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Systems Engineering Procedia 5 (2012) 150 – 155

International Symposium on Engineering Emergency Management 2011

Logarithm Utility Maximization Portfolio Engineering with Bankruptcy Control: a Nonparametric Estimation Framework Qing-hua Ma, Hai-xiang Yao *, Shi-Yang Li School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, China

Abstract Under the assumption that investors have the logarithm utility function, this paper adopts the methodology of nonparametric estimation and the expected utility maximization (EUM) model to explore a portfolio engineering problem with bankruptcy control. First, we obtain the nonparametric estimated calculation formula for expected utility by using the nonparametric estimation. Then, sequential quadratic programming (SQP) algorithm for the optimal investment strategy of the EUM model is given. Finally, a numerical portfolio engineering example based on real data of Chinese stock market is presented. © 2012 2011 Published Elsevier Ltd.Ltd. Selection and peer-review under responsibility of Desheng Dash Wu Dash Wu. © Publishedbyby Elsevier Selection and peer-review under responsibility of Desheng Open access under CC BY-NC-ND license. Keywords: Portfolio engineering; Bankruptcy control; Logarithm utility maximization; Nonparametric estimation;

1. Introduction Expected utility maximization (EUM) model initiated by [1] and mean-variance model established by [2] are the two principal areas for studying the portfolio selection (engineering) problem in portfolio engineering. Using the EUM model, [3] and [4] have study the optimal investment-consumption strategies problem in discrete time and continuous time, respectively. On the other hand, [5] and [6] expanded the mean-variance model to cases of discrete time and continuous time, respectively. However, in portfolio engineering, investors would often be confronted with the risk of bankruptcy when they use directly the investment strategies of EUM model or mean-variance model. For this reason, using discrete time meanvariance model, [7] and [8] investigated the portfolio engineering problem with bankruptcy control. [9] solved a continuous-time mean-variance portfolio selection problem with bankruptcy prohibition. On the other hand, logarithm utility function is always the frequently-used utility function in the EUM model. One reason is that it is easy to compute optimal strategies explicitly, see, for example, [10] and [11]. The other reason is that the logarithm optimal strategy also maximizes the long term growth rate in an almost-sure sense (see [12]), that is the goal for most investor to pursue. [13] studied the optimal long term growth rate of wealth. However, most of the abovementioned literatures are under the assumption that return on assets obey some specific probability distribution type, such as normal distribution or geometry Brownian movement. But it is well known that the finance market is complex and changeable. It is very difficult for us to know the probability distribution type of asset return. On the other hand, nonparametric estimation method need not make any hypothesis about the distribution type, and the calculated results are completely driven by market sample data (see [14]), which can makes it adaptable in the context of capricious financial market. So in recent years, some scholars began to adopt nonparametric estimation

2211-3819 © 2012 Published by Elsevier Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. Open access under CC BY-NC-ND license. doi:10.1016/j.sepro.2012.04.024

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method to study the calculation of financial risk; we refer [15-16] to the readers. But to the best knowledge of the authors, there is no literature to consider the EUM problem by using nonparametric estimation method. Basing on logarithm EUM model and using nonparametric estimation method, we will investigate a portfolio engineering (selection) problem with risk control over bankruptcy in this paper. This paper is organized as follows. In section two, a EUM problem with risk control over bankruptcy is established. In section three, we obtain the estimated calculation formula of expected utility by using nonparametric estimation of the portfolio return’s density function. In section four, we give the sequential quadratic programming (SQP) algorithm for obtaining the optimal investment strategy. Finally, we indicate a portfolio engineering numerical example basing on real data of Chinese stock market to show the validity and the practicability of our results. 2. EUM model with risk control over bankruptcy & Suppose that there are n assets with return vector [ ([1 , [ 2 ,[ n )c for investors to invest. Let W ( w1 , w2 ,  , wn )c denote the portfolio of the assets. Here Ac denotes the transpose of matrix A . Then, return of n & portfolio is [ p : ¦ wi[i W c[ . In reality, there is one another problem for us to consider, namely to avoid i

