Theory of Portfolio and Information Value

Based on Incremental Entropy

**Abstract **Replacing arithmetic mean return and standard deviation
adopted by M. H. Markowitz with geometric mean return as criterion of assessing a
portfolio, we get *incremental entropy*: one of generalized entropies. It indicates
the increasing speed of capital and is a more objective and testable measure. Most
different from Markowitz's theory is that new theory emphasizes that for given probability
of returns there is an objectively optimal portfolio ,which can be tested by playing coins
or by computer simulation. This paper provides some simple and applicable formulas for
optimizing investment ratios and some samples showing the optimization of a portfolio with
multi-securities by computer. The paper also presents a information-value measure based on
the new portfolio theory, analyses difference and similarity between this measure and K.J.
Arrow's information value measure, and discusses how to optimize forecasts with new
information-value measure as criterion ,

**Keywords**: portfolio, criterion of optimization, incremental entropy, information
value evaluation and optimization of forecasts

1 Introduction

Several years ago, the author presented a generalized information theory, and proved that the generalized information measure also had its coding meaning(Lu,1992,1993,1994). Later, he found that new generalized entropy, i.e. incremental entropy, could be used to optimize portfolios. The new portfolio theory carries on some aspects of Markowitz's theory(1959,1991), but has obvious differences. It emphasizes there is an objective criterion that is incremental speed of capital. On this criterion, for given probability forecast of returns, we can obtain the optimal ratios of investments in different securities or assets. Combining the new portfolio theory and the general information theory, we can approach a meaning-explicit measure of information value, which designates the increment of capital-increasing speed after information is provided while one always makes decision in optimal way. The author has applied the portfolio theory to investments in markets of stocks and futures. The practices show that the theory has obvious superiority. Now, the new theories are presented here for criticisms and discussions.

2 Portfolio Optimization

——Talking about from Gambling with Playing Coins

First of all, we talk about the question of portfolios from a simple sample.

Sample 1. Assume there is an investment or bet whose return is uncertain and determined by playing a coin. If A-side appears you will win 200% of your wager and if B-side appears you will lose 100%. You have only 100 dollars and have no way to borrow if you lose . The question is, for many times of the gambling, what ratio of your capital the wager occupies is best so that your capital increases fastest, or say, you become millionaire fastest?

Staking all of your money every time is obvious infeasible because once B-side appears you will lost all of money and the opportunity of your becoming rich forever. Staking 10% of your money is not too bad since you might not lose up forever. After two turns in average your return (yield plus 1) will be (1-1*0.1)(1+2*0.1)=1.08 times. But in this way, whether is the capital-increasing speed too slow? Whether can we find the best investment ratio?

In more general cases, the number of securities or items to be invested in is greater than one, and their future returns are uncertain. For given joint probability distribution of their future returns, how do we determine optimal investment ratios in different securities?

The portfolio theory initiated by Markowitz (1959) has had great achievements. For this reason, M. H. Markowitz, W.F.Shape and M.Miller won 1990's Nobel Prize. The theory tell us that the larger the expected return, the better the investment; the less the standard deviation of return, which means risk, the better; we can reduce the standard deviation of the return or risk by combining anti-covariant securities together. As for how to assess the utility of a portfolio with given expected return and standard deviation, there is no objective criterion, or say, the criterion changes from persons to persons.

The efficient portfolio provided by Markowitz's theory is one that has the smallest risk for a given level of expected return or the largest expected return for a given level of risk. However, it is not a portfolio we need for the fastest increment of capital. So, we need new mathematical method.

**3 A Mathematical Model for Portfolios and Derivation of Incremental Entropy**

Let the prices of *N *securities in a portfolio form a *N*-dimension vector
and the price of the *k-*th security has *n _{k }*possible
value,

(3.1)

where *q _{k }*,

(3.2)

where *d* is handling charges for exchange of each dollar, *q _{k}*' is the former ratio of the investment in the

Assume that a price vector *r _{i}*
happens

(3.3)

the average return for each period is

(3.4)

The author understood until recent time that the above mathematical model was first
proposed by Henry A.L.atane and Donald L. Tuttle(1967) and was called *wealth-maximizing
model. *However, the following research results in this paper are entirely different
from those proposed. One reason is that expectation and standard deviation no longer play
important roles.

Taking the logarithmic value and letting *N* ?8,
we have

(3.5)

We call *H* *incremental entropy*, which is one of generalized entropies(see
Lu, 1993,1994), and has the same metric as information has. It is suggested that we use 2
as the base of the logarithm; then *H *means the times of doubling capital.

4. Comparing Optimization of Portfolio with that of communication coding

In comparison of the generalized entropies(Lu, 1993), the incremental entropy lacks a negative symbol. If let , then means how many dollars as capital now is needed for future one dollar after the investment. Hence we have a incremental entropy with negative symbol:

(4.1)

In communication theory, the formula of average coding length is

(4.2)

where *c _{i}* is coding length for
letter

(4.3)

Comparing with (3.5) and (4.1), we can see that *r _{i}*
matches

If a communication channel transmits a code per unit of time and those codes are
uncorrelated each other, then *H* means the speed of information transmission.
Similarly, in terms of portfolios,* H* means the incremental speed of capital. Let *r _{g}=*2

, i.e. (4.4)

Comparing with physical formula: time=distance/speed, we can see it clearer that
*H *has the meaning of speed.

