In particular,
we offer methods by which the continuum (in risk and expected return) of portfolios
(known as the Efficient Frontier) which consists of the portfolios with the minimum
risk for a given value of the expected return which can be constructed from a given
collection of assets. Moreover, from this collection of Portfolios a unique optimal
portfolio can be selected with respect to a given value of the expected return, risk
(see SolveFrontier) or an investors risk - return profile given in terms
of a utility function.
Overview of Functionality Offered
The Efficient Frontier is the collection of portfolios constructed from
the given set of assets which have the lowest possible risk for a given level of
the expected return. Note that the weights of the assets making up the portfolio may
themselves be subject to constraints (for example, no one asset can have a
weighting more than 20 percent or less the 5 percent).
Once the Efficient Frontier is known, we are able to select from this continuum
a unique portfolio which represent the optimal portfolio with respect to an investors
risk - return profile. The return profile of the investor may be given in three
distinct ways and the correspond optimal portfolio can them be constructed. The
by one of the following means:
Maximum Risk - the investor gives the (maximum) risk which they are prepared
to accept and then the corresponding portfolio with the highest expected
return for that given level of risk is constructed.
Expected Return - the investor gives the expected return which they desire and
the portfolio with the least risk for that given level of expected return is
constructed.
Construction of the Efficient Frontier
The Efficient Frontier is constructed by the following steps:
Evaluated the Efficient Frontier at a finite number of points. That
is find the portfolio which exhibit the lowest risk for a given expected return.
Interpolate about these points using cubic spline (or some other method)
in order to construct the Efficient Frontier.
The points on the Efficient Frontier are portfolios constructed from the
set of assets considered which exhibit the lowest risk for a given expected return.
These portfolio are characterized by the following three characteristics:
Expected Return - The expected return of the portfolio which is estimated
from the historical returns of the assets within the portfolio.
Total Risk - The total risk of the portfolio which is estimated from the
historical returns of the assets within the portfolio.
Asset Weights - the weights of the collection of assets from which the portfolio
can be constructed.
It is important to point out that the Efficient Frontier in monotonically increasing
function in risk and expected return. This means that if we are given a value of
the expected return then there will correspond a unique portfolio on the Efficient
Frontier with a given total risk. Conversely, if we are given the total risk of the
portfolio then there will exist a unique portfolio on the Efficient Frontier with a
corresponding value its expected return.
The three means of Selecting the Optimal Portfolio
Of the above three means of describing the investors risk - reward profile which
are all sufficient for determining a unique optimal portfolio, the first method involving
the utility function allows to most detailed information concerning the investors risk - reward
profile to be taken into account. However, in defining an investors utility function you will
need to know detailing information concerning the investors preferences. For this reason you
may wish to determine the optimal portfolio from either the maximum risk or the required expect
return of the investor. In each of the three approaches you will need to discover as least
aspects of the investors risk - reward profile, with regard to this matter we refer the
reader to the PDF documentation for more details and practical suggestions as to how this
can be achieved.
The utility function may determine one, many or zero optimal portfolios. Please see the
PDF documentation for further explanation.
The Efficient Frontier is a monotonically increasing function for the expected return
against the (total) risk of the portfolio on the Efficient Frontier. Therefore, if the
optimal portfolio is selected by the maximum risk or expected return then a unique portfolio
on the Efficient Frontier will be selected.
Effects of using Absolute of Relative Historical Values
Within the application of portfolio theory the following two quantities
will need to use the corresponding units of measurement throughout the computation:
Historical Values: This is the source data which is given in absolute
or relative terms.
Expected Returns: The expected return of the investment over the period
considered which should be given and will be returned in the units used (i.e.
absolute or relative) by the historical values.
The units used within these two quantities will effect the following objects:
Utility Function: The values of the expected returns provided within the
definition of the utility function should be in accordance with the units
used to describe the historical values.
Efficient Frontier: The values of the expected return which are either
evaluated or given will be or will need to be in accordance with the units
used within the historical values.
