Halaman 13.docx

Halaman 13 2.2 mean-semivariance project portofolio selection problem In this section we formally the MSVP problem. Consider n risky non-divisble investment project. Let r = (𝑟1 , … , 𝑟𝑛 )𝑇 ∈ ℝ𝑛 denote the uncertain NPVs of the n risky project, which are calculated over a time horizon of T Periods. Let x = (𝑥1 , … , 𝑥𝑛 )𝑇 ∈ {0,1}𝑛 denote a portfolio on these project; that is, the binary variables 𝑥𝑖 take the value of 1 if the company invest in project i and 0 otherwise, for i = 1, ... , n. Thus, the portfolio’s NPV is given by 𝑛 𝑇

𝑇

𝑟 𝑥 = 𝑥 𝑟 = ∑ 𝑥𝑖 𝑟𝑖 𝑖=1

Halaman 14 Writen as the following optimization problem: min 𝔼 (min⁡{ 0, 𝑥 𝑇 𝑟 − 𝔼(𝑥 𝑇 𝑟)}2 ) (2.1) 𝑇

𝑠. 𝑡.⁡⁡⁡⁡⁡𝔼(𝑥 𝑟) ≥ 𝜇0 𝑥 ∈ 𝜒 ∩ {0,1}𝑛 . Halaman 15 𝑚

1 2 min ∑ min{0, 𝑥 𝑇 𝑟 𝑗 − 𝑥 𝑇 𝜇) 𝑚 𝑗=𝑖

(2.2) 𝑇

𝑠. 𝑡.⁡⁡⁡⁡⁡𝔼(𝑥 𝑟) ≥ 𝜇0 𝑥 ∈ 𝜒 ∩ {0,1}𝑛 . Where the vektor µ = (𝜇1 , … , 𝜇𝑛 )𝑇 ∈ ℝ𝑛 of mean project return estimates is obtained by letting 𝑚

1 𝑗 𝜇𝑖 = ∑ 𝑟𝑖 , 𝑚 𝑗=1

(2.3) For i = 1, ... , n. Halaman 16 folio selection problem (2.2) is equivalent to: 𝑚

1 𝑗 min ∑ 𝒴𝑖 𝑚 𝑗=1

𝑇

𝑗

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡⁡𝒴𝑗 ≤ 𝑥 (𝑟 − 𝜇)⁡⁡⁡⁡⁡⁡𝑗 = 1, … , 𝑚 (2.4) 𝒴𝑗 ≤ 0 𝑥 𝑇 𝜇 ≥ 𝜇0 𝑥 ∈ 𝜒 ∩ {0,1}𝑛 .

Halaman 18 To (2.2) 𝑛

𝑛

1 min ∑ ∑ 𝜎̃𝑖𝑘 𝑥𝑖 𝑥𝑘 𝑚 𝑖=1 𝑘=1

(2.5) 𝑇

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡𝑥 𝜇 ≥ 𝜇0 𝑥 ∈ 𝜒 ∩ {0,1}𝑛 , Where 𝑚 𝑗

𝑗

𝜎̃𝑖𝑘 = ∑ min{0, 𝑟𝑖 − 𝜇𝑖 } min{0, 𝑟𝑘 − 𝜇𝑘 } 𝑗=1

(2.6) To see that (2,5) provides a pessimistic approximation to (2,2), let 𝑢 ∈ ℝ be given, and define 𝐼 − = {𝑖: 𝑢𝑖 < 0, 𝑖 = 1, … , 𝑛}, and 𝐼 + = {𝑖: 𝑢𝑖 ≥ 0, 𝑖 = 1, … , 𝑛}. Clearly. 𝑛

𝑛

0 ≥ min {0, ∑ 𝑢𝑖 } = { 𝑖=1

∑ 𝑢𝑖 + ∑ 𝑢𝑖 , 𝑖𝑓⁡ |∑ 𝑢𝑖 | ≥ |∑ 𝑢𝑖 | ; } − − + +

𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

0,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (2.7)

