One embodiment of the present invention makes use of matched filtering concepts for the purpose of picking stocks. These matched filtering concepts were previously used in the field of electrical engineering but were not previously used for the purpose of picking the right mix of stocks in a portfolio.
At least one embodiment of the present invention provides a method for picking stocks, which maximizes gain and simultaneously minimizes risk. This is achieved by maximizing the ratio of gain to risk. The right mix of stocks are selected so as to maximize the gain G while simultaneously minimizing the overall risk σP2. In at least one embodiment the following ratio is maximized:
Equation (25) represents the gain over the portfolio risk. However, in one embodiment of the present invention, instead of maximizing (25), the following ratio is maximized:
subject to the normalization constraint αTe=1.
Clearly, equation (26) represents a more aggressive strategy in term of maximizing gain, but more interestingly, the ratio in equation (26) is the same as the familiar SNR (Signal to Noise Ratio) maximization strategy used in classical receiver design in Communication theory, in Electrical Engineering, where a signal corrupted by interference and noise is presented to a receiver to minimize the effect of output interference plus noise while maximizing the output signal component at the decision instant as referred to in “Signals Analysis”, A. Papoulis, McGraw-Hill Companies, New York, USA, 1977, and also “Digital Communications”, Fourth edition, J. Proakis, McGraw-Hill Companies, New York, USA, 2001. The solution to the SNR maximization problem leads to well known matched filter (MF) solution as referred to in “Signals Analysis”, A. Papoulis.
From (26), with
we get
since by Schwarz's inequality
|αTμ|2=|(R1/2α)T(R−1/2μ)|2≦(αTRα)(μTR−1μ), (28)
α
opt=kR−1μ. (29)
In general the entries of the optimum portfolio mix vector shown in equation (29) can be both positive or negative. Negative entries indicate that the corresponding stock is to be shorted. If short sale strategies are prohibited, for example, as in the case of most of mutual funds, then one needs to maintain α>0, and in that case one can perform a constrained optimization strategy of maximizing equation (26) subject to the non-negativity constraint of α. This leads to a suboptimum solution with all positive or non-negative entries for the vector α that requires no short selling. This strategy can be applied to any given set of stocks and securities that the investor has a-priori selected. In that case the capital will be partitioned according to the entries of the suboptimum vector so obtained and invested in the corresponding stocks.
An alternate strategy is to keep the pool of the desired stocks and securities to be selected as potentially open, and select them from a larger pool of stocks and securities in such a way that the inverse of their covariance matrix R−1 turns out to be a positive matrix. If this condition turns out to be too restrictive or severe especially for a portfolio containing a large number of stocks, one can also settle for the less restrictive new condition
R−1μ>0 (31)
by
(i) the judicious selection of stocks that go into the portfolio and by
(ii) the choice of μ vector in (29) that represent the expected average return.
can be easily maintained with the constant k in equation (29) chosen to be
that maximizes the gain and minimizes volatility. In this case,
and
Equation (33) can be given the whitening followed by matched filtering interpretation as well, as will be shown with reference to
The filter output above is uncorrelated and has unit variance since its covariance matrix equals
and to maximally combine these outputs, the coefficients {bi} in
where
μ
x
=E{x(n)}=R−1/2μ. (39)
b=kμx=kR−1/2μ. (40)
G=bTμx=kμTR−1/2μx=kμTR−1/2R−1/2μ=αTμ (41)
or
α=kR−1μ(42)
as in equation (29).
Interestingly, Equations (33)-(35) can be used to generate a gain-risk plot by varying over all sustainable μs. Following equation (33), an arbitrary μ is said to be sustainable if R−1μ is a positive vector. Using a sustainable μ, one can compute the optimum gain and σP using equations (34)-(35).
Notice that although scaling μ does not affect the variance in equation (35), it does affect the gain in equation (34). Hence to avoid duplication by simple scaling, the first entry μ1 in a sustainable μ may be normalized to unity. As an example,
From
Interestingly, other variations of the stock picking strategy shown by equations (29) and (33) also lead to the same result.
One may use a less aggressive strategy in terms of returns while maintaining low volatility. Then we may maximize:
instead of equation (26). This leads to
On the other hand, one can use a much more aggressive strategy such as maximizing
subject to equation (1).
Notice that equation (46) is weighted more towards higher gains.
In summary, for a variety of optimization strategies, the new portfolio mixing vector in equation (33) represents the optimum strategy for building a portfolio. If short selling stocks is allowed, the above strategy can be applied to any set of stocks; if short selling is not permitted, then the selection of stocks and their number that goes into the actual portfolio becomes important and it must be accomplished so as to maintain the desired portfolio mixing vector to be positive, while maintaining a high yield (gain) with minimum fluctuations (risk).
