## Portfolio Market Risk

Category: Risk Management in Banking

The goal of modelling portfolio risk is to obtain the distribution of the portfolio returns at the horizon, set at the liquidation period. This implies a forward revaluation at the horizon date of all instruments once market parameters change. Since these are the value drivers of individual assets within the portfolio, the prerequisite is to model the market parameter random deviations complying with their correlation structure. The next step is to derive individual asset return distributions from market parameters to get all possible portfolio returns. Loss statistics and loss percentiles providing the market risk Value at Risk (VaR) derive from the portfolio return distribution. To achieve this ultimate goal, techniques range from the Delta VaR technique to full-blown simulations of market parameters and portfolio values.

Portfolios benefit from diversification. The portfolio return volatility decreases with the number of assets, down to a floor resulting from general risk. However, the value of risk relates to portfolio value rather than return. The portfolio value volatility increases with the number of assets, and the incremental volatility for a new asset increases with the average correlation of the portfolio. When dealing with single assets, there is no need to worry about standalone risk. For portfolios of assets, we need to include the effect of the correlation between individual asset returns and between risk drivers, or market parameters, influencing these individual returns.

The Delta VaR model relates linearly asset returns to market parameter returns using instrument sensitivities. The essentials are that the portfolio return is a linear combination of random normals. Therefore it follows a normal distribution. The volatility of the portfolio return applies to any set of linear combinations of random variables. Since we can calculate the volatility of the portfolio return, and since we know that it is normally distributed, we have all that we need to measure VaR. When the assumptions get unrealistic, we need to extend the simple Delta VaR technique or rely on full revaluation at horizon.

Full-blown simulations consist of generating risk driver returns complying with the variance-covariance structure observed in the markets, and revaluating each individual transaction. Revaluation uses pricing models, or simulation techniques for complex derivatives. The portfolio return distribution results from the full revaluation for all trials. Forward looking simulations generate random market parameter values complying with market volatilities and correlations. Intermediate techniques use sensitivities to save the time intensive calculations. Historical simulations use past values of all risk drivers, which effectively embed existing correlations. Other intermediate techniques include Delta-Gamma techniques, or grid simulations.

Because of model risk, modelled returns deviate from actual returns. This necessitates back testing and stress testing modelled VaR to ensure that tracking errors remain within acceptable bounds.

The first section shows how the portfolio return and the portfolio value volatility vary with the number of assets in a simple portfolio. The second section summarizes the Delta VaR technique. The third section expands fully the calculation of the Delta-normal VaR technique using the simple example of a two-asset portfolio. The fourth section reviews intermediate techniques. The fifth section expands the simulation technique and details some intermediate stages before moving to full-blown Monte Carlo simulations. The last section addresses back and stress testing of market risk VaR.