Question 1

  1. Suppose \(X, Y\) are bivariate normal random variables with common means being \(0,\) common standard deviations being \(1,\) and correlation \(.5,\) and let \(Z=X+Y\) and \(W=-X\)
  1. Derive the bivariate copula mpdel for \(X, Z\).
  2. Derive the bivariate copula model for \(X, W\)

(a) \(\mathbf{Solution.}\qquad\) We can write \(F_{W}(w)\) as, \[\begin{align*} F_{W}(w) & =\mathbb{P}(-X\leq w)\\ & =1-F_{X}(-w) \end{align*}\]

Then, \[ F_{W}(-X)=1-F_{X}(X) \]

Next, the copula can be written as, \[\begin{align*} C\left(u_{x},u_{w}\right) & =\mathbb{P}\left(F_{X}(x)\leq u_{x},F_{W}(w)\leq u_{w}\right)\\ & =\mathbb{P}\left(F_{X}(x)\leq u_{x},1-u_{w}\leq F_{X}(x)\right)\\ & =\mathbb{P}\left(1-u_{w}\leq F_{x}(x)\leq u_{x}\right) \end{align*}\]

Since \(F_{X}(x)\sim Unif\left(0,1\right)\), the copula becomes, \[ C\left(u_{x},u_{w}\right)=\begin{cases} u_{x}+u_{u}-1 & \text{ if }\quad u_{x}\geq1-u_{w}\\ 0 & \mathrm{otherwise} \end{cases} \]

   

(b) \(\mathbf{Solution.}\qquad\) We are given that \(X\sim N\left(0,1\right)\) and \(Y\sim N\left(0,1\right)\) with \(\rho=0.5\). Then \(Z\sim N\left(0,3\right)\) since \(\mathrm{Var}\left(X+Y\right)=3\) and \(\mathbb{E}\left[X+Y\right]=0\). We can use the Gaussian copula \(C\left(u_{x},u_{z}\right)\) for \(X,Z\), with correlation, \[ \rho=\frac{3/2}{\sqrt{3}}=\sqrt{\frac{3}{2}} \]

Question 2

  1. In the Data directory are Nasdaq weekly return data and SP400 weekly return data from 1992 to 2012
  1. Carry out a fitting of a multivariate normal distribution to the log-returns (computed from the Adjusted Closing prices) and carry out diagnostic plots - univariate \(\mathrm{QQ}\) and a plot comparing the empirical vs. theoretical bivariate cumulative distribution function.
  2. Same as (a), but now use a multivariate \(t\) distribution - also derive a confidence interval for the degrees of freedom via the method of profile likelihood.
  3. Based on results in (a) and (b), which model do you prefer and why. Compare the two models of multivariate normal vs. multivariate \(t\) using the AIC criteria.
  4. Derive the approximate VaR and expected shortfall for \(q=.005\) and a 10 million dollar portfolio that is evenly split between the two indices.

(a) \(\mathbf{Solution.}\qquad\) Below we show normal QQ plots of the log returns for each asset.

Next we show the contour plot comparing the empirical vs. theoretical bivariate normal distribution.

   

(b) \(\mathbf{Solution.}\qquad\) Below we compute the maximum likelihood estimate for the degrees of freedom to use for our multivariate t-distribution and also show 95% confidence interval on the degrees of freedom.

We have \(\hat{\nu}_{MLE}\) = 2.8 and the 95% CI for \(\nu\) is [2.43, 3.25]. Next, we have QQ plots for t distribution.

Next we show the contour plot comparing the empirical vs. theoretical bivariate t-distribution.

   

(c) \(\mathbf{Solution.}\qquad\) Below we compute the AIC for the multivariate normal and multivariate t distributions. We see that AIC for multivariate t is lower indicating that this could be a better model.

## [1] -11087.61 -10459.97

   

(d) \(\mathbf{Solution.}\qquad\) First we compute VaR and ES for normal distribution.

## [1] 725985.7 817748.4

Next we compute VaR and ES for t-distribution.

## [1]  951636.7 1514780.2

   

Question 3

Utilizing the same data as in \(2 .\), carry out a copula-based fitting of the bivariate distribution of the log-returns via the following parts:

  1. Fit a separate \(t\) -distribution, via MLE, to the Nasdaq weekly log-returns and SP400 weekly log-returns.
  2. After transforming via the estimated \(t\) -cdfs, fit a \(t\) -copula to the data. Compare the fit of this model with the estimated multivariate \(t\) -distribution from Problem \(2-(b)\) above. For this comparison, compare the fits of the estimated bivariate cumulative distribution function with the bivariate empirical cdf and compare the AIC criteria. For the AIC criteria, recall that multivariate copula pdf model is \[ f\left(x_{1}, x_{2}\right)=c\left(F_{1}\left(x_{1}\right), F_{2}\left(x_{2}\right)\right) f_{1}\left(x_{1}\right) f_{2}\left(x_{2}\right) \] where \(c\) is bivariate pdf of the copula, and \(f_{1}, f_{2}\) are pdfs of the marginals.
  3. Based on the models derived in (a) and (b) derive the approximate VaR and expected shortfall for \(q=.005\) and a 10 million dollar portfolio that is evenly split between the two indices.

(a) \(\mathbf{Solution.}\qquad\) Below we show QQ plots for weekly log-returns for each asset. The difference between these plots and the plots from 2b), being that we plot separate t-distributions for each asset with their own corresponding degrees of freedom.

   

(b) \(\mathbf{Solution.}\qquad\) Below we show contour plot which compares our estimated bivariate cumulative distribution function with the bivariate empirical cdf.

Next we compute the AIC for our fitted t-Copula. This AIC is slightly higher than the AIC from from 2c). However, the fit of the contour plot looks better, we can still choose to go with t-Copula model instead.

## [1] -11065.22

   

(c) \(\mathbf{Solution.}\qquad\) Below we compute VaR and ES.

##      0.5%           
##  936189.9 1307996.1