Strategy-2 (in Accordance with Another Embodiment of the Present Invention):
In another embodiment of the present invention, a variation of the maximization of the gain to the risk strategy, to be described below, leads to a somewhat different result in terms of the desired portfolio vector. In the previous method, the risk is defined as the variance of the portfolio under consideration as in equation (9). In a method in accordance with an alternative embodiment of the present invention, this definition is extended as follows:
The actual stocks and securities that go into a portfolio as the primary stocks are identified, and equation (7) represents their returns. In that case the risk defined as in equation (9) represents the variance of the exact combination of the returns of these primary stocks that make up the portfolio. The method next includes identifying through market research and other means another set of stocks as secondary or sympathetic stocks that are correlated to these primary stocks in equation (7). Let R0 represent the covariance matrix of the returns of these secondary stocks that is also defined similar to equation (10). Since the secondary stocks have some influence on the behavior of the primary stocks, the argument here is that their covariance matrix R0 must also contribute to the overall risk of the portfolio. Thus in this approach, a scalar function of R0 is added to the primary risk factor in equation (9). In our case, the trace of R0 (sum of the diagonal entries of the covariance matrix R0) of the secondary returns is used as the scalar function. This gives the modified risk of the portfolio to be
σP2=αTRα+σo2 (48)
where
νo2=tr(R0) (49)
represents the trace of R0. In this case the optimization problem in equation (26) gets modified as
Let the vector α0 represent the optimum nonnegative vector (constrained optimization using the non-negativity condition) that maximizes the ratio in equation (50) and whose elements add up to unity. In this approach, the capital will be partitioned according to the entries of this new vector α0 and invested in the primary stocks.
The nonnegative vector α0 above represents a suboptimum solution, and as in equations (26)-(33) there exists an unconstrained (without the nonnegative condition) optimum vector bopt that maximizes equation (50), and once again the capital can be partitioned according to the entries of the vector bopt and invested in the primary stocks. In this case the strategy can include short sales as well.
where I represents the m×m identity matrix (with ones along the main diagonal and zeros elsewhere), and
∥α∥2=αTα>0 (52)
represents the norm (square of the length) of the vector α. In (51), proceeding as in (27) through (29), we obtain the following solution
(where c is a normalization constant) that suggest the iteration
that can be used to solve for the above optimum vector bopt. The above iteration is seen to converge in a variety of situations.
is determined. At step 310 weighting factors, such as the new vector αk+1 are determined based on the above modified inverse covariance matrix times the means for primary returns vectors divided by the sum of the new vector entries so obtained, such as by equations (54) or (55). At step 314, the difference of the new vector αk+1 and old vector αk is defined as the error vector. At step 316 the error norm is computed as in equation (52) for the error vector, and it is compared with a predetermined threshold value, such as for example 0.001 etc. If the error norm is less than the preset threshold value, the new vector obtained at step 310 is taken as the desired weighing factors at step 318. Otherwise, the old vector is replaced with the contents of the new vector at step 320 and it is fed back to step 310, where the entire cycle is repeated till the desired accuracy is achieved.
Optimum Nonnegative Solution: Interestingly, it is possible to guarantee the solution given by equations (54)-(55) to be nonnegative by treating σo2 in equations (54)-(55) as a free parameter. Recall that σo2 represents a measure of the effect of the correlation of the secondary stocks on the primary stocks, and for a given value of σo2, the optimum vector in (54) can have both positive and negative entries. In such situations, by increasing the value of σo2 the optimum vector can be made nonnegative there by avoiding short sales. In fact, for any given covariance matrix R and nonnegative vector μ, there exists a minimum positive value for the constant σo2 in the equation
α=(R+σ02I)−1μ (56)
for which the vector α becomes nonnegative. The proof follows by expanding equation (56) and noticing that as the constant σo2 becomes large, the perturbation terms to the first term μ in the expansion become of decreasing importance, and hence the vector α becomes nonnegative. Using this approach in equations (54)-(55), it follows that there exists a minimum threshold value for the sympathetic stocks' variance term σo2, above which the optimum vector bopt remains non-negative. Using any value above this threshold value for σo2 in equations (54)-(55) avoids short sales for the optimum portfolio mixing strategy. To determine this threshold value, one may proceed using the iterative steps in
As an example, consider a portfolio containing three stocks whose 3×3 covariance matrix is give by
and let μ=(0.20 0.50 0.40)T represent their the expected return values vector. In that case the solution in equation (33) that maximizes the overall gain to risk ratio is given by R−1μ=(−0.237 0.7531 0.310)T and it has one negative entry and hence it involves short sales. However using σo2=0.034995, the solution in (56) after normalization turns out to be α=(0.0000025 0.683533 0.293664)T. Since the new solution has all nonnegative entries, it avoids short sales. Using any other value above this threshold for σo2 results in all positive values for the solution and it avoids short sales in the optimum portfolio.
In summary, methods in accordance with embodiments of the present invention for determining the optimization strategies for building a new portfolio mixing vector are disclosed. In at least most if not all of these cases, the ratio of the overall portfolio gain function to the portfolio risk is maximized, where the definition of the portfolio risk is extended in one case to include the influence of stocks that are sympathetic to the primary stocks of interest. If short selling stocks is allowed, the above strategies can be applied to any set of stocks; if short selling is not permitted, then the selection of stocks and their number that goes into the actual portfolio becomes important and it must be accomplished so as to maintain the desired portfolio mixing vector to be positive, while maintaining a high yield (gain) with minimum fluctuations (risk). This can also be accomplished by extending the definition of risk to include a free variable term that denotes the effect of a secondary set of stocks, and by increasing this term the desired portfolio mixing vector can be made positive through an iterative procedure.
Although the invention has been described by reference to particular illustrative embodiments thereof, many changes and modifications of the invention may become apparent to those skilled in the art without departing from the spirit and scope of the invention. It is therefore intended to include within this patent all such changes and modifications as may reasonably and properly be included within the scope of the present invention's contribution to the art.