5. Optimization of Investment Ratios

In the later, let *P*(*x _{i}*)
be denoted by

(5.1)

where means excess return. The *q**
can be obtained by solving the above a group of *N* equations. Since *q** is the
function of (*P _{i}* ) and (

Considering the question of optimizing investment ratio in the front of the paper. In
that case *N*=1,* i*=1,2, Formula (5.1) becomes

(5.2)

where *P _{1}* ,

(5.3)

Fig. 4.1 Showing relationship between q* and r1,r2,r0

Note the numerator is just the expectation of the excess return. Put ,=1
into the above formula, we have *q**=0.25. It is said that the optimal investment
ratio is 25%. The return will be (1-1*0.25)(1+2*0.25) =1.125 times in average after every two turns. If and are
both greater or less then 0, the above formula is not suitable. In these cases *q**
should be equal to 1 or -1 if overdraft is not allowable.

From (3.3) we can get a interesting conclusion. Let , *r _{0}* =1;

(5.4)

which means that when gain and loss are equally possible, if the loss might be up to 100% then do not stake more than 50% of your fund no matter how high the possible gain may be. This conclusion is very meaningful to investments in futures, options, and the like. Many new comers of future markets lose all of their money very fast because the investment ratios are not well controlled and generally too large.

From (5.3), *q** >1 and* q** <0 are possible. The *q**>1 means
that you had better skate 100% or *q**=100%, if
lending or overdraft is allowable, of your capital. The *q**<0 means that you had
better skate nothing or |*q**|=100% on oversale
if it is allowable.

For a security with possibility of multi-returns, we can use computer to obtain optimal ratios on (3.5), and can also adopt approximate formulas. One is

(omitted)

**6. Affection of Covariance on Geometric Mean Return (GMR) from the Point of New
Theory **

The affection of covariance of returns of securities on the investment value of a portfolio has be well discussed by Markowitz(1959,1991). Now we explain that by the new theory we can reach similar conclusions and how we reduce risk and improve incremental speed of capital by optimizing investment ratios.

Sample 2. Let the excess return of a future investment be represented by two sides of a
couple of coins. You win 300% if two coins show *A* sides, you lose 200% if two coins
show *B *sides, and you win 100% if two coins show different sides. Assume there are
two futures whose returns are determined as above; but probably one or two coins are
commonly used. Calculate the geometric mean returns as you invest all capital(50% for
each) and invest in optimal ratios.

Table 2 shows the results, in which are random variables taking evenly possible value -0.5 or 2.0; and are returns from two futures respectively, *r*_{0}=1 is assumed, the optimal ratio *q** is for each
future..

Table 2. Affection of covariance on geometric mean return and optimized results(data in
brackets ar*e* approximately optimal results from (5.8))

(omitted)

It is obvious that the larger the covariance coefficient, the worse the effect of portfolio. The covariance coefficient 1 means that two securities become one so that loss is great. The less the covariance coefficient, the better the effect. The covariance coefficient -1 results in the best effect so that the geometric mean return is equal to the arithmetic mean return. The data in the right two columns of Table 2 show that we can improve geometric mean return by properly reducing investment ratios especially in case of that great risk exists.

**7 In Comparison with Markowitz's Theory**

The new theory supports Markowitz 's conclusions about reducing investment risk by effective portfolio. Different is that

1)The new theory refers to geometric mean return as objective criterion for optimizing portfolio and provides optimal investment ratios for the fastest increment of capital in a long term;

2)The new theory employs extents and possibilities of gain and loss or utility function
*u=u(q) *instead of expectation and standard deviation of return as description of
investment value of a security or a portfolio.

Sample 3. Current prices of two securities *A* and *B* are both 1 (dollar);
possible prices of *A* in future is 0 and 2 with probability 1/4 and 3/4; the price
of security *B* in future has the same expectation (1.5) and standard deviation
(0.886) but mirrored probability distribution. The analyses of investment values of the
two securities is shown in Table 3, in which bank-interest is neglected.

Table 3

expectation | Standard variance | Average compound interest when Staking 100% | Optimizing ratio | Average compound interest after optimization | |

Security I | 0.5 | 0.886 | -100% | 50 | 15% |

Security II | 0.5 | 0.886 | 32% | >=100 | >=32% |

The optimal ratio* q**>=100% means that if overdraft is allowable, you had
better overdraw to buy and the more you overdraw, the better. On Markowitz 's theory, *A*
and *B *have the same investment value; but on the new theory, *B* is much
better than* A*. For investments with possibility of loss in large proportion, such
as option, future, loan, and insurance sale, the above defect in Markowitz 's theory is
clearer.