Therefore, whenever wishing to apply our portfolio component you should
decide for the beginning whether you wish to use absolute or relative values
for these three instances.
Assumptions underlying Markowitz Theory
Portfolio theory in the shape of Markowitz Theory makes the following assumptions
concerning the investment market and investors behavior within those markets. We summaries
these assumptions below:
Investors seek to maximize the expected return of total wealth.
All investors have the same expected single period investment horizon.
All investors are risk-adverse, that is they will only accept greater
risk if they are compensated with a higher expected return.
Investors base their investment decisions on the expected return and risk
(i.e. the standard deviation of an assets historical returns).
All markets are perfectly efficient (e.g. no taxes and no transaction costs).
Types of functionality provided
With this XML Web service we offer Markowitz Theory related procedures which enable the evaluation
of the Efficient Frontier and the optimal portfolio to be selected from the Efficient Frontier
from knowledge of its expected return or the investors risk - reward utility function. That is,
within this XML Web service we offer the following:
Efficient Frontier Stateful Methods - The method CalculateEfficientFrontier
first evaluates the Efficient Frontier and sets it within private fields of the class. Note
that if applicable the constraints on the weights of the assets within the portfolios on the
Efficient Frontier should be set using SetConstraints. Once the (possibly
constrained) Efficient Frontier has been set the method EfficientFrontier,
can read of the portfolios with almost no additional computational overhead.
Exposing the Efficient Frontier - The components which make up points of the Efficient
Frontier namely: expected return, asset weights and total risk of portfolios on the Efficient
Frontier are exposed at the finite set of points at which they are evaluated using the methods:
GetEfficientFrontierExpectedReturns,
GetEfficientFrontierAssetWeights,
GetEfficientFrontierPortfolioRisks. We also provide
within this class a general cubic spline interpolation procedure CubicSplinePointwise,
which allows you to interpolate the Efficient Frontier from the known values of the expected
return and total risk of the finite set of known points of the Efficient Frontier.
Efficient Frontier Stateless Methods - The method EfficientFrontier
returns the weights of the optimal portfolio (i.e. lowest risk) for a given expected return
and does not require prior evaluation of any other methods.
Optimal Portfolio Stateful - The unique optimal portfolio is accordance with the investors
(risk - reward) utility function can be selected using one of: OptimalPortfolio,
OptimalPortfolioMaxExpected. Please, note that before either of these methods is called
you are required to set the investors utility function and evaluate the Efficient Frontier.
Notes on the Evaluation of the Efficient Frontier
To calculate the Efficient Frontier, Rosen's gradient projection optimization algorithm
is used. If you directly try to evaluate the optimal portfolio with respect to an investors
utility function then you will need numerous applications of Rosen's algorithm which will
become computationally intensive. Therefore, we designed this XML Web service so that this would not
be necessary by allowing the computation at the beginning a number of points on the Efficient
Frontier, from which the other points will be deduced (in fact, estimated) through the use of
cubic spline interpolation. These interpolation points are determined by
CalculateEfficientFrontier, which must be called prior to
any subsequent method which depends on the Efficient Frontier being known.
Estimation/evaluation of non-observable parameters
A number of the parameters which are required by this classes methods such
as the covariance matrix are not directly observable from the market. However,
the evaluation may be evaluated or estimated directly from market driven
information such as historical asset prices. All methods related to the evaluation
of such parameters have been collected or are referenced within the
AssetParameters XML Web service.
In particular, the AssetParameters XML Web service contains the following
procedures:
Evaluation of the Covariance Matrix - CovarianceMatrix,
CovarianceMatrix
Estimation of the Expected Return - ExpectedReturns,
ExpectedReturns
Estimation of the Volatility - Not directly used within this XML Web service but
its estimated value can act of a reference point when judging the effects of diversification of risk.
Which can be used for the evaluation of utility statistical and risk metrics which will be used
in the application of the main portfolio analysis methods.