This section gradually moves from basic correlation definitions to the more elaborated copula approach in five chapters:

• correlation and variance-covariance matrices (Chapter 30)

• the conditional probability approach to dependencies (Chapter 31)

• factor models, which are very commonly used for modeling dependencies (Chapter 32)

• the modern copula approach that allows addressing dependency between non-normal distributions (Chapter 33)

• the simulation algorithms that apply to random variables, through the factor model approach and the copula approach (Chapter 34).

Correlations and Covariances

The classical method for modeling dependencies relies on correlations and variance-covariance matrices. Section 30.1 defines correlations, variance and covariances. The characteristics of variance-covariance matrices are summarized in Section 30.2.

The classical application of modeling portfolio return with correlated asset returns is used as an example in Section 30.3. The calculation of the portfolio return uses both standard algebra and matrix formulas, which are more compact and general.

The example of portfolio return is a very classical example. But in many important instances, we are interested in the volatility of different target variables. The volatility of the P & L of a trading portfolio is a measure of market risk. The volatility of losses across a credit portfolio is a measure of credit risk. In all cases, whether the target variable is the market return of a portfolio, or the P & L of a trading portfolio, or credit loss volatility, correlations are critical for assessing the diversification effect on risk.

CORRELATIONS AND COVARIANCES

This section uses the classical concepts of expectations, variance, covariance and correlation, applied to a portfolio made of two assets only. We provide here only a reminder of basic definitions. Asset values and asset returns are random variables. In this section, we do not consider the weights assigned to each asset and consider their values, or their returns, as two un-weighted random variables.

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