The resolution of excess investment in Fuzzy portfolio selection using piecewise linear function

Figures & data

Table 1. The portfolio selection guaranteed return rate ${\tilde{p}}_1$.

Proportions Parameters	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{22}$	$x_{32}$	$x_{42}$	$x_{52}$	Total proportions	Expected Return Rates
z0 = 0.13	0.2000	0.2000	0.1000	0.1000	0.2819	0	0	0	0	0.8819	0.1300
z0 = 0.15	0.1806	0.2000	0.2000	0.1194	0.3000	0	0	0	0	1	0.1500
z0 = 0.17	0.1000	0.1000	0.2000	0.3000	0.3000	0	0	0	0	1	0.1642
z0 = 0.19	0.1000	0.1000	0.2000	0.3000	0.3000	0	0	0	0	1	0.1642
z0 = 0.21	0.1000	0.1000	0.2000	0.3000	0.3000	0	0	0	0	1	0.1642
z0 = 0.23	0.1000	0.1393	0.2000	0.3000	0.3000	0	0	0	0.1000	1.1393	0.1800
z0 = 0.25	0.1000	0.2000	0.2000	0.3000	0.3000	0	0	0.0704	0.1704	1.3408	0.2000

Table 2. The portfolio selection guaranteed return rate ${\tilde{p}}_2$.

Proportions Parameters	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{42}$	$x_{52}$	Total Proportions	Expected Return Rates
z0 = 0.13	0.2000	0.2000	0.1000	0.1000	0.2819	0	0	0.8819	0.1300
z0 = 0.15	0.1806	0.2000	0.2000	0.1194	0.3000	0	0	1	0.1500
z0 = 0.17	0.1000	0.2000	0.2000	0.2000	0.3000	0	0	1	0.1577
z0 = 0.19	0.1000	0.2000	0.2000	0.2000	0.3000	0	0	1	0.1577
z0 = 0.21	0.1293	0.2000	0.2000	0.2000	0.3000	0	0	1.0293	0.1600
z0 = 0.22	0.2000	0.2000	0.2000	0.2000	0.3000	0	0.0679	1.1679	0.1700
z0 = 0.23	0.2000	0.2000	0.2000	0.2000	0.3000	0.0883	0.1883	1.3766	0.1800

Table 3. The comparison among fuzzy portfolio models.

	σ (%)	M (%)	C.V.	x1	x2	x3	x4	x5	$\sum _{j=1}^{\text{5}}{x}_j$
Zhang [37]	4%	22.55%	0.1774	0.1	0.1	0.1611	0.1	0.5389	1
	5%	23.16%	0.2159	0.1	0.1	0.1	0.1	0.6	1
	6%	23.16%	0.2591	0.1	0.1	0.1	0.1	0.6	1
	7%	23.16%	0.3022	0.1	0.1	0.1	0.1	0.6	1
Tsaur et al.’s model [15] with a guaranteed return rate ${\tilde{p}}_1$	5%	10.50%	0.4762	0.2127	0.1	0.1	0.1	0.1	0.6127
	8%	17.46%	0.4582	0.4	0.1	0.1	0.1	0.2659	0.9659
	10%	21.59%	0.4632	0.2811	0.1	0.1	0.1	0.4189	1
	12%	25.63%	0.4682	0.121	0.1	0.1	0.1	0.5790	1
	15%	26.17%	0.5732	0.1	0.1	0.1	0.1	0.6	1
Chen & Tsaur’s model	6%	10.65%	0.5634	0.4000	0.1000	0.1142	0.1000	0.1	0.8142
	8%	14.48%	0.5525	0.3632	0.1000	0.3368	0.1000	0.1000	1
	10%	15.77%	0.6341	0.1560	0.1000	0.4000	0.2440	0.1000	1
	12%	16.87%	0.7113	0.1000	0.1000	0.3157	0.1000	0.3843	1
	15%	19.17%	0.7825	0.1000	0.1000	0.1000	0.1000	0.6679	1
Proposed model with a guaranteed return rate ${\tilde{p}}_2$	4% (z0 = 0.13)	13%	0.3077	0.2000	0.2000	0.1000	0.1000	0.2819	1
	4% (z0 = 0.15)	15%	0.2667	0.1806	0.2000	0.2000	0.1194	0.3000	1
	4% (z0 = 0.19)	15.77%	0.2536	0.1000	0.2000	0.2000	0.2000	0.3000	1
	4% (z0 = 0.21)	16%	0.2500	0.1293	0.2000	0.2000	0.2000	0.3000	1
	4% (z0 = 0.22)	17%	0.2353	0.2000	0.2000	0.2000	0.2000	0.3679	1.1393
	5% (z0 = 0.23)	18%	0.2778	0.2000	0.2000	0.2000	0.2610	0.4982	1.3408

Table 4. The fuzzy return of each selected security

${\tilde{r}}_1=\left(0.1420,0.0713,0.1296\right)$	${\tilde{r}}_2=\left(0.1358,0.0490,0.0835\right)$	${\tilde{r}}_3=\left(0.0909,0.0499,0.0919\right)$
${\tilde{r}}_4=\left(0.0808,0.0203,0.0297\right)$	${\tilde{r}}_5=\left(0.0786,0.0162,0.0217\right)$	${\tilde{r}}_6=\left(0.0751,0.0226,0.0362\right)$
${\tilde{r}}_7=\left(0.0657,0.0205,0.0325\right)$	${\tilde{r}}_8=\left(0.0643,0.0244,0.0415\right)$	${\tilde{r}}_9=\left(0.0552,0.0133,0.0190\right)$
${\tilde{r}}_{10}=\left(0.0438,0.0079,0.0102\right)$	${\tilde{r}}_{11}=\left(0.0316,0.0110,0.0181\right)$	${\tilde{r}}_{12}=\left(0.0212,0.0051,0.0075\right)$

Table 5. The portfolio selection guaranteed return rate ${\tilde{p}}_3$.

z0 = 0.07 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.1	0.1	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0.028	0.028	0.028	0.028	0.028	0.028	0.028	0.028	0.028	0	0
z0 = 0.1 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0	0	0	0	0	0	0	0	0	0	0
z0 = 0.12 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0	0	0	0	0	0	0	0	0	0	0
z0 = 0.16 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.15	0.075	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0	0	0	0	0	0	0	0	0	0	0
z0 = 0.17 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.1202	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0.0599	0.0599	0.01	0	0	0	0	0	0	0	0	0

Table 6. The portfolio selection guaranteed return rate ${\tilde{p}}_4$.

z0 = 0.06 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.1	0.1	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0	0	0	0	0
z0 = 0.07 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.1	0.1	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0.0291	0.0291	0.0291	0.0291	0.0291
z0 = 0.11 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0	0	0	0	0
z0 = 0.16 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.15	0.075	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0	0	0	0	0
z0 = 0.17 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.1202	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0.0599	0.0599	0.0099	0	0	0

Yue W, Wang Y, Xuan H. Fuzzy multi-objective portfolio model based on semi-variance–semi-absolute deviation risk measures. Soft Comput. 2019;23(17):8159–8179. doi:10.1007/s00500-018-3452-y

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