Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

184 Optimizing Optimization We shall assume the existence of a unique minimum risk portfolio; if it exists, the minimum risk por ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 185 If we assume, without loss of generality that μ 1 μ ...

186 Optimizing Optimization derived. Second, assuming joint lognormality, where we use either Equation (8.28) or (8.32) and nume ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 187 So , examine the integral in brackets: Jqy yt dy p p ...

188 Optimizing Optimization It q qq q p tppt p () ∂   ∂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎟⎟ ⎛ ⎝ ⎜⎜ σ ⎜⎜ μσσμ σ 2 2 22 2 2 exp Φ ⎜⎜⎜ ⎞ ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 189 Finally , if r 1 and r 2 are any two mean – variance ...

190 Optimizing Optimization thus, ()^1 ()1 0  r 1 P P p y ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ exp (8.59) So that () 11 rry ypi∑ωωiexp()() ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 191 we see that () ()( ) () VaR t pp pp    μ μγ βμ ...

192 Optimizing Optimization Simplifying , and letting Δ  α γ  β 2 0, vvvvv v^22222 22 12 2 2 2 210 () () γθ βθ βθ β θ βθ βθ αθ ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 193 To check that this is a minimum, we compute: u   ...

194 Optimizing Optimization and since γ θ 1 and ( u  v ) 0 at ( u , v ) from Equation (8.89), the mini- mum point is establis ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 195 Step 5: Repeat Step 2 to Step 4, M times. Step 6: Fr ...

196 Optimizing Optimization the mean/standard deviation, the mean/semi – standard deviation, the mean/ value at risk (5%), and t ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 197 other three risk measures. The grey lines represent ...

198 Optimizing Optimization measured in daily percentages. For the mean/standard deviation frontier, we measure the error as the ...

Computing optimal mean/downside risk frontiers: the role of ellipticity 199 Campbell , R. , Huisman , R. , & Koedijk , K. ( ...

This page intentionally left blank ...

© 2009 Elsevier Limited. All rights reserved. Doi:10.1016/B978-0-12-374952-9.00009-9. 2010 Portfolio optimization with “ Thresho ...

202 Optimizing Optimization computational restrictions. In fact, already in the 1950s Markowitz pondered using downside semivari ...

Portfolio optimization with “ Threshold Accepting ” : a practical guide 203 portfolios with the Omega ratio ( Keating & Shad ...