bankruptcy. Following [7] and [8], a bankruptcy occurs when the wealth or portfolio return of the investor falls below a predefined ‘‘disaster’’ level b . Therefore, in the process of investment, we should control the probability Var[[ p ] P([ p  b) of bankruptcy. According to Tchebycheff inequality, we have P([ p  b) d . Then, (E[[ p ]  b) 2 controlling the risk of bankruptcy can be achieved by setting a small value d such that

Var[[ p ] (E[[ p ]  b) 2

d d , i.e.

Var[[ p ] d d (E[[ p ]  b) 2  Therefore, EUM portfolio selection model with a bankruptcy control can be represented as

the following optimization problem

& ­max E[U (W c[ )] ° ® & 2 °¯ s.t. W c1 1, Var[[ p ] d d (E[[ p ]  b) ,

(1)

& where 1 denotes a column vector whose n entries are all ones, U (˜) is the utility function corresponding to investor. In order to guarantee the optimal solution exist, U (˜) often needs to satisfy some mathematical properties, such as concavity and monotonic increase. We also suppose that U (˜) is logarithm function, i.e. U ( x) ln x.

3. EUM model base on nonparametric estimation framework In the paper, we will obtain the estimated formula of expected utility by adopting the nonparametric method to & estimate the distribution of [ p or [ , and make further efforts to investigate the EUM portfolio selection problem. & Since the asset’s returns vector [ is a multidimensional random vector, if we adopt nonparametric method to estimate its density function, the convergence rate would be very slow, which sometimes referred to as the “curse of dimensionality” (see [14]). For the sake of overcoming the problem of “curse of dimensionality”, we will obtain the & nonparametric estimation for E[U (W c[ )] by estimating the density of return of portfolio, that is only one dimension. Now we introduce some preliminary knowledge (details see in [14]). Nonparametric estimation of probability density function (PDF) p ( x) of univariate random variable X with sample set { X 1 , X 2 , , X T } is T

pˆ ( x) T 1h 1 ¦ k ( i 1

Xi  x ), h

(2)

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Qing-hua Ma et al. / Systems Engineering Procedia 5 (2012) 150 – 155

where k () is the kernel function, which, for example, can be chosen as k (v)



2S



1

2

e 0.5v . G (v)

³

v

f

k (t )dt ,

and h h(T ) is a smoothing parameter (or alternatively called bandwidth). From [14] we know that the kernel estimator pˆ ( x) defined in (2) is a consistent estimator of p ( x) when kernel function k () and bandwidth h() satisfy the following conditions i) k () is nonnegative and bounded, ³ k (v)dv 1 , k (v)

k (v) (that mean

³

f

f

vk (v)dv

0 ),

2

³ v k (v)dv

N2 ! 0 ;

ii) h(T ) o 0 and Th(T ) o f as T o f . Throughout this paper we always assume that k () and h() satisfy the conditions above. Both theoretical and practical settings show that the choice of bandwidth h() is crucial. There are many methods for selecting h() , here we only introduce the rule-of-thumb as follows. Rule-of-thumb: h | 1.06V T 0.2 , where V is the deviation of X . V can be estimated by the samples:

Vˆ

1

T

T

T  1 ¦ ( X i  X )2

T 1 ¦ X i .

