Financial Modeling of the Equity Market

pp. 100-147

Eileen Rodríguez HernándezEconometría

Prof. Balbino García

Index• Chapter 4: (Continuation, pp. 100-113)– Portfolio constraints commonly used in practice

• Chapter 5: Incorporating higher moments and extreme risk measures (pp.114-147)– Dispersion and downside measures– Portfolio selection with higher moments through

expansions of utility– Polynomial goal programming for portfolio optimi

zation with higher moments– Some remarks on the estimation of higher mome

nts– The approach of Malevergne and Sornette

• Bibliography

Chapter 4 (Continuation): Portfolio constraints commonly used in practice

• A portfolio manager be restricted on how concentrated the investment portfolio can be in a particular industry or sector. These restriction, and many more, can be modeled by adding constraints to the original formulation.

• This section denote the current portfolio weights by w0 and the targeted portfolio weights by w, so that the amount to be traded is x=w-w0.

• Linear and quadratic Constraints– Some of the more commonly used linear and quadratic

constraints are described below. • Long- only constraints– When short- selling is not allowed we require that w≥0. This is a

frequently used constraints, as many funds and institutional investord are prohibited from selling stocks short.

• Tornover constraints– High portfolio turnover can resilt in large transaction costs that

make portfolio rebalancing inefficient. The most commom turnover constraints limit turnover on each individual asset

or on the whole portfolio

• Holding Constraints– A well-doversified portfolio should not exhibit large concentration

in any specific assets, industries, sector, or countries. Maximal holdings in an individual asset can be controlled by the constraint

where li and Ui are vectors representing the lower and upper bounds of the holdings of asset i. To constrain the exposure to a specific set Ii of the available investment universe l, we can introduce constraints of the form

where Li and Ui denote the minimum and maximum exposure to li.

• Risk Factor Constraints– The portfolio managers use factor models to control for

different risk exposure to risk factors such as market, size, and style. Let us assume that security returns have a factor structure with risk factors, that is

– To limit a portfolio’s exposure to the k-th risk factor, we can impose the constraint

Where UK denotes maximum exposure allowed. – To construct a portfolio that is neutral to the k-th risk

factor we would use the constraint

• Benchmark exposure and tracking error constraints– A portfolio manager might choose to limit the

desviations of the portfolio weights from the benchmark weights by imposing

denote the market capitalization weights. For a specific industry requite that

• General linear and quadratic constraints– The linear or quadratic can be cast either as

Or as

• Combinatorial and Integral constraints– This binary decision variable is useful in describing some

combinatorial constraints:

Where denotes the portfolio weight of the asset

• Minimum holding and transaction size constraints– The classical mean-variance optimization problem often results in

a few large and many small positions. In a practice, due to transaction costs to eliminate small holdings, threshold constraints of the following form are often used

where Lwi is the smallest holding size allowed for asset i.

– A portfolio manager might also want to eliminate new trades, x, smaller than a pre-specified amount.

• Cordinality constraints– The cardinality constraint isto restrict the number of

assets in a portfolio:

where K is a positive integer significantly less than the number of assets in the investment universe, N.

• Round lot constraints– The portfolio weights can be represented as

Chapter 5: Incorporating higher moments and extreme risk measures.

• The main objective of portfolio selection is the construction of portfolios that minimize expects return at a certain level of risk. In this chapter we examine some of the most common alternative portfolio risk measures used in practice for asset allocation. In general, we study techniques based upon expansions of the utility function in higher moments such as its mean, variance, skew, and kurtosis.

Dispersion and downside measures

• Dispersion measure– Are measure of uncertainty. Entail positive and negative deviation

from the mean and consider those deviations as equally risky. • Mean standard deviation and the mean-variance approach

– Is probably the most well-known dispersion measure. • Mean absolute deviation

– The dispersion measure is based on the absolution deviations from the mean:

where Ri and are the portfolio return, the return on asset i, and the

expected return on asset i, respectively.

• Mean-absolute Moment– The mean-absolute moment (MAMq) of order q is defined by

and is a straightforward generalization of the mean standard deviation (q=2) and the mean absolute deviation (q=1) approaches. • Downside Measure

– The objective in these models is the maximization of the probability that the portfolio return is above a certain minimal acceptable level, often also referred to as the benchmark level or disaster level. By the estimation of downside risk measure we only use a portion of the original data and hence the estimation error increase.

• Roy’s Safety-First– Which laid the seed for the development of downside risk measure. Roy argue that an

investor rather than thinking in terms of utility functions, first want to make sure that a certain amount of the principal is preserved. Roy pointed out that an investors prefers the investment opportunity with the smallest probability of going below a certain target return or disaster level. In essence, this investors choose this portfolio by solving the following optimization problem

subject to

where P is the probability function.

