3. Financial Modeling
1
Matrix Notation
Generally a necessity with other than small amounts of
data.
Often best to switch to matrix notation when more than
two variables.
2
Notation
 E (r1 ) 
 E (r ) 
 2 




E
(
r
)
 n 
 11  12

 22
21




 n1
 1n 




 nn 
 x1 
x 
x   2
 
 
 xn 

n  n covariance matrix
 ij
covariance of the returns of securities i and j
 ij   ji
 ii
variance of the return of security i
x
vector of weights (where xi is the proportion of capital
to be invested in security i )
E (ri ) expected return of security i
3
In Brealey, Myers and Allen
Chapter 8, pp. 190-191
Port Var = x12 12  x22 22  2( x1 x2 12 1 2 )
 x12 11  x22 22  2 x1 x2 12
 x1 11 x1  x2 22 x2  x1 12 x2  x2 21 x1

 x1
 11  12   x1 
x2  
 x 


 21
22   2 
 xT  x
4
Maximizing Expected Portfolio Return
Consider 5 stocks whose betas are 1.43, 0.91, 1.68, 0.48, 0.81 and whose
forecasted monthly expected returns are 2.4%, 1.2%, 1.9%, 0.9%, 2.6%.
Formulate an LP to solve for the portfolio that maximizes expected portfolio
return given that:
• portfolio beta does not exceed 1.05
• stocks 2 and 5 together do not make up more than 40% of the portfolio
• stocks 1 and 3 together make up at least 20% of portfolio
• no individual stock can be more than 40% of the portfolio
• no short selling is allowed
x1
x2
x3
x4
x5
obj
2.4
1.2
1.9
0.9
2.6
max
s.t.
1
1
1
1
1
=
1
1.43
0.91
1.68
0.48
0.81
<=
1.05
1
<=
0.40
>=
0.20
1
1
up bnds
0.40
1
0.40
0.40
0.40
0.40
all xi >=
0
5