Halaman 19 Also 𝑛

∑ min{0, 𝑢𝑖 } = 0 + ∑ 𝑢𝑖 . 𝑖=1

𝑖∈𝐼

(2.8) Using (2.7) and (2.8) in two cases |∑𝑖∈𝐼− 𝑢𝑖 | ≥ |∑𝑖∈𝐼+ 𝑢𝑖 | and |∑𝑖∈𝐼− 𝑢𝑖 | ≤ |∑𝑖∈𝐼+ 𝑢𝑖 | it follows that 𝑛

𝑛

𝑛

0 ≥ min {0, ∑ 𝑢𝑖 } ⁡𝑎𝑛𝑑 min {0, ∑ 𝑢𝑖 } ≥ ∑ 𝑢𝑖 min{0, 𝑢𝑖 }, 𝑖=1

𝑖=1

𝑖=1

(2.9) And therefore: 𝑛

2

2

𝑛

(min {0, ∑ 𝑢𝑖 }) ≤ (∑ 𝑢𝑖 min{0, 𝑢𝑖 }) . 𝑖=1

𝑖=1

(2.10) With (2.10), and letting 𝑟̃ ≔ 𝑟 − 𝜇, 𝑗 = 1, … , 𝑚, we have that objective function of (2.2) can be bounded from abov as follows: 𝑗

𝑚

𝑛

𝑗=1

𝑖=1

2

𝑗

𝑚

𝑛

𝑗=1

𝑖=1

2

1 1 𝑗 𝑗 ∑ min {0, ∑ 𝑥𝑖 𝑟̃𝑖 } ≤ ∑ (∑ min{0, 𝑥𝑖 𝑟̃𝑖 }) = 𝑚 𝑚

𝑚

𝑛

𝑛

1 𝑗 𝑗 ∑ ∑ ∑ min{0, 𝑥𝑖 𝑟̃𝑖 } min{0, 𝑥𝑖 𝑟̃𝑘 } = 𝑚 𝑗=1 𝑖=1 𝑘=1

(2.11) 𝑚

𝑛

𝑛

1 𝑗 𝑗 ∑ ∑ 𝑥𝑖 𝑥𝑘 (∑ min{0, 𝑟̃𝑖 } min{0, 𝑟̃𝑘 }) = 𝑚 𝑖=1 𝑘=1

𝑖=1 𝑚

𝑛

1 ∑ ∑ 𝜎̃𝑖𝑘 𝑥𝑖 𝑥𝑘 𝑚 𝑖=1 𝑘=1

The first inqulity follows from (2.10), and the second to last equality follows from 𝑥𝑖 ≥ 0, 𝑖 = 1, … , 𝑛. Hence, (2,5) is a presmistic approximation to (2.2) because its objective overestimates the expected squared downside deviations: that is, the semivariance. Notice that (2.5) will be equivalent to (2.2) whenever the second inequlity in (2.9) holds with equlity when replacing 𝑢 = 𝑥 𝑇 𝑟̃ 𝑗 , for all j = 1, ..., m. The second inequality in (2.9) holds with equlity when 𝐼 − = {1, … , 𝑛}⁡𝑜𝑟⁡𝐼 + = {1, … , 𝑛}. That is, problem (2,5) and (2.5) will be 𝑗 𝑗 equivalent if 𝐼 𝑗+ ≔ {𝑖 ∈ {1, … , 𝑛} ∶ 𝑥𝑖 𝑟̃𝑖 ≥ 0, } = {1, … , 𝑛}⁡𝑜𝑟𝐼 𝑗− ≔ {𝑖 ∈ {1, … , 𝑛} ∶ 𝑥𝑖 𝑟̃𝑖 ≥ 0, } = {1, … , 𝑛},⁡for all j = 1, ..., m.

