dynamic portofolio optimization and asset pricing

7/27/2019 Dynamic Portofolio Optimization and Asset Pricing

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Acknowledgments

I am grateful to my Lord for all his blessings and for having given me the

opportunity to meet my supervisor Mike Sherris who is an example of endless

dynamism and a source of pertinent ideas. Thank you Mike for this great

personality. I would also like to thank Dr. John van der Hoek for his early

directions in this area and my co-supervisor Rob Womersley who is always

happy to meet up and have good discussion. The friendly sta in Actuarial

Studies deserve many thanks for the nice ambiance they created.

Thanks also go to my family, especially to my father Professor Farouk

Hamada for his useful advice and remarkable viewpoints. Thank you my

Lord for this nice environment and for making this intellectual adventure a

success. To You this work is dedicated.

Sydney, November 15, 2001.1

1The research was funded by the International Postgraduate Research Scholarship

(IPRS), the Faculty of Commerce and Economics Scholarship and the Actuarial Stud-

ies Grant A79803988

iv


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Initial Wealth

b Stand Mean

(risk aversion) 1 2 3 4 5 6 7 Deviat

101 3000 108.75% -6.46% -0 .38 % 6.75% -2.16 % -1.11% -5.4 0% 1.3 5% 5.63%

102 6000 106.57% -4.91% -0 .07 % 4.70% -1.53 % -0.79% -3.9 7% 1.0 3% 5.73%

103 9000 104.39% -3.37% 0 .24 % 2.64% -0.90 % -0.46% -2.5 4% 0.7 4% 5.83%

105 2000 102.21% -1.83% 0 .55 % 0.58% -0.27 % -0.14% -1.1 1% 0.5 3% 5.93%

106 5000 100.03% -0.28% 0 .87 % - 1.47% 0.36 % 0 .19% 0.3 2% 0.5 1% 6.04%

1078000 97.85% 1.26% 1 .18% - 3.53% 0.99% 0 .51% 1.75% 0.69% 6.14%

1091000 95.66% 2.81% 1 .49% - 5.59% 1.62% 0 .83% 3.17% 0.96% 6.24%

1104000 93.48% 4.35% 1 .80% - 7.65% 2.25% 1 .16% 4.60% 1.28% 6.34%

1117000 91.30% 5.90% 2 .11% - 9.70% 2.88% 1 .48% 6.03% 1.61% 6.44%

1130000 89.12% 7.44% 2.42% -11.76% 3.52% 1.80% 7.46% 1.94% 6.54%1143000 86.94% 8.99% 2.73% -13.82% 4.15% 2.13% 8.89% 2.28% 6.64%

115 6000 84.76% 10.53% 3 .04 % -1 5.87% 4.78 % 2.45% 10.3 2% 2.6 2% 6.75%

116 9000 82.57% 12.08% 3 .35 % -1 7.93% 5.41 % 2.77% 11.7 5% 2.9 6% 6.85%

118 2000 80.39% 13.62% 3 .66 % -1 9.99% 6.04 % 3.10% 13.1 7% 3.3 1% 6.95%

119 5000 78.21% 15.17% 3 .97 % -2 2.04% 6.67 % 3.42% 14.6 0% 3.6 5% 7.05%

120 8000 76.03% 16.71% 4 .28 % -2 4.10% 7.30 % 3.75% 16.0 3% 4.0 0% 7.15%

122 1000 73.85% 18.26% 4 .59 % -2 6.16% 7.93 % 4.07% 17.4 6% 4.3 5% 7.25%

123 4000 71.67% 19.80% 4 .90 % -2 8.21% 8.56 % 4.39% 18.8 9% 4.6 9% 7.35%

124 7000 69.48% 21.35% 5 .21 % -3 0.27% 9.20 % 4.72% 20.3 2% 5.0 4% 7.46%

126 0000 67.30% 22.89% 5 .52 % -3 2.33% 9.83 % 5.04% 21.7 5% 5.3 9% 7.56%

1 27 30 00 6 5.12 % 2 4.44 % 5 .8 3% -3 4.39 % 1 0.4 6% 5 .3 6% 2 3.1 7% 5 .7 3% 7 .6 6%

1 28 60 00 6 2.94 % 2 5.98 % 6 .1 4% -3 6.44 % 1 1.0 9% 5 .6 9% 2 4.6 0% 6 .0 8% 7 .7 6%

