Essentials of Investments: Chapter 6 - Efficient Diversification

Essentials of Investments: Chapter 6 - Efficient Diversification includes Two-Security Portfolio, Correlation Coefficients, Three Security Portfolio, Minimum Variance Combination, Extending Concepts to All Securities, Optimal Risky Portfolios.

Chapter 6

Efficient Diversification

r p = W 1 r 1 + W 2 r 2

Two-Security Portfolio: Return

W

S

n

= 1

Security 1 and Security 2

Security 1 and Security 2

Two-Security Portfolio: Risk

Correlation Coefficients: Possible Values

If r = 1.0, the securities would be perfectly positively correlated

If r = - 1.0, the securities would be perfectly negatively correlated

-1.0 < r < 1.0

+ W 3 2 s 3 2

+ W 3 2 s 3 2

Cov(r 1 r 3 ) Cov(r 1 r 3 )

Three-Security Portfolio

r p = Weighted average of the

Portfolio Risk/Return Two Securities: Correlation Effects

• Relationship depends on correlation coefficient

W 1 =

(.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3)

Minimum Variance Combination: r = -.3

Extending Concepts to All Securities

• The optimal combinations result in lowest level of risk for a given return

• The optimal trade-off is described as the efficient frontier

• These portfolios are dominant

Optimal Risky Portfolios

Portfolio Risk behavior

Variance

# Assets systematic risk

Unsystematic risk

After a certain number of securities, portfolio variance can no longer be reduced

Portfolios of Two Risky Assets

• case (1): Assume c=1.0

stand dev 0.12 0.16

0.08

0.2 1

2 0.105

Stand Dev.

Portfolio Return/Risk

c=-1 c=0.3

2 Return

S p

Max (r p -r f )/s p

{w}

Example of optimal portfolio

The optimal weight in the less risky

asset will be:

w 1 = (r1-rf)s

2

2 -(r 2 -r f )cov(1,2) (r 1 -r f )s 2 2 +(r 2 -r f )s 2 1 -(r 1 -r f +r 2 -r f )cov(1,2)

w 2 =1-w 1

Given:

r 1 =0.1, s 1 =0.2

r 2 =0.3, s 2 =0.6 c(coeff of corr)=-0.2 Then: cov=-0.24

w 1 =0.68

w 2 =1-w 1 =0.32

Lending v.s Borrowing

2 Return

S p U

Assume two portfolios (p, r ), weight

Lending

p

Markowitz Portfolio Selection

• Three assets case return and variance formula for the portfolio

• N-assets case Return and variance formula for the portfolio

risky assets

Efficient frontier

Global minimum variance

variance frontier

Individual assets

Extending to Include Riskless Asset

• The optimal combination becomes linear

• A single combination of risky and riskless assets will dominate

CAL (Global minimum variance)

CAL (A) CAL (P)

Dominant CAL with a Risk-Free Investment (F)

CAL(P) dominates other lines it has the best risk/return or the largest slope

Slope = (E(R) - Rf) / s [ E(R P ) - R f ) / s P ] > [E(R A ) - R f ) / s A ] Regardless of risk preferences

combinations of P & F dominate

Single Factor Model - CAPM

r i = E(R i ) + ß i F + e

to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns

Assumption: a broad market index like the

Single Index Model

Risk Prem Market Risk Prem

or Index Risk Prem

i = the stock’s expected return if the market’s excess return is zero

ß i (r m - r f ) = the component of return due to

movements in the market index

f

AR it is the abnormal for security i at time t

- it is the error term of the market model calculated on an out of sample basis

- it is basically the forecast error

returns abnormal

the compute then

ˆ and ˆ

get window, estimation

in the OLS

run

mt i

i it

it R R

AR    ˆ   ˆ

Let: R i = (r i - r f )

R m = (r m - r f )

Risk premium format

R i =  i + ß i (R m ) + e i

Risk Premium Format

Excess Returns (i)

Security Characteristic Line