Halaman 20 The objective funcion of (2.5) can be linearized by introducing approciate extra continuous variables. Let 𝐼𝜎+ ≔ {(𝑖, 𝑘) ∶ ⁡ 𝜎̃𝑖𝑘 > 0, 𝑖 = 1, … , 𝑛, 𝑘 = 1, … , 𝑛}, and 𝐼𝜎− ≔ {(𝑖, 𝑘) ∶ ⁡ 𝜎̃𝑖𝑘 > 0, 𝑖 = 1, … , 𝑛, 𝑘 = 1, … , 𝑛}. Then problem (2.5) is equivalent to te following MILP problem: 1 min ∑ 𝜎̃𝑖𝑘 𝑦𝑖𝑘 𝑚 + (𝑖,𝑘)∈𝐼𝜎 𝑇

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡𝑥 𝜇 ≥ 𝜇0 𝑦𝑖𝑘 ≥ 𝑥𝑖 + 𝑥𝑘 − 1⁡𝑓𝑜𝑟⁡𝑎𝑙𝑙⁡(𝑖, 𝑘) ∈ 𝐼𝜎+ 𝑦𝑖𝑘 ≥ 0⁡𝑓𝑜𝑟⁡𝑎𝑙𝑙⁡(𝑖, 𝑘) ∈ 𝐼𝜎+ (2.12) {0,1}𝑛

𝑥∈𝜒∩ . The equivalence between (2.5) and (2.12) follows fom the next observatins. First, from (2.6) it follows that and 𝐼𝜎− ≔ {(𝑖, 𝑘) ∶ ⁡ 𝜎̃𝑖𝑘 > 0, 𝑖 = 1, … , 𝑛, 𝑘 = 1, … , 𝑛}. Secons, if (𝑖, 𝑘) ∈ 𝐼𝜎+ , then 𝑦𝑖𝑘 ≥ 𝑥𝑖 + 𝑥𝑘 − 1 and 𝑦𝑖𝑘 ≥ 0 imply that 𝑦𝑖𝑘 ≥ 𝑥𝑖 𝑥𝑘 , but since 𝜎̃𝑖𝑘 > 0, then at any optimal solution of (2.2), 𝑦𝑖𝑘 would be at its lower bound 𝑦𝑖𝑘 ≥ 𝑥𝑖 𝑥𝑘 . Halaman 21 Presentation more succint, we re-state (2.4) as follows: 1 min⁡⁡⁡⁡ 𝑦 𝑇⁡ (𝐼)𝑦 𝑚 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑦 ≤ 𝑅̃ 𝑥 𝑦≤0 (S) 𝑥∈𝜒∩

{0,1}𝑛

,

Where 𝑦 ≔ [𝑦𝑗 ]j = 1, ...., m, I is the m×m identity matrix, 𝑅̃ is a m×n matrix, whose row j is 𝑇 given by [𝑅̃ ⁡]𝑗 ≔ (𝑟 𝑗 − 𝜇) , 𝑗 = 1, … , 𝑚, 𝑎𝑛𝑑⁡𝜒 ′ ≔ 𝜒 ∪ {𝑥 ∈ ℝ𝑛 : 𝑥 𝑇 𝜇 ≥ 𝜇0 }.

Halaman 22 Obtain the problem: 1 𝑇⁡ 𝑦 (𝐼)𝑦 𝑚 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑦 ≤ −𝑅̃ 𝑥⁡⁡⁡(𝑢) min⁡⁡⁡⁡

(2.13) 𝑦 ≤ 0⁡⁡⁡⁡⁡(𝑢0 ) Where 𝑢 ∈ ℝ are the dual variables associated to the return constraints and 𝑢0 ∈ ℝ𝑚 are the dual variables associated to the nin-negativity constraints in (2.13). the (convex) quadratic program in (2.13) corresponds to the benders subproblem, whose Wolfe dual is given by (see, e.g., Nocedal and Wright (2006, Chapter 12)): 1 max ⁡⁡⁡⁡⁡−𝑢𝑇 𝑅̃ 𝑥̂ − 𝑦 𝑇 (𝐼)𝑦 𝑚 2 𝑆. 𝑡.⁡⁡⁡⁡⁡ − 𝑦 + 𝑢 + 𝑢0 = 0 𝑚 (2.14) 𝑢, 𝑢0 ≥ 0, Problem (2.14) is equivalent to: 𝑚 max ⁡⁡⁡⁡⁡−𝑢𝑇 𝑅̃ 𝑥̂ − (𝑢 + 𝑢0 )𝑇 (𝑢 + 𝑢0 ) 4 (2.15) 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑢, 𝑢0 ≥ 0. In any optimal solution of (2.15) we have 𝑢0 = ⃗0,⁡so (2.15) is equivalent to: 𝑚

