228 Optimizing Optimization
straightforward to see that as λ ranges from 0 to ∞ , we move down the fron-
tier from the maximum expected return portfolio to the global minimum variance
portfolio. This framework is widely used in finance, see Sharpe (1981) , Grinold
and Kahn (1999) , and Scherer (2002).
Our first-order condition is:
∂
∂U
bbi b i
()() ( )
ωμλω θ ωμλ θ
ΩΩˆ ˆ 0 or ˆ^1 ^1 ˆUsing i ( ω b ) 0, we see that^1 λ()βθˆγ^0 , where β i Ω 1 μ , γ i Ω ^1 i
and we set a μ Ω ^1 μ. Thus, ωμˆ ()
λ
β
γ
bi^1 ΩΩ^11 and hence active
returns α can be computed as:
αμωβ
γλγβ
γ λ ()ˆ baa121 2⎛⎝⎜⎜
⎜⎜⎞⎠⎟⎟
⎟⎟⎟⎛⎝⎜⎜
⎜⎜⎞⎠⎟⎟
⎟⎟⎟
(10.1)Other terms of interest can be calculated. For example, we have:
σˆ
λμ
β
γμ
β
γλ2
2(^11111)
1
⎛ΩΩΩΩΩii
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
22
22
2
1 2
a
a
β
γ
β
γ
λ
β
γ
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
(10.2)
and we will focus on σˆ the tracking error or standard deviation of relative
returns. Finally,
EU
aa
()
1
2
(^21)
2
2
λ
γβ
γ
λ
λ
γβ
γ
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜⎜
⎞⎞
⎠
⎟⎟
⎟⎟
⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
1
2
2
λ
γβ
γ
a
(10.3)