Portfolio Theory I

1

Portfolio Theory

Slide 2

Premise of Portfolio Theory

• Any asset can go up or down depending on the market conditions.

• When these assets are put together to form a portfolio, their interaction reduces overall volatility which then contributes to the stability of the portfolio.

• Any portfolio is designed with returns in mind. One can then choose an expected return and then seek to minimize the risk.

Slide 3

Assumptions of Portfolio Theory • Asset returns are normal distributed. • Correlations between the asset returns (not raw

prices) are fixed and constant over a period of time.

• Investors seek to maximize their overall return. • All players in the market are rational and risk

averse • Common information is available to all players in

the market • Many other assumptions …….

Slide 4

Topics

• Two risky assets - with specified expected return - global minimum variance portfolio

• One risky asset & one risk-free asset

• Two risky assets & one risk-free asset

• Matrix Formulation

Slide 5

Motivation

• Suppose investment in a certain stock offers an expected return of 12% while the expected return on a certain bond is only 8%. Would you put all your money in the stock ?

• If not, what is the optimal combination of these two assets ?

Slide 6

Data Set for Demonstration

• The following data will be repeatedly used to illustrate the techniques :

μA μB σA2 σB

2 σA σB σAB ρAB 0.175 0.055 0.067 0.013 0.258 0.115 -0.004875 -0.164

Slide 7

Portfolio with 2 Risky Assets

• Suppose a percentage of wA of the capital is invested in security A and wB in security B.

• Thus wA + wB = 1 with wA , wB ≥ 0

• Rate of return rp = wArA + wBrB

• Expected return of portfolio

E(rp) = wAE(rA) + wBE(rB)

Slide 8

Portfolio with 2 Risky Assets (cont’d)

• Variance of rate of return of portfolio

σp2 = var(rp)

= var(wArA + wBrB)

= wA2var(rA) + wB

2var(rB) + 2wAwBcov(rA, rB)

= wA2σA

2 + wB2 σB

2 + 2wAwB ρAB σAσB

where ρAB is the correlation coefficient between rA and rB.

Portfolio with 2 Risky Assets – Example 1

• Consider a portfolio with wA = wB = 0.5.

– μP = (0.5)(0.175) + (0.5)(0.055) = 0.115

– σp2 = (0.5)2(0.067) + (0.5)2(0.013) + 2(0.5)(0.5)(-0.004875)

= 0.01751

σp = 0.1323

The portfolio expected return is the average of the expected returns of assets A and B The portfolio s.d. is less than the average of the asset s.d. This shows diversification reduces risk.

Slide 9

Portfolio with 2 RiskyAssets – Example 2

• Consider a long-short portfolio with wA = 1.5 and wB = -0.5.

– μP = E(rp) = (1.5)(0.175) + (-0.5)(0.055) = 0.235

– σp2 = (1.5)2(0.067) + (-0.5)2(0.013) + 2(1.5)(-0.5)(-0.004875)

= 0.1604

σp = 0.4005

This portfolio has a higher expected return and a higher s.d. than both asset A and asset B.

Slide 10

Slide 11

Shape of Portfolio Frontier : ρ ≠ ±1

( )

( ) ( )( ) ( ) ( )

2 2 2 2 2

22 2 2

2 2 2 2 2

2

1 2 1

2 2 2

P A A B B A B A B

P A A B B A B A B

A A A B A A A B

A B A B A B A B A B

w w w

w w w w

w w w w

w w

µ µ µ µ µ µ

σ σ σ ρσ σ

σ σ ρσ σ

σ σ ρσ σ σ ρσ σ σ

= + = − +

= + +

= + − + −

= + − + − + +

Portfolio lies on the quadratic curve joining A and B.

Slide 12

Shape of Portfolio Frontier : ρ = 1

2 2 2 2 2 2

2 ( )

P A A B B

P A A B B A B A B A A B B

P A A B B

w ww w w w w ww w

µ µ µ

σ σ σ σ σ σ σσ σ σ

= +

= + + = +⇒ = +

Portfolio lies on the line joining A and B.

Slide 13

Shape of Portfolio Frontier : ρ = -1

The portfolio is riskless when :

( )

( )( )

( )

( )

22 2 2 2 2

2

0

1

µ µ µ µ

σ σ σ σ σ σ σ

σ σ σ σ σ σ

σσ σ σ

σ σ

σσ σ σ

σ σ

− +

= + − = −

⇒ − +

++

++

≤ ≤

≤ ≤

=

= = −

− += −

A B A B

P A A B B A B A B A A B B

P A A B B A B A B

BA B A B A

A B

BA B A B A

A B

P w

w w w w w w

w w w

w w

w w

if

if

, 1BA B A

A B

w w wσσ σ

= = −+

Slide 14

Portfolio Frontier : ρ = -1 Example

The portfolio is riskless when :

115

373115

373

0

1

0.12

0.373

0.373

µ µ

σσ

σ

+

⇒

≤ ≤

≤ ≤

=

− + −

=A A

A A

BA

B

B

P

P

w

w w

w w

if

if

115 258, 373 373

= =A Bw w

μA μB σA σB 0.175 0.055 0.258 0.115

Slide 15

Portfolio Frontier : ρ = -1 Example (cont’d)

The portfolio is riskless when :

258

373258

373

0

1

0.12

0.373

0.373

µ µ

σσ

σ

+

⇒

≤ ≤

≤ ≤

= −

− + −

=A

A A

A

A B

B

B

P

P

w

w w

w w

if

if

258 115, 373 373

= =A Bw w

μA μB σA σB 0.055 0.175 0.115 0.258

Shape of Portfolio Frontiers for Various ρ

Efficient Portfolio & Efficient Frontier

• Portfolios with the highest expected return for a given level of risk are called efficient portfolios.

• The curve passing through a collection of efficient portfolios is called the efficient frontier (EF).

Slide 17

Efficient Portfolios & EF with 2 Risky Assets : ρ ≠ ±1

Slide 18

Efficient portfolios in green Inefficient portfolios in red

Global Minimum Variance Portfolio

Slide 19

This portfolio has the smallest variance among all efficient portfolios

Global Minimum Variance Portfolio

• To find this portfolio, one has to solve the constrained minimization problem

• It can be shown that

Slide 20

2 2 2 2 2

, 2

s.t. 1

minA B

P A A B B A B ABw w

A B

w w w w

w w

σ σ σ σ= + +

+ =

2min

2 2

min min

2

1

B ABA

A B AB

B A

w

w w

σ σσ σ σ

−=

+ −

= −

Global Minimum Variance Portfolio - Example

• Using the data, we have

The expected return, variance and s.d. of this portfolio are

Slide 21

min min0.013 ( 0.004875) 0.1992 0.80080.067 0.013 2( 0.004875)A Bw w− −

= = ⇒ =+ − −

2 2 2

(0.1992)(0.175) (0.8008)(0.055) 0.0789

(0.1992) (0.067) (0.8008) (0.013) 2(0.1992)(0.8008)( 0.004875) 0.00944

0.00944 0.09716

P

P

P

µ

σ

σ

= + =

= ++ − =

= =