• Lower Partial Moment– The lower partial moment risk measure provides a natural

generalization of semivariance. The lower partial moment with power index q and the target rate of return R0 is given by

where

is the portfolio return. • Value- at – Risk (VaR)– VaR is related to the percentile of loss distributions and

measures the predicted maximum loss at a specified probability level over a certain time horizon. Formally, VaR is defines as

where P denoted the probability function. Typical values of α that comonly are considered are 90%, 95%, and 99%

• Conditional Value- at- risk– Is a coherent risk measure defined by the

formula

– The function

can be used instead of CVaR – To ilustrate the mean-CVaR optimization

approach we consider an example. We considered two-week returns for all the stocks in the S&P 100 Index over the period July 1, 1997 to July 8, 1999 for scenario generation. In Exhibit 5.1 we see three different mean-CVaR efficient frontiers corresponding to α=90%, 95% and 99%. The risk es 7% and α is 95%, this means that we allow for no more than a 7% loss of the initial value of the portfolio with a probability of 5%. We observed from the exhibit that as the CVaR constraint decreases the rate of return increase.

• In exhibit 5.2 we can see a comparison between the two approaches for α=95%. The same data set is used as in the illustration above. In return/CVaR coordinates the mean-CVaR efficient frontier lies above the mean- variance efficient frontier.

Portfolio selection with higher moments through expansions of utility

• Skew in stock returns is relevant to portfolio selection. If asset returns exhibit nondiversifiable coskew, investors must be rewarded for it, resulting in increased expected returns.

• In the presence of positive skew, investors may be willing to accept a negative expected return.

• To illustrate the effect of skew and kurtosis in the portfolio selection process, we consider three two-asset portfolios: Australia/Singapure, Australia/United Kingdom, and Australia/United States. For each portfolios, the mean, standard deviation, skew, and kurtosis is computed based on the empirical return distribution over the period January 1980 through May 2004 and depicted in Exhibit 5.3– First: While the return is a linear function of

the weight, w, of the first asset and the standard deviation is convex, the qualitative behavior of the skew and the kurtosis is very different for the three portfolios. The skew and kurtosis are highly nonlinear functions that can exhibit multiple maxima and minima

– Second: In the case of Australia of Singapure, the portfolio that minimizes the standard deviation also approximately minimize the skew and minimize the kurtosis. In the case of Australia/United States, the minimum-variance portfolio comes closer to achieving a more desirable objective of minimizing variance and Kurtosis, and maximizing skew.

• P 132-133

• The Mathematics of Portfolio selection with highter moments– Is convenient to have similar formulas for the skew and

kurtosis as for the portfolio mean and standard deviation

where μ and Σ are the vector of expected returns and the

covariance matrix of returns of the assets. Each moment of an random vector can be mathematically represented as a tensor. In the case of the second moment tensor is the familiar N X N covariance matrix, whereas the third moment tensors, the so-called skew tensor, intuitively be seen as a three- dimensional cube with height, width, and depth of N.

– For example, when N=3 the skew matrix takes the form

Polynomial goal programming for portfolio optimization with higher moments

• An investors may attempt to, on the one hand, maximize expected portfolio return and skewness, while on the other, minimize portfolio variance and kurtosis. Mathematically, we can express this by the multiobjective optimization problem:

• The basic idea behind goal programing is to break the overall problem into smaller solvable elements and then interatively attempt to find solutions that preserve, as closely as posible, the individual goals.

Some remarks on the estimation of higher moments

• When models involve estimated quantities, it is important to understand how accurate these estimates really are. It is well know that the sample mean and variance, computed via averaging, are very sensitive to outliers. The measure of skew and kurtosis of returns,

where

are also based upon averages.

The approach of Malevergne and Sornette

• The technique developed by Malevergne and Sornette resolve the problem of make stronger assumptions on the multivariate distribution of the asset return.

• For a complete description of the returns and risks associated with a portfolio of N assets we would meed the knowledge of the multivariate distribution of the returns.

• If the joint distribution of returns is Gaussian, that is,

with μ and Σ being the mean and the covariance of the returns r.

The approach of Malevergne and Sornette

• The one- dimensional case– Let us assume that the

probability density function of an asset’s return r is given by p(r). The transformation q(r) that produces a normal variable q from r is determined by the conservation of probability:

If we solve for q, we obtain

• The multidimensional case– We can map each component ri of

the random vector r into a standard normal variable qi. If these variables were all independient, we could simply calculate the joint distribution as the product of the marginal distribution. Given the covariance matrix Σq , using a classical result of information theory the best distribution of q in the sense of entropy maximization is given by:

Bibliography

• Fabozzi, F.J., Focardi, S. M., Kolm, P. N. (2006) Financial Modeling of the Equity Market.