𝑚

max ∑ (−(𝑥̂ 𝑇 𝑟̃ 𝑗 )𝑢𝑗 − 𝑗=1

𝑚 2 𝑢 ) 4 𝑗 (2.16)

𝑠. 𝑡.⁡⁡⁡⁡⁡𝑢𝑗 ≥ 0, 𝑗 = 1, … , 𝑚. Notice that problem (2.16) decomposes into m independent problems: 𝑚 max −( 𝑥̂ 𝑇 𝑟̃ 𝑗 )⁡𝑢𝑗 − 𝑢𝑗2 4 (2.17) 𝑠. 𝑡.⁡⁡⁡⁡⁡𝑢𝑗 ≥ 0, Halaman 23 For j = 1, ...., m; which can be solved by inspection: if (𝑥̂ 𝑇 𝑟̃ 𝑗 ) ≥ 0, then the optimal solution of (2.7) is 𝑢𝑗∗ = 0. If (𝑥̂ 𝑇 𝑟̃ 𝑗 ) ⁡ < 0,then we get a concave quadratic objective in (2.17) 𝑚 |(𝑥̂ 𝑇 𝑟̃ 𝑗 )|𝑢𝑗 − 𝑢𝑗2 4 2 ∗ 𝑇 𝑗 That has a maximun at 𝑢𝑗 = 𝑚 |(𝑥̂ 𝑟̃ )|. So the optimal solution 𝑢∗ (𝑥̂) ∈ ℝ𝑚 ⁡of the benders dual subproblem (2.14) can be obtained inclosed-from as follows:

0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓(𝑥̂ 𝑇 𝑟̃ 𝑗 ) ≥ 0 𝑢𝑗∗ (𝑥̂) = { 2 |(𝑥̂ 𝑇 𝑟̃ 𝑗 )|⁡⁡⁡⁡⁡⁡𝑖𝑓⁡(𝑥̂ 𝑇 𝑟̃ 𝑗 ) < 0 𝑚 (2.18) For j = 1, ..., m. With the benders dua subproblem solution, the benders master problem is constructed as follows. Given a finite index set K, and a set of feasible portfolios 𝜒̂ K′ = {𝑥̂𝑘 ∈ 𝜒 ′ ∩ {0,1}𝑛 ∶ 𝑘 ∈ K}, consider the Benders master problem min 𝑞 𝑚

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡𝑞 ≥ ∑ −( 𝑥̂ 𝑇 𝑟̃ 𝑗 )⁡𝑢𝑗∗ ⁡(𝑥̂𝑘 ) − ⁡ 𝑗=1

𝑚 ∗ 𝑢 (𝑥̂ )2 ;⁡∀𝑥̂𝑘 ∈ 𝜒̂ K′ ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(ℳ(𝜒̂ K′ )) 4 𝑗 𝑘

