1. Introduction

Ethier et al, [2] studied the rate of return on investment in discrete time gambling method, where the total return on the individual gambler is assumed to follow a random walk. They also showed that the asymptotic distribution of the return, as the mean increment in the random walk goes to zero is a gamma distribution. Also Kelly (1956) in Thorp, [11] observed the relationship between the logarithm of wealth and expected asymptotic rate at which wealth compounds. The object is to let one know how one should invest in each equities of ones highly diversified stocks portfolio to maximize the capital growth. In finance, the rate of return (ROR) which is known as return on investment (ROI) is the ratio of money gained or lost whether realized or unrealized on an investment, relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss or net income/loss. There are several ways to determine ROI, but the most frequently used method is net gain divided by total assets. Constant proportions investment strategies also play a fundamental role in portfolio theory. Under these policies, an investor follows a dynamic trading strategy that continually rebalances the portfolio so as to always allocate fixed constant proportions of the investor’s wealth across the investment opportunities. These strategies are widely used in practice and are also referred to as continuously rebalanced strategy [9]. Given the fundamental nature of policies in theoretical portfolio practice, it is of interest to know what the stochastic behavior of the rate of return on investment (RROI), defined as the net of gain over the cumulative investment [3]. Merton, [5] introduced the setting in the continuous time financial model as used in Black-Scholes, [1].

In this paper, we obtain some limit theorems for RROI which allows us to compare and derive some specific optimality properties for certain portfolio strategies. We also established and proved that the return on investment for such policies converges to a limiting stochastic distribution and the result provides a basis upon which to compare different strategies on Hausdorff measure prior to the heat equation to locate market crises and give early warning.

2. The Rate of Return from Total Investment

The rate of return on investment is defined as the ratio of net gain in wealth over the cumulative investment. Our interest here is the rate of return from the total investment (RROI), which for the fixed policy will be denoted by the process

Hence

(2.1)

which is equivalent to the least square estimator studied by Hu, et al [4] on the continuous parametric estimation of observed fractional Ornstein-Uhlenbeck process defined as

(2.2)

is a fractional Brownian motion with Hurst parameter given as

(2.3)

Where is a measure of the wealth it takes to finance a gain. If is large; it means that the investor is accumulating gains at a faster rate than if it is small. Note that if we divide the numerator and the denominator by t in equation (2.1) we also interpret as the average net gain over the average wealth level.

Theorem 2.1

If the returns of contingent claim follow geometric Brownian motion , then the resulting distribution is Gamma distribution, that is,

Proof

Recall that then

(2.4)

From geometric Brownian motion,

(2.5)

and

(2.6)

Substitute (2.5) and (2.6) in (2.1) to get

which implies

and simplifies to

(2.7)

If we have that from equation (2.7) it shows that for that policy under which the total wealth is always invested in the risk-free asset is

But so that

If then is the risk free interest rate as expected but if the rate of return on investment (RROI) process is complicated and does not yield to a simple direct analysis.

NOTE: a random variable gamma means that X is a random variable with density function

with and the

And for any fixed proportion that satisfies all , the (RROI) process converges in distribution to random variable which has a gamma distribution, where is the constant vector given by

If is a constant vector for all such a policy is called constant proportion policy which is the optimal investment policy for any interesting objective function. A constant vector is also optimal policy for other objective criteria, such as minimizing the expected time t to reach a given level of wealth as well.

Specifically, as we have

(2.8)

where denotes convergence in distribution.

Therefore to get the expectation of we have

(2.9)

The expectation in (2.9) should not be confused with the ratio of the expected gain to the expected total investment which for any is equal to

(2.10)

Theorem 2.2

If the returns as defined in (2.1) follow Weibull randomvariates, then the resulting distribution follows asymptotic power-law.

Proof

Let be distributed according to the following probability density function

(2.11)

where are the mean and the shape parameters of the Weibull distribution (2.11).

If has the Weibull density function, then has the exponential density function with as (using the formula in [7]);

(2.12)

Thus the optimal investment strategy is (see [8]);

(2.13)

That is

(2.14)

But (where is as in (2.4) or (2.6)), so that

Hence the optimal strategy is the asymptotic power-law;

(2.15)

where is the fractal exponent given by (see [6]) with a Bessel function given as and the singularity strength

3. Optimal Growth Policy and Stochastic Dominance

Here we can see that the quantity (2.8) is maximized by the strategy that invests as much as possible in the risky asset. The mean of (2.7), is maximized at a finite value

(3.1)

which is the same policy that is optimal for maximizing logarithmic utility of wealth at a fixed terminal time and hence for maximizing exponential rate of growth. Notice that the mean of the limiting distribution of (RROI) process is maximized at the value with resulting mean for this strategy the RROI, satisfies

(3.2)

In fact, the distribution characterization of the limiting RROI allows for some-what stronger statement, in terms of stochastic orderings. Suppose that for two random variables, X,Y we say that

This is equivalent to say that for all increasing convex function and is referred to as the increasing convex ordering. We also say that (provided the expectations are finite) this is equivalent to saying that for all increasing concave function and is hence referred to as the increasing concave ordering.

