Section 9. Dependencies and Portfolio Risk

The modeling of portfolio risk is a core building block in risk management. The issue is to measure and quantify the diversification effect. The higher the dependency between positions, the lower the diversification is. Furthermore, quantification of portfolio risk requires the distribution of the portfolio values, which is highly sensitive to dependencies.


Diversification of risk has been the key principle that allows financial institutions to bear a much lower risk than the sum of individual risks. Combining risks does not follow the usual arithmetic rules, unlike income. The summation of two risks, each equal to 1, is not 2. It is usually lower because of diversification. Individual risks are said to be "sub-additive." Measuring dependencies with correlations shows that the sum of two individual risks of same size is in the range of 0 to 2. This is the essence of diversification.

Standalone risk is the risk of a single position. For example, it can be measured by the volatility of returns of a single market instrument, or as the probability weighted loss under default of a single loan. For market risk, adding the adverse effects of individual transactions does not make sense because it would assume that all transactions would move adversely jointly. When the market moves, some positions gain while others lose value, and only the net effect after offsetting gains and losses makes sense. For credit risk, the dependencies across firms are generally positive because all firms are generally adversely affected when economic conditions deteriorate. But not all firms will default together. The positive dependency between the credit standing of all firms determines how likely joint defaults are. Again, summing up all standalone risks of loans or bonds within a portfolio assume that all positions default together and ignore diversification effects.

When the position belongs to a portfolio, some of its risk is diversified away by other assets held within the portfolio. Under a risk prospective, summing the standalone risks of each position, whatever the measure used, does not make sense. The diversification effect is the difference between the (arithmetic) sum of individual transaction risks and the risk of the sum. Portfolio risk results from individual risks and from how dependent these risks are. Dependencies are critical for assessing risk of portfolios, both trading and banking portfolios, because they allow measuring the diversification effect. There are multiple measures of dependencies: correlation, factor models, joint and conditional probabilities, and copula functions.

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