Briefly, duration, delta, beta, and volatility are all examples of risk metrics. Any procedures by which we calculate these are risk measures. The distinction is important because there is not a one-to-one correspondence between risk measures and risk metrics. There are, for example, different ways that the volatility of a portfolio might be calculated. Each represents a different risk measure for the single risk metric volatility.
Have someone take off his shoes and stand with his back against a wall that is marked with a scale of inches. Note the number of inches correspond to the top of the person's head. This process is a measure. We interpret the number obtained as the person's height. "Height" is a metric.
Here is another example. A hollow glass cylinder is a meter long and has an internal diameter of a centimeter. It is open at one end, sealed at the other and graded with millimeter measurements. A bath of mercury is maintained at a temperature of . The cylinder is immersed in the bath, allowing it to fill with mercury. While keeping the open end immersed, the cylinder is held vertically, with the sealed end about 950 centimeters above the surface of the mercury. A column of mercury is suspended within the cylinder. Its height is measured in centimeters and is multiplied by 0.01316. The result is recorded. This procedure is a measure. We interpret the recorded number as air pressure measured in atmospheres. The interpretation is a metric.
When we apply a measure, the number obtained is a measurement.
Measures are employed to quantify many things: height, temperature, aptitude, speed, consumer confidence, etc. All of these notions being quantified are metrics. The operations with which we quantify them are measures. There are many metrics of risk—volatility, delta, gamma, duration, convexity, beta, etc. We call these risk metrics. A measure that supports a risk metric is called a risk measure. The value obtained from applying a risk measure is a risk measurement.
Risk measures tend to be categorized according to the risk metrics they support. There are measures of duration, measures of delta, etc. This is an important point. We do not categorize risk measures according to the specific operations they entail. Operationally, there are many different ways we might arrive at a measurement of a portfolio's volatility. Irrespective of the actual operations, all of them are measures of volatility. All support volatility as a risk metric.
The distinction between risk measures and risk metrics was first made by Glyn Holton in his book Value-at-Risk: Theory and Practice.