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Problems in higher dimensional cases:
Standard multivariate copulas, such as elliptical copulas and Archimedian copulas
Hardly accommodate asymmetry of a random vector.
The dependence structure is not flexible enough.
So a multivariate copula called vine copula fix this gap.
Portfolio Selection based on
S&P 500 index
AAPL, AMZN, CAN, GS,
CCE, AKS, YHOO, XOM,
TIF, PG, DELL, Citibank
April 1st, 2011 --April 1st, 2013
12-dimensional data set
Mean-risk efficient frontiers for different risk measures
We used the R-vine copula model after the MLE procedure to simulate new standardized residuals’ CDF.
In order to invert those CDFs into estimated standardized residuals, we used the method of interpolation.
By substituting these values into the marginal time series models, we get the estimated stock price for the 12 companies.
Generalized Autoregressive Conditional Heteroskedastic model
The GARCH (p,q) process introduced in
Bollerslev (1986) is the extension of the
ARCH process and the mixture of MA-GARCH
and ARMA-GARCH model are similar to the
mixture of AR-GARCH model proposed
in (Wong etal., 1998).
The GARCH process can be expressed as
Furthermore, each residual term is
assumed white noise with variance
denoted by GARCH model
Mean-risk efficient frontiers
Once the model has been built, one-step ahead of prediction can be done via taking expectation.
Student-t distribution to account for heavy tails
For all four time series models we then use the Akaike Info Criterion to find the most appropriate models.
Since the Akaike Info Criterion sometimes lacks power, we also consider the Schwarz Criterion for all residuals
We chose the models that have the lowest Akaike info criterion value with a relatively low Schwarz criterion value.
We then estimated the residuals by the models we selected, and did the standardization to the residuals.
Marginal time series models for the
log returns of the considered 12 stocks
Consider a finite set of arbitrary financial assets: with returns
Decide on the positions in these assets such that
The return of the portfolio at the end of the period is
The expected return of the portfolio is
Suppose is the risk measures (standard deviation, VaR and CVaR), for given minimal expected return ,consider
By the GARCH models we selected, we can estimate the residuals of each stock.
Then we did the standardization to the residuals, which can be used to fit Vine Copula.
Following the classical Markowitz-type approach (Markowitz 1952)
provides the best return for a given value of risk
risk quantities are minimized for different values of minimal return u by solving former equations
investor with specific risk preference can choose the target value of risk (i.e.VaR) and select portfolio on the efficient frontier
Classical mean-variance efficient frontier & Mean-VaR efficient frontier
Choose six stocks with largest expected return:
use stock price to reflect real portfolio
Optimal portfolio based on risk measure
Efficient frontier comparison
Standard deviation (StDev)
the StDev optimization is equivalent to the variance optimization, convex quadratic program
where is the average return vector
Smoothed VaR (Alexei 2004)
smooth out the local noisy component of VaR and extract the well behaved global component
import the smoothed quantile function
is the expectation of return conditional on not exceeding the quantile
CVaR can be represented following (Uryasev & Rockafellar 1999)
comparison is performed on computational plane using the stock market data
VaR differs from both variance and CVaR and seems better than the others
mean-VaR feasible set and the respective efficient frontier are not convex, while their distance from convex shape is relatively small
we can expect, all frontiers approximate each other fairly well for high risk portfolios because for large return values all portfolios converge to portfolio consisting of only one asset with largest return
estimate the residuals of daily log-returns from GARCH model
forecast stock price by building copula on residuals
select optimal portfolio by comparison of risk measures using estimated price from GARCH-Copula model