LAMPIRAN 5
Penurunan Persamaan 28 menjadi Persamaan 29. Dari Persamaan 28:
1 2
2
1 1
1 21 2
r W EU
r
γ
λ α
ασ γ λ ρ
γ α μ γ ασ
−
+ =
⎛ ⎞
− + − −
− − −
− ⎜
⎟ ⎝
⎠
1
2 2
2
1 1
1 1
1 1
2 2
r W EU
r
γ
λ α
α σ γ λ ρ
γ μα γ α
γ γ ασ
−
+ =
⎛ ⎞
− + − −
+ − + −
− −
⎜ ⎟
⎝ ⎠
. e.1 Turunan pertama dari
EU α terhadap :
α
1
2 2
2 2
2 2
2
1 2
1 1
1 1
2 2
2 1
1 1
1 1
1 2
2 r W
r EU
r
γ
γ ασ
λ γ μ
γ γ
ασ α
α σ γ λ ρ
γ μα γ α
γ γ ασ
−
⎛ ⎞
− +
− − − + −
+ −
− ⎜
⎟ ⎜
⎟ ⎝
⎠ =
⎛ ⎞
− + − −
+ − + −
− −
⎜ ⎟
⎝ ⎠
1
1
2 2
2 2
2 2
2 2
2 2
1 1
1 1
1 1
2 2
1 1
1 1
1 1
2 2
r W r
EU r
r W r
EU r
γ
γ
λ γ μ
ασ γ ασ
α ασ
γ λ ρ γ α μ
γ ασ λ
γ μ ασ
γ ασ α
ασ γ λ ρ
γ α μ γ ασ
−
−
+ − − −
− − + −
= ⎛
⎞ −
+ − − − −
− −
⎜ ⎟
⎝ ⎠
+ −
− − + −
= ⎛
⎞ −
+ − − − −
− −
⎜ ⎟
⎝ ⎠
1
1
2 2
2 2
2 2
2 2
2
1 1
1 1
2 2
. 1
1 1
2 2
r W r
EU r
r W r
EU r
γ
γ
λ μ
ασ γ ασ
α ασ
λ ρ γ α μ
γ ασ λ
μ γασ
α ασ
λ ρ γ α μ
γ ασ
−
−
+ − −
+ − =
⎛ ⎞
⎛ ⎞
+ − − − −
− −
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
+ − −
= ⎛
⎞ ⎛
⎞ + − −
− − −
− ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠
e.2
LAMPIRAN 6 Penurunan Persamaaan 29 menjadi Persamaan 30.
Dari Persamaan 29:
1
2 2
2 2
1 1
1 2
2 r W
r EU
r
γ
λ μ
γασ α
ασ λ ρ
γ α μ γ ασ
−
+ − −
= ⎛
⎞ ⎛
⎞ + − −
− − −
− ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ f.1
Nilai α yang maksimum dicapai saat
EU α = :
1
2 2
2 2
1 1
1 2
2 r W
r r
γ
λ μ
γασ ασ
λ ρ γ α μ
γ ασ
−
+ − −
= ⎛
⎞ ⎛
⎞ + − −
− − −
− ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ . f.2
Karena λ dan
1
γ
≠
maka
1
r W
γ
λ
−
+ ≠
, sehingga:
2
. r
r r
μ γασ
γασ μ
μ α
γσ
− − =
− = −
− −
=
f.3
LAMPIRAN 7 Penurunan Persamaaan 29 menjadi Persamaan 31.
