Data processing: financial – business practice – management – or co – Automated electrical financial or business practice or... – Finance
Reexamination Certificate
2000-02-04
2002-06-11
Trammell, James P. (Department: 2161)
Data processing: financial, business practice, management, or co
Automated electrical financial or business practice or...
Finance
C705S035000
Reexamination Certificate
active
06405179
ABSTRACT:
1. INVENTOR
George J. Rebane
2. BACKGROUND
2.1 Field of Invention
The invention relates generally to the field of methods and software products for financial analysis risk management, and more particularly to methods and software products for investment portfolio design and the selection, analysis of investments and the allocation of investment assets among investments.
2.2 Background of the Invention
2.2.1 CAPM & CML Background
In this section we present sufficient background of the Capital Asset Pricing Model (CAPM) and the Capital Market Line (CML) to establish the departure points for derivation of the present invention: Risk Direct Asset Allocation (RDAA) and Risk Resolved CAPM (RR/CAPM). A complete tutorial on modern asset allocation methods, particularly the CAPM and the related Arbitrage Pricing Theory, may be found in any one of a number of good texts on corporate finance [4] (A bibliography of reference is found at the end of this disclosure).
The practical application of any quantitative method of portfolio design based on securities' covariance requires the selection of a ‘short list’ of N risky stocks or other securities. Several studies have shown that the investor begins to gain “almost all the benefits of (portfolio) diversification” at N≦8, “virtually no risk reduction” for N>15 [14], and measurable liabilities increasing beyond N=30 [15]. The nomination of the short list may be approached as a formal problem in multi-attribute utility [1]. We proceed here with a specified candidate set of N risky securities whose singular utility to the investor is their ability to contribute to a successful portfolio design.
The motivation for going beyond the CAPM, with its ever-present companion query as to “whether variance is the proper proxy for risk” [21], is in the answer that variance is only the progenitor of risk and not its final measure. Between the two there is a road, unique to each investor, to be traveled that lets us individually answer the question “how much of each of the N securities should I—not he and not she—buy and/or hold?” This question is answered by the present invention.
2.2.1.1 Risk
In the CAPM risk is measured by the rate performance dispersion of a security as expressed by its historical rate standard deviation. A primary problem with the CAPM is that once established, this ‘sigma’ is applied uniformly to all investors independent of the amount they intend to invest or their individual aversion to the possible loss of investment assets. Thus the CAPM has a very egalitarian view of risk, and treats all investors equally, regardless of their total investment assets available for investment and net worth. The levels of risk and the concordant performance of a set of risky securities are quantified by their covariance matrix usually computed from specified historical data.
Suppose we have a candidate portfolio of N risky securities S
i
, i=[1,N]. We select a past performance epoch T
PE
and compute the symmetrical N×N covariance matrix [5] for the securities as
cov S=E{(
s
−
&mgr;
)(
s
−
&mgr;
)
T
} (1)
where
s
is the rate of return (column) N-vector of the securities and
&mgr;
is the vector of mean or expected returns computed over a past epoch T
PE
where
&mgr;
i
=ŝ
i
, i=1,N. (2)
We now allocate a portfolio fraction ƒ
i
to each of the N securities with the elements of
ƒ
summing to one. The total rate of return variance of such a portfolio is then given by
&sgr;
2
s
(
ƒ
)=
ƒ
T
cov S
ƒ
(3)
which shows the dependence of the portfolio's return variance on the allocation vector
ƒ
. In modern portfolio theory [4] it is &sgr;(
ƒ
) from (3) that gives the uniform measure of portfolio risk for all investors, and thus constrains CAPM to treat all investors equally.
The expected rate of return for each risky security over the investment horizon (T
i
) is predicted on the basis of its beta (&bgr;) computed with respect to ‘the market’ (e.g. S&P500) as follows.
β
i
=
σ
i
,
M
2
σ
M
2
=
cov
(
s
1
,
R
M
)
σ
M
2
(
4
)
The familiar beta is further represented as the slope of a straight line relationship between market variation and the security in question. The frequently omitted alpha (&agr;) parameter defines the intercept of the least squares regression line that best fits a set of security and market return rates. A method for predicting a stock's price {circumflex over (R)}
M
from a prediction of market performance {circumflex over (R)}
M
over T
i
then yields
ŝ
i
=&agr;
i
+&bgr;
i
{circumflex over (R)}
M
(5)
The classical CAPM formula for ŝ
i
[4] generates the Security Market Line
s
^
i
=
R
RF
+
β
i
×
(
historical
market
risk
premium
)
=
R
RF
+
β
i
(
R
^
M
-
R
RF
)
(
6
)
where R
RF
is the current risk free lending rate (the historical market risk premium has been calculated at 8.5%) and {circumflex over (R)}
M
is the expected return on the market over the investment horizon.
Keeping in mind the ability here to use other predictive security return models, in the remainder we will use the more straightforward (6) for predicting the performance of a security and understand the quoted ‘sigma’ (standard deviation) of such a security to derive from the regression fit of K points [5] over T
PE
.
σ
i
=
1
K
-
1
∑
k
=
1
K
{
s
k
-
(
α
i
+
β
i
R
M
,
k
)
}
2
(
7
)
Combining a security's expected rate of return and its standard deviation then yields the needed parameters for its assumed probability density function (p.d.f.) which fully characterizes the performance of the individual security with respect to the specified future performance of the market {circumflex over (R)}
M
over T
i
.
For the RR/CAPM and RDAA developments below we additionally acknowledge an uncertain future market and express this by its variance &sgr;
M
2
to reflect the dispersion about the predicted mean return {circumflex over (R)}
M
. This additional uncertainty will be reflected in a given security's ‘sigma’ to yield its total standard deviation as
&sgr;
T,i
={square root over (&sgr;
i
2
+L +&bgr;
i
2
+L &sgr;
M
2
+L )} (8)
2.2.1.2 The Feasible and Efficient Sets
From corporate finance texts [4] we learn that a set of points termed the feasible set can be represented in 2-space where expected portfolio return {circumflex over (R)}
P
is plotted (
FIG. 1
) against the standard deviation s
s
of the portfolio given in (3). The expected return of the ‘risky’ portfolio allocated according to
ƒ
is simply
R
^
P
=
∑
i
=
1
N
f
t
s
^
i
(
9
)
The efficient set is defined as the upper boundary of the feasible set drawn upward from the ‘minimum variance point’ (MVP) since it is not reasonable to choose portfolios with the lesser expected gains for the same ‘risk’ as measured by the portfolio's &sgr;
s
. Therefore, according to the CAPM the optimal portfolios are all represented by the infinite set of optimal allocation vectors {
ƒ
*} that define this upper boundary. The CAPM proceeds to resolve the problem further by introducing the risk free lending option which gives rise to the Capital Market Line.
2.2.1.3 The Capital Market Line
As shown in
FIG. 1
, when we introduce the risk free lending option at rate R
RF
, we add the (N+1)th instrument and increase the dimension of the investor's decision space to N. The CAPM argues that the optimum portfolio now lies along a line—the Capital Market Line (CML)—that originates from (0, R
RF
) and is tangent to the efficient set at some point E for which a unique
ƒ
* can be discovered. Selecting a point between (0, R
RF
) and E defines what fraction should be invested risk free with the re
Ganz Law P.C.
Ganz, Esq. Bradley M.
Hayes John W.
Saddle Peak Systems
Trammell James P.
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