1 29 90 00 6 0.76 % 2 7.53 % 6 .4 5% -3 8.50 % 1 1.7 2% 6 .0 1% 2 6.0 3% 6 .4 3% 7 .8 6%

1 31 20 00 5 8.58 % 2 9.07 % 6 .7 6% -4 0.56 % 1 2.3 5% 6 .3 4% 2 7.4 6% 6 .7 8% 7 .9 6%

1 32 50 00 5 6.40 % 3 0.62 % 7 .0 7% -4 2.61 % 1 2.9 8% 6 .6 6% 2 8.8 9% 7 .1 2% 8 .0 6%

1 33 80 00 5 4.21 % 3 2.16 % 7 .3 8% -4 4.67 % 1 3.6 1% 6 .9 8% 3 0.3 2% 7 .4 7% 8 .1 7%

1 35 10 00 5 2.03 % 3 3.71 % 7 .6 9% -4 6.73 % 1 4.2 5% 7 .3 1% 3 1.7 4% 7 .8 2% 8 .2 7%

1 36 40 00 4 9.85 % 3 5.25 % 8 .0 0% -4 8.78 % 1 4.8 8% 7 .6 3% 3 3.1 7% 8 .1 7% 8 .3 7%

1 37 70 00 4 7.67 % 3 6.80 % 8 .3 1% -5 0.84 % 1 5.5 1% 7 .9 5% 3 4.6 0% 8 .5 2% 8 .4 7%

1 39 00 00 4 5.49 % 3 8.34 % 8 .6 2% -5 2.90 % 1 6.1 4% 8 .2 8% 3 6.0 3% 8 .8 6% 8 .5 7%

1 40 30 00 4 3.31 % 3 9.88 % 8 .9 4% -5 4.96 % 1 6.7 7% 8 .6 0% 3 7.4 6% 9 .2 1% 8 .6 7%

1 41 60 00 4 1.12 % 4 1.43 % 9 .2 5% -5 7.01 % 1 7.4 0% 8 .9 2% 3 8.8 9% 9 .5 6% 8 .7 7%

1 42 90 00 3 8.94 % 4 2.97 % 9 .5 6% -5 9.07 % 1 8.0 3% 9 .2 5% 4 0.3 2% 9 .9 1% 8 .8 8%

1 44 20 00 3 6.76 % 4 4.52 % 9 .8 7% -6 1.13 % 1 8.6 6% 9 .5 7% 4 1.7 4% 1 0.2 6% 8 .9 8%

1 45 50 00 3 4.58 % 4 6.06 % 1 0.1 8% -6 3.18 % 1 9.2 9% 9 .9 0% 4 3.1 7% 1 0.6 1% 9 .0 8%

1468000 32.40% 47.61% 10.49% -65 .24% 19.93% 10.22% 44.60% 10.95% 9 .18%

1481000 30.22% 49.15% 10.80% -67 .30% 20.56% 10.54% 46.03% 11.30% 9 .28%

1494000 28.03% 50.70% 11.11% -69 .35% 21.19% 10.87% 47.46% 11.65% 9 .38%

1507000 25.85% 52.24% 11.42% -71 .41% 21.82% 11.19% 48.89% 12.00% 9 .48%

1520000 23.67% 53.79% 11.73% -73 .47% 22.45% 11.51% 50.32% 12.35% 9 .59%

1533000 21.49% 55.33% 12.04% -75 .52% 23.08% 11.84% 51.74% 12.70% 9 .69%

1546000 19.31% 56.88% 12.35% -77 .58% 23.71% 12.16% 53.17% 13.04% 9 .79%

1559000 17.13% 58.42% 12.66% -79 .64% 24.34% 12.49% 54.60% 13.39% 9 .89%

1572000 14.94% 59.97% 12.97% -81 .70% 24.97% 12.81% 56.03% 13.74% 9 .99%

1585000 12.76% 61.51% 13.28% -83 .75% 25.61% 13.13% 57.46% 14.09% 10.09%1598000 10.58% 63.06% 13.59% -85 .81% 26.24% 13.46% 58.89% 14.44% 10.19%

1611000 8 .40% 64.60% 13.90% -87 .87% 26.87% 13.78% 60.31% 14.79% 10.30%

1624000 6 .22% 66.15% 14.21% -89 .92% 27.50% 14.10% 61.74% 15.14% 10.40%

1637000 4 .04% 67.69% 14.52% -91 .98% 28.13% 14.43% 63.17% 15.48% 10.50%

1650000 1 .85% 69.24% 14.83% -94 .04% 28.76% 14.75% 64.60% 15.83% 10.60%

Asset Allocation

1,000,000

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112/241

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117/241


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118/241


-4.876

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119/241

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120/241

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121/241

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frlqflghv zlwk wkh lqlwldo zhdowk jlyhq dv dq lqsxw wr wkh sureohp +719,1