𝑥 ∈ 𝜒 ′ ∩ {0,1}𝑛 Note that the right-hand side of the first set of constraints in (ℳ(𝜒̂ K′ )) is closely related to the objective function of the Benders dual subproblem (2.15). Halaman 24 Algorithm 1 Benders linear scheme for the MSVP problem 1: procedure MSVP _BENDERS (𝜖 > 0) 2: K ← ∅, 𝑘 = 1,⁡GAP= ∞ 3: while GAP> 𝜖 do 4: compute 𝑥̂𝑘 , 𝑧𝑘 , the optimal solution and objective of (ℳ(𝜒̂ K′ )) 5: compute 𝑢∗ (𝑥̂𝑘 ) using (2.18) 6: K ← K ∪ 𝑘, 𝑘 ←= 𝑘 + 1 𝑚 7: UPPBound← ∑𝑚 ̂ 𝑇 𝑟̃ 𝑗 )⁡𝑢𝑗∗ ⁡(𝑥̂𝑘 ) − ⁡ 𝑢𝑗∗ (𝑥̂𝑘 )2 LowBOUNDk ← 𝑧𝑘 𝑗=1 −( 𝑥 4 8: GAP←|UPPBOUND−LOWBOUNDk|/|LOWBOUNDk| 9: end while 10: end procedure If we let 𝑥 ∗ ≔ ⁡ 𝑎𝑟𝑔𝑚𝑖𝑛𝑥 {𝑺} be the optimal mean-semivariance project portfolio, then 𝑆𝑉(𝑥𝜖∗ ) − 𝑆𝑉(𝑥 ∗ ) < 𝜖, 𝑆𝑉(𝑥 ∗ ) (2.19) 𝑛 Where SV represent semivariance of any portfolio of projects 𝑥 ∈ {0,1} , given by 𝑚

1 2 𝑆𝑉⁡(𝑥) ≔ ∑ min{0, 𝑥 𝑇 𝑟 𝑗 − 𝑥 𝑇 𝜇} . 𝑚⁡ 𝑗=1

Halaman 44 The Value-at-risk (VaR) of a potrfolio measures its exposure to hig losses. Specially, for given 𝛼 ∈ (0,1) (typically 0.01 ≤ 𝛼 ≤ 0.05), the VaR of a portfolio is defined as the 1 − 𝛼 quantile of the portfolio’s losses or equivalently as the a quantile of the portfolio’s return. Here follow the letter definition. Following Gaivoroski and Pflug, given 𝛼 ∈ (0,1), the α-level Var of the Portfolio is defined as follows: 𝑉𝑎𝑅𝛼 (𝑥 𝑇 𝜉) = ℚ𝛼 (𝑥 𝑇 𝜉), (3.1)

Where ℚ𝛼 (𝑥 𝑇 𝜉) is the α quantile of the portfolio’s return distribution; that is, ℚ𝛼 (𝑥 𝑇 𝜉) = inf{𝑣 ∶ ⁡ℙ𝑟(𝑥 𝑇 𝜉 ≤ 𝑣) > 𝛼}. Also, the expected portfolio return from t = 0 to to = T is given by 𝔼(𝑥 𝑇 𝜉). Above, ℙ𝑟(. )𝑎𝑛𝑑⁡𝔼(. ) respectively indicate probability and expectation. Halaman 45 Linear diversification contraints. Formally, the VaR portfolio problem is: min ⁡⁡⁡⁡−𝑉𝑎𝑅𝛼 (𝑥 𝑇 𝜉) (3.2) 𝑇

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡⁡𝔼(𝑥 𝜉) ≥ 𝜇0 𝑥𝑇 𝟙 = 1 𝑥𝜖𝜒 ⊆ ℝ𝑛+ Halaman 46 (2.1) leads to the VaR portfolio problem (3.2) being written as: 𝑧𝑉𝑎𝑅 ≔ ⁡ min ⁡⁡−𝑣 (3.3) ⌊𝑎𝑚⌋+1

{𝑥 𝑇 1

𝑇 𝑚}

𝑠. 𝑡.⁡⁡⁡⁡⁡𝑣 = 𝑚𝑖𝑛⁡ 𝜉 ,…,𝑥 𝜉 𝑇 𝑥 𝜇 ≥ 𝜇0 𝑥𝑇 𝟙 = 1 𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ , 𝑣 ∈ ℝ, Where v represents the VaRα (𝑥 𝑇 𝜉), the vector of mean return estimates is, for simplicity, 𝑗 considered t be givwn by 𝜇 ≔ (1⁄𝑚) ∑𝑚 𝑗=1 𝜉 . However, our result are indepnedent of this coice, and a variety of other estimation methods can be used (see, e.g., Black and Litterman 2001, Meucci 2007). Also, for 𝑘 ∈ {1, … , 𝑚}, 𝑎𝑛𝑑⁡𝑢 𝑗 ∈ ℝ, 𝑗 = 1, … , 𝑚, the k-th smallest element in {𝑢1 , … , 𝑢𝑚 } is denoted by 𝑚𝑖𝑛𝑘 ⁡{𝑢1 , … , 𝑢𝑚 } (i.e., 𝑚𝑖𝑛𝑘 ⁡{𝑢1 , … , 𝑢𝑚 } is the k-th order statistic 𝑢(𝑘) in {𝑢1 , … , 𝑢𝑚 }). Problem (3.3) is equivalent (see, e.g., Benati and Rizzi 2007, Feng et al. 2015) to the following mixed-integer linear program (MILP): 𝑧𝑉𝑎𝑅 = max ⁡⁡⁡⁡⁡⁡𝑣 𝑚