, where X

i 1

i 1

& W c[ also has the sample set

& Given investment strategy W and the sample set {R1 , R2 , , RT } of [ , then [ p

{W cR1 , W cR2 , , W cRT } . Adopting the method of nonparametric estimation, nonparametric PDF estimation of [ p is T § W cRi  x · pˆ [ p ( x) T 1h 1 ¦ k ¨ ¸, h © ¹ i 1 & c After having estimated the PDF of [ p , we can estimate E[U (W [ )] as

& ˆ U (W c[ )] E[

(3)

T f § W cRi  x · U ( x) pˆ [ p ( x)dx T 1h 1 ¦ ³ U ( x)k ¨ ¸ dx . f f h © ¹ i 1 Since the nonparametric estimation is insensitive to the choice of kernel function (details see in [17]), for convenience of research, we choose the Epanechnikov kernel function ­°0.75(1  v 2 ), v d 1, k (v) 0.75(1  v 2 )1^ v d 1` ® 0, other. °¯

³

f

(4)

(5)

Substituting (5) into (4) and noting that U ( x) ln x , we have T & f W cRi  h 1 3 2 2 ˆ U (W c[ )] ˆ g (W , h) : E[ x p x dx T h (ln ) ( ) 0.75 ¦ [ ³ ³ (ln x) ª¬h  (W cRi  x) º¼ dx p f

T

^

0.75T 1h 3 ¦ h 2  W cRi i 1

2

W cRi  h

i 1

³

W cRi  h

W cRi  h

ln xdx  ³

W cRi  h

W cRi  h

x 2 ln xdx  2 W cRi ³

W cRi  h

W cRi  h

`

x ln xdx

W cRi  h W cRi  h T ­ ½° W cRi  h x3 § 1· x2 § 1· 2 ° 0.75T 1h 3 ¦ ® h 2  W cRi x(ln x  1)  ¨ ln x  ¸  2 W cRi ¨ ln x  ¸ ¾ . W cRi  h 3© 3 ¹ W cR  h 2 © 2 ¹ W cR  h ° i 1 ° i i ¯ ¿ After simplifying, we have 3 2 T ­§ 2 W cRi 16 ½° 2 ° W cRi W cRi · W cRi  h 2 1 2 ¸ ln   ln W cRi  h   ¾ . g (W , h) 0.75T ¦ ®¨ ¨ h 9° 3h3 ¸¹ W cRi  h 3 3h 2 i 1 ° ¯© ¿ On the other hand, expectation E[[ p ] and variance Var[[ p ] can be estimated by sample as







T

ˆˆ [ ] T 1 ¦ W cR E[ p i i 1



T

T

i 1

i 1

W c(T 1 ¦ Ri ) W cR , Var[[ p ] (T  1) 1 ¦ (W cRi  W cR ) 2

(T  1) 1 . cM 0 .

T

where R

T 1 ¦ Ri , .

(W cR1 , W cR2 ,  , W cRT )c ƒW , ƒ

i 1

identity matrix and :

(T  1) 1 ƒcM 0 ƒ.

( R1 , R2 ,  , RT )c ˈ M 0

(6)

W c:W ,

& & I  T 1 1 ˜ 1c , I denotes the

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Qing-hua Ma et al. / Systems Engineering Procedia 5 (2012) 150 – 155

ˆ [[ ] Thus if using rule-of-thumb to select bandwidth h , we have h 1.06T 0.2 Var p a 1.06T

1  5

a W c:W , where

! 0 is a constant. Since the domain of ln x is ^ x | x ! 0` and notice that h ! 0 , g (W , h) is meaningful

if only if W cRi  h ! 0 , i 1, 2, , T . Therefore, EUM portfolio selection model with a risk control over bankruptcy can be rewritten as ­°max g (W , h) W ,h (7) ® & °¯ s.t. W c1 1; h a W c:W ; W cRi  h ! 0, i 1, 2, , T ; W c:W d d (W cR  b) 2 . 4. Solving the EUM model base on nonparametric estimation method Since most optimization algorithm is aimed at minimization problem, it is obvious that max g (W , h) is W ,h

equivalent to min ^ g (W , h)` . What is more, only when the inequality constraints W cRi  h t 0 , i 1, 2, , T hold, W ,h

