
and the correlation coefficient can be expressed as
ρ
R
A
R
B
=
Cov {R
A
, R
B
}
σ
A
σ
B
,
=
L
2
[a
1
75 + a
2
(1000) + a
3
(2525)] − (80d
3
+ 30d
2
) (80d
1
+ 50d
2
)
*
(d
3
+ d
2
)
2
100 + d
2
3
25
*
d
2
1
100 + (d
1
+ d
2
)
2
25
.
If d
1
= 2, d
2
= 8 and d
3
= 2, then
µ
A
=
80d
3
+ 30d
2
L
= 33.33
µ
B
=
80d
1
+ 50d
2
L
= 46.67.
Note that L = d
1
+ d
2
+ d
3
. The variances are
σ
2
A
=
1
L
2
&
(d
3
+ d
2
)
2
100 + d
2
3
25
'
= 70.14
σ
2
B
=
1
L
2
&
d
2
1
100 + (d
1
+ d
2
)
2
25
'
= 20.14.
Also,
a
1
=
1
L
2
(
2d
1
d
3
+ d
2
d
3
+ d
1
d
2
+ d
2
2
)
= 0.7222
a
2
=
1
L
2
[d
1
(d
3
+ d
2
)] = 0.1389
a
3
=
1
L
2
[d
3
(d
1
+ d
2
)] = 0.1389.
So
E{R
A
R
B
} = a
1
75 + a
2
(1000) + a
3
(2525) = XX.
The covariance is then given by
Cov {R
A
, R
B
} = a
1
75 + a
2
(1000) + a
3
(2525)
−
80d
3
+ 30d
2
L
80d
1
+ 50d
2
L
= (0.7222)75 + (0.1389) (1000) + (0.