Data processing: financial – business practice – management – or co – Automated electrical financial or business practice or... – Finance
Reexamination Certificate
1998-09-09
2001-08-28
Hafiz, Tariq R. (Department: 2163)
Data processing: financial, business practice, management, or co
Automated electrical financial or business practice or...
Finance
Reexamination Certificate
active
06282520
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates to computer systems and methods for allocation of investments and distribution of investment returns based on a risk return analysis of modern portfolio theory.
BACKGROUND OF THE INVENTION
Individuals today may allocate their investment resources among a variety of asset classes; for example, equity, fixed income, international, emerging markets, etc. Within each asset class are a great number of individual assets to analyze and select. Investors may diversify and obtain professional management of their investment resources by investing in a professionally managed mutual fund. However, there are literally thousands of mutual funds to choose from offering a bewildering array of different investment philosophies. Many individuals do not have the time, inclination or expertise to manage their investments optimally or even choose the best mutual fund for their investment goals. Optimal investment of resources among a variety of assets is a complicated statistical and computationally intensive process beyond the reach of most investors.
Ideally an investor should allocate his or her investment to achieve a maximum expected rate of investment return consistent with the investor's tolerance for risk. A portfolio that is suitable for a particular investor can be constructed by combining assets with different expected rates of return and different levels of risk.
The expected rate of return for a combined portfolio of assets with different expected rates of return is the sum of the expected rates of return of each individual asset in the portfolio weighted by its proportion to the total portfolio:
r
_
T
=
∑
i
=
1
N
r
_
i
w
i
where:
{overscore (r)}
T
is the expected rate of return of the portfolio of combined assets;
{overscore (r)}
i
is the expected rate of return of the i
th
asset;
w
i
is the proportion of the value of the i
th
asset to the total portfolio value,
(
∑
i
=
1
N
w
i
=
1
)
and
N is the total number of assets in the portfolio.
For example, consider a stock, A, with an expected rate of return {overscore (r)}
1
=0.1 and a bond, B, with an expected rate of return {overscore (r)}=0.05. The expected rate of return of a portfolio consisting of 40% stock A and 60% bond B will be:
{overscore (r)}
T
=0.1×0.4+0.05×0.6=0.07
Risk may be characterized in different ways. Probably the most common measure of risk is volatility, measured by standard deviation. Standard deviation is the square root of the variance of the returns of an asset or portfolio of assets. The variance is a measure of the extent to which the return on an asset or portfolio of assets deviates from an expected return. An asset with a higher standard deviation will be considered more risky than an asset with a lower standard deviation. Other measures of risk include semi-variance about a target return, which is a measure of the extent to which the return of an asset or portfolio of assets will fall below a target level of return. Another measure of risk is “value at risk,” which is a measure of how much an asset or portfolio of assets can lose in value with a given probability.
The risk level of a combined portfolio of assets will depend on the risk measure used. For example, consider the risk associated with a combined portfolio using variance, or equivalently, standard deviation as the measure of risk. The standard deviation of the returns of a risk-free asset is zero whereas the standard deviation of the returns of a risky asset is greater than zero. Standard deviation is the square root of the variance. The variance is:
E
{(
r−{overscore (r)}
)
2
}
where:
r is a random variable representing the rate of return on an asset or portfolio;
{overscore (r)} is the expected value of r; and
E denotes the expectation operator.
Combining a plurality of risky and risk-free assets in a portfolio will result in a portfolio with a standard deviation that is equal to or less than the weighted sum of the standard deviations of the component assets. For example, when two risky assets with variances &sgr;
1
2
and &sgr;
2
2
, respectively, are combined into a portfolio with portfolio weights w
1
, and w
2
, respectively, the portfolio variance, &sgr;
T
2
, is given by:
σ
T
2
=
w
1
2
σ
1
2
+
w
2
2
σ
2
2
+
2
w
1
w
2
cov
(
r
1
,
r
2
)
where cov (r
1
, r
2
) =the covariance of the two assets.
The covariance is a measure of how much the returns on the two assets move in tandem, and is defined as follows:
cov (
r
1
,R
2
)=&sgr;
12
=E
[(
r
1
−{overscore (r)}
1
)(
r
2
−{overscore (r)}
2
)]
A positive covariance means that the asset returns move together; if one has a positive deviation from its mean, they both do. A negative covariance means that asset returns move in opposite directions; if one has a positive deviation from its mean, the other has a negative deviation from its mean. The correlation coefficient, &rgr;
12
, is the covariance of the two assets divided by the product of their standard deviations (i.e., &rgr;
12
=&sgr;
12
/(&sgr;
1
&sgr;
2
)). The correlation coefficient, &rgr;
12
may range from −1 (indicating perfect negative correlation) and +1 (indicating perfect positive correlation). Thus, the magnitude of the correlation coefficient, |&rgr;
12
|, is always less than or equal to 1.
The equation for the variance of the portfolio, &sgr;
T
2
, shows that a positive covariance increases portfolio variance beyond &Sgr; w
i
2
&sgr;
i
2
. A negative covariance decreases portfolio variance. By investing in two assets that are negatively correlated, if one asset has a return greater than its expected return, that positive deviation should be offset by the extent to which the return of the other asset falls below its expected return.
The equation for &sgr;
T
further shows that the standard deviation of the portfolio is always equal to (in the case &rgr;
12
=1) or less than (in the case |&rgr;
12
|<1) the weighted sum of the standard deviations of the component assets. That is:
&sgr;
T
≦w
1
&sgr;
1
+w
2
&sgr;
2
Since the return of the combined portfolio is the weighted average of the returns of the component assets, portfolios of less-than-perfectly correlated assets always offer better risk return opportunities than the individual component securities. See, e.g., “Investments, 3rd Edition,” p. 197, Bodie, Kane & Marcus, Irwin, McGraw Hill (1996). These results are true generally for a combined portfolio comprising numerous risky assets, for which the variance is given by:
σ
T
2
=
∑
i
=
1
N
w
i
2
σ
i
2
+
∑
i
=
1
i
≠
j
N
∑
j
=
1
N
w
i
w
j
cov
(
r
i
,
r
j
)
Thus, since the magnitude of the correlation coefficient, |&rgr;
12
|, for any two different assets, (a
i
, a
j
), is less than or equal to 1, &sgr;
T
2
is always less than or equal to
(
∑
i
=
1
N
w
i
σ
i
)
2
.
Thus, &sgr;
T
is always less than or equal to
∑
i
=
1
N
w
i
σ
i
Given a set of imperfectly correlated risky assets, an innumerable set of combined portfolios can be constructed, each comprising different proportions of the component assets. An optimum portfolio is one in which the proportion of each asset comprising the portfolio results in the highest expected return for the combined portfolio for a given level of risk. Alternatively, an optimum portfolio is one in which the proportion of each asset comprising the portfolio minimizes the risk of the combined portfolio for any targeted expected return. See, e.g., “Investments, 3rd Edition,” Bodie, Kane & Marcus, Irwin, McGraw Hill (1996).
This is illustrated in
FIG. 1
, using variance, or equivalently, standard deviation, as the risk measure.
FIG. 1
is a graph of the minimum variance frontier o
Fulbright & Jaworski LLP
Hafiz Tariq R.
Meinecke-Diaz Susanna
Metropolitan Life Insurance Company
LandOfFree
Computer system and methods for allocation of the returns of... does not yet have a rating. At this time, there are no reviews or comments for this patent.
If you have personal experience with Computer system and methods for allocation of the returns of..., we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Computer system and methods for allocation of the returns of... will most certainly appreciate the feedback.
Profile ID: LFUS-PAI-O-2541508