# 14

# Modelling Earnings-at-Risk

Chapter 13 established a basic model for capital, which suggested:

*Economic capital is the amount required which, invested at the risk-free rate, covers the potential downside in earnings*.

Such a model provides a good rule of thumb for management when determining the proper amount of capital for the bank as a whole, but will it work at the level of individual businesses? This chapter examines an approach which attempts to answer the question; it can be classified as an “earnings-at-risk” approach, in contrast to the “value-at-risk” approach seen in Chapter 9. Like the top-down model demonstrated in the previous chapter, the definition of economic capital is derived from the observed volatility of earnings. After examining the methodology for calculating earnings at risk, various models for converting this into an economic capital measure are introduced.

## BASIC ECONOMIC-CAPITAL MODEL

The basic model which underlies earnings-volatility-based approaches is a definition of earnings-at-risk (EAR) using some measure of the extent to which revenues or earnings deviate either side of the mean. A generic definition of EAR is thus:

where *k* is a constant, and *σ*_{r} refers to the standard deviation of the revenues or earnings of the bank or business. In plain English, all this says is that earnings-at-risk are defined as a certain number (*k*) of standard deviations of the distribution of earnings. The generic model can use revenues, contribution (some ...