At paper we investigate problem of the investment portfolio selection from one risky asset and one risk-free asset. We use the probability criterion for the investment portfolio selection. The possibility of rebalancing of the investment portfolio is used for diversication of the portfolio. We nd an approximate analytical solution of the problem using the law of total probability. The investment portfolio is selected for various distributions of returns. We give an example.
The problem of optimal investment is the problem of assets selection for
investment, so that the investor’s capital will be maximal at the some instant of time
in future. Therefore the cost function is investor’s capital. Due to the fact that
at the some instant of time prices of some assets are unknown, returns from
these assets are random variables. Consequently, investor’s capital will become
a random variable with the purchase of these assets. In such a way optimization
is possible after applying some criterions to the cost function: for an example,
probabilistic [
The two-step problem is considered for allowing rebalancing of the investment
portfolio at the some instant of time. As shown in [
Searching for optimal control, as done in [
We will choose one risk-free asset and some risky asset (having non-zero variance) for the investment portfolio selection. Obviously, the structure of the investment portfolio depends on distributions parameters and distribution law. We will analyze the structure of the investment portfolio for various distributions of risky asset return. 2
Assume that the investor have two assets for investment at the each trading period: risk-free asset with constant return b0 and risky asset (stock, bond) with return 1 + X~1 at the rst step and risky asset with return 1 + X~2 at the second step. Variables X1 and X~2 are independent identically distributed ~ random variables with existing rst and second moments and density function f (x). Variables X~1, X~2 characterize the ratio of sale price of risky asset to the purchase price. State some realizations of sample mean xn and sample variance s2n of 1+X~1. Assume u0;i is the fraction of the investor’s capital invested in riskfree asset at the trading period i and u1;i is the fraction of the investor’s capital invested in risky asset at the trading period i, C1 is initial investor’s capital and ' is amount desired by the investor. Assume short-sales are banned and investor’s capital is invested fully at the each trading period. The dynamics of investor’s capital is then described by
Cj+1 = Cj 1 + u0;j b0 + u1;j ( 1 + X~j ) ; j = 1; 2; and control uj d=ef col(u0;j ; u1;j ) is selected at the each step from set U d=ef fy0; y1 : y0 + y1 = 1; y0 0; y1 0g: We consider next sequence of segments which are all set of possible values of investor’s capital C2 s0 d=ef ( 1; C1(N )); s1 d=ef [C1(N ); C2(N )); s2 d=ef [C2(N ); C3(N )); : : : ; sN d=ef [CN (N ); CN+1(N )); sN+1 d=ef [CN+1(N ); +1); where and
Ci(N ) = 2C1(i
m 1) ; i = 1; N + 1;
N +1 Z where N is a priori specied natural number, which characterizes the number of segments of the decomposition, m is expectation of random variable X~1. Such selection of values Ci(N ) is connected with the fact that neness of si tends to zero subject to N ! 1. Segments s0 and sN+1 is saved constant, as segment s0 characterizes infeasible on practice values of investor’s capital, as without debts we cannot obtain capital is less than zero. Segment sN+1 characterizes values with small probability because obtaining return higher than 100 per cents is almost impossible in practice.
