3. Copula model selection
Our model aims to capture the type of asymmetric dependence found in international equity and bond markets. For asset allocation, we need to reduce the fluctuations in invested assets, especially downside co-movement. By means of regime-switching copula, we want to distinct more fluctuations in indices to maximize the portfolio returns.
Therefore, we need to allow for asymmetry in tail dependence, regardless of the possible marginal asymmetry or skewness. Copulas, also known as dependence functions, are an adequate tool to achieve this aim. And simultaneously, for marginal distribution of asset returns, we also try to obtain the evidence of asymmetry.
3.1 marginal distribution
It ‘s well known that the residuals obtained from a GARCH model are generally non-normal. This observation has exposed possibility of fat-tailed distribution.
Hansen(1994) proposed a new density to model the GARCH model residuals, which is a extension of the Student-t distribution with skewed factors. The Hansen’s skewed Student-t distribution is defined by
(3.1)
and and λ denote the degree-of-freedom parameter and the asymmetry parameter respectively. We will write Z~ST( , λ) which means the random variable Z has the
1
‧
After the introduction of the residuals distribution, we continue to finish our models. For marginal distributions for the returns of given assets, we use
ARMA(1,1)-GARCH(1,1)-Skewed-t model to fit asset returns:
(3.2) The variables represents the log returns of equity i , denotes the conditional variances of . The parameters of the marginal distributions are grouped into one factor with
3.2 Dependence structure
Our dependence model is characterized by tow regimes, one the normal regime (
s
t ) corresponds to a symmetric dependence where the conditional joint normality 1 can be supported, and a second regime assumed as a worse state (s
t ), corresponds 0 to the asymmetric regime in which markets are strongly more dependent for negative returns than for positive returns. The conditional copula is given by:(3.3) where , and
s
t 0,1 is a state variable that follows a Markov chainprocess with a constant transition probability matrix as below.
1 1
In the previous chapter, we talk about a lot of copula functions. According to the variety features of these functions, we have several choices to fit our model.
‧
Additionally, we will compare the fitness with no regime-switching copula to observe the improvement.
3.3 Estimation
The maximum likelihood method (MLE) could be very computationally intensive, especially in the case of a high dimension, because it is necessary to
estimate jointly the parameters of the marginal distributions and the parameters of the dependence structure represented by the copula. Let us denote the observed data by
where The log likelihood function is given by: Where is the parameters grouped of the copula and the transition matrix. For the time series model of changes in regime, Hamilton(1989, 1994, Chapter 22) presents a filter procedure to perform this kind of evaluation. With
we denote the density function conditionally to the state variable
and it can be integrated to a unconditional density function:1 1 The conditional probability, denoted by
(3.8)
‧
in different regime state can be computed by Hamilton filter. Given a starting value , the optimal inference and forecast for each date t in the sample can be found by the following iterating equations:
(3.9) (3.10)
where ⊙ denotes element-by-element multiplication. Finally, the log likelihood function can be calculated of this algorithm:
Furthermore, from chapter 2, we know that the joint distribution density can be written as a copula density product the margin density. According to the formula, the log likelihood function can be written as4
We see that the log likelihood function can be decomposed into two positive terms: one term involving the copula density and its parameters, and one term involving the margins and all parameters of the copula density. Because of the numerous parameters needed to estimate from marginal distribution and copula function, it is difficult to estimate all parameters at one step. For that reason, our
ˆ1|0
‧
structure allows for a two-step estimation method, proposed by Joe and Xu(1996), called inference for the margins or IFM:
1. As a first step, we estimate the marginal distribution’s parameters:
1,.. 4
2. As a second step, given , we estimate the copula’s parameters:
ˆ arg max
L
c( , ; t)where , represent the sets of all possible values of
, respectively. For the second step, René and Georges(2011) give a proposition of the decomposition of the copula’s log likelihood function.3.4 Portfolio selection
After we have estimated all parameters by two steps, we continue to decide the method how to choose the best weights for our portfolio. The fund invested by the weights and asset returns are denoted by
returns of every asset. Here we try to use two objective functions:1) Quadratic cost function :
By Wang and Huang (2010), we consider the periodical targets at each time t, where t = 1,…,n, where n is the last date. The target value hold at time t is
*( 1) *( )(1 *( )),
F t
F t
r t (3.14) wherer t*( )Ma rx( ,G rm( )t ),r is the guarantee rate, and
Gr t is the minimum return
m( ) of the four assets at time t. The target fund represent we can invest no worse than the‧
target rate, such as risk-free rate, to satisfy we will have a reasonable return. The cost function at time t is defined as
2
( ) *( ) ( ) *( ) ( ) , for 1,..., .
C t
F t
F t
F t
F t t
n (3.15) The former part of the cost function is to control the final fund value, to reduce the risk of asset matching. The second part of the cost function is to hold the downside risk. The higher θ means the achievement of the fund final target is more important.The higher κ means we take more attention to the downside risk than the asset matching.
We want to minimize the future cost to determine our weights per period.
Consequently, we can write the value function as:
(3.16) where ( ) n n ( )
u t
G t
v C uu denote the discounting future costs,F denote the
tinformation we obtain until time t, and t { ( ),
w s s t t
, 1,...,n
represents the 1}investment strategy of the future. By the objective function to control the future cost, we can obtain an optimal weights for every period.
2) Target volatility
For risk-control target, target volatility is one of the most popular methods that we can control the volatility and make the fund value have smaller fluctuation. For the purpose, we can fix our investment not to be so risky, and have a conservative
decision. We restrict the volatility of fund to keep as the historical data, for maintaining the return stability.
( ) min ( ( ) | )
t
t t
V F E G t F