𝑠. 𝑡.⁡⁡⁡⁡⁡ ∑ 𝑦𝑗 = ⌊𝛼𝑚⌋ 𝑗=1 𝑇 𝑗

𝑀𝑦𝑗 ≥ 𝑣 − 𝑥 𝜉 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 1, … , 𝑚 (3.4) 𝑇

𝑥 𝜇 ≥ 𝜇0 𝑥𝑇 𝟙 = 1 𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ , 𝑣 ∈ ℝ, 𝑦𝑗 ∈ {0,1},⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 1, … , 𝑚, Halaman 47 max ⁡⁡⁡⁡⁡⁡𝔼(𝑥 𝑇 𝜉) 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡⁡ − 𝑉𝑎𝑅𝛼 (𝑥 𝑇 𝜉) ≤ 𝑣̃ (3.5) 𝑇

𝑥 𝟙=1

𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ . Halaman 48 ∗ 𝑧𝑉𝑎𝑅 = max ⁡⁡⁡⁡⁡⁡⁡ 𝑥 𝑇 𝜇

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡ ∑ 𝑦𝑗 ≤ ⌊𝛼𝑚⌋ 𝑗∈1

𝑀𝑦𝑗 ≥ 𝑣̃ − 𝑥 𝑇 𝜉𝑗 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 1, … , 𝑚 (3.6) 𝑇

𝑥 𝟙=1 𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ , 𝑣 ∈ ℝ, 𝑦𝑗 ∈ {0,1},⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 1, … , 𝑚, 𝛽(𝛼) = min ⁡⁡⁡⁡⁡⁡𝜙(𝑥) 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡𝑅(𝑥) ≥ 𝑎 𝑥∈𝜒 𝛼(𝑏) = ⁡⁡ max ⁡⁡⁡⁡⁡𝑅(𝑥) (3.8) 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡𝜙(𝑥) ≤ 𝑏 𝑥∈𝜒 Halaman 55 3.4.1 Lower bound for optimal VaR Let us denote [𝑚] ≔ {1, … , 𝑚}. Now, given 𝐽 ⊆ [𝑚],⁡let 𝐽𝑐 ≔ [𝑚]\𝐽, and consider the following problem: 𝑧𝐽 ≔ max ⁡⁡⁡𝑣 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡ ∑ 𝑦𝑗 = ⌊𝛼𝑚⌋ 𝑗∈𝐽 𝑇 𝑗

𝑀𝑦𝑗 ≥ 𝑣 − 𝑥 𝜉 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 ∈ 𝐽 0 ≥ 𝑣 − 𝑥 𝑇 𝜉𝑗 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 ∈ 𝐽𝑐 (3.10) 𝑇

𝑥 𝜇 ≥ 𝜇0 𝑥𝑇 𝟙 = 1 𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ , 𝑣 ∈ ℝ, 𝑦𝑗 ∈ {0,1},⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 𝐽. Note that (3.10) is the optimazation problem obtained from (3.4) by setting 𝑦𝑗 = 0 for al⁡𝑗 ∈ 𝐽𝑐 . Hence 𝑧𝑗 ≤ 𝑧𝑉𝑎𝑅 for all 𝐽 ⊆ [𝑚].⁡ In Algorithm A below, the formulation (3.10), together with an appropriate update of the set J, is used iterativily to obtain near-optimal feasible solution to (3.4). specifically, after setting an initial set 𝐽 = 𝐽0 ⊂ [𝑚] problem (3.10) is solved. Let 𝑦 𝐽 ∈ {0,1}|𝐽| ⁡be optimal value of the binary variables of (3.10). these values are used to construct the linear program below obtained by fixing the binary variables 𝑦 ∈ {0,1}𝑚 in (3.4) such that 𝑦𝐽 = 𝑦 𝐽 ,

