Abstract: The value of portfolio g at time T, denoted by g[w(T)], is given with: (4.1) $$ g[w(T)] = \sum\limits_{i = 1}^I {\theta _i g_i } [w(T)], $$ where g i [w(T)] denotes the value of the i th investment and θ i stands for the quantity of the i th investment (asset, position) held at time 0. The long and short positions of each instruments have been aggregated so that every θ i represents a net investment. Clearly, the value of the portfolio depends on the level of the risk factors at time T and is therefore stochastic. Moreover, the modelling of the portfolio distribution F g requires the valuation of the I financial instruments of the portfolio as functions of the risk factor values. In theory each asset could be modelled as a risk factor. For well diversified portfolios the procedure would require the distribution information of a large set of size I of instruments. Considering the idealised assumption that the I positions are jointly normal distributed, the computation of F g would demand the estimation of I mean and $$ \frac{{I(I + 1)}} {2} $$ variance-covariance parameters. The parameter estimation for new, respectively illiquid instruments, would be very inaccurate, respectively impossible. Moreover, the modelling of instruments like options with non-linear pay-offs and therefore non-symmetric distributions would require the estimation of additional parameters since their distributions can not be completely described in terms of means and variances.
Publication Year: 2001
Publication Date: 2001-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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