8 Information Value Formula Based on Incremental Entropy

In the generalized information theory( Lu, 1993), probability is distinguished into subjective probability, which is subjectively forecasted by somebody, objective probability, in which a event actually happens, and logical probability. Logical probability of a prediction or a proposition is also called confidence level, it takes a value from real interval [0,1].

Assume *X* is a random variable taking a value from a event set and *Y* is a random variable taking a value
from a proposition set ; *Y* transmits
information about *X*.. Events that make a proposition *y _{j }*be true form a subset

(8.1)

(8.2)

where is objective probability, *Q*(*x _{i}*) is prior subjective probability and is posterior subjective probability deduced from

(8.3)

Given a vector probability distribution ** q=**(

(8.4)

It can be seen that and have similar mathematical construction.
Information value of *y _{j}* as

(8.5)

Both information and information value are relative and especially depend on how an information receiver understands a prediction. The relativity of information value is also due to the return matrix () is different to different investors with different limits of investing tools. The asymmetry of information and information value comes from the same reasons.

**9 In Comparison with Arrow 's Formula of Information Value**

Arrow defined information value or information utility as increment of utility after information is provided(1984). The utility function is

(9.1)

where *r _{i}* denotes the

Under the limit , *U *reaches its
maximum as (*a _{i}*)=(

(9.2)

After information is provided, the investor knows which return will exactly happen and hence invests all capital in this return. Then

(9.3)

The information value is just Shannon's entropy, i.e.

(9.4)

The Arthur's definition of information value carries on Arrow's thought: 1)for given forecast of returns, there is an objective optimal investment ratio; 2)information value is equal to the increment of utility after information is provided. However, the investment model employed in this paper is very different. From the author's view-point, the investment model employed by Arrow is very strange. We can invest in a security or a item; but how can we invest in some return of a security or a item? Equation (9.1) demands each is positive, which means investments with possible loss are not allowable, since the logarithm of a negative number is meaningless. Is this investment or gambling common as we see in daily life? Arrow 's model might be suitable to the game of guessing stock index for award, which is generally organized by newspaper seller. But it is still hard to understand that <1 results in negative utility. It is these reasons that the application of Arrow 's formula of information value is difficult.

**10 Applying New Measure of Information Value to Assessments of Forecasts**

Formula (8.4) can be used to optimize forecasts. Let . Given (*P _{i}* ), we seek
probability forecast

Sample 4 A stock index *X* takes a value from the set *A*=[100,110,120,130,...].
A set of predictions about *X* is *B*={ *y _{j}*
="

(10.1)

Current stock index is *x _{0}* . The
prior probability forecast is

Solution. On generalized Bayesian formula(Lu,1993), we have

(10.2)

in which *Q*( *A _{j}* ) means
logical probability of predicate

*Q*( *A _{j}* )= (10.3)

From the incremental entropy

(10.4)

we obtain optimal ratio** ***q**; from the incremental entropy

(10.5)

we obtain optimal ratio *q***. The information value of *y _{j}* is hence

(10.6)

For each *y _{j}* in

(10.7)

reach the maximum is optimal prediction *y**.

**11 Summary**

This paper has provided applicable the theory of portfolio and information value based on incremental entropy, in which some fundamental thoughts of Markowitz and Arrow are carried on. From the author's view-point, as for investment model, Markowitz is correct yet Arrow is wrong; but as for if there is objectively optimal investment ratio for given probability forecast of returns, Arrow is correct yet Markowitz is wrong . This paper is expected to have advantages of the two theories and to unify the theory of portfolio and that of information value that seem independent each other on higher level.

The author has finished a software for simulating portfolio and testing optimal ratios by repeated investments. The software can calculate optimal investment ratios from probability forecast of returns of several securities and matrix of covariance coefficients of the returns. Running the software shows that the more times the investments are repeated, the clearer the ascendancy of computer is; the greater the risk is or the more complicated the covariance is, the clearer the ascendancy of computer is. Further research of the applications is proceeding. Welcome to common explorations.

References

Arrow,K.J.1984: The Economics of Information, Basil Blackwel Limited.

Latane, H. A. and Tuttle D. L., 1967: Criterion for portfolio building, The Journal of Finance, 22(3),359- 373.

LU, Chen-Guang, 1992: Applying a generalized information theory to assessment and optimization of forecasts, Forecasts, 12(3), 54-57.

LU,Chen-Guang, 1993: A Generalized Information Theory, China Science and Technology University Press.

LU,Chen-Guang, 1994: Coding meanings of the generalized entropies and generalized mutual information, Journal of China Institute of Communications, 15(6), 37-44.

Markowitz, M.H.1959: Portfolio Selection, Efficient Diversification of Investments, Yale University Press

Markowitz, M.H 1991: Foundations of Portfolio Theory, The Journal of Finance, 46(2), 469-477

Shannon,C.E., 1948: A mathematical theory of communication, Bell System Technical Journal, 27, 379-429 and 623-656.