 # 1.2 Risk Measures

In the context of risk measurement, we distinguish between:

• a risk measure, which is the operation that assigns a value to a risk, and
• a risk metric, which is the attribute of risk that is being measured.

Just as duration and size are attributes of a meeting that might be measured, volatility and credit exposure are attributes of bond risk that might be measured. Volatility and credit exposure are risk metrics. Other examples of risk metrics are delta, beta and duration. Any procedure for calculating these is a risk measure. For any risk metric, there may be multiple risk measures. There are, for example, different ways that the delta of a portfolio might be calculated. Each represents a different risk measure for the single risk metric called delta.

According to Holton (2004), risk has two components:

1. exposure, and
2. uncertainty.

If we swim in shark-infested waters, we are exposed to bodily injury or death from a shark attack. We are uncertain because we don’t know if we will be attacked. Being both exposed and uncertain, we face risk.

Risk metrics typically take one of three forms:

• those that quantify exposure;
• those that quantify uncertainty;
• those that quantify exposure and uncertainty in some combined manner.

Probability of rain is a risk metric that only quantifies uncertainty. It does not address our exposure to rain, which depends upon whether or not we have outdoor plans.

Credit exposure is a risk metric that only quantifies exposure. It indicates how much money we might lose if a counterparty were to default. It says nothing about our uncertainty as to whether or not the counterparty will default.

Risk metrics that quantify uncertainty—either alone or in combination with exposure—are usually probabilistic. Many summarize risk with a parameter of some probability distribution. Standard deviation of tomorrow’s spot price of copper is a risk metric that quantifies uncertainty. It does so with a standard deviation. Average highway deaths per passenger-mile is a risk metric that quantifies uncertainty and exposure. We may interpret it as reflecting the mean of a probability distribution.

###### Exercises
1.2

Give an example of a situation that entails uncertainty but not exposure, and hence no risk.

1.3

Give an example of a situation that entails exposure but not uncertainty, and hence no risk.

1.4

In our example of the deaths per passenger-mile risk metric, for what random variable’s probability distribution may we interpret it as reflecting a mean?

1.5

Give three examples of risk metrics that quantify financial risks:

1. one that quantifies exposure;
2. one that quantifies uncertainty; and
3. one that quantifies uncertainty combined with exposure.