1. Financial Modeling of the Equity
Market
pp. 100-147
Eileen Rodríguez Hernández
Econometría
Prof. Balbino García
2. 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
optimization with higher moments
– Some remarks on the estimation of higher moments
– The approach of Malevergne and Sornette
• Bibliography
3. 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.
4. • 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
5. • 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.
6. • 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
7. • 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
8. • 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.
9. • 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
11. • 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.
12. 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.
13. • 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.
14. • 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%
15. • 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.
16. • 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.
17. 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.
18. • To illustrate the effect of skew and kurtosis in
the portfolio selection process, we consider
• P 132-133
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.
19. • 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
20. 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.
21. 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.
22. 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.
23. The approach of Malevergne and Sornette
• The one- dimensional case • The multidimensional case
– Let us assume that the – We can map each component ri of
the random vector r into a
probability density function
standard normal variable qi. If
of an asset’s return r is given
these variables were all
by p(r). The transformation independient, we could simply
q(r) that produces a normal calculate the joint distribution as
variable q from r is the product of the marginal
determined by the distribution. Given the covariance
conservation of probability: matrix Σq , using a classical result
of information theory the best
distribution of q in the sense of
entropy maximization is given by:
If we solve for q, we obtain
