consisting of 50% a and 50 % b.


• Modern Portfolio theory was introduced by Harry Markowitz in 1952 in his paper ‘portfolio Selection”.

• According to him, out of entire universe of possible portfolios, certain ones will optimally balance risk and reward.He called these an efficient frontier of portfolios. An investor should select a portfolio that lies on the efficient frontier.


Assumption of portfolio theory• In order to construct the Markowitz

efficient portfolios, some basic assumptions about asset selection behaviour are as follows:-

• 1. Risk and reward

• 2. Mean and variance, investor makes decision on two parameters of risk and returns.


• 3. Homogeneous expectations

• 4. One period investment horizon.


Construction of efficient portfolios and efficient frontier

• Identifying feasible set of portfolios by combining number of securities (negetive correlation)in different proportions.

• From the feasible set, delineate the efficient set of portfolios and efficient frontier.

• Using investor risk return preferences obtain optimum portfolio for that investor from the feasible set.


CHAPTER 9 – The Capital Asset Pricing Model (CAPM)

Attainable Portfolio Combinations for a Two Asset Portfolio

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Standard Deviation of Returns

Exp

ecte

d R

etu

rn o

f th

e P

ort

foli

o


Two Asset Efficient Frontier

• Figure describes five different portfolios (A,B,C,D and E in reference to the attainable set of portfolio combinations of this two asset portfolio.


Portfolio Risk (σp)

ERp

Achievable Portfolio Combinations

More Possible Combinations Created

E

E is the minimum variance portfolio Achievable Set of

Risky Portfolio Combinations

The highlighted portfolios are ‘efficient’ in that they offer the highest rate of return for a given level of risk. Rationale investors will choose only from this efficient set.



9 - 26

Portfolio Risk (σp)

Achievable Set of Risky Portfolio Combinations

ERp

Achievable Portfolio Combinations

Efficient Frontier (Set)

E

Efficient frontier is the set of achievable portfolio combinations that offer the highest rate of return for a given level of risk.



9 - 27

9 - 4 FIGURE

σρ

ER

RF

A2

T

A

B

B2

Capital Market Line

The New Efficient FrontierThe New (Super) Efficient Frontier

The optimal risky portfolio

(the market portfolio ‘M’)

Clearly RF with T (the market portfolio) offers a series of portfolio combinations that dominate those produced by RF and A.

Further, they dominate all but one portfolio on the efficient frontier!

This is now called the new (or super) efficient frontier of risky portfolios.

Investors can achieve any one of these portfolio combinations by borrowing or investing in RF in combination with the market portfolio.


Utility function and indifference curve

• A utility curve shows how much an investor is willing to trade off between expected return and risk.

• Graph displaying the feasible set (the dark area), the efficient set (the borderline between E and S is ) and some indifference curves. The optimal portfolio is marked with O


INVESTMENT OPPORTUNITY FUNDAMENTALS

Expected Rate of Return & RiskExpected Rate of Return & Risk

Figure 13.2 (c)Figure 13.2 (c)


Limitations of Markowitz Model• Large number of input data is required• Computations are complex and large in

numbers• Variance cannot be taken as an appropriate

measure of risk.• Risk preference of investors is assumed to

be given and remain constant.• Practical obstacles restrict the use of model.


• The gist of Modern portfolio theory is that the market is hard to beat and that the people who beat the market are those who take above average risk.

Download - Markowitz

Top Related