We let denote the RRIO obtained from using the policy defined in (2.9) and let be the RROI. From any other constant proportion strategy where c is an arbitrary constant, the following hold

Equation (a) is in effect for investor with greater relative risk aversion while equation (b) is in effect for an investor with less relative risk aversion. Then for a proportional strategy

where is the optimal policy of (2.7), the relationship holds if and only if C satisfies

Showing that the only type of constant proportion policy (other than for which RROI is stochastically dominated by the optimal growth policy is one that is shorting the stock to the degree required by To establish equation (2.1) in section 1 above, we have that the process does not admit a simple direct analysis, there is a related markov process amenable to analysis which holds the key for the limiting behavior of specifically the 0 defined by

(3.3)

If t = 0 we have also we will first show that the limiting behavior of is equivalent to the limiting behavior of But our interest here is to show the limiting behavior of the RROI process We show that the diffusion process whose limiting behavior can be analyzed.

Suppose that for random variable we have Then for any we have

(3.4)

where is the linear Brownian motion defined by

Note that hasa positive drifts which implies that

as well as

Equation (2.1) will be completely establish if we can prove that for some random variable with

Lemma 3.1: For some fixed proportion investment policy the process follows the stochastic differential equation

(3.5)

is a temporary homogeneous diffusion process with drift function

and diffusion function

Proof

Let be the cumulative wealth investment process, also let

Since is a process of bounded variation, its Ito’s rule show that so applying Ito’s rule to gives

which upon substitution gives

(3.6)

The result is equivalent to (3.5). Here, the policy is maximized by a strategy that invests as much as possible in the risky asset and the total return from this policy turns out to have stochastic dominance property as well. This continuous stochastic process can be used for modeling random behavior that evolves over time like fluctuation in an asset price.

4. Application on Logistic Financial Fractal Dispersion Function of the Hausdorff Priorto Crash Market Signal

One of the several distinct techniques of investigating the size of subsets of zero market in is the notion of packing dimension due to Taylor and Triort, [10].

Packing dimension is defined via the packing measure as a class φ of monotone functions h which is non decreasing, right continuous satisfying h(0+) = 0 and for which there is a constant.

(4.1)

We obtain the packing dimension by two definitions. First, we defined a pre-measure

(4.2)

disjoint

Where denotes the open ball centered on X radius r Eq(4.2) is not an outer measure because it is not count ably Sub-additive. However it leads to an outer measure by defining

(4.3)

which can be thought of as a generalization of Hausdorff measure using maximal packing of E by balls, so that if h(s) then h –p( ) on is n-dimensional Hausdorff measure. Thus to measure the Borel subset of we need

(4.4)

So that if there is a unique value of for which the packing measure h- P(E) drops from infinity to zero.

This in turn means that E is less occupied than if it were dimensional. For We define the next Dim E as

(4.5)

(4.6)

which denotes the packing dimension E.

The underlying assets is paying nontrivial continuous dividends with an annualized yield A holder of the underlying asset receives a dividends yield over any time interval with length Paying dividends leads to the asset price decrease

(4.7)

Comparing equations (3.5) and (4.7) implies that After an elapse of time the value of the portfolio will change by the rate in view of the dividend received on h units held. Using Ito’s lemma for we conclude with the equation

(4.8)

Using a new function (where are some constants) and the transformation we have; which is substituted in (4.8) to get ;

(4.9)

where and By setting that (4.9) reduces to finding the solution of the Cauchy problem

(4.10)

Consider a market comprising of unit of wealth in long position (expected) and unit of the wealth in short position (actual), at time the market value is assumed to be after an elapse the value changes by the amount and the Hausdorff measure in this case is to determine which subsets of is the n-dimensional Euclidean space) are of zero heat capacity (that is where there is no market signal and hence market crash) with respect to the heat equation

(4.11)

A solution to the Cauchy problem (4.10) is given by the Green’s formula

(4.12)

Equation (4.12) gives us the idea that the transition density of the Brownian motion in is just the heat kernel (for

which satisfies the heat equation (4.11). The measure zero is equivalent to the heat capacity zero on the hyper plane. It therefore follows that the solution of (4.8) is given as;

(4.13)

The market crash (bubble) source gives rise to a which assigns a value to a portfolio to each point of through a generating kernel so that on a positive market strategy on is the portfolio growth rate if it is finite on a dense subset of

5. Conclusions

Equations (2.8) and (2.15) reveal that: (i) if assets returns follow a geometric Brownian motion, then the limiting distribution is gamma distribution, conversely (ii) if returns follow Weibull distribution, then it results to asymptotic power-law behavior. Furthermore the optimal strategy (2.15) depends on which in turn depends on (the singularity strength). as and (the optimal claims) increases without bound. As and decrease constantly with Given (3.1), we have (the square of the correlation coefficient) and

The policy is maximized by a strategy that invest as much as possible in the risky asset and the total return from this policy turns out to have stochastic dominance property as well. This continuous stochastic process can be used for modeling random behavior that evolves over time like fluctuation in an asset price and With Hausdorff measure and the heat equation via packing dimension and there is no market signal as it tends to zero hence the market is likely to crash at that point indicating shortfall on the wealth investment. The exact packing dimension is determined for subset of for which is unbounded for Where is n-dimensional Euclidean space. (size of the market)

And for analysis of subset of B of of Zero Hausdorff measure, it is appropriate to assume that

So that if but turns out to be either zero or infinity.