Dari Persamaan 29:
1
2 2
2 2
1 1
1 2
2 r W
r EU
r
γ
λ μ
γασ α
ασ λ ρ
γ α μ γ ασ
−
+ − −
= ⎛
⎞ ⎛
⎞ + − −
− − −
− ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠
. g.1
Misalkan:
2 2
1 1
1 2
2 x
r
ασ λ ρ
γ α μ γ ασ
⎛ ⎞
= + − − − −
− −
⎜ ⎟
⎝ ⎠
1
y r W
γ
λ
−
= +
1
2 2
u r W
r y
r
γ
λ μ
γασ μ
γασ
−
= +
− − =
− −
1
2 2
u r W
y
γ
λ γσ
γσ
−
= +
− = −
2 2
2 2
1 1
1 2
2 v
r x
ασ λ ρ
γ α μ γ ασ
⎛ ⎞
⎛ ⎞
= + − −
− − −
− =
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
2 2
2 2
2 2
2 2
1 2
1 1
1 2
2 2
1 1
v r
r x
r
γ ασ ασ
λ ρ γ α μ
γ μ ασ
γ ασ γ μ
ασ γ ασ
⎛ ⎞
− ⎛
⎞ ⎜
⎟ =
+ − − − −
− − −
− − − −
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
= − −
− − − −
maka:
2
vu uv
EU v
α
− =
2 2
2 2
2 2
2 2
2 1
1 x y
y r
x r
EU x
γσ μ
γασ γ μ
ασ γ ασ
α
− −
− − − −
− − − −
=
2 2
2 2
2 2
2 2
2 1
1 x y
x y r
r EU
x
γ σ μ
γασ γ μ
ασ γ ασ
α
− +
− − −
− − + −
=
g.2 dengan:
2 2
1 1
1 2
2 x
r
ασ λ ρ
γ α μ γ ασ
⎛ ⎞
= + − − − −
− −
⎜ ⎟
⎝ ⎠
1
y r W
γ
λ
−
= +
Uji turunan ke-dua pada titik kritis
2
r
μ α
γσ
− =
:
2 2
2 2
2 2
2 2
2 2
2
2 1
1 r
r r
x y xy
r r
EU x
μ μ
μ γ σ
μ γ
σ γ μ
σ γ
σ γσ
γσ γσ
α
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
− −
− −
+ − −
− − −
+ − ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠ =
2 2
2 2
2 2
2 2
2
2 1
1 r
r x y
xy r
r r
EU x
μ μ
γ σ μ
μ γ μ
σ γ
σ γσ
γσ α
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
− −
− +
− − −
− − −
+ − ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠ =
2 2
2 2
2 2
2 2
2
2 1
1 r
r x y
xy r
EU x
μ μ
γ σ γ μ
σ γ
σ γσ
γσ α
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
− −
− +
− − −
+ − ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠ =
2 2
2 2
x y EU
x
γ σ α
− +
=
2 2
2 2
0. x y
EU x
γ σ α
= −
g.3 Dari Persamaan g.3 diperoleh
EU
α , sehingga
α adalah nilai yang optimal.
LAMPIRAN 8 Program untuk menampilkan Gambar 1, 2, 3, 4, dan 5. Penulisan program menggunakan
Software Mathematica 6:
• Pengaruh koefisien CRRA γ terhadap alokasi Aset
Gambar 1 ListPlot[Expand[{{α,1-α}
={{2.5,0.500},{2.75,0.455},{3,0.417},{3.25,0.385},{3.5,0.357},{3.7 5,0.333},{4,0.313},{4.25,0.294},{4.5,0.278},{4.75,0.263},{5,0.250}
,{5.25,0.238},{5.5,0.227},{5.75,0.217},{6,0.208},{6.25,0.2000},{6. 5,0.192},{6.75,0.185},{7,0.179},{7.25,0.172}},{{2.5,1-
0.500},{2.75,1-0.455},{3,1-0.417},{3.25,1-0.385},{3.5,1- 0.357},{3.75,1-0.333},{4,1-0.313},{4.25,1-0.294},{4.5,1-
0.278},{4.75,1-0.263},{5,1-0.250},{5.25,1-0.238},{5.5,1- 0.227},{5.75,1-0.217},{6,1-0.208},{6.25,1-0.2000},{6.5,1-
0.192},{6.75,1-0.185},{7,1-0.179},{7.25,1-0.172}}}], AxesLabel→{γ,α dan 1-α},
PlotMarkers→{+,},PlotRange→{{2.5,6.1},{0,1}}]
• Pengaruh volatilitas σ terhadap alokasi aset
Gambar 2 ListPlot[Expand[{{α,1-α}
={{0.2,0.500},{0.21,0.454},{0.22,0.413},{0.23,0.378},{0.24,0.347}, {0.25,0.320},{0.26,0.296},{0.27,0.274},{0.28,0.255},{0.29,0.238},{