We will nd control at the second step u2 as piecewise-constant strategy depending on segment si. Thus the problem of searching for optimal control is described by
P~'(u1(C1); u2(s0; s1; : : : ; sN ; sN+1)) d=ef def = PfC3(u1(C1); u2(s0; s1; : : : ; sN ; sN+1)) 'g: (1)
(u~1'( ); u~2'( )) = = arg
max u1(C1)2U;u2(s0;s1;:::;sN ;sN+1)2U P~'(u1(C1); u2(s0; s1; : : : ; sN ; sN+1)): (2) 3
N+1 'g = X i=0 PfC3 PfC2 2 sigPfC3 'jC2 2 sig: (3)
There are following nestings for i = 1; N
PfC2 2 sigPfC3 'jC2 2 sig = PffC2 2 sig fC3 'gg = = PffC2 2 sig fC2(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2))g 'g: fC2 2 sig fCi(N )(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) fC2 2 sig fC2(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) 'g 'g fC2 2 sig fCi+1(N )(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) 'g:
PfC2 2 sigPfCi(N )(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) 'g PffC2 2 sig fC2(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) 'g PfC2 2 sigPfCi+1(N )(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) 'g: Note Ci+1(N ) Ci(N ) ! 0 subject to N ! 1, therefore we use the new functional for obtaining approximate value of functional (1), written also as (3) P^'(u1(C1); u2(s0; s1; : : : ; sN ; sN+1)) d=ef
N+1 d=ef X PfC2 2 sigPfCi(N )(1 + u0;2(si)b0 + u1;2(si)( 1 + X~2)) 'g: (4) i=1
(u^1'( ); u^2'( )) = = arg max u1(C1)2U;u2(s0;s1;s2;:::;sN ;sN+1)2U P^'(u1(C1); u2(s0; s1; : : : ; sN ; sN+1)): (5)
4
P'i (u0;2; u1;2) d=ef
def
= PfCi(N )(1 + u0;2b0 + u1;2( 1 + X~2))
'g ! u0;2+u1;2=1;u0;2 0;u1;2 0
max
:
because controls at the rst step and at the second step are independent and
probabilities PfC2 2 sig are non-negative. Solution of last problem we can nd
in [
Pi = u0;2+u1;2=m1;ua0x;2 0;u1;2 0 P'i (u0;2; u1;2) = = (1; 1
' Ci(N )(1 + b0); F ('=Ci(N )); ' > Ci(N )(1 + b0); x
Z F (x) =
f (t)dt:
PfC2 2 sigPi ! u0;1+u1;1=1;u0;1 0;u1;1 0 max Reduce the dimension of the problem by substitution u0;1 = 1 u1;1 and simplify problem (6)
N+1 X PfC1(1 + b0 i=1 u1;1(1 + b0) + u1;1X~1) 2 sigPi ! 0 u1;1 1 max : Note at the point u11 = 0 value of function P'(u0;1; u1;1) is equal to
N+1 P'(1; 0) = X PfC1(1 + b0) 2 sigPi:
i=1 PfC1(1 + b0 = F a u1;1(1 + b0) + u1;1X~1)
ag = C1(1 + b0 u1;1(1 + b0)) ;
C1u1;1 If i = 1; N we have If i = N + 1 we have
PfC1(1 + b0
u1;1(1 + b0) + u1;1X~1) 2 sig = = PfCi(N )
u1;1(1 + b0) + u1;1X~1) < Ci+1(N )g = = F
F
C1(1 + b0 Ci+1(N )
Ci(N )
C1(1 + b0
C1u1;1 C1(1 + b0
C1u1;1 u1;1(1 + b0)) u1;1(1 + b0)) : PfC1(1 + b0
u1;1(1 + b0) + u1;1X^1) 2 sN+1g = = PfCN+1(N )
C1(1 + b0
u1;1(1 + b0) + u1;1X~1) < +1g = Consequently, M[X~1] = , D[X~1] = 2, where M[X~1] is mean, and D[X~1] is variance. Thus using realizations of return we obtain next relations for and 2 If X~1 is log-normal, then density function f (x) is equal to (M[ 1 + X~1] = 1 +
= xn;
D[ 1 + X~1] = 2 = s2n: f (x) =
exp B = 1 + xn + p3sn:
p3sn; = ln(xn + 1) 2 If X~1 is distributed by uniform law, then density function f (x) is equal to subject to x 2 [A; B] and 0, otherwise. Therefore M[X~1] = (A + B=2), D[X~1] = (B A)2=12
Compare for accuracy obtained relations with exact solution provided in [
Note that exact solution was found in positional strategy class. But nonetheless as follows from table No 1 approximate strategy and approximate value of the probability functional is almost identical with exact solution when N 1000.
Now analyze the structure of the optimal investment portfolio obtained with using derived above relations. Assume xn = 0; 1, and sn = 0; 15 and ' = 1; 08, b0 = 0; 03, C1 = 1 again. We nd the investment portfolio for normal distribution at the example No 3, for log-normal distribution at the example No 4, for uniform distribution at the example No 5. We nd approximate strategies of the rst step and the value of probability functional (6) at this strategy for various value N again. Mesh width is 0,01.
As follows from table No 2 growth of value N does not allow to signicantly increase the value of criterion P(C3 ') after N 1000. At the same time the value of P(C3 ') criterion and the structure of the investment portfolio is almost identical although in each example various distributions were used. In this work we have studied the two-step problem of optimal investment with one risky asset using the probability as optimality criterion. We have found function which approximates criterial function and have proposed the algorithm to optimize this function. Various cases of distribution of returns were investigated and we have found the structure of the optimal investment portfolio is almost identical despite of one or another distribution. 7. Benati S. The optimal portfolio problem with coherent risk measure constraints //
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