PORTFOLIO INVESTMENT

Portfolio Investment

Portfolio Investment

This paper will consist of creating a portfolio of small number of securities i.e. n<4. This will be done through Value at Risk model. Value at Risk (VaR) can describe more about extreme events, but it can not aggregate risk in the sense of being subadditive on portfolios (which means risk is diversified). This is a well-known difficulty that is addressed by the concept of a “coherent risk measure” in the sense of Artzner, Delbaen, Eber, and Heath(1999). A popular example of a coherent risk measure is the expected shortfall (ES), though VaR is still more commonly seen in practice. The construction of an efficient frontier - portfolios with minimum risk for a given return - depends on two inputs: the choice of risk measure (such as standard deviation, VaR, or ES), and the probability distribution used to model returns.

It turns out, by a result of Embrechts, McNeil, and Straumann (2001), that when the underlying distribution is Gaussian - or more generally any “elliptical” distribution - no matter what positive homogeneous and translation invariant risk measure(such as VaR, or ES), no matter what confidence level, the optimized portfolio composition given a certain return will be the same as the traditional Markowitz style portfolio composition obtained by minimizing standard deviation. Only a difference in distribution leads to different portfolio compositions. As mentioned in last paper, portfolio managers can not neglect the deviation of financial returns series from a multivariate normal distribution. Other heavy tailed elliptical distributions, such as Student t and symmetric generalized hyperbolic distributions, and non-elliptical distributions, such as the skewed t distribution, can be used to model financial returns series.

If the underlying is Gaussian distributed, then the portfolio return is Gaussian distributed too. More generally, if the underlying is generalized hyperbolic distributed, then the portfolio return remains in the same sub-family of generalized hyperbolic distributions since they are closed under linear transformations. The usual parametrization of generalized hyperbolic distributions generally does not exhibit this property. In addition, for Gaussian, Student t and symmetric generalized hyperbolic distributions, the portfolio variance is in quadratic form so that it is easy to minimize. For non-elliptical distributions, different risk measures, or the same risk measure with different confidence levels, lead to differing portfolio compositions given a fixed return. Rockafellar and Uryasev (1999) showed that the minimization of ES does not require knowing VaR first, and construct a new objective function. By minimizing this new objective function, we can get VaR and ES. We carry this out and use Monte Carlo simulation to approximate that new objective function by sampling the multivariate distributions. This allows us to construct efficient frontiers for a variety of distributions.

Data Choice

We construct the portfolio by choosing 4 stocks from the components of the FTSE 250 index. They are TESCO PLC, BRITISH PETROLEUM, GLAXO SMITHKLINE, and INTERTEK GROUP PLC. We use the adjusted close data ranged from 7/1/2002 to 08/04/2005. The daily close data are converted to log return. Figure ...