**Copula**

Background

vine copula

results

GARCH(1,1)-

ARCH(1,1)

GARCH(1,1)

Risk-return opimization

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**

GARCH-Copula Model

GARCH-Copula Model

**GARCH**

**Summary**

Basic Theory

Implementation

Data

Results

S&P 500 index

North America

AAPL, AMZN, CAN, GS,

CCE, AKS, YHOO, XOM,

TIF, PG, DELL, Citibank

April 1st, 2011 --April 1st, 2013

500 observations

12-dimensional data set

**Portfolio Selection**

Risk-return optimization

Mean-risk efficient frontiers for different risk measures

Implementation

Steps

family

par

simulation

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.

par2

vine-copula matrix

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

Implementation

Once the model has been built, one-step ahead of prediction can be done via taking expectation.

step 1

step 2

step 3

-ARMA(1,1) -GARCH(1,1)

-AR(1)-GARCH(1,1)

-MA(1)-GARCH(1,1)

-GARCH(1,1)

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.

step 4

We then estimated the residuals by the models we selected, and did the standardization to the residuals.

Example: AAPL

Marginal time series models for the

log returns of the considered 12 stocks

Portfolio Optimization

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

Risk measures

Suppose is the risk measures (standard deviation, VaR and CVaR), for given minimal expected return ,consider

Risk Measure

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:

AAPL,AMZN,CAN,GS,YHOO, CITI

use stock price to reflect real portfolio

Optimal portfolio based on risk measure

probability: 0.9

Efficient frontier comparison

Conclusion

Q&A

Standard deviation (StDev)

the StDev optimization is equivalent to the variance optimization, convex quadratic program

where is the average return vector

Value-at-Risk

Smoothed VaR (Alexei 2004)

smooth out the local noisy component of VaR and extract the well behaved global component

import the smoothed quantile function

Conditional Value-at-Risk

is the expectation of return conditional on not exceeding the quantile

CVaR can be represented following (Uryasev & Rockafellar 1999)

then

GARCH(1,1)

-AR(1)

GARCH(1,1)

-MA(1)

PortfolioCVaR class

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

**Ye Dong**

Xingwen Lu

Wenjia Tan

Xingwen Lu

Wenjia Tan

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

Thank You!