Halaman 56 And 𝑦𝑗 = 0, for all 𝑗 ∈ 𝐽𝑐 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡𝑀𝑦𝑗𝐽

max ⁡⁡⁡⁡⁡𝑣 ≥ 𝑣 − 𝑥 𝑇 𝜉𝑗 ,⁡⁡⁡⁡⁡⁡⁡𝑗 = 1, … , 𝑚 (3.11) 𝑇

𝑥 𝜇 ≥ 𝜇0 𝑥𝑇 𝟙 = 1 𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ , 𝑣 ∈ ℝ, Halaman 57 3.4.2 Upper bound for optimal return In this section, the aim is to obtain a measure of the closeness to optimally of the feasibl solution 𝑥̃, 𝑣̃, for the VaR portfolio problem obtained by Algorithm A. For this purpose, we first apply proposition 2 to the alternative formulations of the VaR Portfolio problem (3.4) and (3.6). specifically, let 𝛿 > 0 be a specified tolerance, and 𝑥̃ ∈ ℝ𝑛+ be a feasible portfolio for (3.4), with associated VaR 𝑣̃; that is 𝑣̃ = 𝑚𝑖𝑛⌊𝛼𝑚⌋+1 {𝑥̃ 𝑇 𝜉1 , … , 𝑥̃ 𝑇 𝜉 𝑚 }. Then from proposion 2, it follows that if the optimal value of (3.6) statisfics ∗ 𝑧𝑉𝑎𝑅 < 𝜇0 ⇒ 𝑣̃ − 𝛿 ≤ 𝑧𝑉𝑎𝑅 ≤ 𝑣̃ (3.12) 𝑛 That is, the near-optimally of the feasible portfolio 𝑥̃ ∈ ℝ+ to the optimal portfolio corresponding to the VaR portfolio problem (3.4), can be measured in terms of 𝛿 ∈ ℝ++ . ∗ Clearly, directly solving (3.6) to check whether condition 𝑧𝑉𝑎𝑅 < 𝜇0 in (3.12) holds for a 𝑛 feasible portfolio 𝑥̃ ∈ ℝ+ ⁡ of (3.4) is as computationally inefficient as dirctly solving (3.4). therefore, w present an appropriate upper bound for the alternative formulation of the VaR portfolio problem (3.6) that allows to efficiently guarantee the near-optimally of the feasible solutios of the VaR problem obtained after using Algorithm A. Specifically, given 𝐼 ⊆ [𝑚] with |𝐼| ≥ ⌊𝛼𝑚⌋⁡𝑎𝑛𝑑⁡𝑣̃, a lower bound (3.4) Halaman 58 (i.e, 𝑣̃ ≤ 𝑧𝑉𝑎𝑅 ), cosider the problem. 𝜇̅𝐼 ≔ ⁡⁡ max ⁡⁡⁡⁡⁡𝑥 𝑇 𝜇 𝑠. 𝑡.⁡⁡⁡⁡⁡ ∑ 𝑦𝑗 = ⌊𝛼𝑚⌋ 𝑗∈𝐼

𝑀𝑦𝑗 ≥ 𝑣 − 𝑥 𝑇 𝜉𝑗 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 ∈ 𝐼 (3.13) 𝑇

𝑥 𝟙=1 𝑥 ∈ 𝜒 ⊆ ℝ𝑛+ , 𝑦𝑗 ∈ {0,1},⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 𝐼. Halaman 114 1 ̂ 𝑤 − 𝑤 𝑇 𝜇̂ min ⁡⁡𝑤 𝑇 ∑ 𝑦

