On the Computation of the Efficient Frontier of the Portfolio Selection Problem

Algorithm 2

The KTEF-S algorithm.

Inputs 𝐕, 𝐞̂, 𝐥̂, 𝐮, {(𝐱_𝑗,𝐲_𝑗)}^𝑘_𝑗=1.

For each 𝐣=𝟏,...,𝐤

 (a) Let𝐱_𝐣, 𝐞_𝐣, 𝐥_𝐣, 𝐮_𝐣be the vectors obtained from𝐱_𝐣, 𝐞̂, 𝐥̂, 𝐮, respectively, by deleting

 the components corresponding to indexes 𝑖 such that 𝑦_𝑗𝑖=0.

 (b) Calculate 𝐿_𝑗 and 𝑈_𝑗 from 𝐱_𝐣 according to (3.6).

End

Eliminate the terms in the sequence {(𝐱_𝑗,𝐲_𝑗)}^𝑘_𝑗=1 giving rise to repeated terms in the

sequence {(𝐲_𝑗,𝐿_𝑗,𝑈_𝑗)}^𝑘_𝑗=1.

For each 𝐣=𝟏,...,𝐤

 (a) Let 𝐕_𝐣 be the submatrix of 𝐕 obtained by deleting the rows and columns for

 which 𝑦_𝑗𝑖=0.

 (b) Call KTEF(𝐕_𝐣, 𝐞_𝐣, 𝐥_𝐣, 𝐮_𝐣, 𝐿_𝑗, 𝑈_𝑗), which provides an interval [𝑟_min𝑗,𝑟_max𝑗]and

 the coefficients (𝑎_𝑗, 𝑏_𝑗, 𝑐_𝑗) of the equation of an arc of parabola.

  on error (KTEF has stoped in a degenerate case) discard the point.

End

(i) Define the functions 𝑅_𝑗(𝑟) given by (4.3).

(ii) Let 𝐏𝐨𝐢𝐧𝐭𝐬={𝑟_max𝑗|𝑗=1,...,𝑘}∪{𝑟_min}, where 𝑟_min is the minimum of all {𝑟_min𝑗}_𝑗.

 For 𝑗=1 to 𝑘−1

  For 𝑖=𝑗+1 to 𝑘

   (a) Let Roots be the set of real roots of (4.4).

   (b) Append to Points any 𝑟∈𝐑𝐨𝐨𝐭𝐬 satisfying (4.5).

   (c) Let Roots be the set of real roots of (4.6).

   (d) Append to Points any 𝑟∈𝐑𝐨𝐨𝐭𝐬 satisfying 𝑟_min𝑗≤𝑟≤𝑟_max𝑗.

  Next 𝑖.

 Next 𝑗.

(i) Order the vector Points and eliminate repeated entries.

(ii) Let 𝑚_𝑙=(𝐏𝐨𝐢𝐧𝐭𝐬_𝑙+𝐏𝐨𝐢𝐧𝐭𝐬_𝑙+1)/2 for each 𝑙.

(iii) Calculate the vector 𝐓 such that 𝑇_𝑙 is the index 𝑗 where min_𝑗𝑅_𝑗(𝑚_𝑙) is attained.

(iv) Let 𝐆𝐨𝐨𝐝={𝑇₁}, let 𝐂𝐡𝐚𝐧𝐠𝐞={𝑟_min𝑇1}.

For 𝑖=1 to the length of 𝐓

 If 𝑇_𝑖≠𝑇_𝑖+1 append to Good the index 𝑇_𝑖+1 and append to Change

 the value 𝐏𝐨𝐢𝐧𝐭𝐬_𝑖+1

Next 𝑖.

(i) Append to Change the last point of Points.

(ii) Set (𝑎_𝑖,𝑏_𝑖,𝑐_𝑖)=(𝑎_𝑔𝑖, 𝑏_𝑔𝑖, 𝑐_𝑔𝑖) (where 𝑔_𝑖 is a short for 𝐆𝐨𝐨𝐝_𝑖).

Outputs Change, {(𝑎_𝑖,𝑏_𝑖,𝑐_𝑖)}^𝑚_𝑖=1.