the objective function of optimization problem (7) is meaningful. So when we adopt numerical algorithm to solve optimization problem (7), if in the process of iteration it emerges that some inequality constraints W cRi  h ! 0 does not hold, the iteration would not be proceeded and lead to the failure of solving. In order to overcome the difficulty mentioned above, we introduce some parameters si , i 1, 2, , T , such that

si2

W cRi  h t 0 and si z 0 . Then the inequality constraints in optimization problem (7) can be transform to

equality constraint. Notice that h

^ g (W , h)`

a W c:W , then W cRi  h

si2  2a W c:W . Therefore, the objective function

can be rewritten as the function of W , s1 , , sT , i.e. f (W , s1 , , sT ) :  g (W , h) , and there is no limit to

the nature domain of function f (W , s1 , , sT ) . From (6) we have f (W , s1 , , sT ) ­§ · 3 2 2 W cRi W cRi ¸ si2  2a W c:W 2 ª 2 2 16 ½° °¨ W cRi º c ln   ln si si  2a W :W  2  ¾ . (8) ®¨ ¦ 3 ¸ 2 ¼ 3a W c:W 9 3 ¬ si i 1 °¨ a W c:W °¿ 3a 3 (W c:W ) 2 ¸¹ ¯© Therefore, optimization problem (7) is equivalent to f (W , s1 , , sT ) ­°W min , s1 ,, sT (9) ® & °¯ s.t. W c1 1; W cRi  a W c:W si2 , i 1, 2, , T ; d (W cR  b) 2  W c:W t 0. It is need to note that after we obtain the optimal solution W * , si* , we still need to check if si* z 0 for all 3  4T

T





i 1, 2, , T . If si* z 0 for all i 1, 2, , T , then W * , si* are the final optimal solution. Otherwise, the solving fails. Optimization problem (9) is a nonlinear constraint optimization problems and is very difficult to obtain the analytic solution in general. So we can only obtain the numerical solution. At present, there are many numerical methods to deal with problem (9), such as penalty method, Augmented Lagrangian method, gradient projection method and sequential quadratic programming (SQP), and so on. This paper only introduces the most commonly used SQP algorithm. Now we consider a general nonlinear constraint optimization problem ­min z ( x) (10) ® ¯ s.t. ci ( x) 0, i  E{1, 2, , me }; ci ( x) t 0, i F {me  1, me  2, , m}.

where x

( x1 , x2 , , xn )c  n . Define the quadratic programming subproblem in k th iteration 1 ­ ’z ( x k )cd  d cBk d , °min n 2 ® d  ° s.t. c ( x k )  ’c ( x k )cd 0, i  E ; c ( x k )  ’c ( x k )cd t 0, i  F ; i i i i ¯

(11)

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Qing-hua Ma et al. / Systems Engineering Procedia 5 (2012) 150 – 155

where ’ is the gradient operator, Bk is the positive definite approximation of Hesse matrix of Lagrangian function m

L( x k , O k )

z ( x k )  ¦ Oik ci ( x) . Define the cost function in k th iteration i 1

W ( xk , P k )

z ( x k )  ¦ Pik ci ( x k )  ¦ Pik max 0,(ci ( x k )} iE

(12)

iF

1 ­ ½ max ®| Oik |,  Pik 1  | Oik | ¾ , k t 2. 2 ¯ ¿ By [18], we introduce the steps of SQP algorithm for solving optimization problem (10). Step 1: Given initial point ( x 0 , O 0 )  n u m and initial matrix B0  nun (often selected as D1 I , D1 ! 0 ).

where, for i 1, 2, , m , Pik is defined as Pi1 | Oi1 |; Pik

Chosen parameter K  (0, 1 ), U  (0,1) and 0 d H1 , H 2 1 . Set k : 0 . 2 Step 2: Solve quadratic programming subproblem (11) for the solution d k and the corresponding Lagrange multiplier O k 1 . Step 3: If || d k ||1  H1 and

¦ c (x i

iE

k

)  ¦ max 0,(ci ( x k )}  H 2 , stop iteration and output x k . Otherwise turn to iF

