Business — Banking — Management — Marketing & Sales

DEFINITIONS AND NOTATION

In this section, we define the standalone risk of a facility, the risk contribution of an existing facility i within a portfolio P, orRC^p, also named Absolute Risk Contributions (ARC), and the marginal risk contribution of the facility f added to a portfolio P ,or MRC^p+^f.

Definitions

The main definitions are:

• The standalone risk is the loss volatility of a single facility.

• The marginal risk contribution is the change in portfolio loss volatility when adding a facility f to the portfolio P.

• The absolute risk contribution of an existing facility i to a portfolio P is the covariance of the random loss of this single facility i with the random loss aggregated over the entire portfolio (including i ), divided by the loss volatility of this aggregated loss. This risk contribution is also the absolute risk contribution; to avoid confusion it is distinct from the marginal risk contribution.

Among the three definitions above, the two first are intuitive and the third sounds mathematical. The absolute risk contribution captures the risk of a facility given that other facilities diversify away a fraction of its standalone risk.

Risk contributions also depend on the overall measure of portfolio risk to which they contribute. The simplest risk contributions are the contributions to the loss volatility of the portfolio loss. These risk contributions have the attractive property of adding up to the loss volatility of the portfolio. Since it is common to express capital as a multiple of this portfolio loss volatility, risk contributions are converted into capital allocations through the same scaling factor. Since risk contributions sum to the loss volatility, capital allocations also sum to the portfolio capital. Because they add up to the portfolio risk measure post-diversification effects, the absolute risk contributions are a convenient basis for the overall portfolio capital allocation to individual facilities. The concept of risk contribution is intuitive, but calculations use technical formulas requiring us to specify the notation.

Notation

Risk contributions always refer to a facility of obligor i and a reference portfolio P .There are several risk contributions defined below. The portfolio P is made up of N facilities. Each facility i relates to a single obligor. The notation applies to both default models and full valuation mode models. However, all examples use calculations in default mode only for simplicity.

ABSOLUTE AND MARGINAL RISK CONTRIBUTIONS TO PORTFOLIO LOSS VOLATILITY AND CAPITAL

This section summarizes the key definitions of absolute and marginal risk contributions in the first subsection and the key properties of absolute risk contributions to portfolio loss volatility and capital, making them pivotal in the capital allocation system. The demonstration of properties follows.

Risk Contribution Definitions

The standalone risk is the loss volatility of a single facility. The absolute risk contribution to the loss volatility of the portfolio is the contribution of an obligor i to the overall loss volatility LV^P. Since it depends on both the portfolio and the obligor, it is useful to use RC_i^P instead of RCi to make explicit the reference to portfolio P . Absolute risk contributions to the portfolio loss volatility differ from the risk contributions to the portfolio capital, which are the capital allocations. The capital allocations relate to the risk contributions through the portfolio loss volatility. To convert absolute risk contributions to portfolio loss volatility into absolute risk contributions to capital, we multiply them by the ratio m (a) of capital to portfolio loss volatility.

The marginal risk contribution to loss volatility is the change in portfolio loss volatility when adding an additional unit of exposure, a new facility, a new obligor, or a new portfolio. For instance, the marginal risk contribution of an obligor f is the variation of the loss volatility with and without the obligor f (or a subset a of obligors). Marginal risk contributions are MRC_f^P below. The marginal risk contribution of obligor f in a portfolio of N obligors without f ,and N + 1 with f ,is:

The marginal risk contribution to the portfolio loss volatility and the marginal contribution to capital differ. The first is the variation of the portfolio LV^P when adding a

Unless otherwise specified, we use marginal risk contribution as the marginal change in loss volatility of the portfolio. If the multiple m (a) does not change significantly when the portfolio changes, the marginal contribution to loss volatility times the overall ratio of capital to portfolio loss volatility is a proxy of marginal risk contribution to capital. This approximation is not valid whenever the portfolio changes significantly.

Basic Properties of Risk Contributions

The absolute risk contributions serve to allocate capital. The absolute risk contribution to the portfolio loss volatility of a facility i to a portfolio P is the covariance of the random loss of this single facility i with the aggregated random portfolio loss over the entire portfolio (including i ), divided by the loss volatility of this aggregated random loss. The formula for calculating absolute risk contributions results from that of the variance of the portfolio, as explained in subsequent sections: Var(L_P) = Y,_i j oj = S _i j PjOiOj. Absolute risk contributions to loss volatility, times the multiple of overall capital to overall portfolio loss volatility, sum exactly to the portfolio capital. This is the key property making them the foundation for the capital allocation system solving the non-intuitive issue of allocating risks.

Marginal risk contributions serve to make incremental decisions and for risk-based pricing. They provide a direct answer to questions such as: What is the additional capital consumed by an additional facility? What is the capital saved by withdrawing a facility or a subportfolio from the current portfolio? Marginal contributions serve for pricing purposes. We show later that:

• Pricing in such a way that the revenues of an additional facility equal the target hurdle rate of return times the marginal risk contribution of the new facility ensures that the target return of the portfolio on capital remains equal to or above the minimum hurdle rate.

• Marginal risk contributions to the portfolio loss volatility are lower than absolute risk contributions to the portfolio loss volatility. However, marginal risk contributions to the portfolio capital can be higher or lower than absolute contributions to the portfolio capital.

The properties of absolute and marginal risk contributions serve to address different issues. The key distinction is ex post versus ex ante applications. Absolute risk contributions serve for ex post allocations of capital based on effective usage of line, while marginal risk contributions serve to make ex ante risk-based pricing decisions (Table 51.1).

A simple example illustrates these properties before deriving general formulas demonstrating the above properties. Using the same example of a very simple portfolio throughout the chapter facilitates our understanding of the formulas and shows the calculation details. The reader can skip the formal demonstrations of the above properties, once having reviewed the example. However, demonstrations are given in the main text. We detail more absolute risk contributions in what follows.