The Capital Asset Pricing Model
(CAPM) is an extension of the Markowitz model provided within Markowitz
class.
Overview
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.
Nature of the Capital Asset Pricing Model (CAPM)
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:
(Expected Return - Risk Free Rate) / Total Risk
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:
Borrow money at the prevailing market rate in order to take of geared position in
the Market Portfolio. In this case the investors portfolio will have an expected return greater
than the Market Portfolio, and a risk level higher than the Market Portfolio.
Lend Money at the prevailing market rate with a weighting in the Market Portfolio
of less than 100 per cent. In this case the investors portfolio will have an expected return less than
the Market Portfolio, and a risk level less than the Market Portfolio.
No cash holdings or borrowing if the expected return (or risk) you require
is the same as the expected return (or risk) of the Market Portfolio.
Brief Overview of the Functionality Offered
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:
Evaluate the weight of the Market Portfolio and hence the weight of the cash component within
the (optimal) portfolio on the CML for a given expected return using the method WeightCML.
Note that the risk of the (optimal) portfolio can be evaluated using riskCML when
the expected return is given, and using weight2Risk when the weight of the market
portfolio is given.
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.
Detailed Overview of the Functionality Offered
The application of the CAPM will consist of performing the following three steps:
Construction of the Efficient Frontier
Evaluation of the Market Portfolio
Selecting a Portfolio from the CML
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:
Set Constraints on Asset Weights - Constraints can be placed on the weights of the assets
of the portfolios which make up the Efficient Frontier by calling the following methods:
SetConstraints - Sets lower and upper bounds on the asset weights.
Construct the Efficient Frontier - Evaluates the Efficient Frontier at a finite number of points
by calling one of the following methods:
CalculateEfficientFrontier -
Evaluates a finite number of equally dispersed portfolios on the (possibly constrained) Efficient Frontier over its entire range.
(double, double, double[][], double[], int, double) calculateEfficientFrontier(double,...) -
Evaluates a finite number of equally dispersed portfolios on the (possibly constrained) Efficient Frontier over a given range
of the expected return. This method allows higher accuracies to be obtained for a given computational demand when the range
of expected returns in which the market portfolio will lie is known.
Remarks:
The asset weights of the portfolios on the Efficient Frontier by default are constrained to lie
within the interval [0,1].
Since the Market Portfolio is a portfolio on the Efficient Frontier, placing constraints on the
portfolios of the Efficient Frontier will ensure that the Market Portfolio satisfies the same constraints.
The more portfolios on the Efficient Frontier which are evaluated the higher the accuracy to which the
Efficient Frontier will be constructed. However, the computational demands will grow in proportion to the
number of portfolios evaluated.
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.
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:
(Expected Return of Portfolio - Risk Free Rate) / Total Risk of Portfolio
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:
Expected Return of the Portfolio of the CML: Given expected return of portfolio on CML, WeightCML returns the Market Portfolio
Weighting, from which the portfolio risk can be evaluated using Weight2Risk.
Total Risk of the Portfolio on the CML: Given the risk of the Portfolio on the CML, ReturnCML returns the
portfolios expected return.
Weighting of the Market Portfolio within the Portfolio on the CML: Given the weight of the Market Portfolio the risk
can be found using Weight2Risk, from which the expected return can be found using
ReturnCML.
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:
Maximum level of cash which should be held within the portfolio.
Maximum level of borrowing (i.e. gearing) which the portfolio can use.
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.
Using Absolute or Relative Historical Values
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:
Historical Values: Source data which is given in absolute or relative (percentage) terms.
Expected Returns: The expected return of the investment over the period considered should be given in the units
(i.e. absolute or relative) in which you wish the result to be returned. That is, if absolute values are given then
the result will be returned as an absolute value and if relative value are given then the result will be returned
as a relative value.
The units used within these two quantities will determine the units used within the construction of the Efficient
Frontier and Market Portfolio.
Assumptions of the Capital Asset Pricing Model (CAPM)
The CAPM makes all the assumptions of the Markowitz models concerning the
(investment) market and investors behavior, namely that:
Investors seek to maximize the expected return on total wealth.
All investors have the same expected single period investment horizon.
All investors are risk-adverse, and hence will only accept greater
risk if they are compensated with a higher expected return.
All markets are perfectly efficient (e.g. no taxes and no transaction costs).
The CAPM in addition to these assumptions also makes the following provisions:
Lend excess capital at the Market Rate: The investor may lend money at the
prevailing market rate. This constitutes the ability to hold within the portfolio
a risk free asset which will provide the prevailing return available on cash. In practice,
such assets are often referred to as money market accounts which yield a return
in the region of LIBOR (London Interbank Overnight Rate).
Borrow capital at the Market Rate: The investor may borrow money from the
market at the prevailing market rate in order to invest within (risky) assets. This will
increase the expected return of the original capital base and also increase its risk.
In practice, the rate at which money can be lent from the market by a fund will be
in the region of LIBOR (at least if the fund structures the loan as a REPO type agreement).
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:
Efficient Frontier Methods - Construction of the Efficient Frontier.
Efficient Frontier Stateful Methods - The stateful method CalculateEfficientFrontier
constructes the Efficient Frontier which may be required to satisfy asset weight constraints. The constraints on the weights of
the assets within the portfolios on the Efficient Frontier should be set prior to constructing the Efficient Frontier by calling
SetConstraints, SetAssetWeightsInequalityConstraints and/or
SetAssetWeightsEqualityConstraints. Once the computationally intensive construction
the Efficient Frontier is complete you are able to read off the asset weights of the frontiers portfolios with almost no
additional computational overhead.
Market Portfolio Methods - Select the Market Portfolio on the Efficient Frontier.
Select the Market Portfolio - The method MarketPortfolio
selects the market portfolio and returns its assets weights. If the market portfolio is known to lie within a given range
of the expected returns then the method (double, double, double[][], double) marketPortfolio(double,...)
offers a more efficient means by which the asset weights can be evaluated.
Quantitative properties of the Market Portfolio - Once the market portfolio has been selected its expected return can be
evaluated using MarketPortfolioExpectedReturn, and its risk can be evaluated using
MarketPortfolioRisk.
Capital Market Line (CML) Methods - Select the optimal portfolio on the CML from knowledge of the portfolios risk,
expected return or Market Portfolio weighting.
Select by Expected Return - When the expected return of the portfolio on the CML is known its total risk can be evaluated
using RiskCML, and it weighting in the Market Portfolio can be evaluated by WeightCML.
Select by Total Risk - When the total risk is known, the expected return can be evaluated using ReturnCML.
Select by weighting of the Market Portfolio - When the weighting in the
Market Portfolio in known we are able to evaluated the total risk of the portfolio
on the CML using Weight2Risk.
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:
Evaluation of the Covariance Matrix - CovarianceMatrix,
CovarianceMatrix
Estimation of the Expected Return - ExpectedReturns,
ExpectedReturns
Estimation of the Volatility - Not directly used within this class but its estimated value can
act of a reference point when judging the effects of diversification.