Step 4. Step 4: Armijo search. Let N m denote the minimal nonnegative integer N that satisfies the following inequality

W ( x k  U N d k , P k )  W ( x k , P k ) d KU N ’ xW ( x k , P k )cd k . Set D k : U Nm and x k 1 B t tc B z zc Step 5: Update Bk as follows Bk 1 Bk  k k k k  k k , where, tkc Bk tk tkc zk tk

x k 1  x k , yk

’ x L( x k 1 , O k 1 )  ’ x L( x k , O k 1 ), zk

1, ­ ° T k ® 0.8tkc Bk tk °tc B t  tc y , ¯k kk k k Step 6: Set k : k  1 , and turn to Step 2.

xk  D k dk .

T k yk  (1  T k ) Bk tk , T k is defined as tkc yk t 0.2tkc Bk tk , tkc yk  0.2tkc Bk tk .

5. Numerical example We randomly select six stocks from Shenzhen and Shanghai stock exchange in China. These stock codes are 200053, 600583, 600547, 601699, 002207 and 600111. Selecting the historical daily data of these stocks from June 1, 2009, to May 26, 2011, we get T 485 Day gross returns samples {R1 , R2 ,  , RT } , where the unit of return is 

1

1/ 5 . Substituting data and through some calculation, we obtain a 1.06T 5 0.3077 , and § 0.0103 0.0066 0.0057 0.0079 0.0070 0.0068 · ¨ ¸ ¨ 0.0066 0.0170 0.0061 0.0115 0.0087 0.0072 ¸ ƒcM 0 ƒ ¨ 0.0057 0.0061 0.0245 0.0121 0.0093 0.0113 ¸ : ¨ ¸. T 1 ¨ 0.0079 0.0115 0.0121 0.0270 0.0119 0.0118 ¸ ¨ 0.0070 0.0087 0.0093 0.0119 0.0243 0.0097 ¸ ¨¨ ¸¸ © 0.0068 0.0072 0.0113 0.0118 0.0097 0.0308 ¹ When we do not consider the risk control over bankruptcy, namely we cancel the inequality constraint d (W cR  b) 2  W c:W t 0 in optimization problem (9). Adopting SQP algorithm, we obtain the optimal investment portfolio and maximum expected utility are, respectively W * (0.6513, 0.3628, 0.2416, 0.1950, 0.4381, 0.7130)c and g (W * , h* )  f * 0.0095.

*

All numerical results is computed in MatLab6p5 by compiling procedure, the computational results are accurate to 0.0001.

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When we consider the risk control over bankruptcy, and let b 0.2, d 0.02 , namely we control that P([ p  0.2) d 0.02 . Adopting SQP algorithm, we obtain the optimal investment portfolio and maximum expected utility are, respectively W * (0.6321,  0.0816,0.2097,0.0791,  0.2051,0.3658)c and g (W * , h* )  f * 0.0069 . In a similar manner, when b 0.3, d 0.02 , the optimal investment portfolio and maximum expected utility are, respectively W *

(0.6209,0.0431,0.1777,0.0243,  0.0895,0.2234)c and g (W * , h* )

f*

0.0041.

6. Conclusion Adopting the nonparametric estimation method, in this paper we obtained the estimated formula for expected logarithm utility, and embed it into the utility maximization portfolio selection problem with bankruptcy control in portfolio engineering. The SQP algorithm is presented to solve the model. The method introduced in this paper has generality, we also can use it to study the utility maximization portfolio selection problem in other realistic conditions (such as transaction costs, no-shorting constraints) and under other utility function (such as power utility function, exponential utility function).

Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 61104138), Humanity and Social Science Foundation of Ministry of Education of China (No. 10YJC790339), Natural Science Foundation of Guangdong Province (No. S2011010005503), and Philosophy and Social Science Foundation of Guangdong Province (No. 09O-19).

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