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lv lqwurgxfhg lq Vhfwlrq 618151 Ohw xv uvw frqvlghu wkh iroorzlqj wzr0shulrg

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71714 D Wzr0Shulrg H{dpsoh

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D. lq Solvnd +4


126/241


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grzq mxps idfwruv dqg D ' > j2*2Iljxuh +7145, vkrzv wkh rswlpdo frqvxpswlrq dw hdfk qrgh ri wkh odwwlfh dv

d ixqfwlrq ri zhdowk dydlodeoh dw wkh ehjlqqlqj ri wkh wlph shulrg1 Iru d


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71 Vwrfkdvwlf G|qdplf Surjudpplqj yv1 Pduwlqjdoh Phwkrgv 44;

w

C o n su m p tio n .50 1*w

.3 3 5*w w.25 2*w .50 1*w

.20 2*w .3 3 5*w w

.16 8*w .25 2*w .50 1*w

.14 5*w .20 2*w .3 3 5*w w

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.20 2*w .3 3 5*w w

.25 2*w .50 1*w

.3 3 5*w w

.501*w

w

Iljxuh 7145= Rswlpdo Frqvxpswlrq Uxohv xvlqj V|perolf G|qdplf Surjudp0plqj

pdwxulw| ri S prqwkv/ wkh odwwlfh kdv . froxpqv1 Lq odvw shulrg wkh lqyhvwru

frqvxphv wkh wrwdo zhdowk1

Risky Investment .0465*w

.0658*w

.0784*w .0523*w

.0887*w .0739*w

.0979*w .0881*w .0587*w

.107*w .0996*w .083*w

.110*w .0989*w .066*w.112*w .0932*w

.111*w .0741*w

.0105*w

.0832*w

Iljxuh 7146= Rswlpdo Lqyhvwphqw Uxohv xvlqj V|perolf G|qdplf Surjudpplqj

Iljxuh +7146, vkrzv wkh rswlpdo qxpehu ri xqlwv ri wkh ulvn| vhfxulw| wr eh

khog dw hdfk qrgh dv d ixqfwlrq ri zhdowk +,1 Qrwh wkdw wkh lqyhvwphqw vwrsv

dw | ' D vlqfh dw pdwxulw| doo zhdowk lv frqvxphg1 Wklv lpsolhv S froxpqv

lq wkh lqyhvwphqw odwwlfh1 Lq rughu wr rewdlq qxphulfdo ydoxhv ri wkh rswlpdo

frqvxpswlrq dqg lqyhvwphqw vwudwhjlhv iru d vshflhg ydoxh ri lqlwldo zhdowk/ d

iruzdug uhfxuvlrq lv fduulhg rxw iurp wlph f wr Sc xvlqj wkh exgjhw frqvwudlqwv

htxdwlrq +615:,1 Iljxuhv +7147, dqg +7148, vkrz wkh rswlpdo groodu dprxqw ri

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lqlwldo ydoxh ri zhdowk `f ' mff Iljxuh +7149, vkrzv wkh zhdowk uhvxowlqj iurp


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surfhvv lv

_`| ' dE)|E> o n o`| S|o _| n )|j`|_| +7144,

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zhoo ghvfulehg lq Nxvkqhu +4


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zkhuh wkh REc pd| eh lqwhusuhwhg dv wudqvlwlrq suredelolwlhv ri d Pdunry fkdlqjlyhq e|=

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surfhvv lv looxvwudwhg lq Iljxuh +714:,1