0.3,0.222},{0.31,0.208},{0.32,0.195},{0.33,0.184},{0.34,0.173},{0. 35,0.163},{0.36,0.154},{0.37,0.146},{0.38,0.139},{0.39,0.131}},{{0
.2,1-0.500},{0.21,1-0.454},{0.22,1-0.413},{0.23,1-0.378},{0.24,1- 0.347},{0.25,1-0.320},{0.26,1-0.296},{0.27,1-0.274},{0.28,1-
0.255},{0.29,1-0.238},{0.3,1-0.222},{0.31,1-0.208},{0.32,1- 0.195},{0.33,1-0.184},{0.34,1-0.173},{0.35,1-0.163},{0.36,1-
0.154},{0.37,1-0.146},{0.38,1-0.139},{0.39,1-0.131}}}],
AxesLabel→{σ,α dan 1-α}, PlotMarkers→{+,},PlotRange→{{0.2,0.4},{0,1}}]
• Pengaruh tingkat imbal hasil yang diharapkan
μ
terhadap alokasi aset
Gambar 3 ListPlot[Expand[{{α,1-α}
={{0.100,0.500},{0.101,0.510},{0.102,0.520},{0.103,0.530},{0.104,0 .540},{0.105,0.550},{0.106,0.560},{0.107,0.570},{0.108,0.580},{0.1
09,0.590},{0.110,0.600},{0.111,0.610},{0.112,0.620},{0.113,0.630}, {0.114,0.640},{0.115,0.650},{0.116,0.660},{0.117,0.670},{0.118,0.6
80},{0.119,0.690}},{{0.100,1-0.500},{0.101,1-0.510},{0.102,1- 0.520},{0.103,1-0.530},{0.104,1-0.540},{0.105,1-0.550},{0.106,1-
0.560},{0.107,1-0.570},{0.108,1-0.580},{0.109,1-0.590},{0.110,1- 0.600},{0.111,1-0.610},{0.112,1-0.620},{0.113,1-0.630},{0.114,1-
0.640},{0.115,1-0.650},{0.116,1-0.660},{0.117,1-0.670},{0.118,1- 0.680},{0.119,1-0.690}}}],
AxesLabel→{μ,α dan 1-α}, PlotMarkers→{+,},PlotRange→{{0.1,0.12},{0,1}}]
• Pengaruh tingkat imbal hasil konstan r terhadap alokasi aset
Gambar 4 ListPlot[Expand[{{α,1-α}
={{0.05,0.500},{0.051,0.490},{0.052,0.480},{0.053,0.470},{0.054,0. 460},{0.055,0.450},{0.056,0.440},{0.057,0.430},{0.058,0.420},{0.05
9,0.410},{0.060,0.400},{0.061,0.390},{0.062,0.380},{0.063,0.370},{ 0.064,0.360},{0.065,0.350},{0.066,0.340},{0.067,0.330},{0.068,0.32
0},{0.069,0.310}},{{0.05,1-0.500},{0.051,1-0.490},{0.052,1- 0.480},{0.053,1-0.470},{0.054,1-0.460},{0.055,1-0.450},{0.056,1-
0.440},{0.057,1-0.430},{0.058,1-0.420},{0.059,1-0.410},{0.060,1- 0.400},{0.061,1-0.390},{0.062,1-0.380},{0.063,1-0.370},{0.064,1-
0.360},{0.065,1-0.350},{0.066,1-0.340},{0.067,1-0.330},{0.068,1- 0.320},{0.069,1-0.310}}}],
AxesLabel→{r,α dan 1-α}, PlotMarkers→{+,},PlotRange→{{0.05,0.07},{0,1}}]
• Pendapatan dari AVS dan ATS pada tahun ke-t
Gambar 5 ListPlot[{
{t,avs}={{1,17257853},{2,19019643},{3,16516940},{4,20437383}, {5,18931775},{6,18575151},{7,21836479},{8,23038112},{9,21199685},
{10,21189151},{11,24004198},{12,26498664},{13,25693383},
{14,26045278},{15,26333005}}, {t,ats}={{1,17500000},{2,17500000},{3,17500000},{4,17500000},
{5,17500000},{6,17500000},{7,17500000},{8,17500000},{9,17500000}, {10,17500000},{11,17500000},{12,17500000},{13,17500000},
{14,17500000},{15,17500000}}}, AxesLabel→{t,Pendapatan dari ATS dan AVS},
PlotMarkers→{+,},PlotRange→{{0,16},{0,30000000}}]
LAMPIRAN 9 Program untuk membangkitkan nilai
t
B
. Penulisan program menggunakan Software Mathematica 6:
t=Range[15] {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
b=RandomReal[1,Length[t]] {0.125872,0.562768,0.119237,0.565601,0.454345,0.0730034,0.325448,0.670733
,0.4252,0.0021821,0.0278613,0.372868,0.388571,0.297289,0.0859393} For[i=0,b[[i+1]]-b[[i]];
i
≤
Length[b]-1,i++, Print[b[[i+1]]-b[[i]]]]
0.125872 0.436897
-0.443532 0.446364
-0.111256 -0.381341
0.252445 0.345285
-0.245533 -0.423018
0.0256792 0.345007
0.0157028 -0.0912821
-0.211349
OPTIMALISASI ALOKASI ASET DI DALAM KONTRAK ANUITAS VARIABEL
PUTRA SETIAWAN G54103056
DEPARTEMEN MATEMATIKA FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM
INSTITUT PERTANIAN BOGOR BOGOR
2008
ABSTRACT
PUTRA SETIAWAN . Optimal Asset Allocation in a Variable Annuity Contract. Supervised by
SISWADI and DONNY CITRA LESMANA.
Variable annuity contract is a gathering plan of long range asset where entire advantages free from tax before the end of asset gathering phase. In variable annuity contract, retirement is a
period when the asset gathering phase ended. At the time of retirement, all gathering asset in variable annuity account can be altered into the form of annuity.
Asset investment account in variable annuity is divided into risk-free asset sub-account and risky asset sub-account. At the time of retirement, asset investment in risk-free asset sub-account
will be altered into the fixed immediate annuity form and asset investment in risky asset sub- account will be altered into the variable immediate annuity form.
In this paper, a decision strategy was employed to derive the optimal asset allocation in a variable annuity contract to obtain the maximal result at retirement.
ABSTRAK
PUTRA SETIAWAN . Optimalisasi Alokasi Aset di dalam Kontrak Anuitas Variabel. Dibimbing
oleh SISWADI dan DONNY CITRA LESMANA.
Kontrak anuitas variabel adalah suatu rencana pengumpulan aset jangka panjang di mana seluruh keuntungan yang didapat tidak dikenai pajak sebelum tahap pengumpulan aset berakhir. Di
dalam kontrak anuitas variabel, retirement adalah masa ketika tahap pengumpulan aset berakhir. Pada saat retirement, seluruh aset yang terkumpul di dalam rekening anuitas variabel dapat diubah
ke dalam bentuk anuitas.
Investasi aset di dalam rekening anuitas variabel dipisah menjadi dua sub-rekening, yaitu sub- rekening aset bebas risiko dan sub-rekening aset berisiko. Pada saat retirement, investasi aset di
dalam sub-rekening bebas risiko akan diubah ke dalam bentuk anuitas tetap segera dan investasi aset di dalam sub-rekening berisiko akan diubah ke dalam bentuk anuitas variabel segera.
Dalam karya ilmiah ini dimodelkan suatu strategi keputusan dalam mengoptimalkan alokasi aset di dalam kontrak anuitas variabel sehingga diperoleh hasil yang maksimal saat retirement.
I PENDAHULUAN
1.1 Latar Belakang