(5.1) 𝑇

𝑠. 𝑡.⁡⁡⁡⁡⁡⁡⁡⁡𝑤 𝑒 = 1 𝑤≥0 𝑁 Where 𝑤 ∈ ℝ+ is the vektor of portfolio weights, 𝑤 𝑇 𝜇̂ is the sample mean of the portfolio ̂ 𝑤 is the sample variance of the portfolio returns, with ∑ ̂ denoting the returns, and 𝑤 𝑇 ∑ sample covariance matrix of the asset returns. The constaint ⁡𝑤 𝑇 𝑒 = 1, where 𝑒 ∈ ℝ𝑛 is the vector of all ones, ensures that the portfolio weights sum to one, and the constraint 𝑤 ≥ 0 ensures that there is no short-selling. ̂𝑤 min ⁡⁡ 𝑤 𝑇 ∑ (5.2) 𝑇 𝑤 𝑒=1 𝑤≥0 Halaman 118 5.2.7 Mean-Variance Portfolio Optimization with a Bench-March return constraint Mean-variance models can be formuated in multiple ways (Braga 2016). We provided the one with the risk aversion parameter in model (5.1). to tes wether mean-variance formulation could return the lower risk whil achieving the same portfolio return achieved by HRP strategy, we reformulate the model (5.1) with a benchmark return constraint and define the brenchmark return value as the HRP portfolio return given by the following equation. 𝑇 𝜇(𝐻𝑅𝑃) = 𝑤(𝐻𝑅𝑃) 𝜇 Where 𝑤𝐻𝑅𝑃 are the weight obtained by HR allocation and µ is the sample average of the asset returns. The mean-variance model with the benchmark return constraint then can be formulated by following model (5.3) ̂𝑤 min ⁡⁡ 𝑤 𝑇 ∑ (5.3) 𝑇 𝑠. 𝑡.⁡⁡⁡⁡⁡⁡𝑤 𝜇 = 𝜇𝐻𝑅𝑃 𝑤𝑇𝑒 = 1 𝑤≥0 Halaman 121 Sharpe ratio Out-of-sample Sharpe ratio of strategy k, defined as the sample mean of out-of-sample excess returns (over the risk-free asset) 𝜇̂ 𝑘 , devided by their sample standard deviation, 𝜎̂𝑘 . To test whethe the sharpe ratios of two strategies are statistically distinguishable, we also compute the p-value of the difference, using the approach introduced by (Jobson and Korkie 1981) and referenced in (DeMidguel et al. 2009). Specifically, given two portfolio i and j, with 𝜇̂ 𝑖 , 𝜇̂ 𝑗 , 𝜎̂𝑖 , 𝜎̂𝑗 , 𝜎̂𝑖𝑗⁡ as their estimated means, variances, and covariances over a sample of size 𝑇−𝑀 𝑅

̂ 𝜇

̂ 𝜇

, the test of the hypothesis 𝐻0 = 𝜎̂𝑖 − 𝜎̂𝑗 = 0 is obtained via the test statistic 𝑖

𝑧̂𝐽𝐾 : Where

𝑖

𝜇̂ 𝑖 ∗ 𝜎̂𝑗 − 𝜇̂ 𝑗 ∗ 𝜎̂𝑖 √𝑣̂

𝑣̂:

1 ̂𝜇 ̂ 𝜇 (2𝜎̂𝑖2 𝜎̂𝑗2 − 2𝜎̂𝑖 𝜎̂𝑗 , 𝜎̂𝑖𝑗 + 0,5𝜇𝑖2 𝜎𝑗2 + 0,5𝜇𝑗2 𝜎𝑖2 − 𝜎̂𝑖 𝜎̂𝑗 𝜎̂𝑖𝑗⁡ 2 ) 𝑖 𝑗 (𝑇 − 𝑀)⁄𝑅

Turnover 𝑁

𝑇𝑂𝑘𝑡 = ∑(|𝑤 ̂ 𝑘,𝑗,𝑡+𝑅 − 𝑤 ̂ 𝑘,𝑗,𝑡+ |) 𝑗=1