186 Optimizing Optimization
derived. Second, assuming joint lognormality, where we use either Equation
(8.28) or (8.32) and numerical methods.
We see, by manipulating the above, that if we can compute the marginal
probability density function of r p , the results will simplify considerably. For the
case of normality, and for general elliptical distributions, the pdf ( r p ) is known.
We proceed to compute the ()μθpp,^2 frontier under normality.
Proposition 8.2: Assuming that
r
r(^1) N
2
1
2
1
2
12
12 2
2
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟∼
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜⎜
⎞
⎠
μ ⎟⎟
μ
σσ
σσ
, ⎟⎟⎟
⎟
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(8.33)
the mean/semivariance frontier can be written as:
θσ μ
μ
σ
μσ φ
μ
σ
pp p
p
p
pp
p
p
t
t
t
22 2 t
()() ()
⎛
⎝
⎜⎜
⎜⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
⎛
⎝
Φ ⎜⎜
⎜⎜⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
(8.34)
where φ ( · ) and Φ ( · ) are the standard normal density and distribution functions
respectively, μ (^) p is given by Equation (8.26), and σωσ ωωσp^22 12 2 () 1 12
()1ωσ^222. Moreover, if r 1 and r 2 are any two mean – variance efficient portfo-
lios, the above result will hold for any N 2.
Proof : Consider the integral (letting θp^2 ()tIt () ),
I t t rppppdf r dr
t
()( ) ( )
2
∫∞
(8.35)
where rNppp∼ ()μσ,^2 , so that
pdf r
r
p
p
pp
p
()
()
1
22
2
σπ^2
μ
σ
exp
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
(8.36)
Transform r p y t r p ⇒ rp t y and drp dy
It y
yt
dy
p
p
p
()
()
2 ^2
0 2
1
σπ 22
μ
σ
exp
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
∞
∫
(8.37)
It
q
qy
yt
dy
p
p
p
() ( )
()
∂
∂
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
(^2) ∞
2
2
0 2
1
22
exp exp
σπ
μ
σ
∫∫
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
q (^0)
(8.38)