Applies the Capital Asset Pricing Model (CAPM) to find the market portfolio and construct the Capital Market Line (CML).
For a list of all members of this type, see CapitalMarket Members.
System.Object
CapitalMarket
The Capital Asset Pricing Model (CAPM) is an extension of the Markowitz model provided within Markowitz class.
Within the CAPM an investors portfolio is constructed from a collection of risky assets, which is then either leveraged or de-leveraged with borrowings or cash deposits. Within the CAPM one assumes that the rate at which monies are lent or borrowed at the prevailing risk free rate. The ability the borrow or deposit capital reflects the real world management of portfolios where excess capital is typically lent or borrowed around LIBOR for GBP funds, or the Fed Funds rate for USD funds.
Remark: The CAPM does not take into account the credit worthiness of the leveraged portfolios. However, assuming the capital can be borrowed at the risk free rate is not unreasonable if the portfolio is not highly leveraged. Since in such instances a repo agreement would allow capital to be borrowed at close to the prevailing risk free rate.
The introduction of the risk free asset, namely cash along with the risky assets transforms the Efficient Frontier curve of the Markowitz Theory on which the optimal portfolios lie into a straight line, known as the Capital Market Line (CML). The CML consists of (optimal) portfolios which are constructed from a weighting or geared position in the Market Portfolio (explained below) with the net cash/debt balance lent or respectively borrowed from the market.
The Market Portfolio is the portfolio on the Efficient Frontier (see Markowitz class or the PDF documentation for more details) which offers the greatest return over the risk free rate per unit of risk. That is, the portfolio on the Efficient Frontier which maximizes:
The Market Portfolio purchased with cash or debt represents the cheapest means in terms of risk to increase the return of the portfolio above the risk free rate. The weighting of the Market Portfolio (and cash/debt) will depend on the level of expected return required. In particular, the portfolios on the Capital Market Line (CML) will fall into the following cases:
As described above the introduction of a risk free asset within the Capital Market Theory transforms the Efficient Frontier into the Capital Market Line (CML) which as the name suggests is a straight line when plotted within the risk vs expected return plane. Portfolios correspond to points on the CML consist of multiples of a basket of assets known as the Market Portfolio and a positive or negative cash balance which is either borrowed from or lend at the risk free interest rate. This class enables the construction of the Market Portfolio from which the corresponding Capital Market Line (CML) will be derived.
The steps required to construct the CML, from which the optimal portfolio with respect to the CAPM can be found, will almost always proceed along the following lines:
Remark: For a given set of assets with possibly constrained asset weights there will correspond a range of values of the expected return over which the (constrained) Efficient Frontier exists and in which the Market Portfolio will lie. You are able to evaluate this continuous range by applying the methods MinFrontierReturn and MaxFrontierReturn, which evaluate the upper and lower bounds respectively of the expected return over which the Efficient Frontier exists. If you are considering the constrained case then the constraints should be set (by calling SetConstraints) before the range of the expected returns are evaluated.
The application of the CAPM will consist of performing the following three steps:
1) Construction of the Efficient Frontier
In order to construct the Market Portfolio it is necessary to evaluate the (possibly constrained) Efficient Frontier which consists of optimal portfolios of risky assets. The (possibly constrained) Efficient Frontier is constructed by the following two steps:
Remarks:
[0,1]
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2) Evaluation of the Market Portfolio
The Efficient Frontier is the collection of portfolios constructed from the given set of available assets which exhibit the lowest risk for a given value of the expected return. Note that the weights of the assets from which the portfolios are constructed may be required to satisfy linearly constraints.
The linear constraints must be set using SetConstraints, SetAssetWeightsInequalityConstraints and SetAssetWeightsEqualityConstraints, before the Efficient Frontier is constructed using CalculateEfficientFrontier(double[][],...), or (double, double, double[][], double[], int, double) calculateEfficientFrontier(double,...).
Once the Efficient Frontier has been constructed, the next step is to find the Market Portfolio. The Market Portfolio is the (unique) portfolio on the Efficient Frontier which maximizes:
The Market Portfolio is selected from the Efficient Frontier by calling one of the following methods:
Each of these methods returns an array containing the (possibly constrained) weights of the Market Portfolio.
3) Selecting a Portfolio on the CML
The Capital Market Line (CML) is the collection of portfolios which are constructed by either deleveraging and gearing the Market Portfolio by lending or borrowing (risk free) cash at the prevailing market rate. These portfolios in accordance with the CAPM offer the lowest risk for a given level of the expected return.
A portfolio on the CML can be selectecd from any one of the following criteria:
Therefore, given any one of the above properties (total risk, expected return and Market Portfolio weighting) the other two properties can be deduced. Hence the methods RiskCML, WeightCML, ReturnCML, and Weight2Risk are complete.
Constraints on the level of Cash Holdings and/or Borrowings with a Portfolio
In practice (as detiled within the PDF documentation) a given fund will often have limits regarding:
Such constraints on the level of cash or borrowings, imply constraints on the weighting of the Market Portfolio within the portfolio on the CML. By applying Weight2Risk one is able to evaluate the range of the total risk of the portfolio on the CML which satisfies these constraints. From which the range of the expected returns can be found using ReturnCML.
Within the application of the CAPM you should decuide from the beginning whether to use absolute or relative values for the historical values and expected returns. In particular, 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. That is:
The units used within these two quantities will determine the units used within the construction of the Efficient Frontier and Market Portfolio.
The CAPM makes all the assumptions of the Markowitz models concerning the (investment) market and investors behavior, namely that:
The CAPM in addition to these assumptions also makes the following provisions:
Note that the rate of which money can be borrowed or lend from or to the market within the CAPM is assummed to be the same.
Summary of Functionality provided
With this class we offer the following functionality:
Estimation/evaluation of non-observable parameters
A number of the parameters which are required by the main methods of this class (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 within the AssetParameters class.
In particular, the AssetParameters class contains the following procedures:
Namespace: WebCab.Libraries.Finance.Portfolio
Assembly: WebCab.Libraries.PortfolioDemo (in WebCab.Libraries.PortfolioDemo.dll)
CapitalMarket Members | WebCab.Libraries.Finance.Portfolio Namespace