D julg vl}h ri ' fff zdv xvhg/ zlwk vwdwh yduldeoh ` wdnlqj ydoxhv ri

iurp 3 wkurxjk 53133/ lq vwhsv ri ' ff21 Dv qrwhg lq Sxufdo dqg Sljjrww +5334/

^;;`,/ ydoxhv dw wkh wrs dqg erwwrp ri wkh julg duh lqdffxudwh/ zkloh ydoxhv lq wkh

plggoh duh dffxudwh1 D }rrp dw wkh plggoh edqg ri wkh julg iru glhuhqw wlphv

ehwzhhq | ' f dqg A ' S prqwkv/ duh jlyhq lq Iljxuh +714;,1 Lq wklv jxuh/

zkhuh wkh lqyhvwru kdv d orjdulwkplf xwlolw|/ iru hdfk ydoxh ri srvvleoh zhdowk/

ydoxhv ri rswlpdo frqvxpswlrq/ sursruwlrq ri zhdowk wr eh lqyhvwhg lq wkh ulvn|

vhfxulw| dqg wkh fruuhvsrqglqj ydoxh ixqfwlrq duh frpsxwhg


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13.94 -200.3 13.94 -200.04

13.92 -22.01 13.92 -22.02

13.9 -200 13.9 -200.01

13.88 -199.8 13.88 -200

13.86 -199.5 13.86 -199.9

13.84 -199.3 13.84 -199.7

13.82 -199 13.82 -199.5

Value Function

Wealthat t=10.000

Value Function

Wealthat t=10.002

One-step backwards

recursion

Iljxuh 714:= H{solflw Ilqlwh Glhuhqfh Phwkrg lq Glvfuhwl}lqj wkh KM E Htxd0wlrq


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wealth value consum- proportion wealth value consum- proportion wealth value consum- proportion

function ption in risky function ption in risky function ption in risky

10.20 2.87 6.93 1.2482 10.20 2.79 7.32 1.2483 10.20 2.43 9.44 1.2487

10.18 2.86 6.92 1.2482 10.18 2.78 7.30 1.2483 10.18 2.42 9.42 1.2487

10.16 2.86 6.91 1.2482 10.16 2.78 7.29 1.2483 10.16 2.42 9.41 1.2487

10.14 2.86 6.89 1.2482 10.14 2.78 7.28 1.2483 10.14 2.42 9.39 1.2487

10.12 2.85 6.88 1.2482 10.12 2.78 7.26 1.2483 10.12 2.42 9.37 1.2487

10.10 2.85 6.87 1.2482 10.10 2.77 7.25 1.2483 10.10 2.42 9.35 1.2487

10.08 2.85 6.85 1.2482 10.08 2.77 7.23 1.2483 10.08 2.41 9.33 1.2486

10.06 2.84 6.84 1.2482 10.06 2.77 7.22 1.2483 10.06 2.41 9.31 1.2486

10.04 2.84 6.82 1.2482 10.04 2.77 7.20 1.2483 10.04 2.41 9.29 1.2486

10.02 2.84 6.81 1.2482 10.02 2.76 7.19 1.2483 10.02 2.41 9.28 1.2486

10.00 2.84 6.80 1.2482 10.00 2.76 7.18 1.2483 10.00 2.41 9.26 1.2486

9.98 2.83 6.78 1.2482 9.98 2.76 7.16 1.2483 9.98 2.40 9.24 1.2486

9.96 2.83 6.77 1.2482 9.96 2.75 7.15 1.2482 9.96 2.40 9.22 1.2486

9.94 2.83 6.76 1.2482 9.94 2.75 7.13 1.2482 9.94 2.40 9.20 1.2486

9.92 2.82 6.74 1.2482 9.92 2.75 7.12 1.2482 9.92 2.40 9.18 1.2486

9.90 2.82 6.73 1.2482 9.90 2.75 7.10 1.2482 9.90 2.39 9.17 1.2486

9.88 2.82 6.72 1.2482 9.88 2.74 7.09 1.2482 9.88 2.39 9.15 1.24869.86 2.82 6.70 1.2481 9.86 2.74 7.07 1.2482 9.86 2.39 9.13 1.2486

9.84 2.81 6.69 1.2481 9.84 2.74 7.06 1.2482 9.84 2.39 9.11 1.2486

9.82 2.81 6.67 1.2481 9.82 2.73 7.05 1.2482 9.82 2.39 9.09 1.2486

9.80 2.81 6.66 1.2481 9.80 2.73 7.03 1.2482 9.80 2.38 9.07 1.2486

t = 0 t =1 month t = 5 month

Iljxuh 714;= Plggoh ri Julg Vroxwlrq dw Glhuhqw Prphqwv gxulqj wkh Lqyhvwphqw


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719 Frpsdulvrq ehwzhhq

UQFDdqg G|0

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Xvlqj UQFDc wkh rswlpdo frqvxpswlrq lv d ixqfwlrq ri lqlwldo zhdowk/

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rswlpdo wudglqj vwudwhj| dw hdfk qrgh1


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dynamic portofolio optimization and asset pricing

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