For YourBusinessTutor- Essentials of Investment

Chapter 6 EFFICIENT DIVERSIFICATION

1) Project Groups

2) Review of HW for chapter 5

3) EFFICIENT DIVERSIFICATION

Asset Allocation with Two Risky Assets

For the two asset portfolio:

E(rp)=W1*r1+ W2*r2

E(rp) = Expected Return on Two Risky Assets

W1 = Proportion of funds in Security 1 (weight, percentage)

W2 = Proportion of funds in Security 2

r1 = Expected return on Security 1

r2 = Expected return on Security 2

1 2

= Variance of Security 1

2 2

= Variance of Security 2

Cov(r1,r2) = Covariance of returns for Security 1 and Security 2

Cov(r1,r2)= ρ*1*2

Relevant formulas for n securities are as follows:

In the two-asset case it is fairly easy to calculate the minimum variance weight with the

following equations:

Once the weights are known, the minimum variance portfolio expected return and risk can be

calculated

),(2 2121

2

2

2

2

2

1

2

1

2 rrCovWWWW

p  

portfolio the in securities # n ;rW)rE( n

1i

iip   

Wi i=1

n

= 1Wi i=1

n

WiWi i=1i=1

n

= 1

  

 n

1I

n

1J

JIJI

2

p )]r,Cov(r W[Wσ

1. Example: The parameters of the opportunity set are:

E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%,  = 0.15, rf = 5.5%

From the standard deviations and the correlation coefficient we generate the covariance matrix [note

that Cov(rS, rB) = SB]:

Bond

s

Stocks

Bonds 529.0 110.4

Stocks 110.4 1024.0

The minimum-variance portfolio proportions are:

wMin(S) = 

－

 

－ =

－

－ = .3142

wMin(B) = 1 – .3142 = .6858

The mean and standard deviation of the minimum variance portfolio are:

E(rMin) = ( .3142  15%) + ( .6858  9%)  10.89%

Min = [



+



+ 2 wS wB Cov(rS, rB)]1/2

= [( .3142 2  1024) + ( .6858

2  529) + (2  .3142  .6858  110.4)]1/2

= 19.94%